MULTIPLE CHOICE
QUESTIONS IN
ENGINEERING MATHEMATICS
BY
BESAVILLA
1. Evaluate the lim (x^2-16)/(x-4).
a. 1
b. 8
c. 0
d. 16
2. Evaluate the limit (x-4)/(x^2-x-12) as
x approaches 4.
a. undefined
b. 0
c. infinity
d. 1/7
3. What is the limit of cos (1/y) as y
approaches infinity?
a. 0
b. -1
c. infinity
d. 1
4. Evaluate the limits of lim (x^3-2x+9)
/(2x^3-8).
a. 0
b. -9/8
c. α
d. ½
5. Evaluate the limit of (x^3-2x^2-x+2)
/(x^2-4) as x approaches 2.
a. α
b. ¾
c. 2/5
d. 4/7
6. Evaluate the limit of √(x-4)/√(x^2-16)
as x approaches 4.
a. 0.262
b. 0.354
c. 0
d. α
7. Evaluate the limit of (x^2-x-6)/(x^24x+3) as x approaches 3.
a. 3/2
b. 3/5
c. 0
d. 5/2
8. Evaluate the limit of (4x^2-x)/
(2x^2+4) as x approaches α.
a. 2
b. 4
c. α
d. 0
9. Evaluate the limit of (x-2)/(x^3-8) as
x approaches 2.
a. α
b. 1/12
c. 0
d. 2/3
10. Evaluate the limit of θ/(2 sinθ) as θ
approaches 0.
a. 2
b. ½
c. 0
d. α
11. Evaluate the limit of (1-sec^2 (x)/
cos (x)-1 as x approaches 0.
a. -2
b. α
c. 0
d. 1
12. Evaluate the limit (x^3-27)/(x-3) as x
approaches to 3.
a. 0
b. infinity
c. 9
d. 27
13. Evaluate the limit (3x^3-4x^2-5x+2)/
(x^2-x-2) as x approaches to 2.
a. α
b. 5
c. 0
d. 7/3
14. Evaluate the limit of (4 tan^3 (x)/
2sin(x)-x as x approaches 0.
a. 1
b. 0
c. 2
d. α
15. Evaluate the limit of 8x/(2x-1) as x
approaches α.
a. 4
b. 3
c. 2
d. -1
16. Evaluate the limit of (x^2-1)/
(x^2+3x-4) as x approaches 1.
a. 2/5
b. 1/5
c. 3/5
d. 4/5
17. Evaluate the limit of (x+2)/(x-2) as x
approaches α.
a. α
b. -1
c. 1
d. 4
18. Evaluate the limit of (1-cosx)/(x^2)
as x approaches 0.
a. α
b. ½
c. 1
d. 0
19. Find the limit of [sqrt(x+4)-2]/x as x
approaches 0.
a. α
b. ¼
c. 0
d. ½
20. Find the limit [sqrt(x+9)-3]/x as x
approaches 0.
a. α
b. 1/6
c. 0
d. 1/3
21. Evaluate the limit (x^2+x-6)/(x^2-4)
as x approaches to 2.
a. 6/5
b. 5/4
c. 4/3
d. 3/2
22. Evaluate the limit (x^4-81)/(x-3) as x
approaches to 3.
a. 108
b. 110
c. 122
d. 100
23. Evaluate the limit (x+sin2x)/
(x-sin2x) as x approaches to 0.
a. -5
b. -3
c. 4
d. -1
24. Evaluate the limit (ln sin x)/(ln tan x)
as x approaches to 0.
a. 1
b. 2
c. ½
d. α
25. Compute the equation of the vertical
asymptote of the curve y=(2x-1)/(x+2).
a. x+2=0
b. x-3=0
c. x+3=0
d. x-2=0
26. Compute the equation of the
horizontal asymptote of the curve y=(2x1)/(x+2).
a. y=2
b. y=0
c. y-1=0
d. y-3=0
27. The function y=(x-4)/(x+2)
discontinuous at x equals?
a. -2
b. 0
c. 1
d. 2
is
28. An elliptical plot of garden has a
semi-major axis of 6m and a semi-minor
axis of 4.8meters. If these are increased
by 0.15m each, find by differential
equations the increase in area of the
garden in sq.m.
a. 0.62π
b. 1.62π
c. 2.62π
d. 2.62π
29. The diameter of a circle is to be
measured and its area computed. If the
diameter can be measured with a
maximum error of 0.001cm and the area
must be accurate to within 0.10sq.cm.
Find the largest diameter for which the
process can be used.
a. 64
b. 16
c. 32
d. 48
30. The altitude of a right circular
cylinder is twice the radius of the base.
The altitude is measured as 12cm. With
a possible error of 0.005cm, find the
approximately error in the calculated
volume of the cylinder.
a. 0.188 cu cm
b. 0.144 cu cm
c. 0.104 cu cm
d. 0.126 cu cm
31. What is the allowable error in
measuring the edge of a cube that is
intended to hold a cu m, if the error in
the computed volume is not to exceed
0.03 cu m?
a. 0.002
b. 0.0025
c. 0.003
d. 0.001
32. If y=x^(3/2) what is the approximate
change in y when x changes from 9 to
9.01?
a. 0.045
b. 0.068
c. 0.070
d. 0.023
33. The expression for the horsepower of
an engine is P=0.4 n x^2 where n is the
number of cylinders and x is the bore of
cylinders.
Determine
the
power
differential added when four cylinder car
has the cylinders rebored from 3.25cm to
3.265cm.
a. 0.156 hp
b. 0.210 hp
c. 0.319 hp
d. 0.180 hp
34. A surveying instrument is placed at a
point 180m from the base of a bldg on a
level ground. The angle of elevation of
the top of a bldg is 30 degrees as
measured by the instrument. What would
be error in the height of the bldg due to
an error of 15minutes in this measured
angle by differential equation?
a. 1.05m
b. 1.09m
c. 2.08m
d. 1.05m
35. If y=3x^2-x+1, find the point x at
which dy/dx assume its mean value in
the interval x=2 and x=4.
a. 3
b. 6
c. 4
d. 8
36. Find the approximate increase by the
use of differentials, in the volume of the
sphere if the radius increases from 2 to
2.05.
a. 2.51
b. 2.25
c. 2.12
d. 2.86
37. If the area of a circle is 64π sq mm,
compute the allowable error in the area
of a circle if the allowable error in the
radius is 0.02 mm.
a. 1.01 sq mm
b. 1.58 sq mm
c. 2.32 sq mm
d. 0.75 sq mm
38. If the volume of a sphere is 1000π/6
cu mm and the allowable error in the
diameter of the sphere is 0.03 mm,
compute the allowable error in the
volume of a sphere.
a. 6.72 cu mm
b. 4.71 cu mm
c. 5.53 cu mm
d. 3.68 cu mm
39. A cube has a volume of 1728 cu mm.
If the allowable error in the edge of a
cube is 0.04 mm, compute the allowable
error in the volume of the cube.
a. 17.28 cu mm
b. 16.88 cu mm
c. 15.22 cu mm
d. 20.59 cu mm
40. Find the derivative of y=2^(4x).
a. 3^(4x+2) ln 2
b. 2^(4x+2) ln 2
c. 6^(3x+2) ln 2
d. 4^(4x+2) ln 2
41. Find the derivative of h with respect
to u if h=π^(2u).
a. π^(2u)
b. 2u ln π
c. 2π^(2u) ln π
d. 2π^(2u)
42. Find y’ if y=ln x
a. 1/x
b. ln x^2
c. 1/ln x
d. x ln x
43. Find y’ if y=arc sin (x)
a. √(1-x^2)
b. 1/√(1-x^2)
c. 1/(1+x^2)
d. (1+x)/√(1-x^2)
44. Find the derivative of loga u with
respect to x.
a. log u du/dx
b. u du/ln a
c. loga e/u
d. log a du/dx
45. Find the derivative of arc cos (2x).
a. -2/√(1-4x^2)
b. 2/√(1-4x^2)
c. 2/(1+4x^2)
d. 2/√(2x^2-1)
46. Find the derivative of 4 arc tan (2x).
a. 4/(1+x^2)
b. 4/(4x^2+1)
c. 8/(1+4x^2)
d. 8/(4x^2+1)
47. Find the derivative of arc csc (3x).
a. -1/[x√(9x^2-1)]
b. 1/[3x√(9x^2-1)]
c. 3/[x√(1-9x^2)]
d. 3/[x√9x^2-1)]
48. Find the derivative of arc sec (2x)
a. 1/[x√(4x^2-1)]
b. 2/[x√(4x^2-1)]
c. 1/[x√(1-4x^2)]
d. 2/[x√(1-4x^2)]
49. If ln (ln y) + ln y = ln x, find y’.
a. x/(x+y)
b. x/(x-y)
c. y/(x+y)
d. y/(x-y)
56. What is the derivative with respect to
x of sec^2 (x)?
a. 2x sec^2 (x) tan^2 (x)
b. 2x sec (x) tan (x)
c. sec^2 (x) tan^2 (x)
d. 2 sec^2 (x) tan^2 (x)
50. Find y” if y=a^u.
a. a^u ln a
b. u ln a
c. a^u/ln a
d. a ln u
57. The derivative with respect to x of
2cos^2 (x^2+2).
a. 4 sin (x^2+2) cos (x^2+2)
b. -4 sin (x^2+2) cos (x^2+2)
c. 8x sin (x^2+2) cos (x^2+2)
d. -8x sin (x^2+2) cos (x^2+2)
51. Find the derivative of y with respect
to x if y = x ln x – x.
a. x ln x
b. ln x
c. (ln x)/x
d. x/ln x
52. If y=tanh x, find dy/dx.
a. sech^2 (x)
b. csch^2 (x)
c. sinh^2 (x)
d. tanh^2 (x)
53. Find the derivative of y=x^x.
a. x^x (2+ln x)
b. x^x (1+ln x)
c. x^x (4-ln x)
d. x^x (8+ln x)
54. Find the derivative of y=loga 4x.
a. y’=(loga e)/x
b. y’=(cos e)/x
c. y’=(sin e)/x
d. y’=(tan e)/x
55. What is the derivative with respect to
x of (x+1)^3 – x^3.
a. 3x+3
b. 3x-3
c. 6x-3
d. 6x+3
58. Find the derivative of [(x+1)^3]/x.
a. [3(x+1)^2]/x – [(x+1)^3]/x^2
b. [2(x+1)^3]/x – [(x+1)^3]/x^3
c. [4(x+1)^2]/x – [2(x+1)^3]/x
d. [(x+1)^2]/x – [(x+1)^3]/x
59. Determine the slope of the curve
y=x^2-3x as it passes through the origin.
a. -4
b. 2
c. -3
d. 0
60. If y1=2x+4 and y2=x^2+C, find the
value of C such that y2 is tangent to y1.
a. 6
b. 5
c. 7
d. 4
61. Find the slope of (x^2)y=8 at the
point (2,2).
a. 2
b. -1
c. -1/2
d. -2
62. What is the first derivative dy/dx of
the expression (xy)^x=e.
a. –y(1-ln xy)/x^2
b. –y(1+ln xy)/x
c. 0
d. x/y
63. Find y’ in the following equation
y=4x^2-3x-1.
a. 8x-3
b. 4x-3
c. 2x-3
d. 8x-x
64.
Differentiate
the
y=(x^2)/(x+1).
a. (x^2+2x)/(x+1)^2
b. x/(x+1)
c. 2x^2/(x+1)
d. 1
equation
65. If y=x/(x+1), find y’.
a. 1/(x+1)^3
b. 1/(x+1)^2
c. x+1
d. (x+1)^2
66. Find dy/dx in the equation
y=(x^6+3x^2+50)/(x^2+1) if x=1
a. -21
b. -18
c. 10
d. 16
67. Find the equation of the curve whose
slope is (x+1)(x+2) and passes through
point (-3, -3/2).
a. y=x^2+2x-4
b. y=(x^3)/3+(3x^2)/2+2x
c. y= 3x^2+4x-8
d. y=(3x^2)/2+4x/3+2
68. Find the equation of the curve whose
slope is 3x^4-x^2 and passes through
point (0,1).
a. y=(3x^5)/5-(x^3)/3+1
b. y=(x^4)/4-(x^3)+1
c. y=(2x^5)/5-2x+1
d. y=(3x^5)-(x^3)/3+1
69. What is the slope of the tangent to
y=(x^2+1)(x^3-4x) at (1,-6)?
a. -8
b. -4
c. 3
d. 5
70. Find the coordinate of the vertex of
the parabola y=x^2-4x+1 by making use
of the fact that at the vertex, the slope of
the tangent is zero.
a. (2,-3)
b. (3,2)
c. (-1,-3)
d. (-2,-3)
71. Find the slope of the curve
x^2+y^2-6x+10y+5=0 at point (1,0).
a. 2/5
b. ¼
c. 2
d. 2
72. Find the slope of the ellipse
x^2+4y^2-10x+16y+5=0 at the point
where y=2+8^0.5 and x=7.
a. -0.1654
b. -0.1538
c. -0.1768
d. -0.1463
73. Find the slope of the tangent to the
curve y=2x-x^2+x^3 at (0,2).
a. 2
b. 3
c. 4
d. 1
74. Find the equation of the tangent to
the curve y=2e^x at (0,2).
a. 2x-y+3=0
b. 2x-y+2=0
c. 3x+y+2=0
d. 2x+3y+2=0
75. Find the slope of the curve
y=2(1+3x)^2 at point (0,3).
a. 12
b. -9
c. 8
d. -16
81. Determine the point on the curve
x^3-9x-y=0 at which slope is 18.
a. x=3, y=0
b. x=4, y=5
c. x=2, y=7
d. x=5, y=6
76. Find the slope of the curve
y=x^2(x+2)^3 at point (1,2).
a. 81
b. 48
c. 64
d. 54
82. Find the second derivative of
y=(2x+1)^2+x^3.
a. 8+6x
b. (2x+1)^3
c. x+1
d. 6+4x
77. Find the slope of the curve
y=[(4-x)^2]/x at point (2,2).
a. -3
b. 2
c. -2
d. 3
83. Find the second derivative of
y=(2x+4)^2 x^3.
a. x^2(80x+192)
b. 2x+4
c. x^3(2x+80)
d. x^2(20x+60)
78. If the slope of the curve y^2=12x is
equal to 1 at point (x,y), find the value of
x and y.
a. x=3, y=6
b. x=4, y=5
c. x=2, y=7
d. x=5, y=6
84. Find the second derivative of
y=2x+3(4x+2)^3 when x=1.
a. 1728
b. 1642
c. 1541
d. 1832
79. If the slope of the curve x^2+y^2=25
is equal to -3/4 at point (x,y) find the
value of x and y.
a. 3,4
b. 2,3
c. 3,4.2
d. 3.5,4
80. If the slope of the curve
25x^2+4y^2=100 is equal to -15/8 at
point (x,y), find the value of x and y.
a. 1.2,4
b. 2,4
c. 1.2,3
d. 2,4.2
85. Find the second derivative of
y=2x/[3(4x+2)^2] when x=0.
a. -1.33
b. 1.44
c. 2.16
d. -2.72
86. Find the second derivative of
y=3/(4x^-3) when x=1.
a. 4.5
b. -3.6
c. 2.4
d. -1.84
87. Find the second derivative of y=x^-2
when x=2.
a. 0.375
b. 0.268
c. 0.148
d. 0.425
88. Find the first derivative
y=2cos(2+x^2).
a. -4x sin (2+x^2)
b. 4x cos (2+x^2)
c. x sin (2+x^2)
d. x cos (2+x^2)
of
89. Find the first derivative of
y=2 sin^2 (3x^2-3).
a. 24x sin (3x^2-3) cos (3x^2-3)
b. 12 sin (3x^2-3)
c. 6x cos (3x^2-3)
d. 24x sin (3x^2-3)
90. Find the first derivative of y=tan^2
(3x^2-4).
a. 12xtan(3x^2-4)sec^2(3x^2-4)
b. x tan (3x^2-4)
c. sec^2 (3x^2-4)
d. 2 tan^2(3x^2-4)csc^2(3x^2-4)
91. Find the derivative of arc cos 4x
a. -4/(1-16x^2)^0.5
b. 4/(1-16x^2)^0.5
c. -4/(1-4x^2)^0.5
d. 4/(1-4x^2)^0.5
92. The equation y^2=cx is the general
equation of.
a. y’=2y/x
b. y’=2x/y
c. y’=y/2x
d. y’=x/2y
93. Find the slope of the curve
y=6(4+x)^1/2 at point (0,12).
a. 1.5
b. 2.2
c. 1.8
d. 2.8
94. Find the coordinate of the vertex of
the parabola y=x^2-4x+1 by making use
of the fact that at the vertex, the slope of
the tangent is zero.
a. (2,-3)
b. (3,2)
c. (-1,-3)
d. (-2,-3)
95. Find dy/dx by implicit differentiation
at the point (3,4) when x^2+y^2=25.
a. -3/4
b. ¾
c. 2/3
d. -2/3
96. Find dy/dx by implicit differentiation
at point (0,0) if (x^3)(y^3)-y=x.
a. -1
b. -2
c. 2
d. 1
97. Find dy/dx by implicit differentiation
at point (0,-2) if x^3-xy+y^2=4.
a. ½
b. -2
c. -2/3
d. ¾
98. Find the point of inflection of
f(x)=x^3-3x^2-x+7.
a. 1,4
b. 1,2
c. 2,1
d. 3,1
99. Find the point of inflection of the
curve y=(9x^2-x^3+6)/6.
a. 3,10
b. 2,8
c. 3,8
d. 2,10
100. Find the point of inflection of the
curve y=x^3-3x^2+6.
a. 1,4
b. 1,3
c. 0,2
d. 2,1
a. 22.36
b. 24.94
c. 20.38
d. 18.42
101. Locate the point of inflection of the
curve y=f(x)=(x square)(e exponent x).
a. -2 plus or minus (sqrt of 3)
b. 2 plus or minus (sqrt of 2)
c. -2 plus or minus (sqrt of 2)
d. 2 plus or minus (sqrt of 3)
107. Find the radius of curvature of the
curve y=2x^3+3x^2 at (1,5).
a. 97
b. 90
c. 101
d. 87
102. The daily sales in thousands of
pesos of a product is given by
S=(x^2-x^3+6)/6 where x is the
thousand of pesos spent on advertising.
Find the point of diminishing returns for
money spent on advertising.
a. 5
b. 4
c. 3
d. 6
108. Compute the radius of curvature of
the curve x=2y^3-3y^2 at (4,2).
a. -97.15
b. -99.38
c. -95.11
d. -84.62
103. y=x to the 3rd power -3x. Find the
maximum value of y.
a. 2
b. 1
c. 0
d. 3
104. Find the curvature of the parabola
y^2=12x at (3,6).
a. -√2/24
b. √2/8
c. 3√2
d. 8√2/3
105. Locate the center of curvature of
the parabola x^2=4y at point (2,2).
a. (-2,6)
b. (-3,6)
c. (-2,4)
d. (-3,7)
106. Compute the radius of curvature of
the parabola x^2=4y at the point (4,4).
109. Find the radius of curvature of a
parabola y^2-4x=0 at point (4,4).
a. 22.36
b. 25.78
c. 20.33
d. 15.42
110. Find the radius of curvature of the
curve x=y^3 at point (1,1).
a. -1.76
b. -1.24
c. 2.19
d. 2.89
111. A cylindrical boiler is to have a
volume of 1340 cu ft. The cost of the
metal sheets to make the boiler should
be minimum. What should be its
diameter in feet?
a. 7.08
b. 11.95
c. 8.08
d. 10.95
112. A rectangular corral is to be built
with a required area. If an existing fence
is to be used as one of the sides,
determine the relation of the width and
the length which would cost the least.
a. width=twice the length
b. width=1/2 length
c. width=length
d. width=3 times the length
113. Find the two numbers whose sum is
20, if the product of one by the cube of
the other is to be minimum.
a. 5 and 15
b. 10 and 10
c. 4 and 16
d. 8 and 12
114. The sum of two numbers is 12. Find
the minimum value of the sum of their
cubes.
a. 432
b. 644
c. 346
d. 244
115. A printed page must contain 60 sq
m of printed material. There are to be
margins of 5cm on either side and
margins of 3cm on top and bottom. How
long should the printed lines be in order
to minimize the amount of paper used?
a. 10
b. 18
c. 12
d. 15
116. a school sponsored trip will cost
each students 15 pesos if not more than
150 students make the trip, however the
cost per student will reduced by 5
centavos for each student in excess of
150. How many students should make
the trip in order for the school to receive
the largest group income?
a. 225
b. 250
c. 200
d. 195
117. A rectangular box with square base
and open at the top is to have a capacity
of 16823 cu cm. Find the height of the
box that requires minimum amount of
materials required.
a. 16.14 cm
b. 14.12 cm
c. 12.13 cm
d. 10.36 cm
118. A closed cylindrical tank has a
capacity of 576.56 cu m. Find the
minimum surface area of the tank.
a. 383.40 cu m
b. 412.60 cu m
c. 516.32 cu m
d. 218.60 cu m
119. A wall 2.245 m high is x meters
away from a building. The shortest
ladder that can reach the building with
one end resting on the ground outside
the wall is 6m. What is the value of x?
a. 2m
b. 2.6m
c. 3.0m
d. 4.0m
120. With only 381.7 sq m of materials,
a closed cylindrical tank of maximum
volume is to be the height of the tank, in
m?
a. 9m
b. 7m
c. 11m
d. 13m
121. If the hypotenuse of a right triangle
is known, what is the ratio of the base
and the altitude of the right triangle
when its are is maximum?
a. 1:1
b. 1:2
c. 1:3
d. 1:4
122. The stiffness of a rectangular beam
is proportional to the breadth and the
cube of the depth. Find the shape of the
stiffest beam that can be cut from a log
of given size.
a. depth=√3 breadth
b. depth=breadth
c. depth=√2 breadth
d. depth=2√2 breadth
123. What is the maximum length of the
perimeter if the hypotenuse of a right
triangle is 5m long?
a. 12.08 m
b. 15.09 m
c. 20.09 m
d. 8.99 m
124. An open top rectangular tank with
square s bases is to have a volume of 10
cu m. The material fir its bottom is to
cost 15 cents per sq m and that for the
sides 6 cents per sq m. Find the most
economical dimensions for the tank.
a. 2 x 2 x 2.5
b. 2 x 5 x 2.5
c. 2 x 3 x 2.5
d. 2 x 4 x 2.5
125. A trapezoidal gutter is to be made
from a strip of metal 22m wide by
bending up the sides. If the base is 14m,
what width across the top gives the
greatest carrying capacity?
a. 16
b. 22
c. 10
d. 27
126. Divide the number 60 into two pats
so that the product P of one part and the
square of the other is maximum. Find the
smallest part.
a. 20
b. 22
c. 10
d. 27
127. The edges of a rectangular box are
to be reinforced with a narrow metal
strips. If the box will have a volume of 8
cu m, what would its dimensions be to
require the least total length of strips?
a. 2 x 2 x 2
b. 4 x 4 x 4
c. 3 x 3 x 3
d. 2 x 2 x 4
128. A rectangular window surmounted
by a right isosceles triangle has a
perimeter equal to 54.14m. Find the
height of the rectangular window so that
the window will admit the most light.
a. 10
b. 22
c. 12
d. 27
129. A normal window is in the shape of
a rectangle surrounded by a semi-circle.
If the perimeter of the window is 71.416,
what is its radius and the height of the
rectangular portion so that it will yield a
window admitting the most light?
a. 10
b. 22
c. 12
d. 27
130. Find the radius of a right circular
cone having a lateral area of 544.12 sq m
to have a maximum volume.
a. 10
b. 20
c. 17
d. 19
131. A gutter with trapezoidal cross
section is to be made from a long sheet
of tin that is 15cm wide by turning up
one third of its width on each side. What
width across the top that will give a
maximum capacity?
a. 10
b. 20
c. 15
d. 13
132. A piece of plywood for a billboard
has an area of 24 sq ft. The margins at
the top and bottom are 9 inches and at
the sides are 6 in. Determine the size of
plywood for maximum dimensions of
the painted area.
a. 4 x 6
b. 3 x 4
c. 4 x 8
d. 3 x 8
133. A manufacturer estimates that the
cost of production of x units of a certain
item is C=40x-0.02x^2-600. How many
units should be produced for minimum
cost?
a. 1000 units
b. 100 units
c. 10 units
d. 10000 units
134. If the sum of the two numbers is 4,
find the minimum value of the um of
their cubes.
a. 16
b. 18
c. 10
d. 32
135. If x units of a certain item are
manufactured, each unit can be sold for
200-0.01x pesos. How many units can be
manufactured for maximum revenue?
What is the corresponding unit price?
a. 10000, P100
b. 10500, P300
c. 20000, P200
d. 15000, P400
136. A certain spare parts has a selling
price of P150 if they would sell 8000
units per month. If for every P1.00
increase in selling price, 80 units less
will be sold out pr month. If the
production cost is P100 per unit, find the
price per unit for maximum profit per
month.
a. P175
b. P250
c. P150
d. P225
137. The highway department is
planning to build a picnic area for
motorist along a major highway. It is to
be rectangular with an area of 5000 sq m
is to be fenced off on the three sides not
adjacent to the highway. What is the
least amount of fencing that ill be
needed to complete the job?
a. 200m
b. 300m
c. 400m
d. 500m
138. A rectangular lot has an area of
1600 sq m. Find the least amount of
fence that could be used to enclose the
area.
a. 160m
b. 200m
c. 100m
d. 300m
139. A student club on a college campus
charges annual membership due of P10,
less 5 centavos for each member over
60. How many members would give the
club the most revenue from annual dues?
a. 130 members
b. 420 members
c. 240 members
d. 650 members
140. A company estimates that it can sell
1000 units per weak if it sets the unit
price at P3.00, but that its weekly sles
will rise by 100 units for each P0.10
decrease in price. Find the number of
units sold each week and its unit price
per max revenue.
a. 2000, P2.00
b. 1000, P3.00
c. 2500, P2.50
d. 1500, P1.50
141. In manufacturing and selling x units
of a certain commodity, the selling price
per unit is P=5-0.002x and the
production cost in pesos is C=3+1.10x.
Determine the production level that will
produce the max profit and what would
this profit be?
a. 975, P1898.25
b. 800, P1750.75
c. 865, P1670.50
d. 785, P1920.60
142. ABC company manufactures
computer spare parts. With its present
machines, it has an output of 500 units
annually. With the addition of the new
machines the company could boosts its
yearly production to 750 units. If it
produces x parts it can set a price of
P=200-0.15x pesos per unit and will
have a total yearly cost of C=6000+6x0.003x in pesos. What production level
maximizes total yearly profit?
a. 660 units
b. 237 units
c. 560 units
d. 243 units
143. The fixed monthly cost for
operating a manufacturing plant that
makes transformers is P8000 and there
are direct costs of P110 for each unit
produced. The manufacturer estimates
that 100 units per month can be sold if
the unit price is P250 and that sales will
in crease by 20 units for each P10
decrease in price. Compute the number
of units that must be sold per month to
maximize the profit. Compute the unit
price.
a. 190, P205
b. 160, P185
c. 170, P205
d. 200, P220
144. The total cost of producing and
marketing x units of a certain
commodity is given as C=(80000x400x^2+x^3)/40000. For what number x
is the average cost a minimum?
a. 200 units
b. 100 units
c. 300 units
d. 400 units
145. A wall 2.245m high is 2m away
from a bldg. Find the shortest ladder that
can reach the bldg with one end resting
on the ground outside the wall.
a. 6m
b. 9m
c. 10m
d. 4m
146. If the hypotenuse of a right triangle
is known, what is the relation of the base
and the altitude of the right triangle
when its area is maximum?
a. altitude=base
b. altitude=√2 base
c. altitude=√2 base
d. altitude=2 base
147. The hypotenuse of a right triangle is
20cm. What is the max possible area of
the triangle in sq cm?
a. 100
b. 170
c. 120
d. 160
148. A rectangular field has an area of
10,000 sq m. What is the least amount of
fencing meters to enclose it?
a. 400
b. 370
c. 220
d. 560
149. A monthly overhead of a
manufacturer of a certain commodity is
P6000 and the cost of material is P1.0
per unit. If not more than 4500 units are
manufactured per month, labor cost is
P0.40 per unit, but for each unit over
4500, the manufacturer must pay P0.60
for labor per unit. The manufacturer can
sell 4000 units per month at P7.0 per
unit and estimates that monthly sales
will rise by 100 for each P0.10 reduction
in price. Find the number of units that
should be produced each month for
maximum profit.
a. 4700 units
b. 2600 units
c. 6800 units
d. 9900 units
150. Find two numbers whose product is
100m and whose sum is minimum.
a. 10, 10
b. 12, 8
c. 5, 15
d. 9, 11
151. Find two numbers whose sum is 36
if the product of one by the square of the
other is a maximum.
a. 12, 24
b. 13, 23
c. 20, 16
d. 11, 25
152. Find the minimum amount of thin
sheet that can be made into a closed
cylinder having a volume of 108 cu in.
in square inches.
a. 125.5
b. 127.5
c. 123.5
d. 129.5
153. A buyer is to take a plot of land
fronting street, the plot is to be
rectangular and three times its frontage
added to twice its depth is to be 96
meters. What is the greatest number of
sq m be may take?
a. 384 sq m
b. 352 sq m
c. 443 sq m
d. 298 sq m
154. A company has determined that the
marginal cost function for the production
of a particular cost function for the
production of a particular commodity is
given as y”=125+10x-(x^2)/9 where y is
the cost of producing x units of the
commodity. If the fixed cost is 250
pesos, what is the cost of producing 15
units?
a. 250
b. 225
c. 300
d. 200
155. A pig weighing 300lb gains 8
pounds per day and cost 6 pesos per day
to maintain. The market price for the pig
is seven pesos and fifty centavos per
pound but is decreasing 10 centavos per
day. When should the pig be sold?
a. 15 days
b. 18 days
c. 20 days
d. 10 days
156. It costs a bus company P125 to run
a bus on a certain tour, plus P15 per
passenger. The capacity of the bus is 20
persons and the company charges P35
per ticket if the bus is full. For each
empty seat, however, the company
increases the ticket price by P2.0. For
maximum profit how many empty seats
would the company like to see?
a. 5
b. 3
c. 6
d. 4
157. A book publisher prints the pages
of a certain book with 0.5 inch margins
on the top, bottom and one side and a
one inch margin on the other side to
allow for the binding. Find the
dimensions of the page that will
maximize the printed area of the page if
the area of the entire page is 96 sq
inches.
a. 8 inches
b. 7 inches
c. 9 inches
d. 10 inches
158. The cost of manufacturing an
engine parts is P300 and the number
which can be sold varies inversely as the
fourth power of the selling price. Find
the selling price which will yield the
greatest total net profit.
a. 400
b. 350
c. 450
d. 375
159. The price of the product in a
competitive market is P300. If the cost
per unit of producing the product is
160+x where x is the number of units
produced per month, how many units
should the firm produce and sell to
maximize its profit?
a. 70
b. 80
c. 60
d. 50
160. If the cost per unit of producing a
product by ABC company is 10+2x and
if the price on the competitive market is
P50, what is the maximum daily profit
that the company can expect of this
product?
a. 200
b. 300
c. 400
d. 600
161. An entrepreneur starts new
companies and sells them when their
growth is maximized. Suppose the
annual profit for a new company is given
by P(x)=22-x/2-18/(x+1) where P is in
thousand of pesos and x is the number of
years after the company is formed. If the
entrepreneur wants to sell the company
before profit begins to decline, after how
many years would the company be sold?
a. 5
b. 4
c. 6
d. 7
162. The profit function for a product is
P(x)=5600x+85x^2-x^3-x-200000. How
many items will produce a maximum
profit?
a. 80
b. 60
c. 70
d. 40
163. The following statistics of a
manufacturing company shows the
corresponding values for manufacturing
x units.
Production cost=60x+10000 pesos
Selling price/unit=200-0.02x pesos
How many units must be produced for
max profit?
a. 3500
b. 3300
c. 4000
d. 3800
164. The cost per unit of production is
expressed as (4+3x) and the selling price
on the competitive market is P100 per
unit. What maximum daily profit that the
company can expect of this product?
a. P768
b. P876
c. P657
d. P678
165. A certain unit produced by the
company can be sold for 400-0.02x
pesos where x is the number of units
manufactured. What would be the
corresponding price per unit in order to
have a max revenue?
a. P200
b. P220
c. P150
d. P180
166. Given the cost equation of a certain
product as follows C=50t^2-200t+10000
where t is in years. Find the maximum
cost from the year 1995 to 2002.
a. P9,800
b. P6,400
c. P7,200
d. P10,600
167. The total cost of production a
shipment of a certain product is
C=5000x+125000/x where x is the
number of machines used in the
production. How many machines will
minimize the total cost?
a. 5
b. 20
c. 10
d. 15
168. The demand x for a product is
x=10000-100P where P is the market
price in pesos per unit. The expenditure
for the two product is E=Px. What
market price will the expenditure be the
greatest?
a. 50
b. 60
c. 70
d. 100
169. Analysis of daily output of a factory
shows that the hourly number of units y
produced after t hours of production is
y=70t+(t^2)/2-t^3. After how many
hours will the hourly number of units be
maximized?
a. 5
b. 6
c. 7
d. 8
170. An inferior product with large
advertising budget sells well when it is
introduced, but sales fall as people
discontinue use of the product. If the
weekly
sales
are
given
by
S=200t/(t+1)^2 where S is in millions of
pesos and t in weeks. After how many
weeks will the sales be maximized?
a. 1
b. 2
c. 3
d. 4
171. In the coming presidential election
of 1998, it is estimated that the
proportions P of votes that recognizes a
certain presidentiables name t months
after the campaign is given by
P=[7.2t/(t^2+16)]+0.20. After how many
months is the proportional maximized?
a. 4
b. 3
c. 5
d. 6
172. A car manufacturer estimates that
the cost of production of x cars of a
certain model is C=20x-0.01x^2-800.
How many cars should be produced for a
minimum cost?
a. 1000
b. 1200
c. 900
d. 1100
173. Analysis of daily output of a factory
shows that the hourly number of units y
produced after t hours of production is
y=70t+(t^2)/2-t^3. After how many
hours will the hourly number of units be
maximized and what would be the
maximum hourly output?
a. 5hrs, 237.5
b. 4hrs, 273.6
c. 6hrs, 243.5
d. 3hrs, 223.6
174. A time study showed that on
average, the productivity of a worker
after t hours on the job can be modeled
by the expression P=27+6t-t^3 where P
is the number of units produced per
hour. What is the maximum productivity
expected?
a. 36
b. 34
c. 44
d. 40
175. The sum of two numbers is equal to
S. Find the minimum sum of the cube of
the two numbers/
a. (S^3)/4
b. S/4
c. (S^2)/4
d. (S^3)/5
176. Given the cost equation of a certain
product
as
follows:
C=50t^2-
200t+10000 where t is in years. Find the
maximum cost from year 1995 to 2002.
a. P9000
b. P9800
c. P8500
d. P7300
177. A manufacturer determines that the
profit derived from selling x units of a
certain
item
is
given
by
P=0.003x^2+10x. Find the marginal
profit for a production of 50 units.
a. P10.30
b. P12.60
c. P15.40
d. P17.30
178. The total cost of production spare
parts of computers is given as C=4000x100x^2+x^3 where x is the number of
units of spare parts produced so that the
average cost will be minimum?
a. 50
b. 10
c. 20
d. 4
179. A viaduct is traversed by a truck
running at 15mph at the same time that
another truck traveling at a speed of
30mph on the street 22ft below and at
right angle to the viaduct, approached
the point directly below the viaduct from
a distance of 55ft. Find the nearest
distance between the trucks.
a. 33 ft
b. 44 ft
c. 29 ft
d. 39 ft
180. A sector is cut out of a circular disk
of radius √3 and the remaining part of
the disk I bent up so that the two edges
join and a cone is formed. What is the
largest volume for the cone?
a. 2π/3
b. π/3
c. 3π/4
d. π/4
181. Four squares are cut out of a
rectangular cardboard 50cm by 80 cm. in
dimension and the remaining piece is
folded into a closed, rectangular box
with two extra flaps trucked in. What is
the largest possible volume for such a
box?
a. 9000
b. 6000
c. 7000
d. 8000
182. An isosceles triangle with equal
sides of 20cm has these sides at a
variable equal angle with the base.
Determine the max area of the triangle.
a. 200 sq cm
b. 250 sq cm
c. 300 sq cm
d. 280 sq cm
183. Formerly, for a package to go by
parcel post, the sum of its length and
girth could not exceed 120cm. Find the
dimensions of the rectangular package of
greatest volume that could be sent.
a. 20 x 20 x 40
b. 20 x 20 x 20
c. 20 x 40 x 10
d. 40 x 20 x 30
184. The cross-section of a trough is an
isosceles trapezoid. If the trough is made
by bending up the sides of s strip of
metal 12cm wide, what would be the
angle of inclination of the sides and the
width across the bottom if the crosssectional area is to be a maximum?
a. 60 degrees
b. 120 degrees
c. 45 degrees
d. 75 degrees
185. Find the minimum amount of thin
sheet that can be made into a closed
cylinder having a volume of 108cu
inches in square inches.
a. 125.5
b. 127.5
c. 123.5
d. 129.5
186. Compute the abscissa of the min
point of the curve y=x^3-12x-9.
a. 2
b. -2
c. -1
d. 1
187. What value of x does a maximum
of y=x^3-3x occur?
a. -1
b. 1
c. 2
d. -2
188. Determine the point on the curve
y^2=8x which is nearest to the external
curve (4,2).
a. (2,4)
b. (4,3)
c. (3,5)
d. (6,8)
189. The LRT system runs from the
Bonifacio Monument to Baclaran for a
total distance of 15km. The cost of
electric energy consumed by a train per
hour is directly proportional to the cube
of its speed and is P250 per hour at
50kph. Other expenses such as salaries,
depreciation, overhead, etc. amounts to
P1687.50 per hour. Find the most
economical speed of the train in kph.
a. 75
b. 80
c. 65
d. 60
190. A businessman found out that his
profit varies as the product of the
amount spent for production and the
square root of the amount spent for
advertisement. If his total budget for
these expenses is P1.5 million, how
much
must
be
allocated
for
advertisement to maximize his profit?
a. 0.5M
b. 0.7M
c. 0.8M
d. 1.0M
194. Postal regulations require that a
parcel post package shall be not greater
than 600cm in the sum of its length and
girth (perimeter of the cross-section).
What is the volume in cu cm of the
largest package allowed by the postal
regulations if the package is to be
rectangular in cu cm?
a. 2 x 10^6
b. 3 x 10^6
c. 1.5 x 10^6
d. 4 x 10^6
191. A steel girder 16m long is moved
on rollers along a passageway 8m wide
and into a corridor at right angles with
the passageway. Neglecting the width of
thr girder, how wide must the corridor
be?
a. 3.6 m
b. 1.4 m
c. 1.8 m
d. 2.8 m
195. Divide 60 into 3 parts so that the
product of the three parts will be a
maximum, find the product.
a. 8000
b. 4000
c. 6000
d. 12000
192. A can manufacturer receives an
order for milk cans having a capacity of
100 cu cm. Each can is made from a
rectangular sheet of metal by rolling the
sheet into a cylinder; the lids are
stamped out from another rectangular
sheet. What are the most economical
proportions of the can?
a. 2.55
b. 2.59
c. 2.53
d. 3.67
193. A triangle has a variable sides x, y
and z subject to the constraint that the
perimeter P is fixed to 18cm. What is the
maximum possible area for the triangle?
a. 15.59 sq cm
b. 18.71 sq cm
c. 14.03 sq cm
d. 17.15 sq cm
196. Find the radius of the circle inscribe
in a triangle having a max area of
173.205 sq cm.
a. 3.45 cm
b. 5.77 cm
c. 4.96 cm
d. 2.19 cm
197. The area of a circle inscribe in a
triangle is equal to 113.10 sq cm. Find
the max area of the triangle.
a. 186.98 sq cm
b. 156. 59 sq cm
c. 175.80 sq cm
d. 193. 49 sq cm
198. Find the perimeter of a triangle
having a max area that is circumscribing
a circle of radius 8cm.
a. 83.13 cm
b. 85.77 cm
c. 84.96 cm
d. 92.19 cm
199. Suppose y is the number of workers
in the labor force neededtp produce x
units of a certain commodity and
x=4y^2. If the production of the
commodity this year is 25000 units and
the production is increasing at the rate
and the production is increasing at the
rate of 18000 units per year, what is the
current rate at which the labor force
should be increased?
a. 9
b. 7
c. 10
d. 15
200. Sugar juice is filtering through a
conical funnel 20cm, deep and 12cm
across top, into a cylindrical container
whose diameter is 10cm. When the
depth of the juice in the funnel is 10cm,
determine the rate at which its level in
the cylinder is rising.
a. 0.45
b. 1.25
c. 0.75
d. 0.15
201. An airplane, flying horizontally at
an altitude of 1km, passes directly over
an observer. If the constant speed of the
plane is 240kph, how fast is its distance
from the observer increasing 30seconds
later?
a. 214.66 kph
b. 256.34 kph
c. 324.57 kph
d. 137.78 kph
202. A metal disk expands during
heating. If its radius increases at the rate
of 20 mm per second, how fast is the
area of one of its faces increasing when
its radius is 8.1 meters?
a. 1.018 sq m per sec
b. 1.337 sq m per sec
c. 0.846 sq m per sec
d. 1.632 sq m per sec
203. The structural steel work of a new
office building is finished. Across the
street 20m from the ground floor of the
freight elevator shaft in the building, a
spectator is standing and watching the
freight elevator ascend at a constant rate
of 5 meters per second. How fast is the
angle of elevation of the spectator’s line
of sight to the elevator increasing 6
seconds after his line of sight passes the
horizontal?
a. 1/13
b. 1/15
c. 1/10
d. 1/12
204. A boy rides a bicycle along the
Quezon Bridge at a rate of 6m /s. 24m
directly below the bridge and running at
right angles to it is a highway along
which an automobile is traveling at the
rate of 80m/s. How far is the distance
between the boy and the automobile
changing when the boy is 6m, past the
point directly over the path of the
automobile and the automobile is 8m
past the point directly under the path of
the boy?
a. 26 m/s
b. 20 m/s
c. 28 m/s
d. 30 m/s
205. A point moves on the parabola
y^2=8 in such a way that the rate of
change of the ordinate is always 5 units
per sec. How fast is the abscissa
changing when the ordinate is 4?
a. 5
b. 4
c. 3
d. 7
206. An air traffic controller spots two
planes at the same altitude converging
on a point as they fly at right angles to
one another. One plane is 150miles from
the point and is moving at 450 mph. The
other plane is 200 miles from the point
and has the speed of 600 mph. How
much time does the traffic controller
have to get one of the planes on a
different flight path?
a. 20 min
b. 25 min
c. 30 min
d. 15 min
207. An LRT train 6 m above the ground
crosses a street at a speed of 9 m/s, at the
instant that a car approaching at a speed
of 4 m/s is 12 m up the street. Find the
rate of the LRT train and the car are
separating one second later.
a. 3.64 m/s
b. 4.34 m/s
c. 6.43 m/s
d. 4.63 m/s
208. A street light is 8m from a wall and
4m from a point along the path leading
to the shadow of the man 1.8m tall
shortening along the wall when he is 3m
from the wall. The man walks towards
the wall at the rate of 0.6m/s.
a. -0.192 m/s
b. -1.018 m/s
c. -0.826 m/s
d. -0.027 m/s
209. A mercury light hangs 12 ft above
the island at the center of Ayala Avenue
whish is 24 ft wide. A cigarette vendor
5ft tall walks along the curb of the street
at a speed of 420 fpm. How fast is the tip
of the shadow of the cigarette vendor
moving at the same instant?
a. 12 fps
b. 15 fps
c. 10 fps
d. 14 fps
210. The sides of an equilateral triangle
are increasing at the rate of 10m/s. What
is the length of the sides at the instant
when the area is increasing 100 sq
m/sec?
a. 20/√3
b. 22/√3
c. 25/√3
d. 15/√3
211. Water is the flowing into a conical
vessel 15cm deep and having a radius of
3.75cm across the top. If the rate at
which water is rising is 2cm/s, how fast
is the water flowing into the conical
vessel when the depth of water is 4cm?
a. 6.28 cu m/min
b. 4 cu m/min
c. 2.5 cu m/min
d. 1.5 cu m/min
212. Two sides of a triangle are 5 and 8
units respectively. If the included angle
is changing at the rate of one radian pr
second, at what rate is the third side
changing when the included angle is 60
degrees?
a. 4.95 units/sec
b. 5.55 units/sec
c. 4.24 units/sec
d. 3.87 units/sec
213. The two adjacent sides of a triangle
are 5 and 8 meters respectively. If the
included angle is changing at the rate of
2 rad/sec, at what rate is the area of the
triangle changing if the included angle is
60 degrees?
a. 20 sq m/sec
b. 25 sq m/sec
c. 15 sq m/sec
d. 23 sq m/sec
214. A triangular trough is 12m long, 2m
wide at the top and 2m deep. If water
flows in at the rate of 12 cu m per min,
find how fast the surface is rising when
the water is 1m deep.
a. 1
b. 2
c. 3
d. 4
215. A man starts from a point on a
circular track of radius 100m and walks
along the circumference at the rate of
40m/min. An observer is stationed at a
point on the track directly opposite the
starting point and collinear with the
center of the circular track. How fast is
the man’s distance form the observer
changing after one minute?
a. -7.95 m/min
b. -6.48 m/min
c. 8.62 m/min
d. 9.82 m/min
216. A plane 3000ft from the earth is
flying east at the rate of 120mph. It
passes directly over a car also going east
at 60mph. How fast are they separating
when the distance between them is
5000ft?
a. 70.4 ft/sec
b. 84.3 ft/sec
c. 76.2 ft/sec
d. 63.7 ft/sec
217. A horseman gallops along the
straight shore of a sea at the rate of
30mph. A battleship anchored 3 miles
offshore keeps searchlight trained on
him as he moved along. Find the rate of
rotation of the light when the horseman
is 2 miles down the beach?
a. 6.92 rad/sec
b. 4.67 rad/sec
c. 5.53 rad/sec
d. 6.15 rad/sec
218. Find the point in the parabola
y^2=4x at which the rate of change of
the ordinate and abscissa are equal.
a. (1,2)
b. (-1,4)
c. (2,1)
d. (4,4)
219. Water flows into a vertical
cylindrical tank, at the rate of 1/5 cu
ft/sec. The water surface is rising at the
rate of 0.425ft/min. What is the diameter
of the tank?
a. 6 ft
b. 10 ft
c. 8 ft
d. 4 ft
220. The radius of a sphere is changing
at a rate of 2 cm/sec. Find the rate of
change of the surface area when the
radius is 6cm.
a. 96π sq cm/sec
b. 78π sq cm/sec
c. 84π sq cm/sec
d. 68π sq cm/sec
221. The radius of a circle is increasing
at the rate of 2cm/min. Find the rate of
change of the area when r=6cm.
a. 24 π sq cm/sec
b. 36 π sq cm/sec
c. 18 π sq cm/sec
d. 30 π sq cm/sec
222. All edges of a cube are expanding
at the rate of 3cm/sec. How fast is the
volume changing when each edge is
10cm long?
a. 900 cu cm/sec
b. 800 cu cm/sec
c. 600 cu cm/sec
d. 400 cu cm/sec
223. A spherical balloon is inflated with
gas at the rate of 20 cu m/min. How fast
is the radius of the balloon changing at
the instant the radius is 2cm?
a. 0.398
b. 0.422
c. 0.388
d. 0.498
224. The base radius of a cone is
changing at a rate of 3cm/sec. Find the
rate of change of its volume when the
radius is 4cm and its altitude is 6cm.
a. 48 π cu cm/sec
b. 24 π cu cm/sec
c. 18 π cu cm/sec
d. 36 π cu cm/sec
225. The edge of cube is changing at a
rate of 2 cm/min. Find the rate of change
of its diagonal when each edge is 10cm
long.
a. 3.464 cm/min
b. 5.343 cm/min
c. 2.128 cm/min
d. 6.283 cm/min
226. The radius of a circle is changing at
a rate of 4cm/sec. Determine the rate of
change of the circumference when the
radius is 6cm.
a. 8 π cm/sec
b. 6 π cm/sec
c. 10 π cm/sec
d. 4 π cm/sec
227. When a squares of side x are cut
from the corners of a 12cm square piece
of cardboard, an open top box can be
formed by folding up the sides. The
volume of this box is given by V=x(122x)^2. Find the rate of change of volume
when x=1cm.
a. 60
b. 40
c. 30
d. 20
228. As x increases uniformly at the rate
of 0.002 ft/sec, at what rate is expression
(1+x) to the third power increasing when
x becomes 8ft?
a. 0.486 cfs
b. 0.430 cfs
c. 0.300 cfs
d. 0.346 cfs
229. A trough 10m long has as it ends
isosceles trapezoids, altitude 2m, lower
base, 2m upper base 3m. If water is let in
at a rate of 3 cu m/min, how fast is the
water level rising when the water is 1m
deep?
a. 0.12
b. 0.18
c.0.21
d. 0.28
230. a launch whose deck is 7 m below
the level of a wharf is being pulled
toward the wharf by a rope attached to a
ring on the deck. If a winch pulls in the
rope at the rate of 15 m/min, how fast is
the launch moving through the water
when there are 25m of rope out?
a. -15.625
b. 14.525
c. -14.526
d. 15.148
231. An object is dropped freely from a
bldg. having a height of 40m. An
observer at a horizontal distance of 30m
from a bldg is observing the object is it
was dropped. Determine the rate at
which the distance between the object
and the observer is changing after 2sec.
a. -11.025
b. 12.25
c. -10.85
d. 14.85
232. Car A moves due east at 30kph at
the same instant car B is moving S
30deg E.with a speed of 30kph. The
distance from A to B is 30km. Find how
fast is the speed between them are
separating after one hour.
a. 45 kph
b. 36 kph
c. 40 kph
d. 38 kph
233. Water is flowing into a frustum of a
cone at a rate of 100 liter/min. The upper
radius of the frustum of a cone is 1.5m
while the lower radius is 1m and a height
of 2m. If the water rises at the rate of
0.04916 cm/sec, find the depth of water.
a. 15.5cm
b. 10.3cm
c. 13.6cm
d. 18.9cm
234. Water is flowing into a conical
vessel 18 cm deep and 10 cm across the
top. If the rate at which the water surface
is rising is 27.52 mm/sec, how fast is the
water flowing into the conical vessel
when the depth of water is 12cm?
a. 9.6 cu cm/sec
b. 7.4 cu cm/sec
c. 8.5 cu cm/sec
d. 6.3 cu cm/sec
235. Sand is falling off a conveyor onto
a conical pile at the rate of 15 cu
cm/min. The base of the cone is
approximately twice the altitude. Find
the height of the pile if the height of he
pile is changing at the rate of 0.047746
cm/min.
a. 10 cm
b. 12 cm
c. 8 cm
d. 6 cm
236. A company is increasing its
production of a certain product at the
rate of 100 units per month. The monthly
demand function is given by P=100x/800. Find the rate of change of the
revenue with respect to time in months
when the monthly production is 4000.
a. P9000/month
b. P8000/month
c. P6000/month
d. P4000/month
237. A machine is rolling a metal
cylinder under pressure. The radius of
the cylinder is decreasing at the rate of
0.05 cm/sec and the volume V is 128π
cu cm. At what rate is the length h
changing when the radius is 2.5cm?
a. 0.8192 cm/sec
b. 0.7652 cm/sec
c. 0.6178 cm/sec
d. 0.5214 cm/sec
238. Two sides of a triangle are 15cm
and 20cm long respectively. How fast is
the third side increasing if the angle
between the given sides is 60 degrees
and is increasing at the rate of 2deg/sec?
a. 0.05 cm/s
b. 2.70 cm/s
c. 1.20 cm/s
d. 3.60 cm/s
239. Two sides of a triangle are 30cm
and 40cm respectively. How fast is the
area of the triangle increasing if the
angle between the sides is 60 degrees
and is increasing at the rate of 4deg/sec?
a. 20.94
b. 29.34
c. 14.68
d. 24.58
240. A man 6ft tall is walking toward a
building at the rate of 5ft/sec. If there is
a light on the ground 50ft from a bldg,
how fast is the man’s shadow on the
bldg growing shorter when he is 30ft
from the bldg?
a. -3.75 fps
b. -7.35 fps
c. -5.37 fps
d. -4.86 fps
241. The volume of the sphere is
increasing at the rate of 6 cu cm/hr. At
what rate is its surface area increasing
when the radius is 50 cm (in cu cm/hr)?
a. 0.36
b. 0.50
c. 0.40
d. 0.24
242. A particle moves in a plane
according to the parametric equations of
motions: x=t^2, y=t^3. Find the
magnitude of the acceleration when
t=2/3.
a. 4.47
b. 5.10
c. 4.90
d. 6.12
243. A particle moves along the righthand part of the curve 4y^3=x^2 with a
speed Vy=dy/dx=constant at 2. Find the
speed of motion when y=4.
a. 12.17
b. 14.10
c. 15.31
d. 16.40
244. The equations of motion of a
particle moving in a plane are x=t^2,
y=3t-1 when t is the time and x and y are
rectangular coordinates. Find the speed
of motion at the instant when t=2.
a. 5
b. 7
c. 9
d. 10
245. A particle moves along the parabola
y^2=4x with a constant horizontal
component velocity of 2m/s. Find the
vertical component of the velocity at the
point (1,2).
a. 2 m/s
b. 7 m/s
c. 5 m/s
d. 4 m/s
246. The acceleration of the particle is
given by a=2+12t in m/s^2 where t is the
time in minutes. If the velocity of this
particle is 11 m/s after 1min, find the
velocity after 2minutes.
a. 31 m/s
b. 45 m/s
c. 37 m/s
d. 26 m/s
247. A particle moves along a path
whose parametric equations are x=t^3
and y=2t^2. What is the acceleration
when t=3sec.
a. 18.44 m/sec^2
b. 15.93 m/sec^2
c. 23.36 m/sec^2
d. 10.59 m/sec^2
248. A vehicle moves along a trajectory
having coordinates given as x=t^3 and
y=1-t^2. The acceleration of the vehicle
at any point of the trajectory is a vector
having magnitude and direction. Find the
acceleration when t=2.
a. 12.17
b. 13.20
c. 15.32
d. 12.45
249. The search light of a lighthouse
which is positioned 2km from the
shoreline is tracking a car which is
traveling at a constant speed along the
shore. If the searchlight is rotating at the
rate of 0.25 rev per hour, determine the
speed of the car when it is 1km away
from the point on the shore nearest to the
lighthouse.
a. 3.93 kph
b. 4.16 kph
c. 2.5 kph
d. 1.8 kph
250. A light is at the top of a pole 80 ft
high. A ball is dropped at the same
height from a point 20 ft from the light.
Assuming that the ball falls according to
S=16t^2, how fast is the shadow of the
ball moving along the ground 1 second
later?
a. -200 ft/sec
b. -180 ft/sec
c. -240 ft/sec
d. -140 ft/sec
251. Water is poured at the rate of 8 cu
ft/min into a conical shaped tank, 20 ft
deep and 10 ft diameter at the top. If the
tank has a leak in the bottom and the
water level is rising at the rate of 1
inch/min, when the water is 16 ft deep,
how fast is the water leaking?
a. 3.81 cu ft/min
b. 4.28 cu ft/min
c. 2.96 cu ft/min
d. 5.79 cu ft/min
252. An airplane is flying at a constant
speed at an altitude of 10000ft on a line
that will take it directly over an observer
on the ground. At a given instant the
observer notes that the angle of elevation
of the airplane is π/3 radians and is
increasing at the rate of 1/60 rad/sec.
Find the speed of the airplane.
a. -222.22 ft/sec
b. -232.44 ft/sec
c. -332.22 ft/sec
d. -432.12 ft/sec
253. A horizontal trough is 16 m long
and its ends are isosceles trapezoids with
an altitude of 4m lower base of 4m and
an upper base of 6m. If the water level is
decreasing at the rate of 25 cm/min,
when the water is 3m deep, at what rate
is water being drawn from the trough?
a. 22 cu m/min
b. 25 cu m/min
c. 20 cu m/min
d. 30 cu m/min
254. The sides of an equilateral triangle
is increasing at rate of 10 cm/min. What
is the length of the sides if the area is
increasing at the rate of 69.82 sq
cm/min?
a. 8 cm
b. 10 cm
c. 5 cm
d. 15 cm
255. The two adjacent sides of a triangle
are 6m and 8m respectively. If the
included angle is changing at the rate of
3 rad/min, at what rate is the area of a
triangle changing if the included angle is
30 degrees?
a. 62.35 sq m
b. 65.76 sq m
c. 55.23 sq m
d. 70.32 sq m
256. Water is pouring into a swimming
pool. After t hours, there are t+√t gallons
in the pool. At what rate is the water
pouring into the pool when t=9hours?
a. 7/6 gph
b. 1/6 gph
c. 3/2 gph
d. ½ gph
257. A point on the rim of a flywheel of
radius cm, has a vertical velocity of 50
cm/sec at a point P, 4cm above the x-
axis. What is the angular velocity of the
wheel?
a. 16.67 rad/sec
b. 14.35 rad/sec
c. 19.95 rad/sec
d. 10.22 rad/sec
258. A spherical balloon is filled with air
at the rate of 2 cu cm/min. Compute the
time rate of change of the surface are of
the balloon at the instant when its
volume is 32π/3 cu cm.
a. 2 cu cm/min
b. 3 cu cm/min
c. 4 cu cm/min
c. 5 cu cm/min
259. The coordinate (x,y) in ft of a
moving particle P are given by x=cos(t)1 and y=2sin(t)+1, where t is the time in
seconds. At what extreme rates in fps is
P moving along the curve?
a. 2 and 1
b. 3 and 2
c. 2 and 0.5
d. 3 and 1
260. A bomber plane is flying
horizontally at a velocity of 440 m/s and
drops a bomb to a target h meters below
the plane. At the instant the bomb was
dropped, the angle of depression of the
target is 45 degrees and is increasing at
the rate of 0.05 rad/sec. Determine the
value of h.
a. 4400 m
b. 2040 m
c. 3500 m
d. 6704 m
261. Glycerine is flowing into a conical
vessel 18cm deep and 10 cm across the
top at the rate of 4 cu cm per min. The
deep of glyerine is h cm. If the rate
which the surface is rising is 0.1146
cm/min, find the value of h.
a. 12 cm
b. 16 cm
c. 20 cm
d. 25 cm
262. Helium is escaping from a spherical
balloon at the rate of 2 cu cm/min. When
the surface area is shrinking at the rate of
sq cm/min, find the radius of the
spherical balloon.
a. 12 cm
b. 16 cm
c. 20 cm
d. 25 cm
263. Water is running into hemispherical
bowl having a radius of 10 cm at a
constant rate of 3 cu cm/min. When the
water is h cm deep, the water level is
rising at the rate of 0.0149 cm/min.
What is the value of h?
a. 4 cm
b. 6 cm
c. 2 cm
d. 5 cm
264. A train, starting noon, travels north
at 40 mph. Another train starting from
the same pint at 2pm travels east at
50mph. How fast are the two trains
separating at 3pm?
a. 56.15 mph
b. 98.65 mph
c. 46.51 mph
d. 34.15 mph
265. An automobile is traveling at 30 fps
towards north is approaching an
intersection. When the automobile is
120ft from the intersection, a truck
traveling at 40fps towards east is 60ft
from the same intersection. The
automobile and the truck are on the
roads that are at right angles to each
other. How fast are they separating after
6 sec?
a. 47.83 fps
b. 87.34 fps
c. 23.74 fps
d. 56.47 fps
266. A train, starting noon, travels north
at 40 mph. Another train starting from
the same point at 2pm travels east at 50
mph. How fast are the trains separating
after a long time?
a. 64 mph
b. 69 mph
c. 46 mph
d. 53 mph
267. At noon a car drives from A
towards the east at 60mph. Another car
starts from B towards A at 30 mph. B
has a direction and distance of N 30
degrees east and 42m respectively from
A. Find the time when the cars will be
nearest each other.
a. 24 min after noon
b. 23 min after noon
c. 25 min after noon
d. 26 min after noon
268. A ferris wheel 15 m in diameter
makes 1 rev every 2 min. If the center of
the wheel is 9m above the ground, how
many fast is a passenger in the wheel
moving vertically when he is 12.5 above
the ground?
a. 20.84 m/min
b. 24.08 m/min
c. 22.34 m/min
d. 25.67 m/min
269. A bomber plane, flying horizontally
3.2 km above the ground is sighting on
at a target on the ground directly ahead.
The angle between the line of sight and
the pad of the plane is changing at the
rate of 5/12 rad/min. When the angle is
30 degrees, what is the speed of the
plane in mph?
a. 200
b. 260
c. 220
d. 240
270.
Two
railroad
tracks
are
perpendicular to each other. At 12pm
there is a train at each track was
approaching the crossing at 50kph, one
being 100km the other 150km away
from the crossing. How fast in kph is the
distance between the two trains changing
at 4pm?
a. 67.08 kph
b. 68.08 kph
c. 69.08 kph
d. 70.08 kph
271. a ball is thrown vertically upward
and its distance from the ground is given
as S=104t-16t^2. Find the maximum
height to which the ball will rise if S is
expressed in meters and t in seconds.
a. 169m
b. 190m
c. 187m
d. 169m
272. If f(x)=ax^3+bx^2+cx, determine
the value of a so that the graph will have
a point of inflection at (1,-1) and so that
the slope of the inflection tangent there
will be -3.
a. 2
b. 5
c. 3
d. 4
273. If f(x)=ax^3+bx^2, determine the
values of a and b so that the graph will
have a point of inflection at (2,16).
a. -1, 6
b. -2, 5
c. -1, 7
d. -2, 8
274. Under what condition is the
inflection point of y=ax^3+bx^2+cx+d
on the y-axis?
a. b=0
b. b=1
c. b=3
d. b=4
275. Find the equation of the curve
whose slope is 4x-5 and passing through
(3,1).
a. 2x^2-5x-2
b. 5x^2-9x-1
c. 5x^2+7x-2
d. 2x^2-8x+5
276. The point (3,2) is on a curve and at
any point (x,y) on the curve the tangent
line has a slope equal to 2x-3. Find the
equation of the curve.
a. y=x^2-3x-4
b. y=x^2-3x+2
c. y=x^2+8x+5
d. y=x^3+3x-3
277. If m is the slope of the tangent line
to the curve y=x^2-2x^2+x at the point
(x,y), find the instantaneous rate of
change of the slope m per unit change in
x at the point (2,2).
a. 8
b. 9
c. 10
d. 11
278. Suppose the daily profit from the
production and sale of x units of a
product is given by P=180x-(x^2)/10002000. At what rate is the profit changing
when the number of units produced and
sold is 100 and is increasing at 10 units
per day?
a. P1798
b. P1932
c. P2942
d. P989
279. The population of a city was found
to be given by P=40500e^(0.03t) where t
is the number of years after 1990. At
what rate is the population expected to
be growing in 2000?
a. 1640
b. 2120
c. 2930
d. 1893
280. A bridge is h meters above a river
which lies perpendicular to the bridge. A
motorboat going 3 m/s passes under the
bridge at the same instant that a man
walking 2 m/s reaches that point
simultaneously. If the distance between
them is changing, at the rate of 2.647
m/s after 3 seconds, find the value of h.
a. 10
b. 12
c. 14
d. 8
281. What is the area bounded by the
curve x^2=-9y and the line y+1=0.
a. 6
b. 5
c. 4
d. 3
282. What is the area bounded by the
curve y^2=x and the line x-4=0?
a. 10
b. 32/3
c. 31/3
d. 11
283. What is the area bounded by the
curve y^2=4x and x^2=4y.
a. 6
b. 7.333
c. 6.666
d. 5.333
284. Find the area bounded by the curve
y=9-x^2 and the x-axis.
a. 25 sq units
b. 36 sq units
c. 18 sq units
d. 30 sq units
285. Find the area bounded by the curve
y^2=9x and its latus rectum.
a. 10.5
b. 13.5
c. 11.5
d. 12.5
286. Find the area bounded by the curve
5y^2=164x and the curve y^2=8x-24.
a. 30
b. 20
c. 16
d. 19
287. Find the area bounded by the curve
y^2=4x and the line 2x+y=4.
a. 10
b. 9
c. 7
d. 4
288. Find the area bounded by the curve
y=1/x with and upper limit of y=2 and a
lower limit of y=10.
a. 1.61
b. 2.61
c. 1.81
d. 2.81
289. By integration, determine the area
bounded by the curves y=6x-x^2 and
y=x^2-2x.
a. 25.60 sq units
b. 21.33 sq units
c. 17.78 sq units
d. 30.72 sq units
290. What is the appropriate total area
bounded by the curve y=sin x and y=0
over the interval 0≤x≤2π (in radians).
a. π/2
b. 2
c. 4
d. 0
291. What is the area between y=0,
y=3x^2, x=0 and x=2?
a. 8
b. 24
c. 12
d. 6
292. Determine the tangent to the curve
3y^2=x^3 at (3,3) and calculate the area
of the triangle bounded by the tangent
line, the x-axis and the line x=3.
a. 3.50 sq units
b. 2.50 sq units
c. 3.00 sq units
d. 4.00 sq units
293. Find the areas bounded by the curve
y=8-x^3 and the x-axis.
a. 12 sq units
b. 15 sq units
c. 13 sq units
d. 10 sq units
294. Find the area in the first quadrant
bounded by the parabola, y^2=4x and
the line x=3 and x=1.
a. 9.535
b. 5.595
c. 5.955
d. 9.955
295. Find the area (in sq units) bounded
by the parabola x^2-2y=0 and x^2=2y+8.
a. 11.7
b. 4.7
c. 9.7
d. 10.7
296. In x years from now, one
investment plan will be generating profit
at the rate of R1(x)=50+x^2 pesos per
yr, while a second plan will be
generating
profit
at
the
rate
R2(x)=200+5x pesos per yr. For how
many yrs will the second plan be more
profitable one? Compute also the net
excess profit if the second plan would be
used instead of the first.
a. 15yrs, P1687.50
b. 12yrs, P1450.25
c. 14yrs, P15640.25
d. 10yrs, P1360.25
297. An industrial machine generates
revenue at the rate R(x)=5000-20x^2
pesos per yr and results in cost that
accumulates
at
the
rate
of
C(x)=2000+10x^2 pesos per yr. For how
many yrs (x) is the use of this machine
profitable? Compute also that net
earnings generated by the machine at
this period.
a. 10yrs, P20000
b. 12yrs, P25000
c. 15yrs, P30000
d. 14yrs, P35000
298. Find the area under one arch of the
curve y=sin(x/2).
a. 4
b. 7
c. 3
d. 5
299. Find the area bounded by the curve
y=arc sin x, x=1 and y=π/2 on the first
quadrant.
a. 0
b. 2
c. 1
d. 3
300. Find the area bounded by the curve
y=8-x^3, x=0, y=0.
a. 12
b. 11
c. 15
d. 13
301. Find the area bounded by the curve
y=cos hx, x=0, x=1 and y=0.
a. 1.175
b. 1.234
c. 1.354
d. 1.073
302. Find the area in the first quadrant
under the curve y-sin hx from x=0 to
x=1.
a. 0.543
b. 0.453
c. 0.345
d. 0.623
303. Find the area of the region in the
first quadrant bounded by the curves
y=sin x, y=cos x and the y-axis.
a. 0.414
b. 0.534
c. 0.356
d. 0.486
304. Find the area of the region bounded
by the x-axis, the curve y=6x-x^2 and
the vertical lines x=1 and x=4.
a. 24
b. 23
c. 25
d. 22
305. Find the area bounded by the curve
y=e^x, y=e^-x and x=1, by integration.
a. [(e-1)^2]/e
b. (e^2-1)/e
c. (e-1)/e
d. [(e-1)^2]/(e^2)
306. Suppose a company wants to
introduce a new machine that will
produce a rate of annual savings
S(x)=150-x^2 where x is the number of
yrs of operation of the machine, while
producing a rate of annual costs of
C(x)=(x^2)+(11x/4). For how many
years will it be profitable to use this new
machine?
a. 7 yrs
b. 6 yrs
c. 8 yrs
d. 10 yrs
307. Suppose a company wants to
introduce a new machine that will
produce a rate of annual savings
S(x)=150-x^2 where x is the number of
yrs of operation of the machine, while
producing a rate of annual costs of
C(x)=(x^2)+(11x/4). What are the net
total savings during the first year of use
of the machine?
a. 122
b. 148
c. 257
d. 183
308. Suppose a company wants to
introduce a new machine that will
produce a rate of annual savings
S(x)=150-x^2 where x is the number of
yrs of operation of the machine, while
producing a rate of annual costs of
C(x)=(x^2)+(11x/4). What are the net
total savings over the entire period of
use of the machine?
a. 771
b. 826
c. 653
d. 711
309. The price in pesos for a certain
product is expressed as p(x)=900-80xx^2 when the demand for the product is
x units. Also the function p(x)=x^2+10x
gives the price in pesos when the supply
is x units. Find the consumer and
producers surplus.
a. P4500; P3375
b. P3400; P4422
c. P5420; P3200
d. P4000; P3585
310. A horse is tied ouside of a circular
fence of radius 4m by a rope having a
length of 4π m. Determine the area on
which the horse can graze.
a. 413.42 sq m
b. 484.37 sq m
c. 398.29 sq m
d. 531.36 sq m
311. A dog is tied to an 8m circular tank
by a 3m length of cord. The cord
remains horizontal. Find the area over
which the dog can move.
a. 16.387 sq m
b. 15.298 sq m
c. 10.286 sq m
d. 13.164 sq m
312. Find the area bounded by the curve
y^2=8(x-4), the line y=4, y-axis and xaxis.
a. 18.67
b. 14.67
c. 15.67
d. 17.67
313. Find the area enclosed by the
parabola y^2=8x and the latus rectum.
a. 32/3 sq units
b. 29/4 sq units
c. 41/2 sq units
d. 33/2 sq units
314. What is the area bounded y the
curve x^2=-9y and the line y+1=0
a. 6 sq units
b. 5 sq units
c. 2 sq units
d. 4 sq units
315. What is the area bounded by the
curve y^2=x and the line x-4=0.
a. 23/4 sq units
b. 32/3 sq units
c. 54/4 sq units
d. 13/5 sq units
321. Find the area of the portion of the
curve y=sin x from x=0 to x=π.
a. 2 sq units
b. 3 sq units
c. 1 sq unit
d. 4 sq units
316. Find the area bounded by
The parabola x^2=4y and y=4.
a. 21.33 sq units
b. 33.21 sq units
c. 31.32 sq units
d. 13.23 sq units
322. Find the area bounded by the curve
r^2=4cos2φ.
a. 8 sq units
b. 2 sq units
c. 4 sq units
d. 6 sq units
317. What is the area bounded by the
curve y^2=-2x and the line x=-2.
a. 18/3 sq units
b. 19/5 sq units
c. 16/3 sq units
d. 17/7 sq units
323. Find the area enclosed by the curve
r^2=4cosφ.
a. 4
b. 8
c. 16
d. 2
318. Find the area enclosed by the curve
x^2+8y+16=0 the x-axis, y-axis and the
line x-4=0.
a. 10.67
b. 9.67
c. 8.67
d. 7.67
324. Determine the period and amplitude
of the function y=2sin5x.
a. 2π/5, 2
b. 3π/2, 2
c. π/5, 2
d. 3π/10, 2
319. Find the area bounded by the
parabola
y=6x-x(square)
and
y=x(square)-2x. Note, the parabola
intersects at point (0,0) and (4,8).
a. 44/3
b. 64/3
c. 74/3
d. 54/3
320. Find the area of the portion of the
curve y=cos x from x=0 to x=π/2.
a. 1 sq unit
b. 2 sq units
c. 3 sq units
d. 4 sq units
325. Determine the period and amplitude
of the function y=5cos2x.
a. π, 5
b. 3π/2, 2
c. π/5, 2
d. 3π/10, 2
326. Determine the period and amplitude
of the function y=5sinx.
a. 2π, 5
b. 3π/2, 5
c. π/2, 5
d. π, 5
327. Determine the period and amplitude
of the function y=3 cos x.
a. 2π, 3
b. π/2, 3
c. 3/2, 3
d. π, 3
328. Find the area of the curve
r^2=a^2cosφ.
a. a^2
b. a
c. 2a
d. a^3
329. Find the area of the region bounded
by the curve r^2=16cosθ.
a. 32 sq units
b. 35 sq units
c. 27 sq units
d. 30 sq units
330. Find the area enclosed by the curve
r=a (1-sinθ).
a. (3a^2)π/2
b. (2a^2)π
c. (3a^2)π
d. (3a^2)π/5
331. Find the surface area of the portion
of the curve x^2=y from y=1 to y=2
when it is revolved about the y-axis.
a. 19.84
b. 17.86
c. 16.75
d. 18.94
332. Find the area of the surface
generated by rotating the portion of the
curve y=(x^3)/3 from x=0 to x=1 about
the x-axis.
a. 0.638
b. 0.542
c. 0.782
d. 0.486
333. Find the surface area of the portion
of the curve x^2+y^2=4 from x=0 to x=2
when it is revolved about the y-axis.
a. 8π
b. 16π
c. 4π
d. 12π
334. Compute the surface area generated
when the first quadrant portion if the
curve x^2-4y+8=0 from x=0 to x=2 is
revolved about the y-axis.
a. 30.64
b. 28.32
c. 26.42
d. 31.64
335. Find the total length of the curve
r=4 (1-sin θ) from θ=90deg to θ=270deg
and also the total perimeter of the curve.
a. 16, 32
b. 18, 36
c. 12, 24
d. 15, 30
336. Find the length of the curve r=4sinθ
from θ=0 to θ=90deg and also the total
length of the curve.
a. 2π; 4π
b. 3π; 6π
c. π; 2π
d. 4π; 8π
337. Find the length of the curve r=a (1cos θ) from θ=0 to θ=π and also the total
length of curve.
a. 4a; 8a
b. 2a; 4a
c. 3a; 6a
d. 5a; 10a
338. Find the total length of the curve
r=a cos θ.
a. πa
b. 2πa
c. 3πa/2
d. 2πa/3
339. Find the length of the curve having
a parametric equations of x= a cos^3 θ
y=a sin^2 θ from θ=0 to θ=2π.
a. 5a
b. 6a
c. 7a
d. 8a
340. Find the centroid of the area
bounded by the curve y=4-x^2 the line
x=1 and the coordinate axes.
a. 1.85
b. 0.46
c. 1.57
d. 2.16
341. Find the centroid of the area under
y=4-x^2 in the first quadrant.
a. 0.75
b. 0.25
c. 0.50
d. 1.15
342. Find the centroid of the area in first
quadrant bounded by the curve y^2=4ax
and latus rectum.
a. 3a/5
b. 2a/5
c. 4a/5
d. 1a
343. A triangular section has coordinates
of A(2,2), B(11,2) and C(5,8). Find the
coordinates of the centroid of the
triangular section.
a. (7, 4)
b. (6, 4)
c. (8, 4)
d. (9, 4)
344. The following cross section has the
following given coordinates. Compute
for the centroid of the given cross
section A(2,2); B(5,8); C(7,2); D(2,0)
and E(7,0).
a. 4.6, 3.4
b. 4.8, 2.9
c. 5.2, 3.8
d. 5.3, 4.1
345. Sections ABCD is a quadrilateral
having the given coordinates A(2,3);
B(8,9); C(11,3); D(11,0). Compute the
coordinates of the centroid of the
quadrilateral.
a. (7.33, 4)
b. (7, 4)
c. (6.22, 3.8)
d. (7.8, 4.2)
346. A cross section consists of a
triangle ABC and a semi circle with AC
as its diameter. If the coordinates of
A(2,6); B(11,9) and C(14,6), compute
the coordinates of the centroid of the
cross section.
a. 4.6, 3.4
b. 4.8, 2.9
c. 5.2, 3.8
d. 5.3, 4.1
347. Locate the centroid of the area
bounded by the parabola y^2=4x, the
line y=4 ad the y-axis.
a. 6/5, 3
b. 2/5, 3
c. 3/5, 3
d. 4/5, 3
348. Find the centroid of the area
bounded by the curve x^2=-(y-4), the xaxis and the y-axis on the first quadrant.
a. ¾, 8/5
b. 5/4, 7/5
c. 7/4, 6/5
d. ¼, 9/5
349. Locate the centroid of the area
bounded by the curve y^2=-3(x-6)/2 the
x-axis and the y-axis on the first
quadrant.
a. 12/5, 9/8
b. 13/5, 7/8
c. 14/5, 5/8
d. 11/5, 11/8
350. Locate the centroid of the area
bounded by the curve 5y^2=16x and
y^2=8x-24 on the first quadrant.
a. x=2.20; y=1.51
b. x=1.50; y=0.25
c. x=2.78; y=1.39
d. x=1.64; y=0.26
355. Find the volume formed by
revolving the hyperbola xy=6 from x=2
to x=4 about the x-axis.
a. 28.27 cu units
b. 25.53 cu units
c. 23.23 cu units
d. 30.43 cu units
351. Locate the centroid of the area
bounded by the parabola x^2=8y and
x^2=16(y-2) in the first quadrant.
a. x=2.12; y=1.6
b. x=3.25; y=1.2
c. x=2.67; y=2.0
d. x=2; y=2.8
356. The region in the first quadrant
under the curve y=sin h x from x=0 to
x=1 is revolved about the x-axis.
Compute the volume of solid generated.
a. 1.278 cu units
b. 2.123 cu units
c. 3.156 cu units
d. 1.849 cu units
352. Given the area in the first quadrant
bounded by x^2=8y, the line y-2 and the
y-axis. What is the volume generated
this area is revolved about the line y2=0?
a. 53.31 cu units
b. 45.87 cu units
c. 28.81 cu units
d. 33.98 cu units
353. Given the area in the first quadrant
bounded by x^2=8y, the line x=4 and the
x-axis. What is the volume generated by
revolving this area about y-axis?
a. 78.987 cu units
b. 50.265 cu units
c. 61.523 cu units
d. 82.285 cu units
354. Given the area in the first quadrant
bounded by x^2=8y, the line y-2=0 and
the y-axis. What is the volume generated
when this area is revolved about the xaxis?
a. 20.32 cu units
b. 34.45 cu units
c. 40.21 cu units
d. 45.56 cu units
357. A square hole of side 2cm is
chiseled perpendicular to the side of a
cylindrical post of radius 2cm. If the axis
of the hole is going to be along the
diameter of the circular section of the
post, find the volume cut off.
a. 15.3 cu cm
b. 23.8 cu cm
c. 43.7 cu cm
d. 16.4 cu cm
358. A hole radius 1cm is bored through
a sphere of radius 3cm, the axis of the
hole being a diameter of a sphere. Find
the volume of the sphere which remains.
a. (64π√2)/3 cu cm
b. (66π√3)/3 cu cm
c. (70π√2)/3 cu cm
d. (60π√2)/3 cu cm
359. Find the volume of common to the
cylinders x^2+y^2=9 and y^2+z^2=9.
a. 241 cu m
b. 533 cu m
c. 424 cu m
d. 144 cu m
360. Given is the area in the first
quadrant bounded by x^2=8y, the line y-
2=0 and the y-axis. What is the volume
generated when this area is revolved
about the line y-2=0.
a. 28.41
b. 26.81
c. 27.32
d. 25.83
365. Find the volume of the solid formed
if we rotate the ellipse (x^2)/9 + (y^2)/4
= 1 about the line 4x+3y=20.
a. 48 π^2 cu units
b. 45 π^2 cu units
c. 40 π^2 cu units
d. 53 π^2 cu units
361. Given is the area in the first
quadrant bounded by x^2=8y, the line
x=4 and the x-axis. What is the volume
generated when this area is revolved
about the y-axis?
a. 50.26
b. 52.26
c. 53.26
d. 51.26
366. The area on the first and second
quadrant of the circle x^2+y^2=36 is
revolved about the line x=6. What is the
volume generated?
a. 2131.83
b. 2242.46
c. 2421.36
d. 2342.38
362. The area bounded by the curve
y^2=12 and the line x=3 is revolved
about the line x=3. What is the volume
generated?
a. 185
b. 187
c. 181
d. 183
363. The area in the second quadrant of
the circle x^2+y^2=36 is revolved about
the line y+10=0. What is the volume
generated?
a. 2218.63
b. 2228.83
c. 2233.43
d. 2208.53
364. The area enclosed by the ellipse
(x^2)/9 + (y^2)/4 = 1 is revolved about
the line x=3, what is the volume
generated?
a. 370.3
b. 360.1
c. 355.3
d. 365.10
367. The area on the first quadrant of the
circle x^2+y^2=25 is revolved about the
line x=5. What is the volume generated?
a. 355.31
b. 365.44
c. 368.33
d. 370.32
368. The area on the second and third
quadrant of the circle x^2+y^2=36 is
revolved about the line x=4. What is the
volume generated?
a. 2320.30
b. 2545.34
c. 2327.25
d. 2520.40
369. The area on the first quadrant of the
circle x^2+y^2=36 is revolved about the
line y+10=0. What is the volume
generated?
a. 3924.60
b. 2229.54
c. 2593.45
d. 2696.50
370. The area enclosed by the ellipse
(x^2)/16 + (y^2)/9 = 1 on the first and
2nd quadrant is revolved about the x-axis.
What is the volume generated?
a. 151.40
b. 155.39
c. 156.30
d. 150.41
line x=4. Locate the centroid of the
resulting solid of revolution.
a. 0.8
b. 0.5
c. 1
d. 0.6
371. The area enclosed by the ellipse
9x^2+16y^2=144 on the first quadrant is
revolved about the y-axis. What is the
volume generated?
a. 100.67
b. 200.98
c. 98.60
d. 54.80
376. The area bounded by the curve
x^3=y, the line y=8 and the y-axis is to
be revolved about the y-axis. Determine
the centroid of the volume generated.
a. 5
b. 6
c. 4
d. 7
372. Find the volume of an ellipsoid
having the equation (x^2)/25 + (y^2)/16
+ (z^2)/4 = 1.
a. 167.55
b. 178.40
c. 171.30
d. 210.20
377. The area bounded by the curve
x^3=y, and the x-axis is to be revolved
about the x-axis. Determine the centroid
of the volume generated.
a. 7/4
b. 9/4
c. 5/4
d. ¾
373. Find the volume of a prolate
spheroid having the equation (x^2)/25 +
(y^2)/9 + (z^2)/9 = 1.
a. 178.90 cu units
b. 184.45 cu units
c. 188.50 cu units
d. 213.45 cu units
374. The region in the first quadrant
which is bounded by the curve y^2=4x,
and the lines x=4 and y=0, is revolved
about the x-axis. Locate the centroid of
the resulting solid of revolution.
a. 8/3
b. 7/3
c. 10/3
d. 5/3
375. The region in the first quadrant
which is bounded by the curve x^2=4y,
and the line x=4, is revolved about the
378. The region in the 2 nd quadrant,
which is bounded by the curve x^2=4y,
and the line x=-4, is revolved about the
x-axis. Locate the cenroid of the
resulting solid of revolution.
a. -4.28
b. -3.33
c. -5.35
d. -2.77
379. The region in the 1 st quadrant,
which is bounded by the curve y^2=4x,
and the line x=-4, is revolved about the
line x=4. Locate the cenroid of the
resulting solid of revolution.
a. 1.25 units
b. 2 units
c. 1.50 units
d. 1 unit
380. Find the moment of inertia of the
area bounded by the curve x^2=4y, the
line y=1 and the y-axis on the first
quadrant with respect to x-axis.
a. 6/5
b. 7/2
c. 4/7
d. 8/7
385. Find the moment of inertia of the
area bounded by the curve y^2=4x, the
line x=1 and the x-axis on the first
quadrant with respect to y-axis.
a. 0.571
b. 0.682
c. 0.436
d. 0.716
381. Find the moment of inertia of the
area bounded by the curve x^2=4y, the
line y=1 and the y-axis on the first
quadrant with respect to y-axis.
a. 19/3
b. 16/15
c. 13/15
d. 15/16
386. Determine the moment of inertia
with respect to x-axis of the region in the
first quadrant which is bounded by the
curve y^2=4x, the line y=2 and y-axis.
a. 1.6
b. 2.3
c. 1.3
d. 1.9
382. Find the moment of inertia of the
area bounded by the curve x^2=8y, the
line x=4 and the x-axis on the first
quadrant with respect to x-axis.
a. 1.52
b. 2.61
c. 1.98
d. 2.36
387. Find the moment of inertia of the
area bounded by the curve y^2=4x, the
line y=2 and the y-axis on the first
quadrant with respect to y-axis.
a. 0.095
b. 0.064
c. 0.088
d. 0.076
383. Find the moment of inertia of the
area bounded by the curve x^2=8y, the
line x=4 and the x-axis on the first
quadrant with respect to y-axis.
a. 25.6
b. 21.8
c. 31.6
d. 36.4
388. Find the moment of inertia with
respect to x-axis of the area bounded by
the parabola y^2=4x and the line x=1.
a. 2.35
b. 2.68
c. 2.13
d. 2.56
384. Find the moment of inertia of the
area bounded by the curve y^2=4x, the
line x=1 and the x-axis on the first
quadrant with respect to x-axis.
a. 1.067
b. 1.142
c. 1.861
d. 1.232
389. What is the integral of
sin^6(φ)cos^4 (φ) dφ if the upper limit is
π/2 and lower limit is 0?
a. 0.0184
b. 1.0483
c. 0.1398
d. 0.9237
390. Evaluate the integral of cos^7 φ
sin^5 φ dφ if the upper limit is 0.
a. 0.1047
b. 0.0083
c. 1.0387
d. 1.3852
391. What is the integral of sin^4 x dx if
the lower limit is 0 and the upper limit is
π/2?
a. 1.082
b. 0.927
c. 2.133
d. 0.589
392. Evaluate the integral of cos^5 φ dφ
if the lower limit is 0 and the upper limit
is π/2.
a. 0.533
b. 0.084
c. 1.203
d. 1.027
393. Evaluate the integral (cos3A)^8 dA
from 0 to π/6.
a. 27π/363
b. 35π/768
c. 23π/765
d. 12π/81
394. What is the integral of sin^5 x
cos^3 x dx if the lower limit is 0 and the
upper limit is π/2?
a. 0.0208
b. 0.0833
c. 0.0278
d. 0.0417
395. Evaluate the integral of 15sin^7 (x)
dx from 0 to π/2.
a. 6.857
b. 4.382
c. 5.394
d. 6.139
396. Evaluate the integral of 5 cos^6 x
sin^2 x dx if the upper limit is π/2 and
the lower limit is 0.
a. 0.307
b. 0.294
c. 0.415
d. 0.186
397. Evaluate the integral of 3(sin x)^3
dx from 0 to π/2.
a. 2
b. π
c. 6
d. π/2
398. A rectangular plate is 4 feet long
and 2 feet wide. It is submerged
vertically in water with the upper 4 feet
parallel and to 3 feet below the surface.
Find the magnitude of the resultant force
against one side of the plate.
a. 38 w
b. 32 w
c. 27 w
d. 25 w
399. Find the force on one face of a right
triangle of sides 4 m, and altitude of 3m.
The altitude is submerged vertically with
the 4m side in the surface.
a. 58.86 kN
b. 53.22 kN
c. 62.64 kN
d. 66.27 kN
400. A plate in the form of a parabolic
segment of base 12m and height of 4m is
submerged in water so that the base is in
the surface of the liquid. Find the force
on the face of the plate.
a. 502.2 kN
b. 510.5 kN
c. 520.6 kN
d. 489.1 kN
401. A circular water main 4 meter in
diam. is closed by a bulkhead whose
center is 40 m below the surface of the
water in the reservoir. Find the force on
the bulkhead.
a. 4931 kN
b. 5028 kN
c. 3419 kN
d. 4319 kN
402. A plate in the form of parabolic
segment is 12m in height and 4m deep
and is partly submerged in water so that
its axis is parallel to end 3m below the
water surface. Find the force acting on
the plate.
a. 993.26 kN
b. 939.46 kN
c. 933.17 kN
d. 899.21 kN
403. A cistern in the form of an inverted
right circular cone is 20 m deep and 12
m diameter at the top. If the water is 16
m deep in the cistern, find the work done
in Joules in pumping out the water. The
water is raised to a point of discharge 10
m above the top cistern.
a. 68166750 Joules
b. 54883992 Joules
c. 61772263 Joules
d. 76177640 Joules
404. A bag containing originally 60 kg
of flour is lifted through a vertical
distance of 9m. While it is being lifted,
flour is leaking from the bag at such rate
that the number of pounds lost is
proportional to the square root of the
distance traversed. If the total loss of
flour is 12 kg find the amount of work
done in lifting the bag.
a. 4591 Joules
b. 4290 Joules
c. 5338 Joules
d. 6212 Joules
405. According to Hooke’s law, the
force required to stretch a helical spring
is proportional to the distance stretched.
The natural length of a given spring is 8
cm. a force of 4kg will stretch it to a
total length of 10 cm. Find the work
done in stretching it from its natural
length to a total length of 16 cm.
a. 6.28 Joules
b. 5.32 Joules
c. 4.65 Joules
d. 7.17 Joules
406. The top of an elliptical conical
reservoir is an ellipse with major axis
6m and minor axis 4m. it is 6m deep and
full of water. Find the work done in
pumping the water to an outlet at the top
of the reservoir.
a. 554742 Joules
b. 473725 Joules
c. 493722 Joules
d. 593722 Joules
407. A bag of sand originally weighing
144 kg is lifted at a rate of 3m/min. the
sand leaks out uniformly at such rate that
half of the sand is lost when the bag has
been lifted 18m. find the work done in
lifting the bag of sand at this distance.
a. 6351 Joules
b. 4591 Joules
c. 5349 Joules
d. 5017 Joules
408. A cylindrical tank having a radius
of 2m and a height of 8m is filled with
water at a depth of 6m. Compute the
work done in pumping all the liquid out
of the top of the container.
d. 6.29 Joules
a. 3 698 283 Joules
b. 4 233 946 Joules
c. 5 163 948 Joules
d. 2 934 942 Joules
409. A right cylindrical tank of radius
2m and a height 8m is full of water. Find
the work done in pumping the tank.
Assume water to weigh 9810 N/m^3.
a. 3945 kN . m
b. 4136 kN . m
c. 2846 kN . m
d. 5237 kN . m
410. A conical vessel 12m across the top
and 15m deep. If it contains water to a
depth of 10m find the work done in
pumping the liquid to the top of the
vessel.
a. 12 327.5 kN . m
b. 24 216.2 kN . m
c. 14 812.42 kN . m
d. 31 621 kN . m
411. A hemispherical vessel of diameter
8m is full of water. Determine the work
done in pumping out the top of the tank
in Joules.
a. 326 740 pi
b. 627 840 pi
c. 516 320 pi
d. 418 640 pi
412. A spring with a natural length of
10cm is stretched by 1/2 cm by a
Newton force. Find the work done in
stretching from 10 cm to 18cm. Express
your answer in joules.
a. 7.68 Joules
b. 8.38 Joules
c. 7.13 Joules
413. A 5 lb. monkey is attached to a 20
ft hanging rope that weighs 0.3 lb/ft. the
monkey climbs the rope up to the top.
How much work has it done?
a. 160
b. 170
c. 165
d. 180
414. A bucket weighing 10 Newton
when empty is loaded with 90 Newton
of sand and lifted at 10 cm at a constant
speed. Sand leaks out of a hole in a
bucket at a uniform rate and one third of
sand is lost by the end of the lifting
process in Joules.
a. 850 Joules
b. 900 Joules
c. 950 Joules
d. 800 Joules
415. A conical vessel is 12 m across the
top and 15 m deep. If it contains water to
a depth of 10m find the work done in
pumping the liquid to a height 3m above
the top of the vessel.
a. 560pi w N.m
b. 660 pi w N.m
c. 520 pi w N.m
d. 580 pi w N.m
416. A small in the sack of rice cause
some rice to be wasted while the sack is
being lifted vertically to a height of 30m.
The weight lost is proportional to the
cube root of distance traversed. If the
total loss was 16 kg, find the work done
in lifting the said sack of rice which
weighs 110kg.
a. 2940 kg.m
b. 2369 kg.m
c. 3108 kg.m
d. 2409 kg.m
421. Find the differential equations of
the family of lines passing through the
origin.
417. A hemispherical tank of diameter
20 ft is full of oil weighing 20pcf. The
oil is pumped to a height of 10 ft, above
the top of the tank by an engine of 1/2
horsepower. How long will it take the
engine to empty the tank?
a. ydx – xdy = 0
b. xdy – ydx = 0
c. xdx + ydy = 0
d. ydx + xdy = 0
a. 1 hr. 44.72 min
b. 1 hr. 15.47 min
c. 1 hr. 24.27 min
d. 2 hrs.
418. A full tank consists of a hemisphere
of radius 4m surmounted by a circular
cylinder of the same radius of altitude
8m. Find the work done in pumping the
water to an outlet of the top of the tank.
a. (2752/3) pi w
b. (2255/3) pi w
c. (2527/3) pi w
d. (5722/3) pi w
419. Determine the differential equation
of a family of lines passing thru (h, k).
a. (y-k) dx – (x-h) dy = 0
b. (x-h) + (y-k) = dy/dx
c. (x-h) dx – (y-k) dy = 0
d. (x+h) dx – (y-k) dy = 0
420. What is the differential equation of
the family of parabolas having their
vertices at the origin and their foci on the
x-axis
a. 2x dy – y dx = 0
b. x dy + y dx = 0
c. 2y dx – x dy = 0
d. dy/dx – x = 0
422. The radius of the moon is 1080
miles. The gravitation acceleration of the
moons surface is 0.165 miles the
gravitational acceleration at the earth’s
surface. What is the velocity of escape
from the moon in miles per second?
a. 2.38
b. 1.47
c. 3.52
d. 4.26
423. Find the equation of the curve at
every point of which the tangent line has
a slope of 2x.
a. x = -y^2 + C
b. y = -x^2 + C
c. x = y^2 + C
d. y = x^2 + C
424. The radius of the earth is 3960
miles. If the gravitational acceleration of
earth surface is 31.16 ft/sec^2, what is
the velocity of escape from the earth in
miles/sec?
a. 6.9455
b. 5.4244
c. 3.9266
d. 7.1842
425. Find the velocity of escape of the
Apollo spaceship as it is projected from
the earth’s surface that is the minimum
velocity imparted to it so that it will
never return. The radius of the earth is
400 miles and the acceleration of the
spaceship is 32.2 ft/sec^2.
c. 62.18
d. 59.24
a. 40478 kph
b. 50236 kph
c. 30426 kph
d. 60426 kph
430. The Bureau of Census record in
1980 shows that the population in the
country doubles compared to that of
1960. In what year will the population
trebles assuming that the rate of increase
in the population is proportional to the
population?
426. The rate of population growth of a
country is proportional to the number of
inhabitants. If a population of a country
now is 40 million and expected to
double in 25 years, in how many years is
the population be 3 times the present?
a. 39.62 yrs.
b. 28.62 yrs.
c. 18.64 yrs.
d. 41.2 yrs.
a. 34.60
b. 31.70
c. 45.65
d. 38.45
427. From the given differential equation
xdx+6y^5dy = 0 solve for the constant
of integration when x = 0, y = 2.
431. A tank contains 200 liters of fresh
water. Brine containing 2 kg/liter of salt
enters the tank at the rate of 4 liters per
min, and the mixture kept uniform by
stirring, runs out at 3 liters per min. Find
the amount of salt in the tank after 30
min.
a. 27x dx + 4y^2 dy = 0
b. 58
c. 48
d. 64
a. 196.99 kg
b. 186.50 kg
c. 312.69 kg
d. 234.28 kg
428. Find the equation of the curve
which passes through points (1, 4) and
(0, 2) if d^2 y/ dx^2 = 1
432. In a tank are 100 liters of brine
containing 50 kg total of dissolved salt.
Pure water is allowed to run into the tank
at the rate of 3 liters per minute. Brine
runs out of the tank at rate of 2 liters per
minute. The instantaneous concentration
in the tank is kept uniform by stirring.
How much salt is in the tank at the end
of 1 hour?
a. 2y = x^2 + 3x + 4
b. 4y = 2x^2 + x + 4
c. 5y = x^2 + 2x + 2
d. 3y = x^2 + x + 4
429. The rate of population growth of a
country is proportional to the number of
inhabitants. If a population of a country
now is 40 million and 50 million in 10
years time, what will be its population
20years from now?
a. 56.19
b. 71.29
a. 20.50
b. 18.63
c. 19.53
d. 22.40
433. Determine the general solution of
xdy + ydx=0.
a. xy = c
b. ln xy = c
c. ln x + ln y = c
d. x + y = c
434. The inverse laplace transform of
s/[(square) + (w square)] is:
a. sin wt
b. w
c. (e exponent wt)
d. cos st
435. The laplace transform of cos wt is:
a. s/[(square) + (w square)]
b. w/[(square) + (w square)]
c. w/s + w
d. s/s + w
436. K divided by [(s square) + (k
square)] is inverse laplace transform of:
a. cos kt
b. sin kt
c. (e exponent Ky)
d. 1.0
437. Find the inverse transform of
[2/(s+1)] – [(4/(s+3)] is equal to:
a. [2 e (exp – t) – 4e (exp – 3t)]
b. [e (exp – 2t) + e (exp – 3t)]
c. [e (exp – 2t) – e (exp - 3t)]
d. [2e (exp – t) – 2e (exp - 2t)]
438. What is the laplace transform of
e^(-4t)
a. 1/ (s + 1)
b. 1/ (s + 4)
c. 1/ (s – 4)
d. 1/ (s + t)
439. Determine the laplace transform of
I(S) = 200 / [(s^2) + 50s + 10625]
a. I(S) = 2e^(-25t) sin100t
b. I(S) = 2te^(-25t) sin100t
c. I(S) = 2e^(-25t) cos100t
d. I(S) = 2te^(-25t) cos100t
440. Determine the inverse laplace
transform of (s+a) / [(s+a) ^2 + w^2]
a. e^(-at) sin wt
b. te^(-at) cos wt
c. t sin wt
d. e^(-at) cos wt
441. Determine the inverse laplace
transform of 100/ [(S+10) (S+20)]
a. 10e^(-10t) – 20e^(-20t)
b. 10e^(-10t) + 20e^(-20t)
c. 10e^(-10t) – 10e^(-20t)
d. 20e^(-10t) + 10e^(-20t)
442. A thin heavy uniform iron rod 16m
long is bent at the 10 m mark forming a
right angle L – shaped piece 6m by 10m
of bend. What angle does the 10m side
make with the vertical when the system
is in equilibrium?
a. 28° 12’
b. 19° 48’
c. 24° 36’
d. 26° 14’
443. Three men carry a uniform timber.
One takes hold at one end and the other
two carry by means of a crossbar placed
underneath. At what point of timber
must the bar be placed so that each man
may carry one third of the weight of the
weight of the timber? The timber has a
length of 12 m.
a. 4m
b. 5m
c. 2.5 m
d. 3m
444. A painters scaffold 30m long and a
mass of 300 kg, is supported in a
horizontal position by a vertical ropes
attached at equal distances from the ends
of the scaffold. Find the greatest distance
from the ends that the ropes may be
attached so as to permit a 200 kg man to
stand safely at one end of scaffold.
a. 8m
b. 7m
c. 6m
d. 9m
445. A cylindrical tank having a
diameter of 16 cm weighing 100 kN is
resting on a horizontal floor. A block
having a height of 4 cm is placed on the
side of the cylindrical tank to prevent it
from rolling. What horizontal force must
be applied at the top of the cylindrical
tank so that it will start to roll over the
block? Assume the block will not slide
and is firmly attached to the horizontal
floor.
a. 68.36 kN
b. 75.42 kN
c. 58.36 kN
d. 57.74 kN
446. Two identical sphere weighing 100
kN are each place in a container such
that the lower sphere will be resting on a
vertical wall and a horizontal wall and
the other sphere will be resting on the
lower sphere and a wall making an angle
of 60 degrees with the horizontal. The
line connecting the two centers of the
spheres makes an angle of 30 degrees
with the horizontal surface. Determine
the reaction between the contact of the
two spheres. Assume the walls to be
frictionless.
a. 150
b. 120
c. 180
d. 100
447. The 5 m uniform steel beam has a
mass of 600 kg and is to be lifted from
the ring B with two chains, AB of length
3m, and CB of length 4m. Determine the
tension T in chain AB when the beam is
clear of the platform.
a. 2.47 kN
b. 3.68 kN
c. 5.42 kN
d. 4.52 kN
448. A man attempts to support a stack
of books horizontally by applying a
compressive force of F=120 N to the
ends of the stack with his hands,
determine the number of books that can
be supported in the stack if the
coefficient of friction between any two
books is 0.40.
a. 15 books
b. 20 books
c. 10 books
d. 12 books
449. Two men are just to lift a 300 kg
weight of crowbar when the fulcrum for
this lever is 0.3m from the weight and
the man exerts their strengths at 0.9 m
and 1.5 m respectively from the fulcrum.
If the men interchange positions, they
can raise a 340 kg weight. What force
does each man exert?
a. 25 kg, 40 kg
b. 35 kg, 45 kg
c. 40 kg, 50 kg
d. 30 kg, 50 kg
450. A man exert a maximum pull of
1000 N but wishes to lift a new stone
door for his cave weighing 20 000 N. if
he uses lever how much closer must the
fulcrum be to the stone than to his hand?
a. 10 times nearer
b. 20 times farther
c. 10 times farther
d. 20 times nearer
451. A simple beam having a span of 6m
has a weight of 20 kN/m. It carries a
concentrated load of 20 kN at the left
end and 40 kN at 2m from the right end
of the beam. If it is supported at 2m from
the left end and the right end, compute
the reaction at the right end of the beam.
a. 40 kN
b. 20 kN
c. 50 kN
d. 30 kN
452. When one boy is sitting 1.20 m
from the center of a seesaw another boy
must sit on the other side 1.50 m from
the center to maintain an even balance.
However, when the first boy carries an
additional weight of 14 kg and sit 1.80 m
from the center, the second boy must
move 3m from the center to balance,
Neglecting the weight of the see weight
of the heaviest boy.
point A is hinged on the wall and joint C
is also hinged connecting the links AC
and CB. AC is horizontal while B is
supported by roller acting on the wall
AB.
a. 2700 N
b. 3600 N
c. 300 N
d. 2200 N
454. An airtight closed box of weight P
is suspended from a spring balance. A
bird of weight W is place on the floor of
the bow, and the balance reads W + P. If
the bird flies without accelerating. What
is the balance reading?
a. P + W
b. P
c. P – W
d. P + 2W
455. A tripod whose legs are each 4
meters long supports a load of 1000 kg.
the feet of the tripod are at the vertices
of a horizontal equilateral triangle whose
side are 3.5 meters. Determine the load
of each leg.
a. 386.19 kg
b. 347.29 kg
c. 214.69 kg
d. 446.27 kg
a. 42 kg
b. 35 kg
c. 58 kg
d. 29 kg
456. A uniform square table top ABCD
having sides 4m long is supported by
three vertical supports at A, E and F, E is
midway n the side BC and F is 1m from
D along the side DC. Determine the
share of load in percent carried by
supports at A, E and F.
453. A wire connects a middle of links
AC and AB compute the tension in the
wire if AC carries a uniform load of 600
N/m. AC is 4.5 m long and BC is 7.5 m.
a. A = 29%, E = 42%, F = 29%
b. A = 32%, E = 46%, F = 20%
c. A = 28%, E = 40%, F = 32%
d. A = 36%, E = 32%, F = 32%
457. The square steel plate has a mass of
1800 kg with mass at its center G.
Calculate the tension at each of the three
cables with which the plate is lifted
while remaining horizontal.
a. Ta = Tb = 6.23 kN, Tc = 10.47 kN
b. Ta = Tb = 7.47 kN, Tc = 7.84 kN
c. Ta = Tb = 5.41 kN, Tc = 9.87 kN
d. Ta = Tb = 4.42 kN, Tc = 6.27 kN
458. A horizontal Circular platform of
radius R is supported at three points A,
B and C on its circumference. A and B
are 90 degrees apart and C is 120
degrees from A. The platform carries a
vertical load of 400 kN at its center and
100 kN at a point d on the circumference
diametrically opposite A. Compute the
reaction at C.
a. 253.45 kN
b. 321.23 kN
c. 310.10 kN
d. 287.67 kN
459. A ladder 4m long having a mass of
15kg is resting against a floor and an
wall for which the coefficients of static
friction are 0.30 for the floor to which a
man having a mass of 70 kg can climb
without causing the plank to slip if the
plank makes an angle of 40 degrees with
the horizontal.
a. 2
b. 1
c. 2.5
d. 3
460. A homogenous block having
dimension of 4cm by 8cm is resting on
an inclined plane making an angle of θ
with the horizontal. The block has a
weight of 20 kN. If the coefficient of
friction between the block and the
inclined plane is 0.55, find the value of θ
before the block starts to move. The 8cm
side is perpendicular to the inclined
plane.
a. 26.57°
b. 28.81°
c. 27.7°
d. 23.4°
461. A uniform ladder on a wall at A and
at the floor at B. Point A is 3.6m above
the floor and point B is 1.5m away from
the wall. Determine the minimum
coefficient of friction at B required for a
mass weighing 65 kg to use the ladder
assuming that there is no friction at A.
a. 0.42
b. 0.50
c. 0.48
d. 0.54
462. A block having a mass of 250 kg is
placed on top of an inclined plane
having a slope of 3 vertical to 4
horizontal. If the coefficient of friction
between the block and the inclined plane
is 0.15, determine the force P that may
be applied parallel to the inclined plane
to keep block from sliding down the
plane.
a. 1177.2 N
b. 1088.2 N
c. 980.86 N
d. 1205.30 N
463. A 3.6 m ladder weighing 180 N is
resting on a horizontal floor at A and on
the wall at B making an angle of 30
degrees from the vertical wall. When a
man weighing 800 N reaches a point
2.4m from the lower end (point A), the
ladder is just about to slip. Determine the
coefficient of friction between the ladder
and the floor if the coefficient of friction
between the ladder and the wall is 0.20.
a. 0.35
b. 0.42
c. 0.28
d. 0.56
464. A dockworker adjusts a spring line
(rope) which keeps the ship from drifting
along side a wharf. If he exerts a pull of
200N on the rope, which ahs 1 ¼ turns
around the mooring bit, what force T can
he support? The coefficient of friction
between the rope and the cast-steel
mooring bit is 0.30.
a. 2110 N
b. 1860 N
c. 155 N
d. 142 N
465. Determine the distance “x” to
which the 90 kg painter can climb
without causing the 4m ladder to slip at
its lower end A. The top of the 15 kg
ladder has a small roller, and the ground
coefficient of static friction is 0.25. the
lower end of the ladder is 1.5 m away
from the wall.
a. 2.55 m
b. 3.17 m
c. 1.58 m
d. 0.1 m
466. The uniform pole of length 4m and
mass 100kg is leaned against a vertical
wall. If the coefficient of static friction
between the supporting surfaces and the
ends of the poles is 0.25, calculate the
maximum angle θ at which the pole may
be placed with the vertical wall before it
starts to slip.
a. 28.07°
b. 26.57°
c. 31.6°
d. 33.5°
467. A horizontal force P acts on the top
of a 30 kg block having a width of 25
cm, and a height of 50cm. if the
coefficient of friction between the block
and the plane is 0.33, what is the value
of P for motion to impend?
a. 7.5 kg
b. 5.3 kg
c. 6.6 kg
d. 8.2 kg
468. A 600 N block rests on a 30° plane.
If the coefficient of static friction is 0.30
and the coefficient of kinetic friction is
0.20, what is the value of P applied
horizontally to prevent the block from
sliding down the plane?
a. 141.85 N
b. 183.29 N
c. 119.27 N
d. 126.59 N
469. A 600 N block rests on a 30° plane.
If the coefficient of static friction is 0.30
and the coefficient of kinetic friction is
0.20, what is the value of P applied
horizontally to keep the block moving up
the plane?
a. 527.31 N
b. 569.29 N
c. 427.46 N
d. 624.17 N
470. Solve for the force P to obtain
equilibrium. Angle of friction is 25°
between block and the inclined plane.
a. 96.46 kg
b. 77.65 kg
c. 69.38 kg
d. 84.22 kg
471. A 200 kg crate impends to slide
down a ramp inclined at an angle of
19.29° with the horizontal. What is the
frictional resistance? Use g = 9.81
m/s^2.
a. 648.16 N
b. 638.15 N
c. 618.15 N
d. 628.15 N
472. A 40kg block is resting on an
inclined plane making an angle of 20°
from the horizontal. If the coefficient of
friction is 0.60, determine the force
parallel to the incline that must be
applied to cause impending motion
down the plane. Use g = 9.81
a. 87 N
b. 82 N
c. 72 N
d. 77 N
473. A 40 kg block is resting on an
inclined plane making an angle of θ
from the horizontal. Coefficient of
friction is 0.60, find the value of θ when
force P = 36.23 is applied to cause the
motion upward along the plane.
a. 20°
b. 30°
c. 28°
d. 23°
474. A 40 kg block is resting on an
inclined plane making an angle θ from
the horizontal. The block is subjected to
a force 87N parallel to the inclined plane
which causes an impending motion
down the plane. If the coefficient of
motion is 0.60, compute the value of θ.
a. 20°
b. 30°
c. 28°
d. 23°
475. A rectangular block having a width
of 8cm and height of 20 cm, is reating on
a horizontal plane. If the coefficient of
friction between he horizontal plane and
the block is 0.40, at what point above the
horizontal plane should horizontal force
P will be applied at which tipping will
occur?
a. 10 cm
b. 14 cm
c. 12 cm
d. 8 cm
476. A ladder is resting on a horizontal
plane and a vertical wall. If the
coefficient of friction between the
ladder, the horizontal plane and the
vertical wall is 0.40, determine the angle
that the ladder makes with the horizontal
at which it is about to slip.
a. 46.4°
b. 33.6°
c. 53.13°
d. 64.13°
477. Three identical blocks A, B and C
are placed on top of each other are place
on a horizontal plane with block B on
top of A and C on top of B. The
coefficient of friction between all
surfaces is 0.20. if block C is prevented
from moving by a horizontal cable
attached to a vertical wall, find the
horizontal force in Newton that must be
applied to B without causing motion to
impend. Each block has a mass of 50kg.
a. 294.3 Newtons
b. 274.7 Newtons
c. 321.3 Newtons
d. 280.5 Newtons
478. A car moving downward on an
inclined plane which makes an angle of
θ from the horizontal. The distance from
the front wheel to the rear wheel is
400cm and its centroid is located at 50
cm from the surface of the plane. If only
rear wheels provide breaking, what is the
value of θ so that the car will start to
slide if the coefficient of friction is 0.6?
a. 15.6°
b. 18.4°
c. 16.8°
d. 17.4°
479. A 40kg block is resting on an
inclined plane making an angle of 20°
from the horizontal. If the coefficient of
friction is 0.60, determine the force
parallel to the inclined plane that must
be applied to cause impending motion up
the plane.
a. 355.42 N
b. 354.65 N
c. 439.35 N
d. 433.23 N
480. A block weighing 40 kg is placed
on an inclined plane making an angle of
θ from th horizontal. If the coefficient of
friction between the block and the
inclined plane is 0.30, find the value of
θ, when the block impends to slide
downward.
a. 16.70°
b. 13.60°
c. 15.80°
d. 14.50°
481. A block having a weight W is
resting on an inclined plane making an
angle of 30° from the horizontal. If the
coefficient of friction between the block
and the inclined plane is 0.50. Determine
the value of W is a force 300 N applied
parallel to the inclined plane to cause an
impending motion upward.
a. 321.54 N
b. 493.53 N
c. 450.32 N
d. 354.53 N
482. 40kg block is resting on an inclined
plane making an angle of 20° from the
horizontal. The block is subjected to a
force 87 N parallel to the inclined plane
which causes an impending motion
down the plane. Compute the coefficient
of friction between the block and the
inclined plane.
a. 0.60
b. 0.80
c. 0.70
d. 0.50
483. A 20kg cubical block is resting on
an inclined plane making an angle of 30°
with the horizontal. If the coefficient of
friction between the block and the
inclined plane is 0.30, what force applied
at the uppermost section which is
parallel to the inclined plane will cause
the 20kg block to move up?
a. 134 N
b. 130 N
c. 146 N
d. 154 N
484. The coefficient of friction between
the 60 kN block is to remain in
equilibrium, what is the maximum
allowable magnitude for the force P?
c. 2000 kN
d. 500 kN
a. 15 kN
b. 12 kN
c. 18 kN
d. 24 kN
488. Is the system in equilibrium? If not,
find the force P so that the system will
be in equilibrium.
485. Find the value of P acting to the left
that is required to pull the wedge out
under the 500kg block. Angle of friction
is 20° for all contact surfaces.
a. 253.80 kg
b. 242.49 kg
c. 432.20 kg
d. 120.50 kg
486. The accurate alignment of a heavy
duty engine on its bed is accomplished
by a screw adjusted wedge with a 20°
taper as shown in the figure. Determine
the horizontal thrust P in adjusting screw
necessary to raise the mounting if the
wedge supports one fourth of the total
engine weight of 20 000N. The total
coefficient of friction for all surfaces is
0.25.
a. 4640 N
b. 4550 N
c. 5460 N
d. 6540 N
487. Two blocks connected by a
horizontal link AB are supported on two
rough planes as shown. The coefficient
of friction for block A on the horizontal
plane is 0.40. the angle of friction for
block B on the inclined plane is 15°.
What is the smallest weight of block A
for which equilibrium of the system can
exists?
a. 1000 kN
b. 1500 kN
a. 80 kg
b. 90 kg
c. 100 kg
d. 70 kg
489. A 12 kg block of steel is at rest on a
horizontal cable. The coefficient of static
friction between the block a table is
0.52. What is the magnitude of the force
acting upward 62° from the horizontal
that will just start the block moving?
a. 65.9 N
b. 78.1 N
c. 70.2 N
d. 72.4 N
490. The pull required to overcome the
rolling resistance of a wheel is 90 N
acting at the c enter of the wheels. If the
weight of the wheel is 18 000 N and the
diameter of the wheel is 300mm,
determine the coefficient of rolling
resistance.
a. 0.60 mm
b. 0.75 mm
c. 0.50 mm
d. 0.45 mm
491. A 1000 kN weight is to be moved
by using 50 mm diameter rollers. If the
coefficient of the rolling resistance for
the rollers and floor is 0.08 mm and that
for rollers and weight is 0.02 mm.
determine the pull required.
a. 2000 N
b. 1500 N
c. 2500 N
d. 1000 N
492. A ball is thrown vertically upward
with an initial velocity of 3m/sec from a
window of a tall building. The ball
strikes at the sidewalk at ground level 4
sec later. Determine the velocity with
which the ball hits the ground.
a. 30.86 m/sec
b. 36.24 m/sec
c. 42.68 m/sec
d. 25.27 m/sec
493. A train starts from rest at station P
and stops from station Q which is 10km
from station P. the maximum possible
acceleration
of
the
train
is
15km/hour/min. if the maximum
allowable speed is 60 kph, what is the
least time the train go from P to Q?
a. 15 min
b. 10 min
c. 12 min
d. 20 min
494. A car starting from rest picks up at
a uniform rate and passes three electric
post in succession. The posts are spaced
360 m apart along a straight road. The
car takes 10sec to travel from first post
to sec post and takes 6 sec to go from the
second to the third post. Determine the
distance from the starting point to the
first post.
a. 73.5 m
b. 80.3 m
c. 77.5 m
d. 70.9 m
495. A stone is dropped from the deck of
Mactan Bridge. The sound of the splash
reaches the deck 3 seconds later. If
sound travels 342 m/s in still air, how
high is the deck of Mactan Bridge above
the water?
a. 40.6 m
b. 45.2 m
c. 57.3 m
d. 33.1 m
496. At a uniform rate of 4 drops per
second, water is dripping from a faucet.
Assuming the acceleration of each drop
to be 9.81 m/sec^2 and no air resistance,
find the distance between two successive
drops in mm if the upper drop has been
in motion for 3/8 seconds.
a. 1230 mm
b. 2340 mm
c. 2231 mm
d. 1340 mm
497. A racing car during the Marlboro
Championship starts from rest and has a
constant acceleration of 4m/sec^2. What
is its average velocity during the first 5
seconds of motion?
a. 10 m/s
b. 4 m/s
c. 6 m/s
d. 12 m/s
498. A train is to commute between
Tutuban station and San Andres station
with a top speed of 250 kph but can not
accelerate nor decelerate faster than 4
m/s. What is its min. distance between
the two stations in order for the train to
be able to reach its top speed?
a. 1106.24
b. 1205.48
c. 1309.26
d. 1026.42
499. A block having a weight of 400N
rests on an inclined plane making an
angle of 30° with the horizontal is
initially at rest after it was released for 3
sec, find the distance the block has
traveled assuming there is no friction
between the block and plane. Determine
the velocity after 3 seconds.
a. 14.71 m/sec
b. 15.39 m/sec
c. 14.60 m/sec
d. 13.68 m/sec
500. A car accelerates for 6 sec from an
initial velocity of 10 m/s. the
acceleration is increasing uniformly
from zero to 8 m/s^2 in 6 sec. during the
next 2 seconds, the car decelerates at a
constant rate of m/s^2. Compute the total
distance the car has traveled from the
start after 8 sec.
distance the car has traveled after 12 sec
from the start.
a. 232 m
b. 240 m
c. 302 m
d. 321 m
503. A car moving at 6 m/s accelerates
at 1.5 m/s^2 for 15 sec, then decelerates
at a rate of 1.2 m/s^2 for 12 sec.
Determine the total distance traveled.
a. 558.75 m
b. 543.80 m
c. 384.90 m
d. 433.75 m
504. A train starting at initial velocity of
30 kph travels a distance 21 km in 8 min.
determine the acceleration of the train at
this instant.
a. 169 m
b. 172 m
c. 180 m
d. 200 m
a. 0.0865 m/s^2
b. 0.0206 m/s^2
c. 0.3820 m/s^2
d. 0.0043 m/s^2
501. A train passing at point A at a speed
of 72 khp accelerates at 0.75 m/s^2 from
one minute along a straight path then
decelerates at 1.0 m/s^2. How far from
point a will be 2min after passing point
A.
505. From a speed of 75 kph, a car
decelerates at the rate of 500 m/min^2
along a straight path. How far in meters
will it travel in 45 sec?
a. 6.49 km
b. 7.30 km
c. 4.65 km
d. 3.60 km
502. A car accelerate 8 seconds from
rest,
the
acceleration
increasing
uniformly from zero to 12 m/s^2. During
the next 4 sec, the car decelerates at a
constant rate of -11 m/s^2. Compute the
a. 790.293 m
b. 791.357 m
c. 796.875 m
d. 793.328 m
506. An object experiences rectilinear
acceleration a(t)= 10 – 2t. How far does
it travel in 6 sec if its initial velocity is
10 m/s?
a. 182
b. 168
c. 174
d. 154
507. An object experiences the velocity
as shown in the diagram. How far will it
move in 6 seconds?
a. 40 m
b. 60 m
c. 80 m
d. 100 m
508. An object is accelerating to the
right along a straight path at 2m/s. the
object begins with a velocity 10 m/s to
the left. How far does it travel in 15
seconds?
a. 125 m
b. 130 m
c. 140 m
d. 100 m
509. What is the acceleration of a body
that increases in velocity from 20 m/s to
40 m/s in 3 sec?
Answer in SI.
a. 8.00 m/s^2
b. 6.67 m/s^2
c. 50. m/s^2
d. 7.0 m/s^2
510. A shell is fired vertically upward
with an initial velocity of 2000 fps. It is
timed to burst in 7 sec. Four seconds
after firing the first shell, a second shell
is fired with the same velocity. This shell
is time to burst in 5 sec. An observer
stationed in a captive balloon near the
line of fire hears both burst. At the same
instance what is the elevation or height
of the balloon. Assume velocity of sound
to be 1100 fps.
a. 10 304 ft
b. 18 930 ft
c. 13 400 ft
d. 14 030 ft
511. An object from a height of 92 m
and strikes the ground with a speed of 19
m/s. Determine the height that the object
must fall in order to strike with a speed
of 24 m/s.
a. 146.94 m
b. 184.29 m
c. 110.12 m
d. 205.32 m
512. A ball is dropped from a balloon at
a height of 195 m. if the balloon is rising
29.3 m/s. Find the highest point reached
by the ball and the time of flight.
a. 238.8 m
b. 487.3 m
c. 328.4 m
d. 297.3 m
513. A ball is thrown vertically upward
with an initial velocity of 3m/sec from a
window of a tall building. The ball
strikes at the sidewalk at ground level 4
sec later. Determine the velocity with
which the ball hits the ground and the
height of the window above the ground
level.
a. 36.2 m/s; 66.79 m
b. 24.4 m/s; 81.3 m
c. 42.3 m/s; 48.2 m
d. 53.2 m/s; 36.8 m
514. A ball is dropped freely from a
balloon at a height 195 m. If the balloon
is rising 29.3 m/s. Find the highest point
reached by the ball and the velocity of
the ball as it strikes the ground.
a. 43.76 m; 68.44 m/s
b. 22.46 m; 71.66 m/s
c. 36.24 m; 69.24m/s
d. 12.8 m; 31.2 m/s
515. How far does the automobile move
while its speed increases uniformly from
15 kph to 45 kph in 20 sec?
a. 185 m
b. 167 m
c. 200 m
d. 172 m
519. A stone was dropped freely from a
balloon at a height of 190m above the
ground. The balloon is moving upward
at a speed of 30 m/s. Determine the
velocity of the stone at it hits the ground.
a. 56.43 m/s
b. 68.03 m/s
c. 62.45 m/s
d. 76.76 m/s
516. An automobile is moving at 20 kph
and accelerates at 0.5 m/s^2 for a
peroiud of 45 sec. Compute the distance
traveled by the car at the end of 45 sec.
520. A ball is thrown vertically at a
speed of 20 m/s from a building 100 m
above the ground. Find the velocity and
the position of the ball above the ground
after 5 seconds.
a. 842.62 m
b. 765.45 m
c. 672.48 m
d. 585.82 m
a. 3.34 m, 45.23 m/s
b. 4.54 m, 47.68 m/s
c. 5.67 m, 56.42 m/s
d. 6.23 m, 34.76 m/s
517. A ball is thrown vertically upward
with an initial velocity of 3m/sec from a
window of a tall building, which is 70 m
above the ground level. How long will it
take for the ball to hit the ground?
521. A ball is thrown vertically at a
speed of 30 m/s from a building 200 m
above the ground. Determine the
velocity and the time that it strikes the
ground.
a. 3.8 sec
b. 4.1 sec
c. 5.2 sec
d. 6.1 sec
a. 11.50 sec, 65.80 m/s
b. 11.45 sec, 66.59 m/s
c. 10.30 sec, 67.21 m/s
d. 10.14 sec, 69.45 m/s
518. A ball is thrown vertically upward
with an initial velocity of 3m/sec from a
window of a tall building. The ball
strikes the ground 4 sec later. Determine
the height of the window above the
ground.
522. A ball is thrown vertically with a
velocity of 20 m/s from the top of a
building 100m high. Find the velocity of
the ball at a height of 40 m above the
ground.
a. 66.331 m
b. 67.239 m
c. 54.346 m
d. 72.354 m
a. 39.71 m/s
b. 40.23 m/s
c. 39.88 m/s
d. 39.68 m/s
523. A ball is shot at a ground level at an
angle of 60 degrees with the horizontal
with an initial velocity of 100 m/s.
Determine the height of the ball after 2
seconds.
with the horizontal. Determine the
minimum value of the coefficient of
friction which will prevent slipping.
a. 162.46 m
b. 153.59 m
c. 175.48 m
d. 186.42 m
a. 0.165
b. 0.362
c. 1.028
d. 0.625
524. A ball is shot at an average speed of
200 m/s at an angle of 20° with the
horizontal. What would be the velocity
of the ball after 8 seconds?
528. At what weight “h” above the
billiard table surface should a billiard
ball of radius 3cm be struck by a
horizontal impact in order that the ball
will start moving with no friction
between the ball and the table?
a. 188.21 m/s
b. 154.34 m/s
c. 215.53 m/s
d. 198.37 m/s
525. A projectile has a velocity of 200
m/s acting at an angle 20 degrees with
the horizontal. How long will it take for
the projectile to hit the ground surface?
a. 13.95 sec
b. 15.75 sec
c. 10.11 sec
d. 24.23 sec
526. A solid homogenous circular
cylinder and a solid homogenous sphere
are placed at equal distances from the
end of an inclined plane. Assuming that
no slipping occurs as the two bodies roll
down the plane, which of them will
reach the end of the plane first? Assume
that they have the same weight and
radius.
a. sphere
b. cylinder
c. both cylinder and sphere
d. none of these
527. A homogenous sphere rolls down
as inclined plane making an angle of 30°
a. 4.9 cm
b. 3.4 cm
c. 4.2 cm
d. 5.5 cm
529. A common swing 7.5 m high is
designed for a static load of 1500 N
(tension in the rope is equal to 1500 N).
Two boys each weighing 500 N are
swinging on it. How much many degrees
on each side of the vertical can they
swing without exceeding the designed
load?
a. 41.41°
b. 45.45°
c. 30.35°
d. 54.26°
530. A wooden block weighing 20N
rests on a turn table having a radius of
2m at a distance on 1m from the center.
The coefficient of friction between the
block and the turn table is 0.30. The
rotation of the table is governed by the
equation Ø = 4t^2 where Ø is in radians
and t in seconds. If the table starts
rotating from rest at t=0, determine the
time elapsed before the block will begin
to slip.
a. 0.21 sec
b. 0.55 sec
c. 1.05 sec
d. 0.10 sec
531. A ball at the end of a cord 121 cm
long is swinging with a complete vertical
circle just enough velocity to keep it in
the top. If the ball is released from the
cord where it is at the top point of its
path, where will it strike the ground 245
cm below the center of the circle.
a. 297.61 cm
b. 332.64 cm
c. 258.37 cm
d. 263.63 cm
532. At what RPM is the ferriswheel
turning when the riders feel “weightless”
or zero gravity every time the each rider
is at the topmost part of the wheel 9m in
radius?
a. 9.97 rpm
b. 8.58 rpm
c. 10.73 rpm
d. 9.15 rpm
533. A wooden block having a weight of
50 N is placed at a distance 1.5m from
the center of a circular platform rotating
at a speed of 2 radians per second.
Determine the minimum coefficient of
friction of the blocks so that it will not
slide. Radius of circular platform is 3m.
a. 0.61
b. 0.84
c. 0.21
d. 1.03
534. A 2N weight is swing in a vertical
circle of 1m radius and the end of the
cable will break if the tension exceeds
500 N. Which of the following most
nearly gives the angular velocity of the
weight when the cable breaks?
a. 49.4 rad/sec
b. 37.2 rad/sec
c. 24.9 rad/sec
d. 58.3 rad/sec
535. A weight is attached to a chord and
forms a conical pendulum when it is
rotated about the vertical axis. If the
period of rotation is 0.2 sec, determine
the velocity of the weight if the chord
makes an angle of 25° with the vertical.
a. 0.146 m/s
b. 0.823 m/s
c. 1.028 m/s
d. 0.427 m/s
536. A ball having a weight of 4N is
attached to a cord 1.2 m long and is
revolving around a vertical axis so that
the cord makes an angle of 20° with the
vertical axis. Determine the rpm.
a. 28.17
b. 24.16
c. 22.12
d. 25.18
537. A wheel is rotating at 4000 rpm. If
it experience a deceleration of 20
rad/sec^2 through how many revolutions
will it rotate before it stops?
a. 400
b. 698
c. 520
d. 720
538. Find the maximum acceleration of a
mass at the end of a 2m long string. It
swing like a pendulum with a maximum
angle of 30°.
a. 4.91 m/s^2
b. 3.61 m/s^2
c. 6.21 m/s^2
d. 7.21 m/s^2
539. A turbine started from rest to 180
rpm in 6 min at a constant acceleration.
Find the number of revolution that it
makes within the elapsed time.
a. 550 revolutions
b. 540 revolutions
c. 630 revolutions
d. 500 revolutions
c. 888 x 10^3 J
d. 1100 x 10^3 J
543. A cyclist on a circular track of
radius r = 800ft travelling at 27 ft/s. His
speed at the tangential direction
increases at the rate of 3 ft/s^2. What is
the cyclist’s total acceleration?
a. 2.8 ft/s^2
b. -3.12 ft/s^2
c. -5.1 ft/s^2
d. 3.31 ft/s^2
540. Traffic travels at 65 mph around
banked highway curved with a radius of
3000 feet. What banking angle is
necessary such that friction will not be
required to resist the centrifugal force?
544. An automobile travels on a
perfectly horizontal, unbanked circular
track of radius R. The coefficient of
friction between the tires and the track is
0.3. If the car’s velocity is 15 m/s, what
is the smallest radius it may travel
without skidding?
a. 3.2°
b. 2.5°
c. 5.4°
d. 18°
a. 68m
b. 69.4 m
c. 76.5 m
d. 71.6 m
541. The rated speed of a highway curve
of 60m radius of 50 kph. If the
coefficient of friction between the tires
and the road is 0.60, what is the
maximum speed at which a car can
round a curve without skidding?
545. Determine the angle of super
elevation for a highway curves of 183 m
radius, so that there will be no “slide
thrust” for a speed of 72 kilometer per
hour. At what speed will skidding
impend if the coefficient of friction is
0.3?
a. 93.6 kph
b. 84.2 kph
c. 80.5 kph
d. 105.2 kph
542. A solid disk flywheel (I = 200
kg.m) is rotating with a speed of 900
rpm. What is the rotational kinetic
energy?
a. 730 x 10^3 J
b. 680 x 10^3 J
a. 12.57°; 31.72 m/s
b. 13.58°; 25.49 m/s
c. 15.29°; 34.24 m/s
d. 10.33°; 30.57 m/s
546. A child places a picnic basket on
the outer rim of merry go round that has
a radius of 4.6m and revolves once every
24 sec. How large must the coefficient of
static friction be for the basket to stay on
the merry go round?
a. 0.032
b. 0.024
c. 0.045
d. 0.052
547. A driver’s manual that a driver
traveling at 48kph and desiring to stop as
quickly as possible travels 4m before the
foot reaches the brake. The car travels
and additional 21 m before coming to
rest. What coefficient of friction is
assumed in this calculation?
a. 0.43
b. 0.34
c. 0.56
d. 0.51
b. 6 m/sec^2
c. 4 m/sec^2
d. 3 m/sec^2
551. A 10.7 kN car travelling at 134 m/s
attempts to round an unbanked curve
with a radius of 61 m. What force of
friction is required to keep the car on its
circular path?
a. 3211 N
b. 3445 N
c. 3123 N
d. 4434 N
552. A rotating wheel has a radius of 2
feet and 6 inches. A point on the rim of
the wheel moves 30ft in 2sec. Find the
angular velocity of the wheel.
548. A point on the rim of a rotating
flywheel changes its speed its speed
from 1.5m/s to 9 m/s while it moves 60
m. If the radius of the wheel is 1m,
compute the normal acceleration at the
instant when its speed is 6m/s.
a. 6 rad/sec
b. 2 rad/sec
c. 4 rad/sec
d. 5 rad/sec
a. 36 m/s^2
b. 24 m/s^2
c. 18 m/s^2
d. 20 m/s^2
553. A prismatic AB bar 6m long has a
weight of 500 N. It is pin connected at
one end at A. If it is rotated about a
vertical axis at Ai how fast would it be
rotated when it makes an angle of 30°
with the vertical?
549. The angular speed of a rotating
flywheel a radius of 0.5m, is 180/π rpm.
Compute the value of its normal
acceleration and the tangential speed.
a. 16 m/s^2; 2 m/s
b. 18 m/s^2; 3 m/s
c. 14 m/s^2; 1.5 m/s
d. 12 m/s^2; 1.0 m/s
550. A pulley has an angular velocity of
2 rad/sec, and a tangential speed of 4
m/s. Compute the normal acceleration.
a. 8 m/sec^2
a. 1.68 rad/sec
b. 2.58 rad/sec
c. 1.22 rad/sec
d. 2.21 rad/sec
554. A prismatic bar weighing 25 kg is
rotated horizontally about one of its ends
at a speed of 2.5 rad/sec. Compute the
length of the prismatic bar when it
makes an angle of 45° with the vertical.
a. 6.5 m
b. 3.33 m
c. 6.20 m
d. 7.35 m
555. A bullet enters a 50 mm plank with
a speed of 600 m/s and leaves with a
speed of 24 m/s. Determine the thickness
of the plank that can be penetrated by the
bullet.
a. 55 mm
b. 60 mm
c. 65 mm
d. 70 mm
556. A balikbayan box is placed on top
on a flooring of a delivery truck with a
coefficient of friction between the floor
and the box equal to 0.40. If the truck
moves at 60 kph, determine the distance
that the truck will move before the box
will stop slipping. The box weighs 200
N.
a. 70.8 m
b. 60.8 m
c. 50.8 m
d. 40.8 m
557. At what speed must a 10 kN car
approach a ramp which makes an angle
of 30° with the horizontal an 18m high
at the top such that it will just stop as it
reaches the top. Assume resisting force
of friction, to be 0.60 kN.
a. 71.57 kph
b. 60.46 kph
c. 54.46 kph
d. 82.52 kph
558. A car weighing 10 kN approaches a
ramp which makes a slope of 20° at the
speed of 75 kph. At the foot of the ramp,
the motor is turned off. How far does the
car travel up the inclined before it stops?
a. 64.57 m
b. 46.74 m
c. 74.84 m
d. 54.84 m
559. A car is running up a grade of 1 in
250 at a speed of 28.8 kph when the
engine conk out. Neglecting friction,
how far will the car have gone after 3
minutes from the point where the engine
conk out?
a. 808.2 m
b. 607.8 m
c. 542.4 m
d. 486.8 m
560. A 70 kg man stands on a spring
scale on an elevator. During the first 2
seconds starting from rest, the scale
reads 80 kg. Find the velocity of the
elevator at the end of 2 seconds and the
tension T in the supporting cable fro the
during the acceleration period. The total
weight of the elevator, man and scale is
7000N.
a. 2.8 m/sec; 8000N
b. 3.4 m/sec; 7000N
c. 4.3 m/sec; 9000N
d. 1.5 m/sec; 6000N
561. A cylinder having a mass of 40 kg
with a radius of 0.5 m is pushed to the
right without rotation and with
acceleration 2 m/sec^2. Determine the
magnitude and location of the horizontal
force P if the coefficient of friction is
0.30.
a. 198 N; 20.2 cm
b. 200 N; 32.4 cm
c. 183 N; 15.7 cm
d. 232 N; 34.2 cm
562. A block having a weight of 400 N
rests on an inclined plane making an
angle of 30° with the horizontal is
initially at rest. After it was released for
3sec, find the distance that the block has
traveled assuming that there is no
friction between the block and the plane.
Compute also the velocity after 3 sec.
a. 22.07 m, 14.71 m/s
b. 27.39 m, 15.39 m/s
c. 20.23 m, 14.60 m/s
d. 15.69 m, 13.68 m/s
563. A block having a weight of 200 N
rests on an inclined plane making an
angle of 30° with the horizontal is
initially at rest. If the block is initially at
rest and the coefficient of friction
between the inclined plane and the block
is 0.20, compute the time to travel a
distance of 14.45m, and the velocity of
the block after 3 sec.
b. 11 382 N
c. 9 254 N
d. 12 483 N
566. A body weighing 40 lb starts from
rest and slides down a plane at an angle
of 30° with the horizontal for which the
coefficient of friction f = 0.30. How far
will it move during the third second?
a. 19.63 ft
b. 19.33 ft
c. 18.33 ft
d. 19.99 ft
567. What force is necessary to
accelerate a 3000 lbs railway electric car
at the rate of 1.25 ft/sec^2, if the force
required to overcome the frictional
resistance is 400 lbs.
a. 3 sec, 9.63 m/s
b. 2 sec, 10.12 m/s
c. 4 sec, 12.20 m/s
d. 5sec, 11.20 m/s
a. 1564.596 lbs
b. 1267.328 lbs
c. 1653.397 lbs
d. 1427.937 lbs
564. A 100kg block is released at the top
of 30° incline 10 m above the ground.
The slight melting of ice renders the
surfaces frictionless; calculate the
velocity of the foot of the incline.
568. A freight car having a mass of 15
Mg is towed along the horizontal track.
If the car starts from rest and attains a
speed of 8 m/s after traveling a distance
of 150m, determine the constant
horizontal towing force applied to the
car. Neglect friction and the mass of the
wheels.
a. 20 m/s
b. 15 m/s
c. 25 m/s
d. 22 m/s
565. Starting from rest, an elevator
weighting 9000 N attains an upward
velocity of 5 m/s in 4 sec. with uniform
acceleration. Find the apparent weight of
600 N man standing inside the elevator
during its ascent and calculate the
tension in the supporting cable.
a. 10 823 N
a. 3.2 kN
b. 2.2 kN
c. 4.3 kN
d. 4.1 kN
569. An elevator weighing 2000 lb
attains an upward velocity of 16 fps in 4
sec with uniform acceleration. What is
the tension in supporting= cables?
a. 2250 lb
b. 2495 lb
c. 1950 lb
d. 2150 lb
570. A block weighing 200N rests on a
plane inclined upward to the right at
slope 4 vertical to 3 horizontal. The
block is connected by a cable initially
parallel to the plane passing through a
pulley which is connected to another
block weighing 100 N moving vertically.
The coefficient of kinetic friction
between the 200N block and the inclined
plane is 0.10, which of the following
most nearly give the acceleration of the
system.
a. 2.93 m/sec^2
b. 0.37 m/sec^2
c. 1.57 m/sec^2
d. 3.74 m/sec^2
571. A pick-up truck is traveling forward
a5 m/s the bed is loaded with boxes,
whose coefficient of friction with the
bed is 0.4. What is the shortest time that
the truck can be bought to a stop such
that the boxes do not shift?
a. 4.75
b. 2.35
c. 5.45
d. 6.37
572. Two barges are weighing 40 kN
and the other 80 kN are connected by a
cable in quiet water. Initially the barges
are 100 m apart. If friction is negligible
calculate the distance moved by the 80
kN barge.
a. 20 m
b. 30 m
c. 12 m
d. 25 m
573. Two blocks A and B weighs 150 N
and 200 N respectively is supported by a
flexible cord which passes through a
frictionless pulley which is supported by
a rod attached to a ceiling. Neglecting
the mass and friction of the pulley,
compute the acceleration on the blocks
and the tension on the rod supporting the
frictionless pulley.
a. 1.40 m/s^2, 342.92 N
b. 1.50 m/s^2, 386.45 N
c. 1.80 m/s^2, 421.42 N
d. 2.2 m/s^2, 510.62 N
574. A pendulum with the concentrated
mass “m” is suspended vertically inside
a stationary railroad freight car by means
of a rigid weightless connecting rod. If
the connecting rod is pivoted where it
attaches to the boxcar, compute the
angle of that the rod makes with the
vertical as a result of constant horizontal
acceleration of 2m/s.
a. 11°31’
b. 9°12’
c. 6°32’
d. 3°56’
575. Two 15 N weights A and B are
connected by a massless string hanging
over a smooth frictionless peg. If a third
weight of 15N is added to A and the
system is released, by how much is the
force on the peg increased?
a. 10 kN
b. 12 kN
c. 15 kN
d. 20 kN
576. Three crates with masses A = 45.2
kg, B = 22.8 kg, and C = 34.3 kg are
placed with B besides A and C besides B
along a horizontal frictionless surface.
Find the force exerted by B and C by
pushing to the right with an acceleration
of 1.32 m/s^2.
580. What is the kinetic energy of 4000
lb automobile which is moving at 44
fps?
a. 45.3 kN
b. 54.2 kN
c. 43.2 kN
d. 38.7 kN
a. 1.2 x 10^5 ft-lb
b. 2.1 x 10^5 ft-lb
c. 1.8 x 10^5 ft-lb
d. 3.1 x 10^5 ft-lb
577. Three blocks A, B and C are placed
on a horizontal frictionless surface and
are connected by chords between A, B
and C. Determine the tension between
block B and C when a horizontal tensile
force is applied at C = 6.5 N. Masses of
blocks are A = 1.2 kg, B 2.4 kg, and C =
3.1 kg.
581. A box slides from rest from a point
A down a plane inclined 30° to the
horizontal. After reaching the bottom of
the plane, the box move at horizontal
floor at distance 2m before coming to
rest. If the coefficient of friction between
the box and the plane and the box and
the floor is 0.40, what is the distance of
point “A” from the intersection of the
plane and the floor?
a. 3.50 N
b. 4.21 N
c. 3.89 N
d. 4.65 N
578. A constant force P = 750 N acts on
the body shown during only the first 6 m
of its motion starting from rest. If u =
0.20, find the velocity of the body after it
has moved a total distance of 9m.
a. 3.93 m/s^2
b. 4.73 m/s^2
c. 2.32 m/s^2
d. 3.11 m/s^2
579. A weight 9 kN is initially
suspended on a 150 m long cable. The
cable weighs 0.002 kN/m. If the weight
is then raised 100 m how much work is
done in Joules.
a. 915000
b. 938700
c. 951000
d. 905100
a. 7.24 m
b. 5.21 m
c. 4.75 m
d. 9.52 m
582. A 400 N block slides on the
horizontal plane by applying a horizontal
force of 200 N and reaches a velocity of
20 m/s in a distance of 30m from rest.
Compute the coefficient of friction
between the floor and the block.
a. 0.18
b. 0.24
c. 0.31
d. 0.40
583. A car weighing 40 tons is switched
to a 2 percent of upgrade with a velocity
of 30 mph. If the train resistance is 10
lb/ton, ho9w far up the grade will it go?
a. 1124 ft on slope
b. 2014 ft on slope
c. 1203 ft on slope
d. 1402 ft on slope
584. A car weighing 10 kN is towed
along a horizontal surface at a uniform
velocity of 80 kph. The towing cable is
parallel with the road surface. When the
car is at foot of an incline as shown
having an elevation of 30m, the towing
cable was suddenly cut. At what
elevation in the inclined road will the car
stop in its upward motion?
a. 55.16 m
b. 60.24 m
c. 51.43 m
d. 49.62 m
585. A wooden block starting from rest,
slides 6 m down a 45° slope, then 3m
along the level surface and then up 30°
incline until it come to rest again. If the
coefficient of friction is 0.15 for all
surfaces in contact compute the total
distance traveled.
a. 20 m
b. 11 m
c. 14 m
d. 18 m
586. The block shown starting from rest
and moves towards the right. What is the
velocity of block B as it touches the
ground? How far will block A travel if
the coefficient of friction between block
A and the surface is 0.20? Assume
pulley to be frictionless.
a. 1.44 m
b. 2.55 m
c. 5.22 m
d. 3.25 m
587. After the block in the figure has
moved 3m from rest the constant force
P=600N is removed find the velocity of
the block when it is returned to its initial
position.
a. 8.6 m/s
b. 5.6 m/s
c. 6.4 m/s
d. 7.1 m/s
588. A 10 kg block is raised vertically 3
meters. What is the change in potential
energy? Answer in SI units closest to:
a. 350 kg-m^2/ s
b. 320 J
c. 350N-m
d. 294 J
589. A car weighing 40 tons is switched
2% upgrade with a velocity of 30 mph.
If the car is allowed to run back what
velocity will it have at the foot of the
grade?
a. 37 fps
b. 31 fps
c. 43 fps
d. 34 fps
590. A 200 ton train is accelerated from
rest to a velocity of 30 miles per hour on
a level track. How much useful work
was done?
a. 12 024 845
b. 13 827 217
c. 11 038 738
d. 10 287 846
591. A drop hammer weighing 40 kN is
dropped freely and drives a concrete pile
150 mm into the ground. The velocity of
the drop hammer at impact is 6m/s.
What is the average resistance of the soil
in kN?
a. 542.4
b. 489.3
c. 384.6
d. 248.7
592. A force of 200 lbf act on a block at
an angle of 28° with respect to the
horizontal. The block is pushed 2 feet
horizontally. What is the work done by
this force?
a. 320 J
b. 480 J
c. 540 J
d. 215 J
593. A small rocket propelled test
vehicle with a total mass of 100 kg starts
from rest
at A and moves with
negligible friction along the track in the
vertical plane as shown. If the propelling
rocket exerts a constant thrust T of 1.5
kN from A to position B. where it is shut
off, determine the distance S that the
vehicle rolls up the incline before
stopping. The loss of mass due to the
expulsion of gases by the rocket is small
and may be neglected.
a. 170 m
b. 165 m
c. 160 m
d. 175 m
594. A body that weighs W Newton fall
from rest from a height 600 mm and
strikes a sping whose scale is 7 N/mm. If
the maximum compression of the spring
is 150 mm, what is the value of W?
Disregard the mass of spring.
a. 105 N
b. 132 N
c. 112 N
c. 101 N
595. A 100 N weight falls from rest from
a height of 500 mm and strikes a spring
which compresses by 100 mm. Compute
the value of the spring constant,
neglecting the mass of the spring.
a. 10 N/mm
b. 15 N/mm
c. 12 N/mm
d. 8 N/mm
596. A 200 N weight falls from rest at
height “h” and strikes spring having a
spring constant of 10 N/mm. the
maximum compression of spring is 100
mm, after the weight the weight stikes
the spring. Compute the value of h is
meter.
a. 0.12 m
b. 0.10 m
c. 0.15 m
d. 0.21 m
597. A block weighing 500 N is dropped
from a height of 1.2 m upon a spring
whose modulus is 20 n/mm. What
velocity will the block have at the instant
the spring is deformed 100 mm?
a. 6.55 m/s^2
b. 5.43 m/s^2
c. 4.65 m/s^2
d. 3.45 m/s^2
598. A 50 kg object strikes the
unstretched spring to a vertical wall
having a spring constant of 20 kN/m.
Find the maximum deflection of the
spring. The velocity of the object before
it strikes the spring is 40 m/s.
a. 1 m
b. 2 m
c. 3 m
d. 4 m
599. A large coil spring with a spring
constant k=120 N/m is elongated, within
its elastic range by 1m. Compute the
store energy of the spring in N-m.
a. 60
b. 40
c. 50
d. 120
600. To push a 25 kg crate up a 27°
inclined plane, a worker exerts a force of
120 N, parallel to the incline. As the
crate slides 3.6 m, how much is the work
done by the worker and by the force of
gravity.
a. 400 Joules
b. 420 Joules
c. 380 Joules
d. 350 Joules
601. A train weighing 12 000 kN is
accelerate up a 2% grade with velocity
increasing from 30 kph to 50 kph in a
distance of 500 m. Determine the horse
power developed by the train.
a. 5.394 kW
b. 5.120 kW
c. 4.468 kW
d. 4.591 kW
602. An elevator has an empty weight of
5160 N. It is designed to carry a
maximum load of 20 passengers from
the ground floor to the 25th floor of the
building in a time of 18 seconds.
Assuming the average weight of the
passenger to be 710 N. and the distance
between floors is 3.5 m, what is the
minimum constant power needed for the
elevator motor?
a. 85.5 kW
b. 97.4 kW
c. 94.3 kW
d. 77.6 kW
603. An engine hoist 1.50 m^3 of a
concrete in a 2200 N bucket moves a
distance of 12 m in 20 seconds. If
concrete weighs 23.5 kN/m^3, determine
the engine horsepower assuming 80
efficiency.
a. 32.12 hp
b. 42.23 hp
c. 37.74 hp
d. 28.87 hp
604. A train weighs 15 000 kN. The
train’s resistance is 9 n per kilo-Newton.
If 5000 is available to pull this train up
to 2% grade, what will be its maximum
speed in kph?
a. 46.2 kph
b. 50 kph
c. 40 kph
d. 35 kph
605. An engine having a 40 hp rating is
used as an engine hoist to lift a certain
weight of height 8 m. Determine the
maximum weight it could lift in a period
of 20 sec.
a. 85.5 kW
b. 97.4 kW
c. 74.6 kW
d. 77.6 kW
606. A 100 kg body moves to the right 5
m/s and another 140 kg body moves to
the left at 3 m/s. They collided and after
impact the 100 kg body rebounds to the
left at 2 m/s. Compute the coefficient of
restitution.
a. 0.40
b. 0.50
c. 0.30
d. 0.60
607. A ball is dropped from an initial
height of 6m above a solid floor, how
high will the ball rebound if the
coefficient of restitution is e = 0.92?
a. 5.08
b. 5.52
c. 5.41
d. 5.12
608. A ball strikes the ground at an angle
of 30° with the ground surface; the ball
then rebounds at a certain angle θ with
the ground surface. If the coefficient of
restitution is 0.80, find the value of θ.
a. 24.79°
b. 18.48°
c. 32.2°
d. 26.7°
609. A ball is thrown with an initial
horizontal velocity of 30m/s from a
height of 3m above the ground and 40 m
from a vertical wall. How high above the
ground will the ball strike if the
coefficient of restitution is 0.70?
a. 1.46 m
b. 2.52 m
c. 1.11 m
d. 0.89 m
610. Two cars having equal weights of
135 kN are traveling on a straight
horizontal track with velocities of 3m/s
to the right and 1.5 m/s to the left
respectively. They collide and are
coupled during impact. Neglecting
friction due to skidding, determine their
final common velocity and the gain or
loss in kinetic energy after impact.
a. 7.74 m/s; 69.67 kN-m
b. 1.25 m/s; 66.35 kN-m
c. 2.06 m/s; 57.25 kN-m
d. 3.12 m/s; 77.36 kN-m
611. A man weighing 68 kg jumps from
a pier with horizontal velocity of 6 m/s
onto a boat that is rest on the water. If
the boat weighs 100 kg, what is the
velocity of the boat when the man comes
to rest relative to the boat?
a. 2.43 m/s
b. 3.53 m/s
c. 2.88 m/s
d. 1.42 m/s
612. A man weighing 68 kg jumps from
a pier with horizontal velocity of 5 m/s
onto a 100 kg boat moving towards the
dock at 4m/s What would be the velocity
of the boat after the man lands on it?
a. -0.56 m/s
b. -0.36 m/s
c. -0.78 m/s
d. -1.33 m/s
613. A ball is thrown at an angle of 40°
from the horizontal toward a smooth
foor and it rebounds at an angle of 25°
with the horizontal floor. Compute the
value of coefficient of restitution.
a. 0.56
b. 0.66
c. 0.46
d. 0.76
614. Car B is moving at a speed of 12
m/s and is struck by car A which is
moving at a speed of 20 m/s. The weight
of car A is 14 tons and of car B is 10
tons. Determine the velocities of the car
after impact assuming that the bumpers
got locked after impact. Both cars are
moving in the same direction to the
right.
a. 16.67 m/s
b. 14.25 m/s
c. 15.42 m/s
d. 13.62 m/s
615. Two cars A and B have weights
equal to 12 tons and 8 tons respectively
are moving in opposite directions. The
speed of car A is 22m/s to the right and
that of car B is 18 m/s to the left. Two
cars bumped each other. Determine the
velocity of the cars after impact
assuming the bumpers get locked.
a. 6 m/s
b. 8 m/s
c. 4 m/s
d. 3 m/s
616. A 6000 N drop hammer falling
freely trough a height of 0.9 m drives a
3000N pile 150mm vertically to the
ground. Assuming the hammer and the
pile to cling together after the impact,
determine the average resistance to
penetration of the pile.
a. 32 976 N
b. 42 364 N
c. 30 636 N
d. 28 476 N
617. Two identical balls collide as
shown. What is V2’ if the coefficient of
restitution is 0.8?
a. 4.8 m/s
b. 3.6 m/s
c. 5.6 m/s
d. 2.4 m/s
618. A 16gm mass is moving at 30 cm/s
while a 4gm mass is moving opposite
direction at 50cm/sec. They collide head
on and stick together. Their velocity
after collision is:
a. 0.14 m/s
b. 0.21 m/s
c. 0.07 m/s
d. 0.28 m/s
619. A bullet weighing 0.014 kg and
moving horizontally with a velocity of
610 m/s strikes centrally a block of
wood having a mass of 4.45 kg which is
suspended by a cord from a point 1.2 m
above the center of the block. To what
angle from the vertical will the block
and embedded bullet swing?
a. 31.79°
b. 29.32°
c. 30.12°
d. 28.64°
620. A body having a mass of 100kg and
having velocity of 10m/s to the right
collides with an 80 kg mass having a
velocity of 5 m/s to the left. If the
coefficient of restitution is 0.5,
determine the loss of kinetic energy after
impact.
a. 3750 N-m
b. 4260 N-m
c. 3640 N-m
d. 4450 N-m
621. A 0.44N bullet is fired horizontally
to an 89.18 N block of wood resting on a
horizontal surface which the coefficient
of friction is 0.30. If the block is moved
a distance 375 mm along the surface,
what was the velocity of the bullet
before striking?
a. 303.49 m/s
b. 204.61 m/s
c. 142.52 m/s
d. 414.25 m/s
622. A 60 ton rail car moving at 1 mile
per hour is instantaneously coupled to a
stationary 40 to rail car. What is the
speed of the coupled cars?
a. 0.88 mi/hr
b. 1.0 mi/hr
c. 0.60 mi/hr
d. 0.40 mi/hr
623. What momentum does a 40 lbm
projectile posses if the projectile is
moving 420 mph?
a. 24 640 lbf-sec
b. 16 860 lbf-sec
c. 765 lbf-sec
d. 523.6 lbf-sec
624. A 300 kg block is in contact with a
level of coefficient of kinetic friction is
0.10. if the block is acted upon by a
horizontal force of 50kg what time will
elapse before the block reaches a
velocity of 48.3 m/min from rest? If the
50 kg is then removed, hos much longer
will the block continue to move?
a. 12.08 sec; 8.05 sec
b. 15.28 sec; 9.27 sec
c. 10.42 sec; 7.64 sec
d. 13.52 sec; 10.53 sec
625. A 100kg body moves to the right at
5m/s and another body of mass W is
moves to the left 3m/s. they meet each
other and after impact the 100kg body
rebounds to the left at 2m/s. Determine
the mass of the body if coefficient of
restitution is 0.5.
a. 140 kg
b. 150 kg
c. 100 kg
d. 200 kg
626. A wood block weighing 44.75 N
rests on a rough horizontal plane, the
coefficient of friction being 0.40. if a
bullet weighing 0.25 N is fired
horizontally into the block with the
velocity of 600m/s, how far will the
block be displaced from its initial
position? Assume that the bullet remains
inside the block.
a. 1.41 m
b. 2.42 m
c. 1.89 m
d. 0.98 m
627. The system is used to determine
experimentally the coefficient of
restitution. If ball A is released from rest
an ball B swings through θ=53.1°, after
being struck, determine the coefficient of
restitution. Weight of A is 150 N while
that of b is 100 N.
a. 0.537
b. 0.291
c. 1.083
d. 0.926
628. The ball A and B are attached to
stiff rods of negligible weight. Ball A is
released from rest and allowed to strike
B. If the coefficient of restitution is 0.60,
determine the angle θ through which ball
B will swing. If the impact lasts for 0.01
sec, also find the average impact force.
Mass of A is 15kg and that of B is 10kg.
a. 64.85° ; 6720 N
b. 60.58° ; 6270 N
c. 57.63° ; 7660 N
d. 73.32° ; 7670 N
629. A 1500N block is in contact with a
level plane whose coefficient of kinetic
friction is 0.10. If the block is acted upon
a horizontal force of 250 N, at what time
will elapse before the block reaches a
velocity of 14.5m/s starting from rest?
a. 22.17 sec
b. 18.36 sec
c. 21.12 sec
d. 16.93 sec
630. A 1600N block is in contact with a
level plane whose coefficient of kinetic
friction is 0.20. If the block is acted upon
a horizontal force of 300 N initially
when the block is at rest and the force is
removed when the velocity of the block
reaches 16m/s. How much longer will
the block continue to move?
a. 8.15 sec
b. 6.25 sec
c. 4.36 sec
d. 5.75 sec
631. A bullet weighing 0.30N is moving
660m/s penetrates an 50N body and
emerged with a velocity of 180 m/s. For
how long will the body moves before it
stops? Coefficient of friction is 0.40.
a. 7.34 sec
b. 6.84 sec
c. 5.24 sec
d. 8.36 sec
632. A 1000N block is resting on an
incline plane whose slope is 3 vertical to
4 horizontal. If a force of 1500 N acting
parallel to the inclined plane pushes the
block up the inclined plane, determine
the time required to increase the velocity
of the block from 3m/s to 15m/s.
Coefficient of friction between the block
and the plane is 0.20.
a. 1.65 sec
b. 1.86 sec
c. 2.17 sec
d. 3.64 sec
633. The 9kg block is moving to the
right with a velocity of 0.6 m/s on the
horizontal surface when a force P is
applied to it at t=0. Calculate the
velocity V2 of the block when the t=0.4
sec. the kinetic coefficient of friction is
Mk = 0.30.
a. 1.82 m/s
b. 1.23 m/s
c. 2.64 m/s
d. 2.11 m/s
634. A 50kg block initially at rest is
acted upon by a force P which varies as
shown. Knowing the coefficient of
kinetic friction between the block and
the horizontal surface is 0.20, compute
the velocity of the block after 5 s, and
after 8s.
a. 15.19 m/s; 16.804 m/s
b. 13.23 m/s; 15.534 m/s
c. 10.65 m/s; 17.705 m/s
d. 17.46 m/s; 14.312 m/s
635. The motion of the block starting
from rest is governed by the a-t curve
shown. Determine the velocity and
distance traveled after 9 sec. Neglecting
friction.
a. 66 m/s; 228 m
b. 45 m/s; 233 m
c. 57 m/s, 423 m
d. 72 m/s; 326 m
636. From the V-t curve shown compute
the distance traveled by a car starting
form rest after 6 sec.
a. 50m
b. 60 m
c. 70 m
d. 80 m
637. A bullet weighing 50g is fired into
a block of wood weighing 20 lbs on a
top of a horizontal table. The block
moves 45 cm. the coefficient of friction
between the block and table is 0.30.
What is the speed of the bullet in kph
before hitting the block? Assume that the
bullet is embedded of the block.
a. 1068.77 kph
b. 1843.53 kph
c. 1144.38 kph
d. 1683.78 kph
638. A block weighing 60 N is subjected
to a horizontal force P = (10 + t^3) and a
friction resisting equal to (6 + t^2).
Compute the velocity of the block 2 s
after it has started from rest.
a. 2.4 m/s
b. 1.8 m/s
c. 2.8 m/s
d. 1.4 m/s
639. A price tag of P1200 is specified if
paid within 60 days but offers 3%
discount for cash in 30 days. Find the
rate of interest.
a. 37.11%
b. 38.51 %
c. 40.21 %
d. 39.31 %
640. It is the practice of almost all banks
in the Philippines that when they grant a
loan, the interest the interest for one year
is automatically deducted from the
principal amount upon release of money
to a borrower. Let us that you applied for
a loan with a bank and the P80 000 was
approved with interest rate of 14%
which P11 200was deducted and you
were given a check of P68 800. Since
you have to pay the amount of P80 000
in one year after, what then will be the
effective interest rate?
a. 16.28%
b 16.18%
c. 16.30%
d. 16.20%
641. Mr. J. dela Cruz borrowed money
from a bank. He received from the bank
P1 340 and promised to pay P1 500 at
the end of 9 months. Determine the
simple
interest
rate
and
the
corresponding discount rate or often
referred to as “Banker’s Discount”.
a. 15.92%; 13.73 %
b. 18.28%; 13.12 %
c. 12.95%; 17.33 %
d. 19.25%; 13.33%
642. A man borrowed from a bank with
a promissory note that he signed in the
amount of P25000 for a period of one
year. He received only the amount of
P21, 915 after the bank collected the
interest and additional amount of P85 00
for notarial and inspection fees. What
was the rate of interest that the bank
collected in advance?
a. 13.64%
b. 18.37%
c. 16.43%
d. 10.32%
643. Agnes Abadilla was granted a loan
of P20 000 by her employer CPM
Industrial Fabricator and Construction
Corporation with an interest rate of 6%
for 180 days on the principal collected in
advance. The corporation will accept a
promissory note for P20 000 non-interest
for 180 days. If discounted at once, find
the proceeds on the note.
a. P18 800
b. P19 000
c. P18 000
d. P18 400
644. P400 is borrowed for 75 days at
16% per annum simple interes. How
much wil be due at the end of 75 days?
a. P4168.43
b. P5124.54
c. P4133.33
d. P5625.43
645. Mr. Almagro made a money market
of P1 000 000 for 30 days at 7.5% per
year. If the withholding tax is 20%, what
is the net interest that he will receive at
the end of the month?
a. P3 000
b. P4 000
c. P6 000
d. P5 000
646. L for a motorboat specifies a cost of
P1200 due at the end of 100 days but
offers 4% discount for cash in 30 days.
What is the highest rate, simple interest
at which the buyer can afford to borrow
money in order to take advantage of the
discount?
a. 18.4%
b. 19.6%
c. 20.9%
d. 21.4%
647. In buying a computer disk, the
buyer was offered the options of paying
P250 cash at the end of 30 days or P270
at the end of 120 days. At what rate is
the buyer paying simple interest if he
agree to pay at the end of 120 days.
a. 32%
b. 40%
c. 28%
d. 25%
648. On March 1, 1996 Mr. Almagro
obtains a loan of P1500 from Mr. Abella
and signs a note promising to pay the
principal and accumulated simple
interest at rate of 5% at the end of 120
day. On March 15, 1996, Mr. Abella
discounts the note at the bank whose
discount rate is 6%. What does he
receive?
a. P 2 201.48
b. P1 123.29
c. P1 513.56
d. P938.20
649. A deposit of P110 000 was made
for 31 days. The net interest after
deducting 20% withholding tax is
P890.36. Find the rate of return
annually.
a. 12.25
b. 11.75
c. 12.75
d. 11.95
650. If you borrowed money from your
friend with simple interest of 12%, find
the present worth of P50 000, which is
due at the end of 7 months.
a. P46 200
b. P44 893
c. P46 729
d. P45 789
651. A man borrowed P2000 from a
bank and promises to pay the amount for
one year. He received only the amount
of P1920 after the bank collected an
advance interest of P80. What was the
rate of discount and the rate of interest
that the bank collected in advance.
a. 4%, 4.17%
b. 3%, 3.17%
c. 4%, 4.71%
d. 3%, 3.71%
652. The amount of P12 800 in 4yrs. At
5% compounded quarterly is ______?
a. P 14 785.34
b. P 15 614.59
c. P 16 311.26
d. P 15 847.33
653. A man borrows money from a bank
which uses a simple discount rate of
14%. He signs a promissory note
promising to pay P500 per month at the
end of 4th, 6th, and 7th months
respectively. Determine the amount of
money that he received from the bank.
thrust fund by her guardian until it
amounts to P50 000. When will the girl
receive the money if the fund is invested
at 8% compounded quarterly?
a. 7.98 years
b. 10.34 years
c. 11.57 years
d. 10.45 years
656. A man expects to receive P25 000
in 8 years. How much is that worth now
considering interest at 8% compounded
quarterly?
a. P13 859.12
b. P13 958.33
c. P13 675.23
d. P13 265.83
657. P500 000 was deposited at an
interest of 6% compounded quarterly.
Compute the compounded interest after
4 years and 9 months.
a. 163 475.37
b. 178 362.37
c. 158 270.37
d. 183 327.37
a. P1403.68
b. P1340.38
c. P1102.37
d. P1030.28
658. If the nominal interest rate is 3%,
how much is P5000 worth in 10 years in
a continuous compounded account?
654. A nominal interest of 3%
compounded continuously is given on
the account. What is the accumulated
amount of P10 000 after 10 years?
a. P5750
b. P6750
c. P7500
d. P6350
a. P13 610.10
b. P13 500.10
c. P13 498.60
d. P13 439.16
659. P200 000 was deposited for a
period of 4 years and 6 months and bears
on interest of P85 649.25. What is the
rate of interest if it is compounded
quarterly?
655. By the condition of a will, the sum
of P2000 is left to a girl to be held in
a. 8%
b. 6%
c. 7%
d. 5%
660. How many years will P100 000
earned a compound interest o P50 000 if
the interest rate is 9% compounded
quarterly?
a. 3.25
b. 4.55
c. 5.86
d. 2.11
661. A certain amount was deposited 5
years and 9months ago at an intrest 8%
compounded quarterly. If the sum now is
P315 379.85, how much was the amount
deposited?
a. P200 000
b. P180 000
c. P240 000
d. P260 000
662. Compute the effective annual
interest rate which is equivalent to 5%
nominal annual interest compounded
continuously.
a. 5.13%
b. 4.94%
c. 5.26%
d. 4.9%
663. Find the time required fro a sum of
money to triple itself at 5% per annum
compounded continuously?
a. 21.97 yrs.
b. 25.34 yrs.
c. 18.23 yrs.
d. 23.36 yrs.
664. A man wishes to have P40 000 in a
certain fund at the end of 8years. How
much should he invest in a fund that will
pay 6% compounded continuously?
a. P24 751.34
b. P36 421.44
c. P28 864.36
d. P30 486.42
665. If the effective annual interest rate
is 4%, compute the equivalent nominal
annual interest compounded
continuously.
a. 3.92%
b. 4.10%
c. 3.80%
d. 4.09%
666. What is the nominal rate of interest
compounded continuously for 10 years if
the compounded amount factor is equal
to 1.34986?
a. 3%
b. 4%
c. 5%
d. 6%
667. American Express Corp. charges
1.5% interest per month, compounded
continuously on the unpaid balance
purchases made on this credit card.
Compute the effective rate of interest.
a. 19.72%
b. 20.25%
c. 21.20%
d. 13.19%
668. If the nominal interest is 12%
compounded continuously, compute the
effective rate of annual interest.
a. 12.75%
b. 11.26%
c. 12.40%
d. 11.55%
669. Compute the difference in the
future amount of P500 compounded
annually at nominal rate of 5% and if it
is compounded continuously for 5 years
at the same rate.
a. P3.87
b. P4.21
c. P5.48
d. P6.25
670. If the effective interest rate is 24%,
what nominal rate of interest is charged
for a continuously compounded loan?
a. 21.51%
b. 22.35%
c. 23.25%
d. 21.90%
671. What is the nominal rate of interest
compounded continuously for 8 years if
the pre4sent worth factor is equal to
0.6187835?
a. 4%
b. 5%
c. 6%
d. 7%
672. What is the difference of the
amount 3 yrs. for now for a 10% simple
interest and 10% compound interest per
year?
a. P155
b. P100
c. same
d. P50
673. Find the discount is P2000 is
discounted for 6 months and at 8%
compounded quarterly.
a. P76.92
b. P80.00
c. P77.66
d. P78.42
674. If a sum of money triples in a
certain period of time at a given rate of
interest, compute the value of the single
payment present worth factor.
a. 0.333
b. 3
c. 0.292
d. 1.962
675. If the single payment amount factor
for a period of 5 years is 1.33822. What
is the nearest value of the interest rate?
a. 8%
b. 7%
c. 5.50%
d. 6%
676. If the single payment present worth
factor for a period of 8 years is 0.58201,
compute the nearest value of the rate for
that period.
a. 6%
b. 7%
c. 6.5%
d. 8%
677. If money is worth 8% compounded
quarterly, compute the single payment
amount factor for a period of 6 years.
a. 1.60844
b. 0.62172
c. 1.70241
d. 0.53162
678. Which of these gives the lowest
effective rate of interest?
a. 12.35% compounded annually
b. 11.9% compounded semi-annually
c. 12.2% compounded quarterly
d. 11.6% compounded monthly
679. It takes 20.15 years to quadruple
your money if invested x% compounded
semi-annually. Find the value of x.
a. 8%
b. 6.5%
c. 7%
d. 5%
680. It takes 13.87 years to treble the
money at the rate of x% compounded
quarterly. Compute the value of x.
a. 5%
b. 6%
c. 7%
d. 8%
681. Money was invested at x%
compounded quarterly. If it takes the
money up to quadruple in 17.5 years,
find the value of x.
a. 8%
b. 6%
c. 7%
d. 5%
682. Fifteen years ago P1000 was
deposited in a bank account an today it is
worth P2370. The bank pays semiannually. What was the interest rate paid
on this account?
a. 4.9%
b. 5.8 %
c. 5.0%
d. 3.8%
683. You borrow P3500 for one year
from a friend at an interest rate of 1.5 per
month instead of taking a loan from a
bank at rate of 18% a year. Compare
how much money you will save or lose
on the transaction.
a. You will pay P155 more than if you
borrowed from the bank
b. You will save P55 by borrowing
from your friend
c. You will pay P85 more that if you
borrowed from the bank
d. You will pay P55 less than if you
borrowed from the bank
684. Find the present worth of a future
payment of P100 000 to be made in 10
years with an interest of 12%
compounded quarterly.
a. P30444.44
b. P33000.00
c. P30655.68
d. P30546.01
685. An initial deposit of P80 000 in a
certain bank earns 6% interest per
annum compounded monthly. If the
earnings from the deposit are subject to
20% tax what would be the net value of
the deposit be after three quarters?
a. P95324.95
b. P82938.28
c. P68743.24
d. P56244.75
686. The effective rate of interest of 14%
compounded semi-annually is:
a. 14.49%
b. 14.36%
c. 14.94%
d. 14.88%
687. The amount of P50 000 was
deposited in the bank earning an interest
of 7.5% per annum. Determine the total
amount at the end of % years, if the
principal and interest were not
withdrawn during the period?
a. P71 781.47
b. P72 475.23
c. P70 374.90
d. P78 536.34
688. What is the effective rate
corresponding to 18% compounded
daily? Take 1 year is equal to 360 days.
a. 18.35%
b. 19.39%
c. 18.1%
d. 19.72%
689. If P1000 becomes P1126.49 after 4
years when invested at a certain nominal
rate of interest compounded semiannually determine the nominal rate and
the corresponding effective rate.
a. 3% and 3.02%
b. 4.29% and 4.32%
c. 2.30% and 2.76%
d. 3.97% and 3.95%
690. Convert 12% semi-annually to
compounded quarterly
a. 19.23% compounded quarterly
b. 23.56% compounded quarterly
c. 14.67% compounded quarterly
d. 11.83% compounded quarterly
c. 18.46%
d. 18.95%
692. If P5000 shall accumulate for 10
years at 8% compounded quarterly, find
the compounded interest at the end of 10
years.
a. P6005.30
b. P6000.00
c. P6040.20
d. P6010.20
693. A couple borrowed P4000 from a
lending company for 6 years at 12%. At
the end of 6 years, it renews the loan for
the amount due plus P4000 more for 3
years at 12%. What is the lump sum
due?
a. P14 842.40
b. P16 712.03
c. P12 316.40
d. P15 382.60
694. How long (years) will it take money
if it earns 7% compounded semiannually?
a. 26.30
b. 40.30
c. 33.15
d. 20.15
695. P200 000 was deposited on Jan. 1,
1988 at an interest rate of 24%
compounded semi-annually. How much
would the sum be on Jan. 1, 1993?
691. What is the corresponding effective
interest rate of 18% compounded semiquarterly?
a. P421 170
b. P521 170
c. P401 170
d. P621 170
a. 19.25%
b. 19.48%
696. If P500 000 is deposited at a rate of
11.25% compounded monthly;
determine the compounded interest rate
after 7 years and 9 months.
a. 690 849
b. 670 258
c. 680 686
d. 660 592
697. P200 000 was deposited at an
interest rate of 24% compounded semiannually. After how many years will the
sum be P621 170?
a. 4
b. 3
c. 5
d. 6
698. A bank is advertising 9.5%
accounts that yield 9.84 annually. How
often is the interest compounded?
a. monthly
b. bi- monthly
c. quarterly
d. daily
699. Evaluate the integral of 2dx/(4x+3)
if the upper limit is 5 and the lower limit
is 1.
a. 0.595
b. 0.675
c. 0.486
d. 0.387
700.
Evaluate
the
integral
of
2xdx/(2x^2+4) if the upper limit is 6 and
the lower limit is 3.
a. 0.620
b. 0.675
c. 0.486
d. 0.580
MULTIPLE CHOICE QUESTIONS IN
ENGINEERING MATHEMATICS
FOR
COMPUTERIZED LICENSURE EXAM
Author:
Venancio I. Besavilla, Jr.
Engineering Economics
1.) A price tag of P1200 is specified if paid within 60 days but offers a 3% discount for cash in
30 days. Find the rate of interest.
a.37.11%
c. 40.21%
b.38.51%
d. 39.31%
2.) It is the practice of almost all bank in the Philippines that when they grant a loan, the interest
for 1 year is automatically deducted from the principal amount upon release of money to a
borrower. Let us therefore assume that you applied for a loan with the bank and the P80000 was
approved at an interest rate of 14% of which P11200 was deducted and you were given ac check
of P68800. Since you have to pay the amount of P80000 one year after, what then will be the
effective interest rate?
a.16.28%
c. 17.30%
b.38.51%
d. 39.31%
3.) Mr. J. Dela Cruz borrowed money from the bank. He received from the bank P1,340.00 and
promised to pay P1,500.00 at the end of 9 months. Determine the simple interest rate and the
corresponding discount rate or often referred to as the “Banker’s Discount”.
a.15.92% ; 13.73%
c. 12.95% ; 17.33%
b.18.28% ; 13.12%
d. 19.25% ; 13.33%
4.) A man borrowed from the bank under a promissory note that he signed in the amount of
P25000.00 for a period of 1year. He received only the amount of P21,915.00 after the bank
collected the advance interest and an additional amount of P85.00 for notarial and inspection
fees. What was the rate of interest that the bank collected in advance?
a.13.64%
c. 16.43%
b.18.37%
d. 10.32%
5.) Agnes Abanilla was granted a loan of P20,000 by her employer CPM Industrial Fabricator
and Construction Corporation with an interest at 6% for 180days on the principal collected in
advance. The corporation would accept a promissory note for P20,000 non-interest for 180days.
If discounted at once, find the proceeds in the note.
a. P18,800
c. P18,000
b. P19,000
d. P18,400
6.) P4000 is borrowed for 75days at 16% per annum simple interest. How much will be due at
the end of 75days?
a. P4186.43
c. P4133.33
b. P5124.54
d. P5625.43
7.) Mr. Almagro made a money market placement of P1,000,000 for 30 days at 7.5% per year. If
withholding tax is 20%, what is the net interest that Mr. Almagro will receive at the end of the
month?
a. P3,000
c. P6,000
b. P4,000
d. P5,000
8.) A bill for motorboat specifies the cost as P1,200 due at the end of 100days but offers a 4%
discount for cash in 30days. What is the highest rate, simple interest at which the buyer can
afford to borrow money in order to take advantage of the discount?
a. 18.4%
c. 20.9%
b. 19.6%
d. 21.4%
9.) In buying a computer disk, the buyer was offered the options of paying P250 cash at the end
of 30days or P270 at the end of 120days. At what rate is the buyer paying simple interest if he
agree to pay at the end of 120days?
a. 32%
c. 28%
b.40%
d. 25%
10.) On March 1, 1996 Mr. Almagro obtains a loan of P1500 from Mr. Abella and signs a note
promising to pay the principal and accumulated simple interest at the rate of 5% at the end of
120days. On May 15, 1996, Mr. Abella discounts the note at the bank whose discount rate is 6%.
What does he receive?
a. P2,201.48
b. P1,123.29
c. P1,513.56
d. P938.20
11.) A deposit of P110,000 was made for 31days. The net interest after deducting 20%
withholding tax is P890.36. Find the rate of return annually.
a. 12.25
c. 12.75
b. 11.75
d. 11.95
12.) If you borrowed money from your friend with simple interest of 12%, find the present worth
of P50,000 which is due at the end of 7months.
a. P46,200
c. P46,729
b. P44,893
d. P45,789
13.) A man borrowed P2000 from a bank and promise to pay the amount for 1year. He received
only the amount of P1,920 after the bank collected an advance interest of P80. What was the rate
of discount and the rate of interest that the bank collected in advance?
a. 4% ; 4.17%
c. 4% ; 4.71%
b. 3% ; 3.17%
d. 3% ; 3.71%
14.) An engineer promised to pay P36,000 at the end of 90days. He was offered a 10% discount
if he pays in 30days. Find the rate of interest.
a. 64.6%
c. 66.6%
b. 65.6%
d. 67.6%
15.) A man is required to pay P200 at the end of 160days or P190 at the end of 40days.
Determine the rate of interest.
a. 18.4%
c. 15.8%
b. 19.6%
d. 16.4%
16.) Compute the discount if P2000 is discounted for 6months at 8% simple interest.
a. P29.67
c. P76.92
b. P67.29
d. P92.76
17.) The amount of P12800 in 4years at 5% compounded quarterly is _____.
a. P14,785.34
c. P16,311.26
b. P15,614.59
d. P15,847.33
18.) A man borrows money from a bank which uses a simple discount rate of 14%. He signs a
promissory note promising to pay P500.00 per month at the end of 4th, 6th, and 7th month
respectively. Determine the amount of money that he received from the bank.
a. P1403.68
c. P1102.37
b. P1340.38
d. P1030.28
19.) A nominal interest of 3% compounded continuously is given on the account. What is the
accumulated amount of P10,000 after 10years?
a. P13,610.10
c. P13,498.60
b. P13,500.10
d. P13,439.16
20.) BY the condition of a will, the sum of P2000 is left to a girl to be held in trust fund by her
guardian until it amount to P50000.00. When will the girl receive the money of the fund is
invested at 8% compounded quarterly?
a. 7.98 years
c. 11.57 years
b. 10.34 years
d. 10.45 years
21.) A man expects to receive P25,000 in 8 years. How much is that worth now considering
interest at 8% compounded quarterly?
a. P13,859.12
c. P13,675.23
b. P13,958.33
d. P13,265.83
22.) P500,000 was deposited at an interest of 6% compounded quarterly. Compute the compound
interest after 4 years and 9 months.
a. P163,475.37
c. P158,270.37
b. P178,362.37
d. P183,327.37
23.) If the nominal interest rate is 3%, how much is P5000 worth in 10 years in a continuously
compounded account?
a. P5750
c. P7500
b. P6750
d. P6350
24.) P200,000 was deposited for a period of 4 years and 6 months and bears on interest of
P85649.25. What is the rate of interest if it is compounded quarterly?
a. 8%
c. 7%
b. 6%
d. 5%
25.) How many years will P100,000 earn a compound interest of P50,000 if the interest rate is
9% compounded quarterly?
a. 3.25 years
c. 5.86 years
b. 4.55 years
d. 2.11 years
26.) A certain amount was deposited 5 years and 9 months ago at an interest of 8% compounded
quarterly. If the sum now is P315,379.85, how much was the amount deposited?
a. P200,000
c. P240,000
b. P180,000
d. P260,000
For Problems 27-29:
When compounded Bi-monthly, P15000 becomes P22,318.30 after 5years.
27.) What is the nominal rate of interest?
a. 7%
c. 9%
b. 8%
d. 10%
28.) What is the equivalent rate if it is compounded quarterly?
a. 7.03%
c. 9.03%
b. 8.03%
d. 10.03%
29.) What is the effective rate if it is compounded quarterly?
a. 7.28%
c. 9.28%
b. 8.28%
d. 10.28%
30.) How will it take a money to double itself if invested at 5% compounded annually?
a. 12.2 years
c. 14.2 years
b. 13.2 years
d. 15.2years
31.) Compute the effective annual interest rate which is equivalent to 5% nominal annual interest
compounded continuously.
a. 5.13%
c. 5.26%
b. 4.94%
d. 4.90%
32.) Find the time required for a sum of money to triple itself at 5% per annum compounded
continuously.
a. 21.97 years
c. 18.23 years
b. 25.34 years
d. 23.36 years
33.) A man wishes to have P40,000 in a certain fund at the end of 8 years. How much should he
invest in a fund that will pay 6% compounded continuously?
a. P24,751.34
c. P28,864.36
b. P36,421.44
d. P30,468.42
34.) If the effective annual interest rate is 4%, compute the equivalent nominal annual interest
compounded continuously.
a. 3.92%
c. 3.80%
b. 4.10%
d. 4.09%
35.) What is the nominal rate of interest compounded continuously for 10 years if the compound
amount factor is equal to 1.34986?
a. 3%
c. 5%
b. 4%
d. 6%
36.) American Express Corp. charges 1.5% interest per month, compounded continuously on the
unpaid balance purchases made on this credit card. Compute the effective rate of interest.
a. 19.72%
c. 21.20%
b. 20.25%
d. 19.90%
37.) If the nominal interest is 12% compounded continuously, compute the effective annual rate
if interest.
a. 12.75%
c. 12.40%
b. 11.26%
d. 11.55%
38.) Compute the difference in the future amount of P500 compounded annually at nominal rate
of 5% and if it is compounded continuously for 5 years at the same rate.
a. P3.87
c. P5.48
b. P4.21
d. P6.25
39.) If the effective rate is 24%, what nominal rate of interest is charged for a continuously
compounded loan?
a. 21.51%
c. 23.25%
b. 22.35%
d. 21.90%
40.) What is the nominal rate of interest compounded continuously for 8 years if the present
worth factor is equal to 0.6187835?
a. 4%
c. 6%
b. 5%
d. 7%
41.) What is the difference of the amount 3 years from now for a 10% simple interest and 10%
compound interest per year?(P5000 accumulated)
a. P155
c. same
b. P100
d. P50
42.) Find the discount if P2,000 is discounted for 6 months at 8% compounded quarterly.
a. P76.92
c. P77.66
b. P80.00
d. P78.42
43.) If a sum of money triples in a certain period of time at a given rate interest, compute the
value of the single payment present worth factor.
a. 0.333
c. 0.292
b. 3.000
d. 1.962
44.) If the single payment amount factor for a period of 5 years is 1.33822. What is the nearest
value of the interest rate?
a. 8%
c. 5%
b. 7%
d. 6%
45.) If the single payment present worth factor for a period of 8 years is equal to 0.58201,
compute the nearest value of the rate of interest for that period.
a. 6%
c. 5%
b. 7%
d. 8%
46.) If money is worth 8% compounded quarterly, compute the single payment amount factor for
a period of 6 years.
a. 1.60844
c. 1.70241
b. 0.62172
d. 0.53162
47.) A sum of P1,000 is invested now and left for 8 years, at which time the principal is
withdrawn. The interest has an accrued left for another 8 years. If the effective annual interest is
5%, what will be the withdrawn amount at the end of the 16 th year?
a. P507.42
c. P750.42
b. P705.42
d. P425.07
For problems 48-50:
Compute the interest for an amount of P200,000 for a period of 8 years.
48.) If it was made at a simple interest rate of 16% .
a. P274,000
c. P256,000
b. P265,000
d. P247,000
49.) If it was made at 16% compounded bi-monthly.
a. P507,267.28
c. P407,283.01
b. P507,365.96
d. P459,923.44
50.) If it was made at 16% compounded continuously.
a. P422,462.64
c. P524,242.46
b. P507,233.24
d. P519,327.95
51.) Find the value of x, (F/P, x, 6) if F/P compounded quarterly is equal to 1.612226.
a. 7%
c. 5%
b. 8%
d. 6%
52.) Find the value of y, (P/F, 6%, y) if P/F compounded bi-monthly is equal to 0.787566.
a. 7 years
c. 5 years
b. 6 years
d. 4 years
53.) Find the rate of interest if compound amount factor compounded bi-monthly for 5 years is
equal to 1.487887.
a. 7%
c. 5%
b. 8%
d. 6%
54.) Compute the nominal rate for a period of 6 years for an effective rate of 8.33% if it is
compounded continuously.
a. 7%
c. 5%
b. 8%
d. 6%
55.) Compute the equivalent compound amount factor if it is compounded continuously.
a. 1.61607
c. 1.24532
b. 1.24282
d. 0.24245
56.) Compute the equivalent present worth factor if it is compounded continuously.
a. 1.249347
c. 1.243248
b. 1.214359
d. 0.616783
57.) A man loan P2000 from the bank. How long would it take in years for the amount of the
loan and interest to equal P3280 if it was made at 8% simple interest.
a. 7 years
c. 5 years
b. 8 years
d. 6 years
58.) A man loan P2000 from the bank. How long would it take in years if it was made at 8%
compounded quarterly.
a. 5.25 years
c. 7.25 years
b. 6.25 years
d. 8.25 years
59.) A man loan P2000 from the bank. How long would it take in years if it was made at 8%
compounded continuously.
a. 7.18 years
c. 5.18 years
b. 8.18 years
d. 6.18 years
60.) An amount of P1000 becomes P1608.44 after 4 years compounded bi-monthly. Find the
nominal rate of interest, the effective rate of interest, and the equivalent rate of interest if it is
compounded quarterly.
a. 11% ; 12.623% ; 11.42%
c. 12% ; 11.06% ; 12.724%
b. 12% ; 12.616% ; 12.06%
d. 11% ; 11.664% ; 11.93%
61.) How long would it take your money to double itself if it is invested at 6% simple interest,
compounded semi-quarterly, and compounded continuously?
a. 12.67yrs ; 11.2yrs ; 15.05yrs
c. 18.67yrs ; 11.6yrs ; 11.24yrs
b. 16.67yrs ; 11.6yrs ; 11.55yrs
d. 17.67yrs ; 10.2yrs ; 11.45yrs
For problems 62-64:
An amount of P50,000 was invested for 3 years.
62.) Compute its interest at the end of 3 years if it is invested at 10% simple interest.
a. P16,550.00
c. P15,000.00
b. P17,492.94
d. P14,242.15
63.) Compute for its compound interest if it is invested at 10% compounded annually.
a. P16,550.00
c. P15,000.00
b. P17,492.94
d. P14,242.15
64.) Compute its interest if it is invested at 10% compounded continuously.
a. P16,550.00
c. P15,000.00
b. P17,492.94
d. P14,242.15
For Problems 65-67:
P200,000 was deposited for a period of 4 yrs. And 6 months and bears on interest of P85,659.25.
65.) What is the nominal rate of interest if it is compounded quarterly?
a. 8.00%
c. 7.00%
b. 8.24%
d. 7.96%
66.) What is the actual rate of interest?
a. 8.00%
c. 7.00%
b. 8.24%
d. 7.96%
67.) What is the equivalent nominal rate if it is compounded semi-quarterly?
a. 8.00%
c. 7.00%
b. 8.24%
d. 7.96%
68.) What is the value of (F/P, 8%, 6) if it is compounded semi-quarterly.
a. 0.524273541
c. 1.487886717
b. 0.787566127
d. 1.612226000
69.) What is the value of (P/F, 6%, 4) if it is compounded bi-monthly.
a. 0.524273541
c. 1.487886717
b. 0.787566127
d. 1.612226000
70.) What is the single payment compound amount factor for 8% compounded bi-monthly for 5
years.
a. 0.524273541
c. 1.487886717
b. 0.787566127
d. 1.612226000
For Problems 71-73:
If money is invested at a nominal rate of interest of 8% for a period of 4 years.
71.) What is the effective rate if it is compounded continuously?
a. 8.33%
c. 9.33%
b. 8.93%
d. 9.93%
72.) What is the value of the compound amount factor if it is compounded continuously?
a. 1.377128
c. 1.424231
b. 0.214212
d. 0.122416
73.) What is the value of the present worth factor if it is compounded continuously?
a. 1. 272441
c. 0.272441
b. 1.726419
d. 0.726149
74.) If the single payment amount factor for a period of 5 years is 1.33822, what is the nominal
rate of interest?
a. 6.00%
c. 7.00%
b. 6.92%
d.7.92%
75.) What is the effective rate of interest if it is compounded semi-annually of problem# 74?
a. 5.00%
c. 7.24%
b. 6.92%
d.6.09%
76.) What is the equivalent nominal rate if it is compounded quarterly of problem# 74?
a. 6.12%
c. 4.24%
b. 5.43%
d.5.87%
77.) Money was invested at x% compounded quarterly. If it takes money to quadruple in 17.5
years, find the value of x.
a. 6.23%
c. 8.00%
b. 5.92%
d.9.78%
78.) What is the actual interest rate of problem# 77?
a. 7.24%
c. 7.92%
b. 8.24%
d.8.87%
79.) What is the equivalent rate if problem# 77 is compounded daily?
a. 7.24%
c. 7.92%
b. 8.24%
d.8.87%
For Problems 80-82:
A businessman loaned P500,000 from a local bank that charges an interest rate of 12%.
80.) How much is he going to pay at the end of 5 years if it was made at 12% simple interest?
a. P800,000.00
c. P911,059.20
b. P823,243.09
d. P907,009.21
81.) How much is he going to pay at the end of 5 years if it was made at 12% compound semiquarterly?
a. P800,000.00
c. P911,059.20
b. P823,243.09
d. P907,009.21
82.) How much is he going to pay at the end of 5 years if it was made at 12% compound
continuously?
a. P800,000.00
c. P911,059.20
b. P823,243.09
d. P907,009.21
83.) P60,000 was deposited at 6% compounded quarterly, tax free for 9 years and 3 months. How
much interest was earned at the end of the period?
a. P43,214.24
c. P44,086.60
b. P43.242.24
d. P44,215.60
84.) P100,000 was placed in a time deposit which earned 9% compounded quarterly tax free.
After how many years would it be able to earn a total interest of P50,000?
a. 4.56 years
c. 3.45 years
b. 4.23 years
d. 3.64 years
85.) P200,000 was placed in a time deposit at x% compounded quarterly and was free of taxes.
After exactly 5 years, the total interest earned was P120,000. What is the value of x?
a. 9.5%
c. 6.35%
b. 8.21%
d. 7.12%
86.) Which of these gives the lowest effective rate of interest?
a. 12.35% compounded annually
c. 12.2% compounded quarterly
b. 11.9% compounded semi-annually
d. 11.6% compounded monthly
87.) It takes 20.15 years to quadruple your money if it is invested at x% compounded semiannually. Find the value of x.
a. 8%
c. 7%
b. 6%
d. 5%
88.) It takes 13.87 years to treble the money at the rate of x% compounded quarterly. Compute
the value of x.
a. 5%
c. 7%
b. 6%
d. 8%
89.) Money was invested at x% compounded quarterly. If it takes the money into quadruple in
17.5 years, find the value of x.
a. 8%
c. 7%
b. 6%
d. 5%
90.) Fifteen years ago P1,000.00 was deposited in a bank account, and today it is worth
P2.370.00. The bank pays interest semi-annually. What was the interest rate paid on this
account?
a. 4.9%
c. 5.0%
b. 5.8%
d. 3.8%
91.) You borrow P3,500.00 for one year from a friend at an interest rate of 1.5% per month
instead of taking a loan from a bank at a rate of 18% per year. Compare how much money you
will save or lose on the transaction.
a. pay P155 more if you borrowed from the bank
b. save P55 by borrowing from your friend
c. pay P85 more if you borrowed from the bank
b. save P95 by borrowing from your friend
92.) Find the present worth of a future payment of P1000,000 to be made in 10 years with an
interest of 12% compounded quarterly.
a. P30,444.44
c. P30,655.68
b. P33,000.00
d. P30,546.01
93.) An initial savings deposit of P80,000 in a certain bank earns 6% interest per annum
compounded monthly. If the earnings from the deposit are subject to a 20% tax, what would the
net value of the deposit be after three quarters?
a. P95,324.95
c. P68743.24
b. P82938.28
d. P56244.75
94.) The effective rate of interest of 14% compounded semi-annually is:
a. 14.49%
c. 14.94%
b. 14.36%
d. 14.88%
95.) The amount of P50,000 was deposited in a bank earning an interest of 7.5% per annum.
Determine the total amount at the end of 5 years, if the principal and interest were not withdrawn
during the period.
a. P71,781.47
c. P70,374.90
b. P72,475.23
d. P78,536.34
96.) What is the effective rate corresponding to 18% compounded daily? Take 1 year is equal to
360 days.
a. 18.35%
c. 18.10%
b. 19.39%
d. 19.72%
97.) If P1,000 becomes P1,126.49 after 4 years when invested at a certain nominal rate of interest
compounded semi-annually, determine the nominal rate and the corresponding effective rate.
a. 3.00% and 3.02%
c. 2.30% and 2.76%
b. 4.29% and 4.32%
d. 3.97% and 3.95%
98.) Convert 12% semi-annually to compounded quarterly.
a. 19.23%
c. 14.67%
b. 23.56%
d. 11.83%
99.) What is the corresponding effective interest rate of 18% compounded semi-quarterly?
a. 19.25%
c. 18.46%
b. 19.48%
d. 18.95%
100.) If P5000 shall accumulate for 10 years at 8% compounded quarterly, find the compounded
interest at the end of 10 years.
a. P6,005.30
c. P6,040.20
b. P6,000.00
d. P6,010.20
101.) A couple borrowed P4,000 from a lending company for 6 years at 12%. At the end of 6
years, it renews the loan for the amount due plus P4,000 more for 3 years at 12%. What is the
lump sum due?
a. P14,842.40
c. P12,316.40
b. P16,712.03
d. P15,382.60
102.) How long (in years) will it take the money to quadruple if it earns 7% compounded semiannually?
a. 26.30 years
c. 33.15 years
b. 40.30 years
d. 20.15 years
103.) P200,000 was deposited on Jan. 1,1988 at an interest rate of 24% compounded semiannually. How much would the sum be on Jan. 1, 1993?
a. P421,170
c. P401,170
b. P521,170
d. P621,170
104.) If P500,000 is deposited at a rate of 11.25%
compounded interest after 7 years and 9 months.
compounded monthly, determine the
a. P690,849
c. P680,686
b. P670,258
d. P660,592
105.) P200,000 was deposited at an interest rate of 24% compounded semi-annually. After how
many years will the sum be P621,170?
a. 4 years
c. 5 years
b. 3 years
d. 6 years
106.) A bank is advertising 9.5% accounts that yields 9.84% annually. How often is the interest
compounded?
a. monthly
b. bi-monthly
c. quarterly
d. daily
107.) A marketing company established a program to replace the cars of its sales representatives
at the end of every 5 years. If the present price of the approved type of car is P520,000.00 with a
resale value at the end of 5 years of 60% its present value, how much money must the company
accumulate for 5 years if inflation annually is 10%. Release value will also appreciate at 10%
yearly.
a. P120,289.51
c. P110,297.27
b. P129,382.38
d. P122,265.69
108.) In year zero, you invest P10,000.00 in a 15% security for 5 years. During that time, the
average annual inflation is 6%. How much, in terms of year zero will be in the account at
maturity?
a. P15,386.00
c. P13,382.00
b. P15,030.00
d. P16,653.00
109.) A machine has been purchased and installed at a total cost of P18,000.00. The machine will
be retired at the end of 5 years, at which time it is expected to have a scrap value of P2,000.00
based on current prices. The machine will then be replaced with an exact duplicate. The
company plans to establish a reserve fund to accumulate the capital needed to replace the
machine. If an average annual rate of inflation of 3% is anticipated, how much capital must be
accumulated?
a. P15,030.00
c. P12,382.00
b. P18,548.39
d. P15,386.00
110.) If the inflation rate is 6%, cost of money is 10%, what interest rate will take care of
inflation and the cost of money?
a. 16.6%
c. 17.7%
b. 15.5%
d. 14.4%
111.) A man bought a government bond which cost P1000 and will pay P50 interest each year
for 20 years. The bond will mature at the end of 20 years and he will receive the original P1000.
If there is 2% annual inflation during this period, what rate of return will the investor receive
after considering the effect of inflation?
a. 2.94%
c. 4.25%
b. 3.16%
d. 5.16%
112.) The inflated present worth of P2000 in two years is equal to P1471.07. What is the rate of
inflation if the interest rate is equal to 10%?
a. 6%
c. 7%
b. 5%
d. 4%
113.) 12% rate of interest can take care of the cost of money and inflation. If the nominal rate of
interest is 6%, what is the rate of inflation?
a. 6.62%
c. 7.67%
b. 5.66%
d. 4.64%
114.) An engineer bought an equipment for P500,000. Other expenses including installations
amounted to P30,000. At the end of its estimated useful life of 10 years, the salvage value will be
10% of the first cost. Using straight line method of depreciation, what is the book value after 5
years?
a. P291,500
c. P242,241
b. P282,242
d. P214,242
115.) A small machine costing P80,000 has a salvage value of x at the end of its life of 5 years.
The book value at the end of the 4th year is P22,400. What is the value of x using the straight line
method depreciation?
a. P6000
c. P8000
b. P7000
d. P9000
116.) A machine has a salvage value of P12,000 at the end of its useful life of 6 years. The book
value at the end of 5 years is P30,833.33. Using a straight line method of depreciation, what is
the first cost of the machine?
a. P125,500
c. P125,000
b. P135,500
d. P135,000
117.) A manufacturing plant was built at a cost of P5M and is estimated to have a life of 20 years
with a salvage value of P1M. A certain equipment worth P570,000 was installed at a cost of
P80,000 is expected to operate economically for 15 years with a salvage value of P50,000.
Determine the book value of the plant and equipment after 10 years, use straight line
depreciation method.
a. P3,250,000
c. P4,250,000
b. P3,750,000
d. P4,500,000
118.) A printing equipment costs P73,500 has a life expectancy of 8 years and has a salvage
value of P3,500 at the end of its life. The book value at the end of x years is equal to P38,500.
Using straight line method of depreciation, solve for the value of x.
a. 5 years
c. 6 years
b. 4 years
d. 3 years
119.) The cost of the printing equipment is P500,000 and the cost of handling and installation is
P30,000. If the book value of the equipment at the end of the 3 rd year is P242,000 and the life of
the equipment is assumed to be 5 years, determine the salvage value of this equipment at the end
of 5 years.
a. P50,000
c. P53,000
b. P60,000
d. P64,000
120.) An engineer bought an equipment for P500,000. He spent an additional amount of P30,000
for installation and other expenses. The salvage value is 10% of the first cost. If the book value at
the end of 5 years will be P291,500 using straight line method of depreciation, compute the
useful life of the equipment in years.
a. 10 years
c. 6 years
b. 8 years
d. 15 years
121.) The cost of equipment is P500,000 and the cost of installation is P30,000. If the salvage
value is 10% of the cost of equipment at the end of 5 years, determine the book value at the end
of the fourth year. Use straight line method.
a. P155,000
c. P146,000
b. P140,000
d. P132,600
For Problems 122-124:
The first cost of a machine is P1,800,000 with a salvage value of P300,000 at the end of its life
of 5 years. Determine the total depreciation after 3 years.
122.) Using Straight Line Method
a. P800,000
c. P900,000
b. P600,000
d. P700,000
123.) Using Sum of Years Digit Method
a. P1,150,000
c. P1,300,000
b. P1,200,000
d. P1,350,600
124.) Using Constant Percentage Method
a. P1,355,024.24
c. P1,246,422.53
b. P1,185,769.76
d. P1,432,624.84
125.) An asset is purchased for P9,000.00. Its estimated economic life is 10 years after which it
will be sold for P1,000.00. Find the depreciation in the first three years using straight line
method.
a. P2,500
c. P3,000
b. P2,400
d. P2,000
126.) The purchase of a motor for P6000 and a generator for P4000 will allow the company to
produce its own energy. The configuration can be assembled for P500. The service will operate
for 1600 hours per year for 10 years. The maintenance cost is P300 per year, and cost to operate
is P0.85 per hour for fuel and related cost. Using straight line depreciation, what is the annual
cost for the operation? There is a P400 salvage value for the system at the end of 10 year.
a. P2,710
c. P2,630
b. P2,480
d. P2,670
127.) A machine has an initial cost of P50,00.00 and a salvage value of P10,000.00 after 10
years. What is the straight line method depreciation rate as a percentage of the initial cost?
a. 10%
c. 12%
b. 8%
d. 9%
128.) A machine has an initial cost of P50,00.00 and a salvage value of P10,000.00 after 10
years. What is the book value after 5 years using straight line method depreciation rate?
a. P35,000
c. P15,500
b. P25,000
d. P30,000
129.) A machine has a first of P80,000 and a salvage of P2,000 at the end of its life of 10 years.
Find the book value at the end of the 6th year using straight line method of depreciation.
a. P33,200
c. P34,300
b. P35,400
d. P32,900
130.) An asset is purchased for P90,000.00. Its estimated life is 10 years after which it will be
sold for P1,000.00. Find the book value during the first year if Sum of the Years Digits(SYD)
depreciation is used.
a. P7,545.45
c. P5,245.92
b. P2,980.24
d. P6,259.98
131.) A telephone company purchased a microwave radio equipment for P6M. Freight and
installation charges amounted to 3% of the purchased price. If the equipment shall be depreciated
over a period of 8 years with a salvage value of 5%, determine the depreciation charge during the
5th year using the Sum of Year Digit Method.
a. P756,632.78
c. P652,333.33
b. P957,902.56
d. P845, 935.76
132.) A consortium of international communications companies contracted for the purchase and
installation of a fiber optic cable linking two major Asian cities at a total cost of P960M. This
amount includes freight and installation charges at 10% of the above total contract price. If the
cable depreciated over a period of 15 years with zero salvage value, what is the depreciation
charge during the 8th year using the sum of year digits method?
a. P64M
c. P80M
b. P23M
d. P76M
133.) A machine cost P7,350 has a life of 8 years and has a salvage value of P350 at the end of 8
years. Determine its book value at the end of 4 years using sum years digit method.
a. P3,476.90
c. P6,543.21
b. P2,294.44
d. P5,455.01
134.) A certain equipment costs P7,000 has an economic life of n years and a salvage value P350
at the end of n years. If the book value at the end of 4 years is equal to P2197.22, compute for the
economic life of the equipment using the sum of years digit method.
a. 10 years
c. 8 years
b. 16 years
d. 11 years
135.) A company purchased an asset for P10000 and plans to keep it for 20 years. If the salvage
value is zero at the end of the 20th year, what is the depreciation in the third year? Use sum of
years digit method.
a. P1000
c. P938
b. P857
d. P747
136.) An equipment costs P7000 and has a life of 8 years and salvage value of x after 8 years. If
the book value of the equipment at the 4th year is equal to P2197.22, compute the salvage value x
using the sum of years digit method.
a. P594
c. P350
b. P430
d. P290
137.) ABC Corporation makes it policy that for every new equipment purchased, the annual
depreciation should not exceed 20% of the first cost at any time without salvage value.
Determine the length of service if the depreciation used is the SYD Method.
a. 9 years
c. 12 years
b. 10 years
d. 19 years
138.) A machine having a certain first cost has a life of 10 years and a salvage value of 6.633%
of the first cost at the end of 10 years. If it has a book value of P58,914 at the end of the 6 th year,
how much is the first cost of the machine if the constant percentage of declining value is used in
the computation for its depreciation.(Matheson’s Method)
a. P600,000
c. P100,000
b. P300,000
d. P900,000
139.) A machine costing P720,000 is estimated to have a life of 10 years. If the annual rate of
depreciation is 25%, determine the total depreciation using a constant percentage of the declining
balance method.
a. P679,454.27
c. P532,825.73
b. P432,725.45
d. P764,243.33
140.) An earth moving equipment that cost P90,000 will have an estimated salvage value of
P18,000 at the end of 8 years. Using double-declining balance method, compute the book value
and the total depreciation at the end of the 5th year.
a. P21,357.42 ; P68,642.58
c. P24,362.48 ; P65,637.52
b. P15,830.34 ; P74,169.66
d. P19,442.78 ; P70,557.22
141.) A certain office equipment has a first cost of P20,000 and has a salvage value of P1,000 at
the end of 10 years. Determine the depreciation at the end of the 6th year using Sinking fund
method at 3% interest.
a. P10,720
c. P11,680
b. P12,420
d. P9,840
142.) An equipment which cost P200,000 has a useful life of 8 years with a salvage value of
P25,000 at the end of its useful life. If the depreciation at the first year is P21,875, what method
is used in the calculation of depreciation?
a. Straight Line
c. Declining Balance
b. Sinking Fund
d. Sum of Years Digit
143.) An equipment costs P8,000 has an economic life of 8 years and salvage value of P400 at
the end of 8 years. The first year depreciation amounts to P1,688.89. What method is used in the
calculation of the depreciation?
a. Straight Line
c. Declining Balance
b. Sinking Fund
d. Sum of Years Digit
144.) The original cost of a certain machine is P150,000 and has an economic life of 8 years with
a salvage value of P9,000 at that time. If the depreciation of the first year is equal to P44,475,
what method is used in the calculation of the depreciation?
a. Straight Line
c. Declining Balance
b. Sinking Fund
d. Sum of Years Digit
145.) A machine has a first cost of P140,000 and a life of 8 years with a salvage value of
P10,000 at the end of its useful life. If the depreciation at the first year amounts to P35,000, what
method is used in the calculation of depreciation?
a. Double Declining Balance
b. Declining Balance
c. Straight Line
d. Sum of Years Digit
146.) A hydraulic machine cost P180,000 and has a salvage value of P15,000 at the end of its
useful life which is 12 years. If the depreciation at the first year is P9,780.71, what method is
used in computing the depreciation. Assume money is worth 6% annually.
a. Straight Line
c. Declining Balance
b. Sinking Fund
d. Sum of Years Digit
147.) An equipment costs P480,000 and has a salvage value of 10% of its cost at the end of its
economic life of 35,000 operating hours. In the first year, it was used for 4,000 hours. Determine
its book value at the end of the first year.
a. P430,629.00
c. P418,360.00
b. P380,420.00
d. P376,420.00
148.) An equipment costs P480,000 and has a salvage value of 10% of its cost at the end of its
economic life of 36,000 operating hours in a period of 5 years. In the first year of service, it was
used for 12,000 hours. If at the end of the 2nd year it was used for 15,000 hours, find the
depreciation at the second year.
a. P180,000
c. P190,000
b. P160,000
d. P150,000
149.) A certain machine cost P40,000 and has a life of 4 years and a salvage value of P5000. The
production output of this machine in units per year is 1000 units for the first year, 2000 units for
the second year, 3000 units for the third year, and 4000 units for the fourth year. If the units
produced are in uniform quality, what is the depreciation charge at the end of the fourth year.
a. P14,000
c. P15,000
b. P13,000
d. P16,000
150.) A lathe machine costs P300,000 brand new with a salvage value of x pesos. The machine is
expected to last for 28500 hours in a period of 5 years. In the first year of service it was used for
8000 hours. If the book value at the end of the first year is P220,000, compute for the salvage
value x in pesos.
a. P15,000
c. P12,000
b. P18,000
d. P20,000
151.) A certain machine cost P40,000 and has a life of 4 years and a salvage value of P5000. The
production output of this machine in units per year is first year 1800 units, second year2200
units, third year 3000, and fourth year 4000 units. If the units produced are of uniform quality,
what is the depreciation charge at the end of 4th year?
a. P12,727.27
c. P16,420.43
b. P15,350.23
d. P17,200.98
152.) A lathe machine cost P300,000 with a salvage value of P15,000 is expected to last for
285000 hours in a period of 5 years. In the first year of service it was used for 8000 hours.
Compute the book value of the machine at the end of the first year.
a. P292,000
c. P250,000
b. 200,000
d. P323,000
153.) A machine costing P280,000 has a useful life of 20,000 hrs. at the end of which its salvage
value is P30,000. In the first year, it was used for 2080 hrs, in the second year, 3160 hrs. Find the
second depreciation cost in pesos.
a. P23,520
c. P39,500
b. P25,252
d. P35,400
154.) An equipment costs P400,000 and has a life of 30,000 hrs at the end of which its salvage
value is x pesos. In the first year, it was used for 6240 hrs. The book value at the end of the first
year was P325,120. Find the value of x.
a. P40,629
c. P40,000
b. P30,420
d. P30,000
155.) An engineering firm from purchased, 12 years ago, a heavy planner for P50,000 with no
salvage value. As the life of the planner was 20 years, a straight line depreciation reserve has
been provided on that basis. Now the firm wishes to replace the old planner with a new one
possessing several advantages. It can sell the old planner for P10,000. The new one will cost
P100,000. How much new capital will be required to make the purchase?
a. P60,000
c. P66,000
b. P55,000
d. P57,000
156.) Ten years ago, a contractor was able to purchase a crane whose capacity is 2000 tons
costing P125 per ton. The life was estimated to be 15 years with a salvage value of 10% of the
cost. A market has been found for the old crane for P80,000. If the depreciation has been figured
on a straight line basis what is the difference between the depreciation book value of the old
crane and its sale value.
a. P20,000
c. P15,000
b. P30,000
d. P10,000
157.) What is the nominal value of interest compounded continuously for a period of 5 years of
an equal payment series if the capital recovery factor is equal to 0.2385787.
a. 6%
c. 5%
b. 4%
d. 8%
158.) What is the nominal rate of interest compounded continuously for a period of 5 years of an
equal payment series if the sinking fund factor is equal 0.180519?
a. 5%
c. 6%
b. 4%
d. 8%
159.) Compute the number of years so that the capital recovery factor of a uniform payment
series be equal to 0.218638 if money is worth 3% compounded continuously.
a. 5
c. 6
b. 4
d. 3
160.) A manufacturing firm wishes to give each 80 employee a holiday bonus. How much is
needed to invest monthly for a year at 12% nominal interest rate, compounded monthly, so that
each employee will receive a P2000 bonus?
a. P12,608
c. P12,600
b. P12,615
d. P12,300
161.) An instructor plans to retire in one year and want an account that will pay him P25000 a
year for the next 15 years. Assuming 6% annual effective interest rate, what is the amount he
would need to deposit now? (The fund will be depleted after 15 years)
a. P249,000
c. P248,500
b. P242,806
d. P250,400
162.) Ryan invest P5,000 at the end of each year in an account which gives a nominal annual
interest of 7.5%, compounded continuously. Determine the total worth of his investment at the
end of 15 years.
a. P133,545.58
c. P126,336.42
b. P142,647.28
d. P135,429.64
163.) A car dealer advertise the sale of a car model far a cash price of P280,000. If purchased in
installment, the required down payment is 15% and balance payable in 18 equal monthly
installments at an interest rate of 1.5% per month. How much will be the required monthly
payments?
a. P15,185.78
c. P10,972.33
b. P11,588.72
d. P15,558.12
164.) How much must be deposited at 6% each year beginning Jan 1, year 1, in order to
accumulate P5,000 on the date of the last deposit, Jan 1, year 6?
a. P751
c. P715
b. P717
d. P775
165.) In anticipation of a much bigger volume of business after 10 years, a fabrication company
purchased an adjacent lot for its expansion program where it hopes to put up a building projected
to cost P4,000,000 when it will be constructed 10 years after. To provide for the required capital
expense, it plans to put up a sinking fund for the purpose. How much must the company deposit
each year if the interest to be earned is computed at 15%
a. P197,008.25
c. P177,009.25
b. P199,090.25
d. P179,008.25
166.) Rainer Wandrew borrowed P50,000 from Social Security System, in the form of calamity
loan, with interest at 8% quarterly installments for 10 years. Find the quarterly payments.
a. P1827.79
c. P1287.78
b. P1892.18
d. P1972.36
167.) For having been loyal, trustworthy, and efficient, the company has offered a superior
yearly gratuity pay of P20,000 for 10 years with the first payment to be made one year after his
retirement. The supervisor, instead, requested that he be paid a lump sum, on the date of his
retirement, having less interest that the company would have earned if the gratuity is to be paid
in yearly basis. If interest is 15%, what is the equivalent lump sum that he could get?
a. P100,375.37
c. P101,757.37
b. P100,735.37
d. P100,575.37
168.) If P500 is invested at the end of each year for 6 years, at an annual interest rate of 7%, what
is the total peso amount available upon the deposit of the sixth payment?
a. P3,210
c. P3,000
b. P3,577
d. P4,260
169.) A series of year and payments extending over eight years are as follows: P10,000 for the
first year, P20,000 for the second year, P50,000 for the third year, and P40,000 for each year
from fourth year through the 8th year. Find the equivalent annual worth of these payments if the
annual interest is 8%.
a. P44,800.00
c. P35,650.00
b. P30,563.00
d. P33,563.85
170.) In five years, P18,000 will be needed to pay for a building renovation,. In order to generate
this sum, a sinking fund consisting of three annual payments is established now. For tax
purposes, no further payments will be made after three years. What payments are necessary if
money is worth 15% per annum?
a. P2870
c. P5100
b. P3919
d. P2670
171.) San Miguel Corporation purchases P400,00 worth of equipment in year 1970. It decided to
use straight line depreciation over the expected 20 year life of the equipment. The interest rate is
16%. If the overall tax rate is 35% , what is the present worth of the tax shield?
a. P40,298.68
c. P45,450.28
b. P41,501.89
d. P51,410.37
172.) A local firm is establishing a sinking fund for the purpose of accumulating a sufficient
capital to retire its outstanding bonds and maturity. The bonds are redeemable in 10 years, and
their maturity value is P150,000. How much should be deposited each year if the fund pays
interest at the rate of 3%?
a. P12,547.14
c. P14,094.85
b. P13,084.58
d. P16,848.87
173.) A machine costs P20,000 today and has an estimated scrap value of P2,000 after 8 years.
Inflation is 2% per year. The effective annual interest rate earned on money invested is 8%. How
much money needs to be set aside each year to replace the machine with an identical model 8
years from now?
a. P2,808.88
c. P3,920.00
b. P3,290.00
d. P3,610.00
174.) A machine is under consideration for investment. The cost of the machine is P25,000. Each
year it operates, the machine generates P15,000. Given an effective annual interest rate of 18%,
what is the discounted payback period, in years, on the investment of the machine?
a. 1.75 years
c. 1.67 years
b. 3.17 years
d. 2.16 years
175.) Company A purchases P200,000 of equipment in year zero. It decides to use straight line
depreciation over the expected 20 year life of the equipment. The interest rate is 14%. If the
average tax rate is 40%, what is the present worth of the depreciation tax held?
a. P30,500
c. P39,700
b. P26,500
d. P40,000
176.) Instead of paying P100,000 in annual rest for office space at the beginning of each year for
the next 10 years, an engineering firm has decided to take out a 10 year P100,000 loan for a new
building at 6% interest. The firm will invest P100,000 of the rent saved and earned 18% annual
interest on that amount. What will be the difference between the revenue and expenses?
a. Firm will need P17,900 extra
b. Firm will break even
c. Firm will have P21,500 left over
d. Firm will need P13,000 extra
177.) A man inherited a regular endowment of P100,000 every end of 3 months for x years.
However, he may choose to get a single lump of P3,702,939,80 at the end of 4 years. If the rate
interest was 14% compounded quarterly, what is the value of x?
a. 13 years
c. 12 years
b. 10 years
d. 11 year
178.) A service car whose car price was P540,000 was bought with a down payment of P162,000
and monthly installments of P10,847.29 for 5 years. What was the rate of interest if compounded
monthly?
a. 30%
c. 20%
b. 24%
d. 15%
179.) What is the present worth of a 3 year annuity paying P3000 at the end of each year, with
interest at 8% compounded annually?
a. P7,731.29
c. P7,371.29
b. P9,731.79
d. P9,371.79
180.) A man paid a 10% down payment of P200,000 for a house and lot and agreed to pay the
balance on monthly installments for 5 years at an interest rate of 15% compounded monthly.
What was the monthly installment on pesos?
a. P44,528.34
c. P43,625.92
b. P42,821.87
d. P45,825.62
181.) A man inherited a regular endowment of P100,000 every end of 3 months for 10 years.
However, he may choose to get a single lump sum payment at the end of 4 years. How much is
this lump sum if the cost of money is 14% compounded quarterly?
a. P3,702,939.73
c. P3,502,546.23
b. P3,802,862.23
d. P3,602,431.73
182.) A man paid 10% down payment of P200,000 for a house and lot and agreed to pay the
balance on monthly installments for x years at an interest rate of 15% compounded monthly. If
the monthly installment was P42,821.87, find the value of x.
a. 5 years
c. 8years
b 9 years
d. 7 years
183.) A father wishes to provide P4000 for his son on his 21 st birthday. How much should he
deposit every 6 months in a savings bank which pays 3% compounded semi-annually if the first
deposit is made when the son is 3.5 years old?
a. P84.61
c. P45.76
b. P94.24
d. P78.68
184.) An employee obtained a loan of P100,000 at the rate of 6% compounded in order to build a
house. How much must he pay monthly to amortize the loan within a period of 10 years?
a. P8,322.07
c. P2,494.04
b. P1,101.80
d. P3,452.90
185.) If money is worth 5% compounded semi-annually, find the present value of a sequence of
12 semi-annual payments of P500 each, the first of which is due at the end of 4.5 years.
a. P4,209.51
c. P6,240.62
b. P5,602.62
d. P7,161.42
186.) An annual deposit of P1270 is placed on the fund at the end of each year for 6 years. If the
fund invested has a rate of interest of 5% compounded annually, how much is the worth of this
fund at the end of 9 years?
a. P12,000
c. P11,000
b. P10,000
d. P14,000
187.) A fund for replacement of a machinery in a plant must have P30,000 at the end of 9 years.
An equal deposit of P2,965 was made on the fund at the end of each 6 months for 4 years only.
How much is the rate of the fund invested if it is compounded semi-annually?
a. 3.5%
b. 4.5%
c. 5.5%
d. P6.5%
188.) In purchasing a house, a man makes a cash payment and takes out a mortgage for P10,000
on which he agrees to pay P200 at the end of each month for 5 years. At what interest rate
compounded monthly was interest charged on the mortgage?
a. 8.41%
c. 7.68%
b. 7.42%
d. 9.60%
189.) How much money must you invest in order to withdraw P2000 annually for 10 years if the
interest rate is 9%?
a. P12,853.32
c. P12,835.32
b. P12,881.37
d. P12,385.32
190.) If interest is at rate of 8% compounded semi-annually, what sum must be invested at the
end of each 6 months to accumulate a fund of P10,000 at the end of 8 years?
a. P458.20
c. P498.23
b. P532.11
d. P753.10
191.) A Corporation will invest P5000 in a fund at the end of each 6 months to accumulate
P100,000 to initiate a plant overhaul. If the fund is invested at 6.5% compounded semi-annually,
how may years will the fund contain at least P100,000?
a. 8 years
c. 10 years
b. 6 years
d. 9 years
192.) A piece of machinery can be bought for P10,000 cash, or for P2000 down and payments of
P750 per year for 15 years. What is the annual interest rate for time payments?
a. 4.61%
c. 5.71%
b. 3.81%
d. 11.00%
193.) To accumulate a fund of P8000 at end of 10 years, a man will make equal annual deposit of
P606.94 in the fund at the end of each year. How much is the rate of interest if it is compounded
annually?
a. 6%
c. 10%
b. 8%
d. 12%
194.) A purchasing engineer of a certain firm is to purchase a second hand truck fro P75,000. A
dealer offers cash payment of P5,000 and P6486 per month for 12 months. Another dealer
offered under the same condition with 0.75% interest per month for 12 months of the unpaid
balance. Which offer should the engineer choose and what is the rate of interest?
a. 0.62%
c. 0.75%
b. 1.66%
d. 0.40%
195.) If a low cost house and lot worth P87,000 were offered at 10% down payment and P500
per month for 25 years. What is the effective monthly interest rate on the diminished balance?
a. 0.492%
c. 0.531%
b. 0.687%
d. 0.683%
196.) A house and lot can be acquired with a down payment of P500,000 and a yearly payment
P100,000 at the end of each year for a period of 10 years, starting at the end of 5 years from the
date of purchase. If the money is worth 14% compounded semi-annually, what is the cash price
of the property?
a. P810,000
c. P801,900
b. P808,836
d. P805,902
197.) A man bought a brand new car for P650,000 on installment basis at the rate of 10% oer
annum on the unpaid balance. If he paid a down payment of P120,000 cash and proposed to pay
the balance in equal monthly payment for 2 years, what should be his monthly payment?
a. P54,323.03
c. P24,447.03
b. P34,532.94
d. P83,534.32
198.) A businessman is faced with the prospect of fluctuating future budget for the maintenance
of the generator. During the first 5 years, P1,000 per year will be budgeted. During the second 5
years, the annual budget will be P1500 per year. In addition, P3500 will be budgeted for an
overhaul of the machine at the end of the fourth year and another P3500 for an overhaul at the
end of 8th year. Assuming compounded interest at 6% per annum, what is the equivalent annual
cost of maintenance?
a. P1,888.87
c. P1,777.38
b. P1,738.34
d. P1,999.34
199.) A parent on the day the child is born wishes to have to determine what lump sum would
have to be paid into an account bearing interest at 5% compounded annually, in order to
withdraw P20,000 each on the child’s 18th, 19th, 20th, and 21st birthdays.
a. P35,941.73
c. P30,941.73
b. P33,941.73
d. P25,941.73
200.) If money is worth 5% compounded semi-annually, find the present value of a sequence of
12 semi-annual payments of P500 each, the first of which is due at the end of 4.5 years.
a. P4,209.51
c. P3,958.48
b. P5,038.29
d. P4,936.39
201.) A businessman borrowed P300,000 with interest at the rate of 6% compounded semiannually. He agrees to discharge his obligation by paying a series of 8 equal semi-annual
payments, the first being due at the end of 5.5 years. Find the semi-annual payment.
a. P69,475.53
c. P73.083.59
b. P57,434.78
d. P40,922.40
202.) A man borrowed P300,000 from a lending institution which will be paid after 10 years at
an interest rate of 12% compounded annually. How much should he deposit to a bank monthly in
order to discharge his debt 10 years hence?
a. P2,798.52
c. P4,672.31
b. P3,952.50
d. P5,093.06
203.) What is the accumulated amount of the five year annuity paying P6000 at the end of each
year, with interest at 15% compounded annually?
a. P40,454.29
c. P41,454.29
b. P41,114.29
d. P40,544.29
204.) A man owes P12,000 today and agrees to discharge the debt by equal payments at the
beginning of each 3 months for 8 years, where this payments include all interest at 8% payable
quarterly. Find the quarterly payment.
a. P501.30
c. P498.20
b. P602.40
d. P701.60
205.) A man will deposit P200 with a savings and loan association at the beginning of 3 months
for 9 years. If the association pays interest at the rate of 5.5%quarterly, find the sum to his credit
just after the last deposit.
a. P9236
c. P9563
b. P9363
d. P9684
206.) At what interest rate payable quarterly will payments of P500 at the beginning of each 3
months for 7 years discharge a debt of P12500 due immediately?
a. 3.44%
c. 5.44%
b. 4.33%
d. 6.33%
207.) A P1,000,000 issue of 3% 15 years bonds was sold at 95%. If miscellaneous initial
expences of the financing were P20,000 and yearly expenses of P2,000 is incurred, what is the
true cost the company is paying for the money it borrowed?
a. 3.8%
c. 4.0%
b. 4.2%
d. 2.6%
208.) A man was offered a Land Bank certificate with a face value of P100,000 which is baring
interest of 8% per year payable semi-annually and due in 6 years. If he wants to earn 6% semiannually, how much must he pay the certificate?
a. P90,614.92
c. P90,061.49
b. P96,041.92
d. P99,814.92
209.) The National Irrigation Administration undertakes the construction of an irrigation project
in the province of oriental Mindoro which will cover an area of 10,000 hectares and estimated
cost P10,000,000 which was borrowed from the World Bank at the start of the construction. The
construction will last 30 years with no salvage value. Bonds will be paid at 4% per annum
compounded annually for 30 years. The construction of project will take 4 years. Insurance
operation and maintenance of the system will cost P120,00 per year. Interest on sinking fund is
6%. How much should each hectare be charged?
a. P64.65
c. P60.66
b. P66.56
d. P45.65
210.) A company issued 50 bonds of P1,000 face value each redeemable at par at the end of 15
years to accumulate the funds required fro redemption. The firm established a sinking fund
consisting of annual deposits, the interest rate of the fund being 4%. What was the principal in
the fund at the end of 12th year?
a. P35,983
c. P41,453
b. P38,378
d. P37,519
211.) An oil well which could produce a net income of P15,000,000 per year for 25 years is
being considered to be purchased by a group of businessman. If the return on investment is
targeted to be 20% out of the net income and the sinking fund at 18% interest is to be established
at recover of investment, how much must be paid to the oil well?
a. P73,921,184.58
c. P70,215,276.17
b. P73,297,198.28
d. P75,973,209.26
212.) An investor pays P1,100,000 for a mine which will yield a net income of P200,000 at the
end of each year for 10 years and then will become useless. He accumulated a replacement fund
to recover his capital by annual investments at 4.5%. At what rate(%) does he receive interest on
his investment at the end of each year?
a. 10.04%
c. 11.5%
b. 8.5%
d. 14.5%
213.) A certain marble mine property has an estimated life of 30 years at a projected annual
output of 3000 cubic meters of marble blocks. Estimated management cost per year is placed at
P1,500,000 and operating cost of the quarry and processing plant is P8000 per cubic meter. The
finished products, tiles and slabs, can be sold for P12,000 per cubic meter if exported. Determine
the present valuation of the mineral property if the sinking fund rate of interest is 15% and the
annual dividend rate is to be 12%.
a. P85,854,317.13
c. P85,444,313.27
b. P85,554,371.18
d. P85,345,365.28
214.) The annual dividend from a mine will be P75,000 until the ore is exhausted at the end of 30
days, and the mine becomes useless. Find the price of the mine to yield the investor 6.5%, if he
accumulated a replacement fund to restore his capital by annual investment at 5%.
a. P936,897.63
c. P982,286.29
b. P836,286.39
d. P735,385.53
215.) The privileges of a patent will last for 20 more years and the royalty from it will be
P60,000 at the end of each year during that time. Find the value of his patent rights to an investor
who desires interest at 8% on his investment and will accumulate a capital replacement fund at
5%.
a. P594,297.20
c. P405,384.28
b. P544,254.30
d. P629,289.40
216.) The annual income from the mine is P100,000 and the life of the mine is 20 years. Find the
price that an investor is willing to pay for the mine if he considers that money is worth 5% and if
he is to accumulate a sinking fund at 6% in order to replace the capital he invested.
a. P1,295,595.57
c. P1,529,847.29
b. P1,995,959.97
d. P1,159,287.92
217.) An untreated electric wooden pole that will last 10 years under a certain soil conditions
costs P1200. If a treated pole will last for 20 years, what is the maximum justifiable amount that
can be paid for the treated pole, if the maximum return on investment is 12%? Consider annual
taxes and insurance amount to be 1% at first cost.
a. P1,559.50
c. P1,593.20
b. P1995.28
d. P1,959.30
218.) A company must relocate one of its factories in three years. Equipment for the loading
dock is being considered for purchase. The original cost is P20,000,000, the salvage value of the
equipment after three years is P8,000. The company’s rate of return on money is 10%. Determine
the capital recovery rate per year.
a. P5115
c. P5625
b. P4946
d. P4805
219.) A new engine will cost P12,000 with an estimated life of 15 years and a salvage value of
P800 and guaranteed to have an operating cost of P3500 per year. The new engine is considered
as a replacement of the old one. The old engine had a total annual cost of P5,200 to operate.
Determine the rate of return of the new investment using 6% sinking fund to recover
depreciation, if the old engine could be sold now for P2000.
a. 12.19%
c. 10.47%
b. 14.29%
d. 15.92%
220.) A corporation uses a type of motor truck which costs P5000with a life of 2 years and final
salvage value of P800. How much could the corporation afford to pay for another type of truck
of the same purpose for a life of 3 years with a final salvage value of P1000? Money is worth 4%
a. P8450.66
c. P6398.24
b. P7164.37
d. P9034.56
221.) A granite quarry purchased of P1,600,000 is expected to be exhausted at the end of 4 years.
If the resale value of the land is P100,000, what annual income is required to yield an investment
rate of 12%? Use a sinking fund rate of 3%
a. P551,544
c. P550,540
b. P552,550
d. P553,420
222.) A machine has a first cost of P800,000 and a salvage value of P50,000 at the end of its life
after 10 years. The annual saving for the use of the machine amount s to P124,900.97. If the
annual maintenance of the machine is P4000 and the sinking fund to recover depreciation earns
6%, compute for the rate of return of investment.
a. 8%
c. 6%
b. 7%
d. 9%
223.) The first cost of a certain equipment is P324,000 and a salvage value of P50,000 at the end
of its life of 4 years. If money is worth 6% compounded annually, find the capitalized cost.
a. P1,367,901.15
c. P936,431.16
b. P1,427,846.17
d. P843,916.27
224.) A multi million project can purchase heavy duty trucks for P600,000 each. It is estimated
to have a salvage value of P60000 at the end of its life which is 10 years. Maintenance and
operating cost including the driver is estimated to cost an average of P3000 per year. The
contractor however can hire a similar truck and its operator for P420 per day. If money is worth
12%, how many days per year must the service of the truck be required to justify the buying of
the trucks. Use annual cost method.
a. 252 days
c. 243 days
b. 225 days
d. 255 days
225.) Each removal from a ditch in city streets is accomplished by a machine loading into trucks.
This machine will cost P20,000 with labor, fuel, oil, and maintenance amounting to P5000 per
year. Life of the machine is estimated to be 5 years and no salvage value. The contractor
however can hire a similar machine and its operator at P340 per day. How many days per year
must the services of the machine be required to justify the purchase of their new machine of the
money is worth 10%. Use annual cost method.
a. 248 days
c. 284 days
b. 428 days
d. 482 days
226.) A contractor can purchase a heavy-duty truck for P500,000. Its estimated life is 8 years and
estimated salvage value of P6000. Maintenance is estimated to be P2500 annually including the
cost of driver and fuel maintenance. The contractor can hire a similar unit and driver for P750 a
day. If interest is taken at 8%, how many days per year must be services of a dump truck be
required to justify the purchase of a truck? Use annual cost method.
a. 112 days
c. 132 days
b. 121days
d. 211 days
227.) It cost P50,000 at the end of each year to maintain a section of Kennon road. If money is
worth 10%, how much would it pay to spend immediately to reduce the annual cost to P10,000?
a. P410,000
c. P400,000
b. P554,000
d. P453,000
228.) If money is worth 12% compounded quarterly, what is the present value of the perpetuity
of P1,000 payable monthly?
a. P453,876.80
c. P342,993.70
b. P100,976.23
d. P100,993.78
229.) Find the present value, in peso, of a perpetuity of P15,000 payable semi-annually if money
is worth 8% compounded quarterly.
a. P372,537
c. P373,767
b. P374,977
d. P371,287
230.) A businessman invested in a medium scale business which cost him P47,000. The net
annual return estimated is P14,000 for each of the next 8 years. Compute the benefit cost ratio if
the annual rate of interest is 18%.
a. 1.21
c. 2.23
b. 1.76
d. 1.11
231.) A project costs P100,000. The benefit at the end of each year for a period of 5 years is
equal to P40,000. Assuming money is worth 8% with no salvage value, compute the benefit cost
ratio.
a. 1.597
c. 1.875
b. 2.124
d. 1.125
232.) Compute the benefit cost ratio of the following project:
Project cost = P80,000
Gross income = P25,000 per year
Opening Cost = P6,000 per year
Salvage Value = 0
Life of Project = 10 years
Rate of Interest = 12%
a. 1.34
c. 2.23
b. 1.78
d. 1.11
233.) A local factory assembling calculators produces 100 units per month and sells them at
P1,800 each. Dividends are 8% on the 8000 shares with par value of P250 each. The fixed
operating cost per month is P25,000. Other costs are P1,000 per unit. Determine the break even
point. If only 200 units were produced per month, determine the profit.
a. 48 ; P121,666.67
c. 50 ; P112,656.67
b. 45 ; P122,676.88
d. 48 ; P212,666.67
234.) General Electric Company, which manufactures electric motor, has a capacity of producing
150 motors a month. The variable costs are P4,000 per month, the average selling price of the
motor is P750 per motor. Fixed costs of the company amounts to P78,000 per month which
includes all taxes. Determine the number of motors to be produced per month to break even and
the sales volume in pesos at this point.
a. 110 units
c. 105 units
b. 120 units
d. 115 units
235.) A plywood manufacturer produces a piece of plywood at a labor cost of P0.50 and material
at P3.00. The fixed charges on business are P50,000 a month and the variable cost is P0.50 per
piece. If one plywood sells for P6.00 each, how many pieces must be produced each month for
the manufacturer to break even?
a. 25,000
c. 24,000
b. 27,000
d. 22,000
236.) The profit on a product selling for P8.20 is 10% of the selling price. What percentage
increase in production cost will reduce the profit by 60%?
a. 6.67%
c. 7.66%
b. 6.76%
d. 7.66%
237.) A local company assembling stereo radio cassette produces 300 units per month at a cost of
P800 per unit. Each stereo radio cassette sells for P1,200. If the firm makes a profit of 10% on its
10,000 shares with a par value of P200 per share, and the total fixed cost is P20,000 per month.
What is the break even point and how much is the loss or profit if only 100 units are produced in
a given month?
a. 92 ; P3,333.33
c. 91 ; P4,333.44
b. 90 ; P3,444.33
d. 93 ; P4,444.33
238.) A certain operation is now performed by hand, the labor cost per unit is P0.54 and the
annual fixed charge for tool used is estimated at P100 per year. A machine that is being
considered for this job will cost P2,400, have a salvage value of P100 at any time and a fixed
annual cost of P200. With it, labor cost is P0.22 per unit. For what number of units of product
per year at zero interest and life of 6 years for the machine will the annual cost of the two
methods break even?
a. 1510 units
c. 1150 units
b. 1050 units
d. 1551 units
239.) A shoe manufacturer produces a pair of shoes at a labor cost of P9.00 a pair and a material
cost of P8.00 a pair. The fixed charges on the business is P90,000 a month and the variable cost
is P4.00 a pair. If the shoes sells at P30 a pair, how many pairs must be produced each month for
the manufacturer to break even?
a. 10,000
c. 11,000
b. 12,000
d. 13,000
240.) An item which can be sold for P36.00 per unit wholesale is being produced with the
following cost data; labor cost, P10 per unit; material cost, P15.00 per unit; fixed charges,
P10,000; variable cost, P8.00 per unit. What is the break even point sales volume and the break
even sales volume if one out of every ten units produced is defective and is rejected with only
full recovery on materials?
a. 333.33 ; 397
c. 353.33 ; 333
b. 345.33 ; 379
d. 322.33 ; 377
241.) A certain firm has the capacity to produce 650,000 units of product per year. At present, it
is operating at 62% capacity. The firm’s annual income is P4,160,000. Annual fixed cost is
P1,920,000 and the variable cost is equal to P3.56 per unit of product. What is the firm’s annual
profit or loss and what volume of sales does the firm break even?
a. P805,320 ; P3,354,680
c. P803,550 ; P3,276,398
b. P850,330 ; P3,543,683
d. P800,286 ; P3,186,586
242.) The direct labor cost and material cost of a certain product are P300 and P400 per unit,
respectively. Fixed charges are P100,000 per month and other variable costs are P100 per unit. If
the product is sold at P1,200 per unit, how many units must be produced and sold to break even?
a. 250 units
c. 300 units
b. 200 units
d. 260 units
243.) XYZ Corporation manufacturers book cases that it sells for P65.00 each. It costs XYZ
P35,000 per year to operate its plant. This sum includes rent, depreciation charges on equipment
and salary payments. If the cost to produce one bookcase is P50.00, how many cases must be
sold each year for XYZ to avoid taking a loss?
a. 2334
c. 750
b. 539
d. 2233
244.) A telephone switchboard 100 pair cable can be made up with either enameled wire or
tinned wire. There will be 400 soldered connections. The cost of soldering a connection on the
enameled wire will be P1.65, on the tinned wire, it will be P1.15. A 100 pair cable made up with
enameled wire cost P0.55 per lineal foot and those made up to tinned wire cost P0.76 per lineal
foot. Determine the length of cable run in feet so that the cost of each installation would be the
same.
a. 1121.06 ft
c. 864.92 ft
b. 1001.25 ft
d. 952.38 ft
245.) A company which manufactures electric motors has a production capacity of 200 motors a
month. The variable costs are P150 per motor. The average selling price of the motor is P275.
Fixed costs of the company amounts to P20,000 per month which included taxes. The number of
motors that must be sold each month to break even is closest to:
a. 40
c. 80
b. 150
d. 160
246.) Steel drums manufacturer incurs a yearly fixed operating cost of P200,000. Each drum
manufactured cost P160 to produce and sells for P200. What is the manufacturer’s break even
sales volume in drum per year? If they could manufacture 7,000 drums per year, determine the
amount of profit or loss.
a. 1250 ; P70,000
c. 5,000 ; P80,000
b. 2500 ; P60,000
d. 1,000 ; P75,000
247.) A new Civil Engineer produces a certain construction ,material at a labor cost of P16.20
per piece, material cost of P38.50per piece and variable cost of P7.40 per piece. The fixed
charges on the business is P100,000 a month. If he sells the finished product at P95.00 each, how
many pieces must be manufactured each month to break even?
a. 3040
c. 3004
b. 3400
d. 4300
248.) A manufacturer produces certain items at a labor cost per unit of P315, material cost per
unit of P100, variable cost of P3.00 each. If the item has a selling price of P995, how many units
must be sold to break even if the monthly overhead is P461,600?
a. 800
c. 700
b. 600
d. 900
249.) A cement firm with production capacity of 130 tons per day (24 hrs) of clinker has its
burning zone about 45 tons of magnesium chrome bricks being replaced periodically, depending
on some operational factors and the life of the bricks. If locally produced bricks cost costing
P30,000 per ton and have a life of 6 months, determine the more economical bricks and by how
much?
a. P6,075,000
c. P6,505,000
b. P6,750,000
d. P6,057,000
250.) An equipment installation job in the completion stage can be completed in 40 days of 8
hour day work, with 40 men working. With the contract expiring in 30 days, the mechanical
engineer contractor decided to add 10 men on the job, overtime not being permitted. If the
liquidated damages is P2,000 per day of delay, and the men are paid P80 per day, how much
money would he save if he will add workers?
a. P16,000
c. P16,500
b. P15,500
d. P16,000
251.) A fixed capital investment of P10,000,000 is required for a proposed manufacturing plant
and an estimated working capital of P2,000,000. Annual depreciation is estimated to be 10% of
the fixed capital investment. Determine the rate of return on the total investment and the
minimum pay out period if the annual profit is P2,500,000.
a. 20.83% ; 2.86
c. 23.80% ; 6.28
b. 20.38% ; 2.68
d. 23.08% ; 6.66
252.) A 500kw electric lighting plant cost P95 per kw installed. Fixed charges is 14%, operating
cost is P0.013 per kw-hr. The plant averages 150kw for 5000 hour of the year, 420 kw for 1000
hour and 20kw for the remainder. What is the unit cost of production of electric energy?
a. P0.0184
c. P0.1840
b. P0.1084
d. P0.8104
253.) A mechanical engineer who was awarded a P450,000 contract to install the machineries of
an oil mill failed to finish the work on time. As provided for in the contract, he has to pay a daily
penalty equivalent to one fourth of one percent per day for the next ten days and one percent per
day for every day thereafter. If the total penalty was P60750, how many days was the completion
of the contract delayed?
a. 26 days
c. 30 days
b. 22 days
d. 24 days
254.) By selling balut at P5 per dozen, a vendor gains 20%. The cost of the eggs rises by 12.5%.
If he sells at the same price as before, find his new gain %.
a. 6.6%
c. 7.7%
b. 5.5%
d. 7.6%
255.) In a certain department store, the monthly salary of a saleslady is partly constant and partly
varies as the value of her sales for the month. When the value of her sales for the month is
P10,000, her salary for that month is P900, when her monthly sales goes up to P12,000, her
monthly salary goes up to P1,000. What must be the value of her sales for the month so that her
salary for that month would be P2,000?
a. P32,000
c. P30,000
b. P35,000
d. P40,000
256.) An equipment installation job in the completion stage can be completed in 50 days of 8
hour day work, with 50 men working. With the contract expiring in 40 days, the mechanical
engineer contractor decided to add 15 men on the job, overtime not being permitted. If the
liquidated damages is P5,00 per day of delay, and the men are paid P150 per day, how much
money would he save with the additional workers?
a. P44750
c. P44570
b. P47540
d. P45407
257.) Jojo bought a second hand Betamax VCR and then sold it to Rudy at a profit of 40%; Rudy
then sold the VCR to Noel at a profit of 20%. If Noel paid P2,856 more than it cost Jojo, how
much did Jojo pay for the unit?
a. P4200
c. P2400
b. P4400
d. P2200
258.) Dalisay Corporation’s gross margin is 45% of sales. Operating expenses such as sales and
administration are 15% of sales. Dalisay is in 40% tax bracket. What percent of sales is their
profit after taxes?
a. 18%
c. 24%
b. 5%
d. 0%
259.) A manufacturer of sports equipment produces tennis rackets for which there is a demand of
200 per month. The production setup cost for each lot of racket is O300. IN addition, the
inventory carrying cost for each racket is P24 per year. Using the Economic Order Quantity
(EQQ) model, which is the best production batch size fro the rackets?
a. 245 units
c. 173 units
b. 71 units
d. 346 units
260.) A manufacturing firm maintains one product assembly line to produce signal generators.
Weekly demand for the generators is 35 units. The line operates for 7 hours per day, 5 days per
week. What is the maximum production time per unit in hours required of the line to meet the
demand?
a. 1 hour
c. 3 hours
b. 0.75 hours
d. 2.25 hours
261.) A businessman wishes to earn 7% on his capital after payment of taxes. If the income from
an available investment will be taxed at an average of 42%, what minimum rate or return, before
payment of taxes, must the investment offer to be justified?
a. 12.1%
c. 11.1%
b. 10.7%
d. 12.7%
262.) A 200 hp generator is being considered for purchase. The generator will cost P320,000
with a life expectancy of 10 years, with an efficiency of 82%. The maintenance cost per year is
P5,000. This generator is used for 300 hours per year and the cost of fuel, oil is P0.12 per
kilowatt-hour. (1hp = 0.746kw). Assuming the generator will have no salvage value, what will be
the monthly cost of maintaining the generator?
a. P842.40
c. P786.40
b. P962.52
d. P695.40
263.) An engineer buys a machine costing P500,000. Compute the capitalized cost if the machine
has a life of 5 years and a salvage value of P80,000. Rate of interest is 12% per annum.
a. P2,424,732
c. P2,431,643
b. P1,050,934
d. P5,124,153
264.) A project costing P250,000 yields a yearly benefit of P80,000 for a period of 10 years with
no salvage value at an interest rate of 6%. What is the benefit cost ratio?
a. 4.24
c. 2.36
b. 3.85
d. 4.30
265.) A company constructed its factory with a fixed capital investment of P120M. The net
income after the tax and depreciation is expected to be P23m per year. Annual depreciation cost
is 10 % of fixed capital investment. Determine the payout period in years.
a. 3.04 years
c. 3.43 years
b. 2.54 years
d. 4.85 years
266.) A machinery costing P720,000 is estimated to have a book value of P40,545.73 when
retired at the end of 10 years. Depreciation cost is computed using a constant percentage of the
declining book value. What is the annual rate of depreciation?
a. 20%
c. 30%
b. 25%
d. 35%
267.) An asset is purchased for P9,000. Its estimated economic life is 10 years after which it will
be sold for P1,000. Find the depreciation in the first three years using straight line method.
a. P2,400.00
c. P2,250.00
b. P2,412.34
d. P2,450.00
268.) An engineer bought an equipment for P500,000. He spent an additional amount of P30,000
for installation and other expenses. The estimated useful life of the equipment is 10 years. The
salvage value is x% of the first cost. Using the straight line method of depreciation, the book
value at the end of 5 years will be P291,500. What is the value of x?
a. 20%
c. 30%
b. 40%
d. 10%
269.) The initial cost of a paint sand mill, including its installation is P800,000. The BIR
approved life of this machine is 10 years for depreciation. The estimated salvage value of the
mill is P50,000 and the cost of dismantling is estimated to be P15,000. Using straight line
depreciation, what is the annual depreciation charge?
a. P75,500
c. P76,500
b. P76,000
d. P77,000
270.) The initial cost of a paint sand mill, including its installation is P800,000. The BIR
approved life of this machine is 10 years for depreciation. The estimated salvage value of the
mill is P50,000 and the cost of dismantling is estimated to be P15,000. Using straight line
depreciation, what is book value of the machine at the end of 6 years?
a. P341,000
c. P340,000
b. P343,000
d. P342,000
271.) A unit of welding machine cost P45,000 with an estimated life of 5 years. Its salvage value
is P2,500. Find its depreciation rate by straight line method.
a. 18.89%
c. 19.58%
b. 19.21%
d. 19.89%
272.) A tax and duty free importation of a 30hp sand mill for painting manufacturing cost
P360,000. Bank charges and brokerage cost P5,000. Foundation and installation costs were
P25,000. Other incidental expenses amount to P20,000. Salvage value of the mills estimated to
be P60,000 after 20 years. Find the appraisal value of the mill using straight line depreciation at
the end of 10 years.
a. P234,000
c. P234,500
b. P235,000
d. P235,500
273.) An equipment costs P10,000 with a salvage value of P500 at the end of 10 years. Calculate
the annual depreciation cost by sinking fund method at 4% interest.
a. P721.54
c. P791.26
b. P724.56
d. P721.76
274.) An equipment costs P50,000 with a salvage value of P250 at the end of 10 years. Calculate
the annual depreciation cost by sinking fund method at 8% interest.
a. P3123.53
c. P3434.22
b. P3223.75
d. P3241.24
275.) An asset is purchased for P12,000. Its estimated economic life is 20 years after which it
will be sold for P5,000. Find the depreciation in the first five years using straight line method.
a. P1,750
c. P1,250
b. P1,412
d. P1,450
276.) What is the difference of the amount 5 years from now for a 12% simple interest and 12%
compound interest per year?(P8,000 accumulated)
a. P1298.73
c. P1224.97
b. P1281.24
d. P1862.76
277.) Find the discount if P5,500 is discounted for 9 months at 15% compounded quarterly.
a. P2442.09
c. P1883.66
b. P1248.24
d. P2451.99
278.) Find the ordinary simple interest at 7.5% on P5000 and the corresponding amount at the
end of 59 days.
a. P2145.24
c. P5061.45
b. P2241.12
d. P5123.24
279.) Find the exact simple interest at 7.5% on P5000 and the corresponding amount at the end
of 59 days.
a. P5187.24
c. P5021.45
b. P5221.12
d. P5060.60
280.) If P1050 accumulate P1275 when invested at a simple interest for 3 years. What is the rate
of interest?
a. 4.21%
c. 6.73%
b. 5.85%
d. 7.14%
281.) Determine the exact simple interest on P5,000 investment for the period from January 15,
1996 to October 12, 1996 of the rate of interest in 18%.
a. P666.39
c. P632.40
b. P621.22
d. P636.29
282.) The exact simple interest of P5000, invested from June 21, 1995 to December 25, 1995, is
P100. What is the rate of interest?
a. 2.4%
c. 2.7%
b. 3.2%
d. 3.9%
283.) The amount of P50,000 was deposited in the bank earning at 7.5% per annum. Determine
the total amount at the end of 5 years if the principle and interest were not withdrawn during the
period.
a. P71,781.47
c. P70,024.29
b. P24,257.75
d. P29,240.99
284.) Determine the future amount of P100 for 10.25 years at a rate of 5% compounded monthly.
a. P166.77
c. P163.12
b. P224.09
d. P214.12
285.) Determine the future amount of P100 for 10.25 years at a rate of 5% compounded
quarterly.
a. P124.02
c. P182.42
b. P166.42
d. P175.10
286.) Determine the future amount of P100 for 10.25 years at a rate of 5% compounded semiannually.
a. P153.29
c. P169.22
b. P165.90
d. P173.24
287.) Determine the future amount of P100 for 10.25 years at a rate of 5% compounded daily.
a. P129.24
c. P195.32
b. P166.94
d. P128.87
288.) What is the present worth of a P500 annuity starting at the end of the third year and
continuing to the end of the fourth year, if the annual interest rate is 10%?
a. P717.17
c. P720.24
b. P252.91
d. P287.09
289.) Today a businessman borrowed money to be paid in 10 equal payments for 10 quarters. If
the interest rate is 10% compounded quarterly and the quarterly payment is P2,000, how much
did he borrow?
a. P24214.23
c. P15125.24
b. P24311.53
d. P17504.13
290.) What annuity is required over 12 years to equate with a future amount of P20,000? Assume
i=6% annually.
a. P1121.24
c. P2314.12
b. P1244.22
d. P1185.54
291.) Find the annual payment to extinguish a debt of P10,000 payable for 6 years at 12%
interest annually.
a. P2214.42
c. P2525.24
b. P2432.26
d. P2044.83
292.) A bond issue of P50,000 in 10 years, bonds in P1000 units paying 10% interest in annual
payments, must be retired by the use of sinking fund which earns 8% compounded annually.
What is the total cost for the interest ad retirement of the entire bond issue.
a. P82,241
c. P82,214
b. P84,150
d. P84,510
293.) Determine the amount of interest you would receive per period if you purchase a 6%,
P5000 bond which matures in 10 years with interest payable quarterly.
a. P24
c. P75
b. P98
d. P52
294.) A corporation floats callable bonds amounting to P100,000 each having a par value of
P500, the bond rate is 7.5% and the bonds are to be retired in 5 years, the annual payments being
as nearly equal as possible. What is the total payment for the whole period of 5 years?
a. P123,625
c. P214,124
b. P124,241
d. P923,124
295.) A book store purchased the best selling book at P200. At what price should this book be
sold so that by giving a 20% discount, the profit is 30%.
a. P200
c. P400
b. P300
d. P500
296.) The selling price of a tv set is double that of its net cost. If the tv set is sold to a customer at
a profit of 25% of the net cost, how much discount was given to the customer?
a. 35.8%
c. 34.5%
b. 37.5%
d. 44.5%
297.) A manufacturing firm maintains one product assembly line to produce signal generators.
Weekly demand for the generators is 35 units. The line operates for 7 hours per day, 5 days per
week. What is the maximum production time per unit in hours required of the line to meet the
demand?
a. 1 hour
c. 3 hours
b. 2 hours
d. 4 hours
298.) Mahusay Corporation’s gross margin is 55% of sales. Operating expenses such as sales and
administration are 5% of sales. Mahusay is in 40% tax bracket. What percent of sales is their
profit after taxes?
a. 20%
c. 40%
b. 30%
d. 50%
299.) How much money must you invest in order to withdraw P5000 annually for 20 years if the
interest rate is 12%?
a. P37,347.22
c. P38,243.29
b. P23,325.23
d. P27,124.09
300.) If interest is at rate of 10% compounded semi-annually, what sum must be invested at the
end of each 6 months to accumulate a fund of P12,000 at the end of 8 years?
a. P507.24
c. P247.24
b. P451.24
d. P694.92
Differential Calculus
301.) Find two numbers whose sum is 20, if the product of one by the cube of another is to be the
maximum.
a. 5 and 15
c. 4 and 16
b. 10 and 10
d. 8 and 12
302.) The sum of two numbers is 12. Find the minimum value of the sum of their cubes.
a. 432
c. 346
b. 644
d. 244
303.) A printed page must contain 60sq.m. of printed material. There are to be margins of 5cm.
on either side and the margins of 3cm. on top and bottom. How long should the printed lines be
in order to minimize the amount of paper used?
a. 10
c. 12
b. 18
d. 15
304.) A school sponsored trip will cost each student 15 pesos if not more than 150 students make
the trip. However, the cost will be reduced by 5 centavos for each student in excess of 150. How
many students should make the trip in order for the school to receive the largest group income?
a. 225
c. 200
b. 250
d. 195
305.) A rectangular box with square base and open at the top is to have a capacity of
16823cu.cm. Find the height of the box that requires minimum amount of material required.
a. 16.14cm
c. 12.14cm
b. 14.12cm
d. 10.36cm
306.) A closed cylindrical tank has a capacity of 576.56 cubic meters. Find the minimum surface
area of the tank.
a. 383.40 cubic meters
c. 516.32 cubic meters
b. 412.60 cubic meters
d. 218.60 cubic meters
For Problems 307-309:
Two vertices of a rectangle are on the x axis. The other two vertices are on the lines whose
equations are y=2x and 3x+y=30.
.
307.) If the area of the rectangle is maximum, find the value of y.
a. 8
c. 9
b. 7
d. 6
308.) Compute the maximum area of the rectangle.
a. 30 sq. units
c. 90 sq. units
b. 70 sq. units
d. 40 sq. units
309.) At what point from the intersection of the x and y axes will the farthest vertex of the
rectangle be located along the x axis so that its area is max.
a. 8 units
c. 9 units
b. 7 units
d. 6 units
310.) A wall 2.245m high, is “x” meters away from a building. The shortest ladder that can reach
the building with one end resting on the ground outside the wall is 6m. What is the value of x?
a. 2 m
c. 6 m
b. 4 m
d. 8 m
311.) With only 381.7 square meter of materials, a closed cylindrical tank of maximum volume.
What is to be the height of the tank in m?
a. 9 m
c. 11 m
b. 7 m
d. 13 m
312.) If the hypotenuse of a right triangle is known, what is the ratio of the base and the altitude
of the right triangle when its area is maximum?
a. 1:1
c. 1:3
b. 1:2
d. 1:4
313.) What is the maximum length of the perimeter if the hypotenuse of a right triangle is 5m
long?
a. 12.08 m
c. 20.09 m
b. 15.09 m
d. 8.99 m
314.) An open top rectangular tank with square bases is to have a volume of 10 cubic meters.
The material for its bottom is to cost 15 cents per square meter and that for the sides 6 cents per
square meter. Find the most economical dimension for the tank.
a. 2 x 2 x 2.5
c. 2 x 3 x 2.5
b. 2 x 5 x 2.5
d. 2 x 4 x 2.5
315.) A trapezoidal gutter is to be made from a strip of metal 22m wide by bending up the sides.
If the base is 14m, what width across the top gives the greatest carrying capacity?
a. 16
c. 10
b. 22
d. 27
316.) Divide the number 60 into two parts so that the product P of one part and the square of the
other is the maximum. Find the smallest part.
a. 20
c. 10
b. 22
d. 27
317.) The edges of a rectangular box are to be reinforced with narrow metal strips. If the box will
have a volume of 8 cubic meters, what would its dimension be to require the least total length of
strips?
a. 2 x 2 x 2
c. 3 x 3 x 3
b. 4 x 4 x 4
d. 2 x 2 x 4
318.) A rectangular window surmounted by a right isosceles triangle has a perimeter equal to
54.14m. Find the height of the rectangular window so that the window will admit the most light.
a. 10
c. 12
b. 22
d. 27
319.) A normal widow is in the shape of a rectangle surrounded by a semi-circle. If the perimeter
of the window is 71.416, what is the radius and the height of the rectangular portion so that it
will yield a window admitting the most light.
a. 20
c. 12
b. 22
d. 27
320.) Find the radius of a right circular cone having a lateral area of 544.12 sq. m. to have a
maximum value.
a. 10
c. 17
b. 20
d. 19
321.) A gutter with trapezoidal cross section is to be made from a long sheet of tin that is 15 cm.
wide by turning up one third of its width on each side. What is the width across the top that will
give a maximum capacity?
a. 10
c. 15
b. 20
d. 13
322.) A piece of plywood for a billboard has an area of 24 sq. feet. The margins at the top and
bottom are 9 inches and at the sides are 6 in. Determine the size of the plywood for maximum
dimensions of the painted area.
a. 4x6
c. 4x8
b. 3x4
d. 3x8
323.) A manufacturer estimates that the cost of production of “x” units of a certain item is
C=40x-0.02x2-600. How many units should be produced for minimum cost?
a. 1000 units
b. 100 units
c. 10 units
d. 10000 units
324.) If the sum of the two numbers is 4, find the minimum value of the sum of their cubes.
a. 16
c. 10
b. 18
d. 32
325.) If x units of a certain item are manufactured, each unit can be sold for 200-0.01x pesos.
How many units can be manufactured for maximum revenue? What is the corresponding unit
price?
a. 10000,P100
c. 20000,P200
b. 10500,P300
d. 15000,P400
326.) A certain spare parts has a selling price of P150 if they would sell 8000 units per month. If
for every P1.00 increase in selling price, 80 units less will be sold out per month. If the
production cost is P100 per unit, find the price per unit for maximum profit per month.
a. P175
c. P150
b. P250
d. P225
327.) The highway department is planning to build a picnic area for motorist along a major
highway. It is to be rectangular with an area of 5000 sq. m. is to be fenced off on the three sides
not adjacent to the highway. What is the least amount of fencing that will be needed to complete
the job?
a. 200 m.
c. 400 m.
b. 300 m.
d. 500 m.
328.) A rectangular lot has an area of 1600 sq. m. find the least amount of fence that could be
used to enclose the area.
a. 160 m.
c. 100 m.
b. 200 m.
d. 300 m.
329.) A student club on a college campus charges annual membership dues of P10, less 5
centavos for each member over 60. How many members would give the club the most revenue
from annual dues?
a. 130 members
c. 240 members
b. 420 members
d. 650 members
330.) A monthly overhead of a manufacturer of a certain commodity is P6000 and the cost of the
material is P1.0 per unit. If not more than 4500 units are manufactured per month, labor cost is
P0.40 per unit, but for each unit over 4500, the manufacturer must pay P0.60 for labor per unit.
The manufacturer can sell 4000 units per month at P7.0 per unit and estimates that monthly sales
will rise by 100 for each P0.10 reduction in price. Find the number of units that should be
produced each month for maximum profit.
a. 4700 units
c. 6800 units
b. 2600 units
d. 9900 units
331.) A company estimates that it can sell 1000 units per week if it sets the unit price at P3.00,
but it’s weekly sales will rise by 100 units for each P0.10 decrease in price. Find the number of
units sold each week and its unit price per maximum revenue.
a. 2000 ; P2.00
c. 2500 ; P2.50
b. 1000 ; P3.00
d. 1500 ; P1.50
332.) In manufacturing and selling “x” units of a certain commodity, the selling price per unit is
P=5-0.002x and the production cost in pesos is C=3+1.10x. Determine the production level that
will produce the maximum profit and what would this profit be?
a. 975, P1898.25
c. 865, P1670.50
b. 800, P1750.75
d. 785, P1920.60
333.) ABC company manufactures computer spare parts. With its present machines, it has an
output of 500 units annually. With the addition of the new machines, the company could boost its
yearly production to 750 units. If it produces “x: parts it can set a price of P=200-0.15x pesos per
unit and will have a total yearly cost of C=6000+6x+0.003x2 in pesos. What production level
maximizes total yearly profit?
a. 660 units
c. 560 units
b. 237 units
d. 243 units
334.) The hypotenuse of a right triangle is 20cm. What is the maximum possible area of the
triangle in square centimeters?
a. 100
c. 120
b. 170
d. 160
335.) Sand is falling off a conveyor onto a conical pile at the rate of 15cm3/min. The base of the
cone is approximately twice the altitude. Find the height of the pile if the height of the pile is
changing at the rate 0.047746 cm/min.
a. 10cm
c. 8cm
b. 12cm
d. 6cm
336.) A machine is rolling a metal cylinder under pressure. The radius of the cylinder is
decreasing at the rate of 0.05cm per second and the volume V is 128π cu.cm. At what rate is the
length “h” changing when the radius is 2.5 cm.
a. 0.8192 cm/sec
c. 0.6178 cm/sec
b. 0.7652 cm/sec
d. 0.5214 cm/sec
337.) Two sides of a triangle are 15cm and 20cm long respectively. How fast is the third side
increasing if the angle between the given sides is 60º and is increasing at the rate of 2º/sec.
a. 0.05 cm/sec
c. 1.20 cm/sec
b. 2.70 cm/sec
d. 3.60 cm/sec
338.) Two sides of a triangle are 30cm and 40cm respectively. How fast is the area of the triangle
increasing if the angle between the given sides is 60º and is increasing at the rate of 4º/sec.
a. 20.94 m2/sec
c. 14.68 m2/sec
b. 29.34 m2/sec
d. 24.58 m2/sec
339.) A man 6ft tall is walking toward a building at the rate of 5ft/sec. If there is a light on the
ground 50ft from the building, how fast is the man/s shadow on the building growing shorter
when he is 30ft from the building?
a. -3.75 fps
c. -5.37 fps
b. -7.35 fps
d. -4.86 fps
340.) The volume of the sphere is increasing at the rate of 6cm3/hr. At what rate is its surface
area increasing when the radius is 50cm(in cm3/hr)
a. 20.94 m2/sec
c. 14.68 m2/sec
b. 29.34 m2/sec
d. 24.58 m2/sec
341.) A particle moves in a plane according to the parametric equations of motions: x=t2, y=t3.
Find the magnitude of the acceleration when the t=0.6667.
a. 6.12
c. 4.90
b. 5.10
d. 4.47
342.) The acceleration of the particle is given by a=2+12t in m/s2 where t is the time in minutes.
If the velocity of this particle is 11m/s after 1min, find the velocity after 2mins.
a. 31 m/sec
c. 37 m/sec
b. 45 m/sec
d. 26 m/sec
343.) A particle moves along a path whose parametric equations are x=t3 and y=2t2. What is the
acceleration when t=3sec?
a. 15.93 m/sec2
c. 23.36m/sec2
b. 18.44 m/sec2
d. 10.59 m/sec2
344.) A vehicle moves along a trajectory having coordinates given as x=t 3 and y=1-t2. The
acceleration of the vehicle at any point of the trajectory is a vector, having magnitude and
direction. Find the acceleration when t=2.
a. 13.20
c. 15.32
b. 12.17
d. 12.45
345.) Y = x3 – 3x. Find the maximum value of y.
a. 2
c. 0
b.1
d. 3
346.) Find the radius of curvature of the curve y=2x3+3x2 at (1,5).
a. 90
c. 95
b. 97
d. 84
347.) Compute the radius of curvature of the curve x=2y3-3y2 at (4, 2).
a. -99.38
c. -95.11
b.- 97.15
d. -84.62
348.) Find the radius of curvature of a parabola y2-4x=0 at point (4, 4).
a. 25.78
c. 20.33
b. 22.36
d. 15.42
349.) Find the radius of curvature of the curve x=y3 at point (1, 1).
a. -1.76
c. 2.19
b. -1.24
d. 2.89
350.) Find the point of inflection of the curve y=x3-3x2+6.
a. (1, 4)
c. (0, 2)
b. (1,3)
d. (2, 1)
Integral Calculus
351.) Find the total length of the curve r=4(1-Sinθ) from θ=90º to θ=270º and also the total
perimeter of the curve.
a. 18, 36
c. 12, 24
b. 16, 32
d. 15, 30
352.) Find the length of the curve r=4Sin θ from θ=0º to θ=90º and also the total length of curve.
a. 2π ; 4π
c. π ; 2π
b. 3π ; 6π
d. 4π ; 8π
353.) Find the length of the curve r = a (1-Cosθ) from θ=0º to θ=π and also the total length of the
curve.
a. 4a ; 8a
c. 3a ; 6a
b. 2a ; 4a
d. 5a ; 9a
354.) Find the total length of the curve r = a Cosθ.
a. 2πa
c. 1.5πa
b. πa
d. 0.67πa
355.) Find the length of the curve having a parametric equations of x = a Cos3θ, y = a Sin2θ from
θ=0º to θ=2π.
a. 5a
c. 7a
b. 6a
d. 8a
356.) Find the centroid of the area bounded by the curve y=4-x2, the line x=1 and the coordinate
axes.
a. (0.48, 1.85)
c. (0.24, 1.57)
b. (1.22, 0.46)
d. (2.16, 0.53)
357.) Find the centroid of the area under y=4-x2 in the first quadrant.
a. (0.75, 1.6)
c. (0.74, 1.97)
b. (1.6, 0.95)
d. (3.16, 2.53)
358.) Find the centroid of the area in first quadrant bounded by the curve y2=4ax and the latus
rectum.
a. (0.6a, 0.75a)
c. (0.94a, 2.97a)
b. (1.23a, 0.95a)
d. (1.16a, 0.53a)
359.) A triangular section has coordinates of A(2,2), B(11,2), and C(5,8). Find the coordinates of
the centroid of the triangular section.
a. (7, 4)
c. (8, 4)
b. (6, 4)
d. (9, 4)
360.) The following cross section has the following given coordinates. Compute for the centroid
of the given cross section. A(2,2), B(5,8), C(7,2), D(2,0), and E(7,0).
a. (4.6, 3.4)
c. (5.2, 3.8)
b. (4.8, 2.9)
d. (5.3, 4.1)
361.) Sections ABCD is a quadrilateral having the given coordinates A(2,3), B(8,9), C(11,3), and
D(11,0). Compute for the coordinates of the centroid of the quadrilateral.
a. (7.33, 4)
c. (5.32, 3)
b. (6.23, 4)
d. (8.21, 3)
362.) A cross section consists of a triangle and a semi circle with AC as its diameter. If the
coordinates of A(2,6), B(11,9), and C(14,6). Compute for the coordinates of the centroid of the
cross section.
a. (4.6, 3.4)
c. (5.2, 3.8)
b. (4.8, 2.9)
d. (5.3, 4.1)
363.) A 5m x 5cm is cut from a corner of 20cm x 30cm cardboard. Find the centroid from the
longest side.
a. 10.33m
c. 10.99m
b. 11.42m
d. 12.42m
364.) Locate the centroid of the area bounded by the parabola y2=4x, the line y=4 and the y-axis.
a. (1.2, 3)
c. (0.6, 3)
b. (0.4, 3)
d. (1.33, 3)
365.) Find the centroid of the area bounded by the curve x2=-(y-4), the x-axis and the y-axis on
the first quadrant.
a. (0.75, 1.6)
c. (1.75, 1.2)
b. (1.25, 1.4)
d. (0.25, 1.8)
366.) Locate the centroid of the area bounded by the curve y2 =-1.5(x-6), the x-axis and the y-axis
on the first quadrant.
a. (2.4, 1.13)
c. (2.8, 0.63)
b. (2.6, 0.88)
d. (2.2, 1.38)
367.) Locate the centroid of the area bounded by the curve 5y2=16x and y2=8x-24 on the first
quadrant.
a. (2.20, 1.51)
c. (2.78, 1.39)
b. (1.50, 0.25)
d. (1.64, 0.26)
368.) Locate the centroid of the area bounded by the parabolas x2=8y and x2=16(y-2) in the first
quadrant.
a. (3.25, 1.2)
c. (2.67, 2.0)
b. (2.12, 1.6)
d. (2.00, 2.8)
369.) Given the area in the first quadrant bounded by x2=8y, the line y-2=0 and the y-axis. What
is the volume generated when revolved about the line y-2=0?
a. 53.31m3
c. 26.81m3
b. 45.87m3
d. 33.98m3
370.) Given the area in the first quadrant bounded by x2=8y, the line x=4 and the x-axis. What is
the volume generated by revolving this area about the y-axis?
a. 78.987m3
c. 61.253m3
b. 50.265m3
d. 82.285m3
371.) Given the area in the first quadrant bounded by x2=8y, the line y-2=0 and the y-axis. What
is the volume generated when this area is revolved about the x-axis.
a. 20.32m3
c. 40.21m3
b. 34.45m3
d. 45.56m3
372.) Find the volume formed by revolving the hyperbola xy=6 from x=2 to x=4 about the xaxis.
a. 28.27m3
c. 23.23m3
b. 25.53m3
d. 30.43m3
373.) The region in the first quadrant under the curve y=Sinh x from x=0 to x=1 is revolved
about the x-axis. Compute the volume of solid generated.
a. 1.278m3
c. 3.156m3
b. 2.123m3
d. 1.849m3
374.) A square hole of side 2cm is chiseled perpendicular to the side of a cylindrical post of
radius 2cm. If the axis of the hole is going to be along the diameter of the circular section of the
post, find the volume cutoff.
a. 15.3m3
c. 43.7m3
b. 23.8m3
d. 16.4m3
375.) Find the volume common to the cylinders x2 +y2=9 and y2+z2=9.
a. 241m3
c. 424m3
b. 533m3
d. 144m3
376.) Given is the area in the first quadrant bounded by x2=8y, the line, the line x=4 and the xaxis. What is the volume generated by revolving this area about the y-axis.
a. 50.26m3
c. 53.26m3
b. 52.26m3
d. 51.26m3
377.) The area bounded by the curve y2=12x and the line x=3 is revolved about the line x=3.
What is the volume generated?
a. 185
c. 181
b. 187
d. 183
378.) The area in the second quadrant of the circle x2+y2=36 is revolved about the line y+10=0.
What is the volume generated?
a. 2128.63
c. 2233.43
b. 2228.83
d. 2208.53
379.) The area enclosed by the ellipse 0.11x2+0.25y2=1 is revolved about the line x=3, what is
the volume generated?
a. 370.3
c. 355.3
b. 360.1
d. 365.1
380.) Find the volume of the solid formed if we rotate the ellipse 0.11x2+0.25y2=1 about the line
4x+3y=20.
a. 48 π 2m3
c. 40 π 2m3
b. 45π2m3
d. 53 π 2 m3
381.) The area on the first and second quadrant of the circle x2+y2=36 is revolved about the line
x=6. What is the volume generated?
a. 2131.83
c. 2421.36
b. 2242.46
d. 2342.38
382.) The area on the first quadrant of the circle x2 +y2=25 is revolved about the line x=5. What is
the volume generated?
a. 355.31
c. 368.33
b. 365.44
d. 370.32
383.) The area of the second and third quadrant of the circle x2+y2 =36 is revolved about the line
x=4. What is the volume generated?
a. 2320.30
c. 2327.25
b. 2545.34
d. 2520.40
384.) The area on the first quadrant of the circle x2+y2=36 is revolved about the line y+10=0.
What is the volume generated?
a. 3924.60
c. 2593.45
b. 2229.54
d. 2696.50
385.) The area enclosed by the ellipse 0.0625x2 +0.1111y2=1 on the first and 2 nd quadrant, is
revolved about the x-axis. What is the volume generated?
a. 151.40
c. 156.30
b. 155.39
d. 150.41
386.) The area enclosed by the curve 9x2+16y2=144 on the first quadrant, is revolved about the
y-axis. What is the volume generated?
a. 100.67
c. 98.60
b. 200.98
d. 54.80
387.) Find the volume of an ellipsoid having the equation 0.04x2+0.0625y2+0.25z2=1.
a. 167.55
c. 171.30
b. 178.40
d. 210.20
388.) Find the volume of a spheroid having equation 0.04x2+0.111y2+0.111z2=1.
a. 178.90
c. 188.50
b. 184.45
d. 213.45
389.) The region in the first quadrant which is bounded by the curve y2=4x, and the lines x=4
and y=0, is revolved about the x-axis. Locate the centroid of the resulting solid revolution.
a. 2.667
c. 1.111
b. 2.333
d. 1.667
390.) The region in the first quadrant, which is bounded by the curve x2=4y, the line x=4, is
revolved about the line x=4. Locate the centroid of the resulting solid revolution.
a. 0.8
c. 1.0
b. 0.5
d. 0.6
391.) The area bounded by the curve x3=y, the line y=8 and the y-axis, is to be revolved about
the y-axis. Determine the centroid of the volume generated.
a. 5
c. 4
b. 6
d. 7
392.) The area bounded by the curve y=x3 and the x-axis. Determine the centroid of the volume
generated.
a. 2.25
c. 1.25
b. 1.75
d. 0.75
393.) Find the moment of inertia of the area bounded by the curve x2=4y, the line y=1 and the yaxis on the first quadrant with respect to x-axis.
a. 1.2
c. 0.57
b. 3.5
d. 1.14
394.) Find the moment of inertia of the area bounded by the curve x2=4y, the line y=1 and the yaxis on the first axis with respect to y axis.
a. 6.33
c. 0.87
b. 1.07
d. 0.94
395.) Find the moment of inertia of the area bounded by the curve x2=8y, the line x=4, and the xaxis on the first quadrant with respect to x-axis.
a. 1.52
c. 1.98
b. 2.61
d. 2.36
396.) Find the moment of inertia of the area bounded by the curve x2=8y, the line x=4, and the xaxis on the first quadrant with respect to y-axis.
a. 25.6
c. 31.6
b. 21.8
d. 36.4
397.) Find the moment of inertia of the area bounded by the curve y2=4x, the line x=1, and the xaxis on the first quadrant with respect to x-axis.
a. 1.067
c. 1.861
b. 1.142
d. 1.232
398.) Find the moment of inertia of the area bounded by the curve y2=4x, the line x=1, and the xaxis on the first quadrant with respect to y-axis.
a. 0.571
c. 0.436
b. 0.682
d. 0.716
399.) Find the moment of inertia of the area bounded by the curve y2=4x, the line y=2, and the yaxis on the first quadrant with respect to y-axis.
a. 0.095
c. 0.088
b. 0.064
d. 0.076
400.) Find the moment of inertia with respect to x-axis of the area bounded by the parabola
y2=4x, the line x=1.
a. 2.35
c. 2.13
b. 2.68
d. 2.56
MULTIPLE CHOICE QUESTIONS IN
MATHEMATICS
PERFECTO B. PADILLA JR
AND
DIEGO INOCENCIO TAPANG GILLESANIA
1. What is the allowable error in
measuring the edge of a cube that is
intended to hold 8 cu.m, if the error
of the compound volume is not to
exceed 0.03m3?
a. 0.002
b. 0.001
c. 0.0025
d. 0.0001
2. Find the area bounded by the
parabola
and its latus
rectum.
a.10.67 sq. units
b. 32 sq. units
c. 48 sq. units
d. 16.67 sq. units
3. The effective rate of 14%
compounded semi-annually is:
a. 14.49%
b. 12.36%
c. 12.94%
d. 14.88%
4.
is the equation of
_______?
a. Parallel sides
b. Parabola
c. Circle
d. Ellipse
5. A section in a coliseum has 32 seats
in the 1st row, 34 in the 2nd row, 36 in
the 3rd row, . . and 48 in the 9th row.
From the 10th up to the 20th row, all
have 50 seats. Find the seating
capacity of this section of the
coliseum.
a. 908
b. 900
c. 920
d. 910
6. Smallest term that can be factored
from a number
a. Greater
b. None of these
c. equal
d. lesser
7. How many horsepower are there in
800 kW?
a. 2072.4 hp
b. 746 hp
c. 1072.4 hp
d. 3072.4 hp
8. A man roes downstream at the rate
of 5 mph and upstream at the rate of
2 mph. how far downstream should
he go if he is to return 7/4 hour after
leaving?
a. 2.5 mi
b. 3.3 mi
c. 3.1 mi
d. 2.7 mi
9. Find the angular velocity of a
flywheel whose radius is 20 ft. if it is
revolving at 20 000 ft/min
a. 500 rad/min
b. 750 rad/min
c. 1000 rad/min
d. 800 rad/min
10. Find the area of parabolic segment
whose base is 10 and height of 9
meters.
a. 60 m2
b. 70 m2
c. 75 m2
d. 65 m2
11. A line which a curve approach
infinity but will never intersect.
a.
b.
c.
d.
Parallel line
Assymptote
Inclined line
Skew line
12. An organization that aims to block
the entry of a new comer.
a. Monopoly
b. Cartel
c. Competitor
d. Proprietor
13. The tens digit of a two-digit number
is 1 less than twice the unit’s digit.
They differ by 4. Find the number.
a. 65
b. 95
c. 84
d. 73
14. At the surface of the earth g=9.806
m/s2. Assuming the earth to be a
sphere of radius 6.371x106m.
Compute the mass of the earth.
a. 5.97x1024 kg
b. 5.62 x1024 kg
c. 5.12 x1024 kg
d. 5.97 x1023 kg
15. A material has a modulus of
elasticity of 200 GPa. Find the
minimum cross sectional area of the
said material so as not to elongate by
more than 5mm for every 2m length
when subjected to 10 kN tensile
force.
a. 20 mm2
b. 10 mm2
c. 30 mm2
d. 40 mm2
16. At what temperature is the ˚C and ˚F
numerically the same?
a. 40˚
b. 32˚
c. -40˚
d. -32˚
17. On ordinary day, 400 m3 of air has a
temperature of 30˚C. During El Nino
drought, temperature increased to
40˚C. Find the volume of air of
k=3670x10-6.
a. 416.86 m3
b. 418.86 m3
c. 414.68 m3
d. 416.48 m3
18. A sphere has a volume of 36π cubic
meters. The rate of change in volume
is 9π cubic meters per minute. Find
the rate of change in area of the
sphere.
a. 6 π m2/min
b. 2 π m2/min
c. 3 π m2/min
d. 4 π m2/min
19. Sin A=2.5x, cos A= 5.5x. Find A.
a. 34.44˚
b. 24.44˚
c. 44.44˚
d. 64.44˚
20. A ladder 5 meter long leans on a wall
and makes an angle of 30˚ with the
horizontal. Find the vertical height
from the top to the ground.
a. 2.5 meter
b. 1.5 meter
c. 2.0 meter
d. 2.75 meter
21. A rectangular lot is bounded on its
two adjacent sides by existing
concrete walls. If it is to be fenced
along two remaining sides and the
available fencing material is 30
meters long, find the largest possible
area of the lot.
a. 200 sq. m
b. 225 sq. m
c. 175 sq. m
d. 250 sq. m
22. A tangent line intersects a secant line
to a circle. If the distance from the
point of tangency to the point of
intersection is 6, and the external
distance of the secant line is 4, find
the length of the secant line.
a. 5
b. 7
c. 8
d. 9
23. In an oblique triangle, a=25, b=16,
angle C=94˚06’. Find the measure of
angle A.
a. 54.5˚
b. 45.5˚
c. 24.5˚
d. 54.5˚
26. Find the tangential velocity of a
flywheel whose radius is 14 ft. if it is
revolving at 200 rpm.
a. 17 593 ft/min
b. 18 593 ft/min
c. 19 593 ft/min
d. 12 593 ft/min
27. A ball is thrown vertically upward at
a velocity of 10 m/s. What is its
velocity at the maximum height?
a. 10 m/s
b. 0
c. 5 m/s
d. 15 m/s
28. The volume of a sphere is tripled.
What is the increase in surface area
if the radius of the original sphere is
50 cm.?
a. 34 931.83 sq. units
b. 33 931.83 sq. units
c. 35 931.83 sq. units
d. 36 931.83 sq. units
24. Q=25 when t=0. Q=75 when t=2.
What is Q when t=6?
a. 185
b. 145
c. 150
d. 175
29. Given a right triangle ABC. Angle C
is the right triangle. BC=4 and the
altitude to the hypotenuse is 1 unit.
Find the area of the triangle.
a. 2.0654 sq. units
b. 1.0654 sq. units
c. 3.0654 sq. units
d. 4.0645 sq. units
25. Pipes A and B can fill an empty tank
in 6 and 3 hours respectively. Drain
C can empty a full tank in 24 hours.
How long will an empty tank be
filled if pipes A and B with drain C
open?
a. 1.218 hours
b. 2.182 hours
c. 5.324 hours
d. 3.821 hours
30. Find the equation of a parabola
passing through (3, 1), (0, 0), and (8,
4) and whose axis is parallel to the xaxis.
a.
b.
c.
d.
31. Pedro runs with a speed of 20 kph.
Five minutes later, Mario starts
running to catch Pedro in 20
minutes. Find the velocity of Mario.
a. 22.5 kph
b. 25 kph
c. 27.5 kph
d. 30 kph
32. How much do ten P2000 quarterly
payments amount at present if the
interest rate is 10% compounded
quarterly.
a. P17 771.40
b. P17 504.13
c. P18 504.13
d. P71 504.13
33. A man bought a machine costing
P135 000 with a salvage value of
P20 000 after 3 years. If the man will
sell it after 2 years, how much is the
loss or gain (i.e. the cost of
equipment) if i=10%.
a. P134 350
b. P143 350
c. P153 350
d. P163 350
34. P1000 becomes P1500 in three years.
Find the simple interest rate.
a. 16.67%
b. 15.67%
c. 17.67%
d. 18.67%
35. Form of paper money issued by the
central bank.
a. T-bills
b. Check
c. Cash
d. Stocks
36. _________ is the concept of finding
the derivative of an exponential
expression.
a. Logarithmic derivative
b. Chain rule
c. Trigonometric derivative
d. Implicit derivative
37. The line y=5 is the directrix of a
parabola whose focus is at point (4, 3). Find the length of the latus
rectum.
a. 8
b. 4
c. 16
d. 24
38. 2.25 revolutions are how many
degrees?
a. 810˚
b. 730˚
c. 190˚
d. 490˚
39. The sum of two numbers is 21 and
their product is 108. Find the sum of
their reciprocals.
a.
b.
c.
d.
40. What is the accumulated amount of
five years annuity paying P 6000 at
the end of each year, with interest at
15% compounded annually?
a. P40 454.29
b. P41 114.29
c. P41 454.29
d. P40 544.29
41. Ana is 5 years older than Beth. In 5
years, the product of their ages is 1.5
times the product of their present
ages. How old is Beth now?
a. 25
b. 20
c. 15
d. 30
42. In
, x=
distance in meters, and t= time in
seconds. What is the initial velocity?
a. 2000 m/s
b. 3000 m/s
c. 4000 m/s
d. 5000 m/s
43. The highest point that a girl on a
swing reaches is 7 ft above the
ground, while the lowest point is 3 ft
above the ground. Find its tangential
velocity at the lowest point.
a. 16.05 ft/sec
b. 12.05 ft/sec
c. 20.05 ft/sec
d. 12.05 ft/sec
44. If m=tan25˚, find the value of
˚
˚
in terms of m.
˚
˚
a. -1/m
b.
c.
d. –m
45. A VOM has a current selling price of
P400. If it’s selling price is expected
to decline at the rate of 10% per
annum due to obsolence, what will
be its selling price after 5 years?
a. P236.20
b. P200.00
c. P213.10
d. P245.50
46. Evaluate ∫
a. 1.051
b. 1.501
c. 3.21
d. 2.321
dx
47. Fin the eccentricity of an ellipse
when the length of the latus rectum
is 2/3 the length of the major axis.
a. 0.577
b. 0.477
c. 0.333
d. 0.643
48. What is the book value of an
electronic test equipment after 8
years of use if it depreciates from its
original value of P120 000 to its
salvage value of 13% in 12 years.
Use straight line method.
a. P20 794.76
b. P50 400
c. P40 794.76
d. P50 794.76
49. What is the book value of an
electronic test equipment after 8
years of use if it depreciates from its
original value of P120 000 to its
salvage value of 13% in 12 years.
Use declining balance method.
a. P20 794.76
b. P30 794.76
c. P40 794.76
d. P50 794.76
50. A balloon is released from the
ground 100 meters from an observer.
The balloon rises directly upward at
the rate of 4 meters per second. How
fast is the balloon receding from the
observer 10 seconds later?
a. 1.4856 m/s
b. 2.4856 m/s
c. 3.4856 m/s
d. 5 m/s
51. Divide 120 into two parts so that
product of one and the square of
another is maximum. Find the small
number.
a. 60
b. 50
c. 40
d. 30
52.
. What is the period?
.π
.2 π
.4 π
.3 π
53. A horizontal force of 80 000 N is
applied unto a 120 ton load in 10
seconds. Find its acceleration.
a. 0.67 m/s2
b. 0.75 m/s2
c. 1.05 m/s2
d. 1.35 m/s2
54. A plane is headed due to east with
airspeed 240 mph. if a wind at 40
mph from the north is blowing; find
the groundspeed of the plane.
a. 342 mph
b. 532 mph
c. 243 mph
d. 4123 mph
55. The ratio of radii of cone and
cylinder is 1:2 while the ratio of
radius of cone to its altitude is 1:3. If
lateral surface area of cylinder equals
volume of cone, find the radius of
the cone if the altitude of cylinder is
4.
a. 5
b. 4
c. 3
d. 6
56. If a derivative of a function is
constant, the function is:
a. First degree
b. Exponential
c. Logarithmic
d. Sinusoidal
57. 2700 mils is how many degrees?
a. 151.875˚
b. 270˚
c. 180˚
d. 131.875˚
58. An air has an initial pressure of
100kPa absolute and volume 1 m3. If
pressure will be increased to 120
kPa, find the new volume.
a. 1.2 m3
b. 0.83 m3
c. 0.63 m3
d. 1.5 m3
59. The pistons (A&B) of a hydraulic
jack are at the same level. Pistol A is
100 cm2 while piston B is 500 cm2.
Piston A carries a 500 kg load. Find
the required force F at piston B to
carry the load.
a. 3.5 tons
b. 2.5 tons
c. 4.5 tons
d. 1.5 tons
60. A rectangular dodecagon is inscribed
in a circle whose radius is 1 unit.
Find the perimeter.
a. 5.21
b. 6.21
c. 7.21
d. 8.21
61. In a box, there are 52 coins,
consisting of quarters, nickels, and
dimes with a total amount of $2.75.
If the nickel were dimes, the dimes
were quarters and the quarters were
nickels; the total amount would be
$3.75. How many quarters are there?
a. 16
b. 10
c. 5
d.12
62. A stone is thrown vertically upward
at 12 m/s. Find the time to reach the
ground.
a. 2.45 secs.
b. 1.35 secs.
c. 2.15 secs.
d. 1.95 secs.
63. A regular polygon has 27 diagonals.
Then it is a :
a. Pentagon
b. Heptagon
c. Nonagon
d. Hexagon
67. A hyperbola has its center at point
(1, 2), vertex at (2, 2) and conjugate
vertex at (1, 0). Find the equation.
a. 4x2-y2-8x+4y-4=0
b. x2-4y2-8x+4y-4=0
c. 4x2-y2-8x-4y-4=0
d. x2-4y2+8x-4y-4=0
68. A pipe can fill a tank in 2 hours. A
drain can empty a full tank in 6
hours. If the pipe runs with the drain
open, how long will take to fill-up an
empty tank?
a. 2.5 hrs
b. 4 hrs
c. 3 hrs
d. 3.5 hrs
69. Fin the 7th term in the series: , ,
..
a.
b.
64. A 50 meter cable is divided into two
parts and formed into squares. If the
sum of the areas is 100 sq. meter,
find the difference in length?
a. 21.5
b. 20.5
c. 24.5
d. 0
65. What theorem is used to solve for
centroid?
a. Pappus
b. Varignon’s
c. Castiglliano’s
d. Pascal’s
66. ∫
a.
b.
c.
d.
tan x – x + c
x - tan x + c
sec x
sec x tan x
c.
d.
70. Find the length of the larger base of
the largest isosceles trapezoid if the
legs and smaller base measure 8
units.
a. 8
b. 16
c. 10
d. 20
71. y=arctan ln x. Find y’.
a.
b.
c.
,
d.
72. The general equation of a conic
section whose axis is inclined is
given by
Ax2+Bxy+Cy2+Dx+Ey+F=0. When
B2-4 Ac=0, the curve is a/an _____.
a. Hyperbola
b. Parabola
c. Ellipse
d. Circle
73. The time required for two examinees
to solve the same problem differs by
two minutes. Together they can solve
32 problems in one hour. How long
will it take for the slower problem
solver to solve the problem?
a. 2 min
b. 3 min
c. 4 min
d. 5 min
74. cos4 θ – sin4 θ= ?
a. sin 2θ
b. cos 2θ
c. cos 4θ
d. cos 3θ
75. A function wherein one variable is
not yet readily expressed as function
of another variable is said to be:
a. symmetric
b. implicit
c. explicit
d. exponential
76. Given an ellipse + =1.
Determine the distance between
directrix:
a. 3
b. 4
c. 2
d. 8
77. Three forces 20N, 30N, and 40N are
in equilibrium. Find the angle
between 30N and 40N forces.
a. 28.96˚
b. 25.97˚
c. 40˚
d. 30˚15’25”
78. At the inflection point where x=a
a. f”(a) > 0
b. f”(a) < 0
c. f”(a) = 0
d. f”(a) is no equal to zero
79. A merchant has three items on sale
namely: a radio for $50.00, a clock
for $30.00 and a flashlight for $1.00.
At the end of the day, she has sold a
total of 100 of the three sale items
and has taken in exactly $1, 000.00
on the total sales, how many radios
did she sell?
a. 4
b. 80
c. 16
d. 20
80. Which of the following is true?
a. sin(-θ)= sin θ
b. tan(-θ)= tan θ
c. cos(-θ)= cos θ
d. csc(-θ)= csc θ
81. _______ is the loss of value of the
equipment with use over a period of
time. It could mean a difference in
value between a new asset and the
used asset currently in service.
a. Loss
b. Depreciation
c. Gain
d. Extracted
82. Find the area bounded by the curve
defined by the equation x2=8y and its
latus rectum.
a. 11/3
b. 32/3
c. 16/3
d. 22/3
83. The height of a right circular
cylinder is 50 inches and decreases at
the rate of 4 inches per second.
While the radius of the base is 20
inches and increases at the rate of
one inch per second. At what rate is
the volume changing?
a. 11 130 cu. in/sec
b. 11 310 cu. in/sec
c. 1 275 cu. in/sec
d. 1 257 cu. in/sec
84. This occurs in a situation where a
commodity or service is supplied by
a number of vendors and there is
nothing to prevent additional vendors
entering the market.
a. Elastic demand
b. Perfect competition
c. Monopoly
d. Oligopoly
85. The graphical representation of the
cumulative frequency distribution in
a set statistical data is called?
a. Frequency polygon
b. Mass diagram
c. Ogive
d. Histogram
86. If the product of the slopes of two
straight lines is negative 1, one of
these lines are said to be:
a. Skew
b. Non-intersecting
c. Parallel
d. Perpendicular
87. Pedro can paint a fence 50% faster
than Juan and 20% faster that Pilar
and together they can paint a given
fence in 4 hours. How long will it
take Pedro to paint the same fence if
he had to work alone?
a. 10 hrs
b. 13 hrs
c. 11 hrs
d. 15 hrs
88. If you borrowed money from your
friend with simple interest of 12%,
find the present worth of P50 000,
which is due at the end of 7 months.
a. P46 200
b. 44 893
c. P46 729
d. 45 789
89. The amount of P12 800 in 4 years at
5% compounded quarterly is?
a. P14 785.34
b. P15 614.59
c. P16 311.26
d. P15 847.33
90. What is the effective rate
corresponding to 18% compounded
daily? Take 1 year =365 days.
a. 17.35%
b. 19.72%
c. 17.84%
d. 16.78%
91. In how many ways can 2 integers be
selected from the integers 1 to 100 so
that their difference is exactly 7?
a. 74
b. 81
c. 69
d. 93
92. A 2 lbs liquid has an specific heat of
1.2 Btu/ lb-˚F. How much heat is
required to increase its temperature
by 10˚C?
a. 100BTU
b. 110BTU
c. 120 BTU
d. 130 BTU
93. A machine costing P100 000
depreciates at 10% annually. What is
its book value after 5 years?
a. P59 049
b. P69 049
c. P49 049
d. P79 049
94. Find the length of the latus rectum of
the parabola y2=-8x?
a. 8
b. 9
c. 7
d. 6
95. The property by virtue of which a
body tends to return to its original
size and shape after a deformation
and when the deforming forces have
been removed.
a. Elasticity
b. Malleability
c. Ductility
d. Plasticity
96. A man wants to make 14% nominal
interest compounded semi-annually
on a bond investment. How should
the man be willing to pay now for
12% -P10 000 bond that will mature
in 10 years and pays interest semiannually?
a. P2 584.19
b. P3 118.05
c. P8 940.60
d. P867.82
97. Evaluate ∫
a. -3/2 cos 2 + C
b. -3 cos 2
c. 3/2 cos 2 + C
d. 3 cos 2 + C
98. Find the maximum height which a
cannonball fired at an initial velocity
of 100 m/s at 30˚ above the
horizontal.
a. 127.42 m
b. 172.42 m
c. 137.42 m
d. 177.42 m
99. A man expects to receive P20 000 in
10 years. How much is that money
worth now considering interest at 6%
compounded quarterly.
a. P 12 698.65
b. P11 025.25
c. P17 567.95
d. P15 678.45
100. The area of a rhombus is 24. One
diagonal measures 6 units, find the
length of the other diagonal.
a. 9
b. 7
c. 6
d. 8
101. The area of a rhombus is 24. One
diagonal measures 6 units, find the
length of a side.
a. 5
b. 6
c. 7
d. 8
102. The sum of the coefficients in the
expansion of (x+y-z)8 is:
a. From 2 to 5
b. From 5 to 10
c. Above 10
d. Less than 2
103. A banca traveled at an average speed
of 15 kph downstream and then back
at an average speed of 12 kph
upstream. If the total time of travel is
3 hours, find the total distance traveled
by the banca.
a. 40 km
b. 30 km
c. 60 km
d. 50 km
104. A father is now 41 and his son 9.
After how many years will his age be
just triple his son’s age?
a. 6
b. 5
c. 4
d. 7
105. Find the area of the largest rectangle
which you can inscribe in a semicircle whose radius is 10.
a. 1000 sq. units
b. √ sq. units
c. 100 sq. units
d. 2√ sq. units
106. Given y = 4 cos 2x. Determine its
amplitude.
a. 2
b. 4
c. 8
d. √
107. A central angle of 45˚ subtends an
arc of 12cm. What is the radius of the
circle?
a.
b.
c.
d.
12.58 cm
15.28 cm
15.82 cm
12.85 cm
108. The volume of two spheres is in the
ratio of 27:343 and the sum of their
radii is 10. Find the radius of the
smaller sphere.
a. 6
b. 3
c. 5
d. 4
109. The integral of any quotient whose
numerator is the differential of the
denominator is the:
a. Product
b. Derivative
c. Cologarithm
d. Logarithm
110. Find the sum of the roots 5x2 -10x +
2=0
a. -2
b. 2
c. 1/2
d. -1/2
111. Determine the vertical pressure due
to a column of water 85 cm high.
a. 8.33 x 103 N/m2
b. 8.33 x 104 N/m2
c. 8.33 x 105 N/m2
d. 8.33 x 106 N/m2
112. A rectangular hexagonal pyramid has
a slant height of 4 cm and the length
of each side of the base is 6 cm. find
the lateral area.
a. 52 cm2
b. 62 cm2
c. 72 cm2
d. 82 cm2
113. If a =b, the b = a. This illustrates
which axiom in algebra?
a. Replacement axiom
b. Symmetric axiom
c. Transitive axiom
d. Reflexive axiom
c. P5 637.50
d. P5 937.50
114. If arc tan x + arc tan 1/3 = π/4, find
the value of x.
a. 1/2
b. 1/3
c. 1/4
d. 1/5
119. To compute for the value of the
factorial, in symbolic form (n!) where
n is a large number, we use a formula
called:
a. Matheson formula
b. Diophantine formula
c. Stirlings Approximation
formula
d. Richardson-Duchman
formula
115. It is the measure of relationship
between two variables.
a. Correlation
b. Function
c. Equation
d. Relation
120. Find the distance of the directrix
from the center of an ellipse if its
major axis is 10 and its minor axis is
8.
a. 8.1
b. 8.3
c. 8.5
d. 8.7
116. It is a polyhedron of which two faces
are equal, polygons in parallel planes
and the other faces are parallelograms.
a. Cube
b. Pyramid
c. Prism
d. Parallelepiped
117. What is the distance in cm. between
two vertices of a cube which are
farthest from each other, if an edge
measures 8 cm?
a. 12.32
b. 13.86
c. 8.66
d. 6.93
118. A loan of P5000 is made for a period
of 15 months at a simple interest rate
of 15%. What future amount is due at
the end of the loan period?
a. P 5 842.54
b. P5 900.00
121. A 200 gram apple is thrown from the
edge of a tall building with an initial
speed of 20 m/s. What is the change is
kinetic energy of the apple if it strikes
the ground at 50 m/s?
a. 100 joules
b. 180 joules
c. 81 joules
d. 210 joules
122. When two planes intersect with each
other, the amount of divergence
between the two planes is expressed
by the measure of:
a. Polyhedral angle
b. Dihedral angle
c. Reflex angle
d. Plane angle
123. The median of a triangle is the line
connecting a vertex and the midpoint
of the opposite side. For a given
triangle, the medians intersects at a
pint which is called the:
a. Circumcenter
b. Incenter
c. Orthocenter
d. Centroid
124. A five-pointed star is also known as:
a. Quintagon
b. Pentagon
c. Pentatron
d. Pentagram
125. The altitudes of the sides of a
triangle intersect at the point, which is
known as:
a. Centroid
b. Incenter
c. Orthocenter
d. Circumcenter
126. The arc length equal to the radius of
the circle is called:
a. 1 grad
b. 1 radian
c. π radian
d. 1 quarter circle
127. One gram of ice at 0˚C is placed on a
container containing 2,000,000 cu. m
of water at 0˚C. Assuming no heat
loss, what will happen?
a. The volume of ice will not
change
b. Ice will become water
c. Some part of ice will not
change
d. All of the above
128. The angular bisector of the sides of a
triangle at a point which is known as:
a. Centroid
b. Incenter
c. Orthocenter
d. Centroid
129. A pole cast a shadow of 15 meters
long when the angle of elevation of
the sun is 61˚. If the pole has leaned
15˚ from the vertical directly toward
the sun, what is the length of the pole?
a. 53.24 m
b. 54.25 m
c. 52.43 m
d. 53.25 m
130. Each side of a cube is increased by
1%. By what percent is the volume of
the cube increased?
a. 3%
b. 23.4%
c. 33.1%
d. 34.56%
131. MCMXCIV is a Roman numeral
equivalent to:
a. 2174
b. 3974
c. 2974
d. 1994
132. The sum of the digits of a two digit
number is 11. If the digits are
reversed, the resulting number is
seven more than twice the original
number. What is the original number?
a. 44
b. 83
c. 38
d. 53
133. A regular octagon is inscribed in a
circle of radius 10. Find the area of the
octagon.
a. 288.2
b. 282.8
c. 228.2
d. 238.2
134. Find the probability of getting
exactly 12 out of 30 questions on the
true or false question.
a. 0.04
b. 0.15
c. 0.12
d. 0.08
135. Find the length of the vector (12, 4,
4).
a. 8.75
b. 5.18
c. 7
d. 6
136. According to this law, “The force
between two charges varies directly as
the magnitude of each charge and
inversely as the square of the distance
between them”.
a. Newton’s law
b. Inverse Square law
c. Coulomb’s law
d. Law of Universal Gravitation
137. Mr. J. Reyes borrowed money from
the bank. He received from the back
P1842 and promised to pay P2000 at
the end of 10 months. Determine the
simple interest.
a. 15.7%
b. 16.1%
c. 10.29%
d. 19.45%
138. Evaluate the expression (1 + i2 )10
where I is an imaginary number.
a. -1
b. 10
c. 0
d. 1
139. The amount of heat needed to change
solid to liquid.
a. Latent heat of fusion
b. Solid fusion
c. Condensation
d. Cold fusion
140. Solve for x in the equation: 2 log4 x
– log4 9 = 2
a. 12
b. 10
c. 11
d. 13
141. Two post, one 8m and the other 12 m
high are 15 m apart. If the posts are
supported by a cable running from the
top of the first post to a stake on the
ground and then back to the top of the
second post, find the distance from the
lower post to the stake to use the
minimum amount of wire.
a. 4 m
b. 6 m
c. 8 m
d. 9m
142. A 40 gm rifle bullet is fired with a
speed of 300 m/s into a ballistic
pendulum of mass 5 kg suspended
from a chord 1 m long. Compute the
vertical height through which the
pendulum arises.
a. 29.88 cm
b. 28.89 cm
c. 28.45 cm
d. 29.42 cm
143. If the roots of an equation are zero,
then they are classified as:
a. Trivial solution
b. Hypergolic solution
c. Zeros of function
d. Extraneous roots
144. Of what quadrant is A, if secA is
positive and cscA is negative?
a. IV
b. II
c. III
d. I
145. The reciprocal of bulk modulus of
any fluid is called ______.
a. Volume stress
b. Compressibility
c. Shape elasticity
d. Volume strain
146. Assuming that the earth is a sphere
whose radius is 6,400 km. Find the
distance along 3 deg arc at the equator
of the earth’s surface.
a. 335.10 km
b. 533.10 km
c. 353.10 km
d. 353.01 km
147. Equations relating x and y that
cannot readily solved explicitly for y
as a function of x or for x as a function
of y. Such equation may nonetheless
determine y as a function of x or vice
versa, such as function is called
_____.
a. Logarithmic function
b. Implicit function
c. Continuous function
d. Explicit function
148. What is the integral of (3t-1)3 dt?
a. 1/12 (3t-1)4 + c
b. 1/12 (3t-1)3 + c
c. ¼ (3t-1)3 + c
d. ¼ (3t-1)4 + c
149. If 16 is 4 more than 4x, find x-1
a. 14
b. 3
c. 12
d. 5
150. A frequency curve which is
composed of a series of rectangles
constructed with the steps as the base
and the frequency as the height.
a. Histogram
b. Ogive
c. Frequency distribution
d. Bar graph
151. It is a sequence of numbers such that
successive terms differ by a constant
a. Arithmetic progression
b. Infinite progression
c. Geometric progression
d. Harmonic progression
152. If the second derivative of the
equation of a curve is equal to the
negative of the equation of that same
curve, the curve is:
a. A paraboloid
b. A sinusoid
c. A cissoids
d. An exponential
153. Determine x, so that: a, 2x + 4, 10x –
4 will be a geometric progression.
a. 4
b. 6
c. 2
d. 5
154. The angular distance of a point on
the terrestrial sphere from the north
pole is called its:
a. Co-latitude
b. Altitude
c. Latitude
d. Co-declination
155. If one third of the air in a tank is
removed by each stroke of an air
pump, what fractional part of the air
removed in 6 strokes?
a. 0.7122
b. 0.9122
c. 0.6122
d. 0.8122
156. The linear distance between -4 and
17 on the number line is
a.
b.
c.
d.
13
21
-17
-13
157. Determine the angle of the super
elevation for a 200 m highway curve
so that there will be no side thrust at a
speed of 90 kph.
a. 19.17˚
b. 17.67˚
c. 18.32˚
d. 20.11˚
158. A ball is dropped from a building
100 m high. If the mass of the ball is
10 grams, after what time will the ball
strike the earth?
a. 4.52s
b. 4.42s
c. 5.61s
d. 2.45s
159. Centrifugal force is _____
a. Directly proportional to the
radius of the curvature
b. Directly proportional to the
square of the tangential
velocity
c. Inversely proportional to the
tangential velocity
d. Directly proportional to the
square of the weight of the
object
160. Each of the faces of a regular
hexahedron is a _____
a. Triangle
b. Square
c. Rectangle
d. Hexagon
161. Find the mean proportion of 4 and 36
a. 72
b. 24
c. 12
d. 20
162. Simplify the expression i1999 + i1999
where I is an imaginary number.
a. 0
b. -1
c. 1+1
d. 1-i
163. In a club of 40 executives, 33 likes to
smoke Marlboro and 20 like to smoke
Philip Moris. How many like both?
a. 13
b. 10
c. 11
d. 12
164. The graph of r=a+bcos θ is a :
a. Lemniscates
b. Limacon
c. Cardioids
d. Lituus
165. Solve for A in the equation: cos2A =
1- cos2A
a. 15˚, 125˚, 225˚, 335˚
b. 45˚, 125˚, 225˚, 315˚
c. 45˚, 135˚, 225˚, 315˚
d. 45˚, 150˚, 220˚, 315˚
166. Momentum is the product of velocity
and
a. Acceleration
b. Mass
c. Force
d. Time
167. If 15 people can win prices in a
estate lottery (assuming that there are
no ties). How many ways can these 15
people win first, second,, third, fourth
and fifth prizes?
a. 4,845
b. 116,280
c. 360,360
d. 3,003
168. Find the 30th term of the A.P 4, 7,
10,…
a. 75
b. 90
c. 88
d. 91
169. Mary is 24. She is twice as old as
Ann was when Mary was as old as
Ann now. How old is Ann now?
a. 16
b. 17
c. 12
d. 15
170. Find the ratio of an infinite
geometric series if the sum is 2 and
the first term is ½
a. 1/3
b. 1/2
c. 3/4
d. 1/4
171. Given a cone of diameter x and
altitude of h. What percent is the
volume of the largest cylinder which
can be inscribed in the cone to the
volume of the cone?
a. 44%
b. 46%
c. 56%
d. 65%
172. Find the equation of the curve at
every point of which, the tangent line
has a slope of 2x.
a. x
b. y=x2+c
c. y=x1/2+c
d. x=y2+c
173. csc 520˚ is equal to
a. cos 20˚
b. csc 20˚
c. tan 45˚
d. sin 20˚
174. A rotating wheel has a radius of 2 ft.
and 6 in. A point on the circumference
of the wheel moves 30 ft in 2 seconds.
Find the angular velocity of the wheel.
a. 2 rad/sec
b. 4 rad/sec
c. 6 rad/sec
d. 5 rad/sec
175. It is a series equal payments accruing
at equal intervals of the time where the
first payment is made several periods
after.
a. Deferred annuity
b. Delayed annuity
c. Progressive annuity
d. Simple annuity
176. Exact angle of the dodecagon equal
to ________ deg.
a. 135
b. 150
c. 125
d. 105
177. A load of 100 lb. is hung from the
middle of a rope, which is stretched
between wo rigid walls of 30 ft apart.
Due to the load, the rope sags 4 ft in
the middle. Determine the tension in
the rope.
a. 165 lbs
b. 173 lbs
c. 194 lbs
d. 149 lbs
178. How far does an automobile move
while its speed increases uniformly
from 15 kph to 45 kph in 20 seconds?
a. 185 mi
b. 167 mi
c. 200 mi
d. 172 mi
179. A block weighing 500 kN rest on a
ramp inclined at 25˚ with horizontal.
The force tending to move the block
down the ramp is:
a. 100 kN
b. 211 kN
c. 255 kN
d. 450 kN
180. What is the value of log25+log35?
a. 7.39
b. 3.79
c. 3.97
d. 9.37
181. The distance between the center of
the three circles which are mutually
tangent to each other externally are 10,
12 and 14 units. The area of the largest
circle is
a. 72 π
b. 23 π
c. 64 π
d. 16 π
182. To maximize the horizontal range of
the projectile, which of the following
applies?
a. Maximize velocity
b. Maximize the angle of
elevation and velocity
c. Maximize the angle of
elevation
d. The tangent function of the
angle of trajectory must be
equal to one
183. What is the lowest common factor of
10 and 32?
a. 320
b. 2
c. 180
d. 90
184. The distance that the top surface is
displaced in the direction of the force
divided by the thickness of the body is
known as __________
a. Longitudinal strain
b. Linear strain
c. Shear strain
d. Volume strain
185. It can be defined as the set of all
points on a plane whose sum of
distances of any of which from two
fixed points is constant.
a. Circle
b. Hyperbola
c. Parabola
d. Ellipse
186. A statue 3m high is standing on a
base of 4m high. If an observer’s eye
is 1.5m above the ground, how far
should he stand from the base in order
that the angle suspended bu the statue
is maximum.
a. 3.41 m
b. 3.51 m
c. 3.71 m
d. 4.41 m
187. A baseball is thrown from a
horizontal plane following a parabolic
path with an initial velocity of 100 m/s
at an angle of 30˚ above the
horizontal. How far from the throwing
point well the ball attains its original
level.
a. 882.2 m
b. 8.828 m
c. 288.8 m
d. 82.88 m
188. A balloon is rising vertically over a
point A on the ground a rate of 15
ft/sec. A point B on the ground is level
with and 30 ft from A. When the
balloon is 40 ft from A, at what rate is
its distance from B changing?
a. 13 ft/sec
b. 15 ft/sec
c. 12 ft/sec
d. 10 ft/sec
189. The diameter of a circle described by
9x2 + 9y2 = 16 is ______
a. 4/3
b. 16/9
c. 8/3
d. 4
190. A man finds the angle of elevation of
the top of a tower to be 30 degrees. He
walks 85 m nearer the tower and find
its angle of elevation to be 60 degrees.
What is the height of the tower?
a. 76.31 m
b. 73.31 m
c. 73.16 m
d. 73. 61 m
191. Two electrons have speeds of 0.7c
and x respectively at an angle of 60.82
degrees between each other. If their
relative velocity is 0.65c, find x.
a. 0.02c
b. 0.12c
c. 0.09c
d. 0.25c
192. Arc tan{2 cos(arcsin
to:
a. π/3
b. π/4
c. π/6
d. π/2
) )} is equal
193. Determine B such that 3x + 2y – 7 =
0 is perpendicular to 2x – By + 2 = 0
a. 5
b. 4
c. 3
d. 2
194. Find the point in the parabola y2 = 4
at which the rate of change of the
ordinate and abscissa are equal.
a. (1, 2)
b. (-1, 4)
c. (2, 1)
d. (4, 4)
195. Find the equation of the axis of
symmetry of the function y= 2x2-7x+5
a. 7x+4=0
b. 4x+7=0
c. 4x-7=0
d. 7x-4=0
196. The major axis of the elliptical path
in which the earth moves around the
sum is approximately 186, 000, 000
miles and the eccentricity of the
ellipse is 1/60. Determine the apogee
of the earth
a. 93 000 000 miles
b. 91 450 000 miles
c. 94 335 100 miles
d. 94 550 000 miles
197. The angle of inclination of ascends
of a road having 8.25% grade is _____
degrees.
a. 4.72˚
b. 4.27˚
c. 5.12˚
d. 1.86˚
198. Find the sum of the first term of the
geometric progression 2,4,8,16,…
a. 1 023
b. 2 046
c. 225
d. 1 596
199. Find the sum of the infinite
geometric progression 6, -2, 2/3
a. 9/2
b. 5/2
c. 11/2
d. 7/2
200. Evaluate
(
a. Undefined
b. 0
c. Infinity
d. 1/7
)
201. What is the speed of asynchronous
earth’ satellite situated 4.5x107 m
from the earth
a. 11 070.0 kph
b. 12 000.0 kph
c. 11 777.4 kph
d. 12 070.2 kph
202. A semiconductor company will hire
7 men and 4 women. In how many
ways can the company choose from 9
men and 6 women who qualified for
the position
a. 680
b. 540
c. 480
d. 840
c.
d.
90
in3
in3
30.4
205. Find the 100th term of the sequence,
1.01, 1.00, 0.99, ….
a. 0.05
b. 0.03
c. 0.04
d. 0.02
206. Find the coordinates of the point P(2,
4) with respect to the translated axis
with origin at (1, 3)
a. (1, -1)
b. (-1, -1)
c. (1, 1)
d. (-1, 1)
207. The roots of a quadratic equation are
1/3 and ¼. What is the equation?
a. 12x2+7x+1=0
b. 122-7x+1=0
c. 12x2+7x-1=0
d. 12x2-7x-1=0
208. Covert θ=π/3 to Cartesian equation
a. x=31/2 x
b. 3y=31/2x
c. y=x
d. y=31/2 x
203. The wheel of a car revolves n times
while the car travels x km. The radius
of the wheel in meter is:
a. 10 000x/π n
b. 500 00x/ π n
c. 500x/ π n
d. 5 000x/ π n
209. A piece of wire is shaped to enclose
a square whose area is 169 sq cm. It is
then reshaped to enclose a rectangle
whose length is 15 cm. The area of the
rectangle is:
a. 165 m2
b. 170 m2
c. 175 m2
d. 156 m2
204. The volume of a gas under standard
atmospheric pressure, 76 cm. Hg is
200 in3. What is the volume when the
pressure is 80 cm. Hg, if the
temperature is unchanged?
a. 190 in3
b. 110 in3
210. If (x+3) : 10=(3x-2): 8, find (2x-1).
a. 1
b. 4
c. 2
d. 3
211. In complex algebra, we use a
diagram to represent a complex plane
commonly called:
a. De Moivre’s diagram
b. Argand diagram
c. Funicular diagram
d. Venn diagram
212. The quartile deviation is a measure
of:
a. Division
b. Certainty
c. Central tendency
d. Dispersion
213. The velocity of an automobile
starting from rest is given by
ft/sec. determine its acceleration
after an interval of 10 sec. (in ft/sec2)
a. 2.10
b. 1.71
c. 2.25
d. 2.75
214. An automobile accelerates at a
constant rate of 15 mi/hr to 45 mi/hr in
15 seconds, while traveling in a
straight line. What is the average
acceleration?
a. 2 ft/sec
b. 2.12 ft/sec
c. 2.39 ft/sec
d. 2.93 ft/sec
215. A comfortable room temperature is
72˚F. What is the temperature,
expressed in degrees Kelvin?
a. 290
b. 263
c. 275
d. 295
216. 15% when compounded semiannually will have effective rate of:
a.
b.
c.
d.
15.93%
16.02%
18.78%
15%
217. A non-square rectangle is inscribed
in a square so that each vertex of the
rectangle is at the trisection point of
the different sides of the square. Find
the ratio of the area of the rectangle to
the area of the square.
a. 4:9
b. 2:7
c. 5:9
d. 7:72
218. If the radius of the circle is decreased
by 20%, by how much is its area
decreased?
a. 46%
b. 36%
c. 56%
d. 26%
219. A flowerpot falls off the edge of a
fifth-floor window, just as it passes the
third-floor window someone
accidentally drops a glass of water
from the window. Which of the
following is true?
a. The flowerpot and the glass
hit the ground at the same
instant
b. The flowerpot hits the ground
at the same time as the glass
c. The glass hits the ground
before the flowerpot
d. The flowerpot hits the
ground first with a higher
speed than the glass
220. Is sinA=2.571x, cosA=3.06x, and
sin2A=3.939, find the value of x.
a. 0.100
b. 0.150
c. 0.250
d. 0.350
221. How many terms of the sequence -9,
-6, -3 … must be taken so that the sum
is 66?
a. 12
b. 4
c. 11
d. 13
222. A man in a hot air balloon drops an
apple at a height of 50 meters. If the
balloon is rising at 15 m/s, find the
highest point reached by the apple.
a. 141.45 m
b. 171.55 m
c. 151.57 m
d. 161.47 m
223. If sin A=4/5 and A is in the second
quadrant, sin B= 7/25 and B is in the
first quadrant, find sin (A+B)
a. 3/5
b. 3/4
c. 2/5
d. 4/5
224. If cosθ=-15/17 and θ is in the third
quadrant, find cos θ/2.
a. -1/√
b. -8/√
c. 2/√
d. 3/√
225. What is the maximum moment of a
10 meter simply supported beam
subjected to a concentrated load of
500kN at the mid-span?
a. 1250 kN-m
b. 1520 kN-m
c. 1050 kN-m
d. 1510 kN-m
226. It represents the distance of a point
from the y-axis
a. Ordinate
b. Abscissa
c. Coordinate
d. Polar distance
227. The logarithm of a number to the
base e (2.7182818….0 is called
a. Characteristic
b. Mantissa
c. Briggsian logarithm
d. Napierian logarithm
228. Terms that a differ only in numeric
coefficients are known as:
a. Unequal terms
b. Like terms
c. Unlike terms
d. Equal terms
229. In Plain Geometry, two circular arcs
that together make up a full circle are
called:
a. Conjugate arcs
b. Co-terminal arcs
c. Half arcs
d. Congruent arcs
230. For a particular experiment you need
5 liters of a 10% solution. You find
7% and 12% solution on the shelves.
How much of the 7% solution should
you mix with the appropriate amount
of the 12% solution to get 4 liters of a
10% solution.
a. 1.43
b. 1.53
c. 1.63
d. 1.73
231. A mango falls from a branch 5
meters above the ground. With what
speed in meters per second does it
strike the ground? Assume g=10m/s2.
a.
b.
c.
d.
10 m/sec
14 m/sec
12 m/sec
8 m/sec
232. When two waves of the same
frequency speed and amplitude
traveling in opposite directions are
superimposed.
a. The phase difference is
always zero
b. Distractive waves are
produced
c. Standing waves are
produces
d. Constructive interference
always results
233. The work done by all the forces
except the gravitational force is
always equal to the _____of the
system
a. Total mechanical energy
b. Total potential energy
c. Total kinetic energy
d. Total momentum
234. Ten less than four times a certain
number is 14. Determine the number
a. 7
b. 5
c. 4
d. 6
235. Equal volumes of two different
liquids evaporate at different, but
constant rates. If the first is totally
evaporated in 6 weeks, and the second
in 7 weeks, when will be the second
be ½ the volume of the first.
a. 3.5 weeks
b. 4 weeks
c. 5/42 weeks
d. 42/5 weeks
236. Find the fourth term of the
progression ½ , 0.2, 0.125 …
a. 0.099
b. 1/11
c. 1/10
d. 0.102
237. The time required by an elevator to
lift a weight varies directly through
which it is to be lifted and inversely as
the power of the motor. If it takes 30
seconds for a 10 hp motor to lift 100
lbs through 50 feet. What size of
motor is required to lift 800 lbs in 40
seconds through a distance of 40 feet.
a. 58 hp
b. 48 hp
c. 50 hp
d. 56 hp
238. Find the dimensions of the right
circular cylinder of greatest volume that
can be inscribed in a right circular cone
of radius r and altitude h.
a. Radius=2/3r; altitude=2/3h
b. Radius=1/3r; altitude=1/3h
c. Radius=2/3r; altitude=1/3h
d. Radius=1/3r; altitude=2/3h
239. An angular unit equivalent to 1/400
of the circumference of a circle is
called:
a. Grad
b. Mil
c. Degree
d. Radian
240. A condition where only few
individuals produce a certain product
and that any action of one will lead to
almost the same action of the others.
a. Monopoly
b. Perfect competition
c. Semi-monopoly
d. Oligopoly
241. Ivory soaps floats in water because:
a. The specific gravity of ivory
soap is less than that of
water
b. The specific gravity of ivory
soap is greater than that of
water
c. The density of ivory soap is
unity
d. All matters has mass
242. On a certain test, the average passing
score is 72 while the average for entire
test is 62, what part of the group of
students passed the test?
a. 5/9
b. 6/11
c. 7/13
d. 4/7
243. Ghost images are formed in a TV set
when the signal from the TV
transmitter is received directly at the
TV set and also indirectly after
reflection from a building or other
large metallic mass. In a certain 25
inch TV set, the ghost is about 1 cm,
to the right of the principal image of
the reflected signal arrives 1
microsecond after the principal signal.
What is the difference in the path
length of the reflected and principal
signals in this case?
a. 100 meters
b. 300 meters
c. 200 meters
d. 400 meters
244. A stone is dropped into a well, and
the sound of the splash was heard
three seconds later. What was the
depth of the well?
a. 37 meters
b. 41 meters
c. 53 meters
d. 30 meters
245. Two thermometers, one calibrated in
Celsius and the other in Fahrenheit,
are used o measure the same
temperature, the numerical reading
obtained on the Fahrenheit
thermometer.
a. Is greater than that obtained
on the Celsius thermometer
b. Is less than that obtained on
the Celsius thermometer
c. May be greater or less than
that obtained on the Celsius
thermometer
d. Is proportional to that
obtained on the Celsius
thermometer
246. 1 atm of pressure is equal to
_______.
a. 101300 Pa
b. 14.7 bars
c. 1.013 psi
d. 2117 psi
247.
Find the least number of
years required to double a certain
amount of money at 5% per annum
compound interest to the nearest year
a. 14 years
b. 12 years
c. 18 years
d. 20 years
248. The replacement of the original cost
of an investment
a. Capital recovery
b. Breakeven
c. Payoff
d. Return on investment
249. When comparing leasing against
outright purchase of equipment, which
of the following is not correct?
a. Leasing frees needed
working capital
b. Leasing reduces maintenance
and administrative expenses
c. Leasing offers less flexibility
with respect to technical
obsolescence
d. Leasing offers certain tax
advantages
250. Find the volume of the solid above
the elliptic paraboloid 3x2+y2=z and
below the cylinder x2+z=4
a. 2π cubic units
b. π/4 cubic units
c. π cubic units
d. 4 π cubic units
251. An oil well that yields 300 barrels of
cure oil a month will run dry in 3
years. If is estimated that t months
from now, the price of crude oil will
be P(t)=18 + 0.3√ dollars per barrel.
If the oil is sold as soon as it is
extracted from the ground, what will
be the total future revenue from the oil
well?
a. $253,550
b. $207,612
c. $150,650
d. $190,324
252. A point on the graph of a
differentianble function where the
concavity changes is called a point of
______
a. Inflection
b. Mean value
c. Local minimum value
d. Deflection
253. Find the maximum and minimum
values of 3sinθ for 0˚
a. 3, 1/3
b. 1, 0
c. 2, -2
d. 1, -1
254. The spherical excess of a spherical
triangle is the amount by which the
sum of its angles exceed
a. 180˚
b. 90˚
c. 360˚
d. 270˚
255. the area of three adjacent surfaces of
a rectangular block are 8 sq cm, 10 sq
cm and 20 sq cm. the volume of the
rectangular block is
a. 200 cu m
b. 40 cu m
c. 10 cu m
d. 20 cu m
256. In the story about the crow who
wanted to drink water from a
cylindrical can but could not reach the
water, it is said that the crow dropped
a pebble which was a perfect sphere 3
cm in radius into the can. If the can
was 6 cm radius, what was the rise in
water level inside the can after that
pebble was dropped?
a. 2 cm
b. 1 cm
c. 3 cm
d. 2.5 cm
257. When a line y=mx+b slopes
downwards from left to right, the
slope m is
a. Less than 0
b. Greater than 0
c. Equal to 0
d. Equal to 1
258. A line perpendicular to a plane
a. Is perpendicular to only two
intersecting lines in the plane
b. Makes a right angle in the
plane which passes through
its foot
c. Is perpendicular to every line
is the plane
d. Makes a right angle with
every line is the plane
259. If the area of an equilateral triangle
is 9√ sq cm then its perimeter is
a. 9√ cm
b. 18 cm
c. 18√ cm
d. 12 cm
261. When a certain polynomial p(x) is
divided by (x-1), remainder is 12.
When the same polynomial is divided
by (x-4), the remainder is 3. Find the
remainder when the polynomial is
divided by (x-1)(x-4)
a. x+5
b. -2x-8
c. -3x+15
d. 4x-1
262. The scalar product of A and B is
equal to the product of the magnitudes
of A and B and the ______ of the
angle between them
a. Sine
b. Value in radians
c. Tangent
d. Cosine
263. If the surd (√
x is equal to:
, then
√
a.
√
b.
260. A transport company has been
contracted to transport a minimum of
600 factory workers from a gathering
point in Makati to their working place
in Canlubang daily. The transport
company has nine 5-passenger cars,
six 10-passenger mini buses and 12
drivers. The cars can make 14 trips a
day while the mini busses can make
10 trips a day. How should the
transport company use their cans and
mini buses in order to carry the
maximum number of passengers each
day?
a. 9 cars and 3 mini buses
b. 3 cars and 9 mini buses
c. 6 cars and 6 mini buses
d. 7 cars and 5 mini buses
√ )
c. √
d.
√
√
√
264. A certain electronics company has
16 tons of raw materials, of which 10
tons are stored in warehouse in
Quezon city, and 6 tons are stored in
warehouse in Makati. The raw
materials have to be transported to
three production points in Dasmarinas
Cavite, Canlubang Laguna and
Batangas city in the amounts of 5, 7
and 4 tons respectively, the cost per
ton for transporting the raw materials
from the two warehouses to the three
production points areas as follows
To/Fro
m
Damarin Canluba
as
ng
Batang
as
P 700
P500
P800
P 200
P300
P400
267. Arrange the following surds in
descending order: a=√
√ ,
b=3+√ , c=√
√ , d=√
√
a. c, d, a, b
b. b, a, d, c
c. c, d, b, a
d. d, c, a, b
Q.C
Makati
Find the minimum possible
transportation cost. HINT let a=no of
tons to be shopped from Q.C to
Dasmarinas, b=no of tons to be
shipped ftom Q.C to Canlubang, c=no
of tons to be shipped from Q.C to
Batangas, d= no of tons to be shopped
from Makati to Dasmarinas, e= no of
tons to be shopped from Makati to
Canlubanga and f= no of tons to be
shopped from Makati to Batangas.
a. 7 300.00
b. 8 300.00
c. 9 300.00
d. 10 300.00
268. If
the following relationship is correct?
a. x+z=y
b. x=y+z
c. x+y=z
d. x-y=z
269. evaluate u=
a.
b.
c.
d.
265. Which of the following is a correct
relationship for any triangle whose
sides are a, b, c and the respective
sides are a, b, c and the respective
opposite angles are A, B and C.
a. a2=b2+c2-bc cos A
b. a2=b2+c2-2bc cos A
c. a2=b2+c2-2bc sin A
d. a2=b2+c2-2bc cos B cos C
|
|
N=|
a. |
|
b. |
|
c. |
|
d. |
|
(
)
2
9
6
8
270. Evaluate: I= ∫ ∫
a. 88/3
b. 89
c. 3
d. 79/3
266. find the product MN of the following
matrices
M=|
, which of
271. The probability for the ECE board
examinees from a certain school to
pass the subject in mathematics is 3/7
and for the subject of Communication
is 5/7. If none of those examinees fail
both subjects and there are four
examinees who passed both subjects,
find the number of examinees from
that school who took the examinations
a. 21
b. 14
c. 28
d. 35
272. A number when divided by 6 leaves
a remainder of 5, when divided by 5
leaves a remainder of 4, by 4 leaves a
remainder of 3, by 3 leaves a
remainder of 2, and by 2 leaves a
remainder of 1. Find the smallest
possible value of the number.
a. 29
b. 39
c. 49
d. 59
273. _________ are irrational numbers
involving radical signs
a. Radicals
b. Surd
c. Irrational number
d. Transcendental number
274. When rounded off to two significant
figures, the number 4.371x10 -10
becomes ______
a. 4.4x 10-10
b. 4x10-10
c. 4.3x10-10
d. 4.2x10-10
275. The __________ of a and b is the
smallest positive integer that is a
multiple of both a and b.
a. Least common multiple
b. Least common denominator
c. Least common factor
d. Greatest common factor
276. If soldering lead contains 63% silver,
______ grams of soldering lead can be
made from 520 grams of silver.
a. 852.4
b. 825.4
c. 845.2
d. 842.5
277. In the equation ÿ=mx+b”, m
represents the _______
a. Distance from a point
b. Coordinate of the line
c. Coefficients
d. Slope of the line
278. In the equation “n x m=q”, the
multiplicand is _______
a. n
b. m
c. q
d. none of the choices
279. The hypotenuse of an isosceles right
triangle whose perimeter is 24 inches
is ____ inches.
a. 9.94 inches
b. 7.94 inches
c. 7.03 inches
d. 6.94 inches
280. An arc equal to one-fourth of a circle
is called a ____
a. Quarter circular arc
b. Quarter circle
c. Conjugate circle
d. Complimentary circle
281. If angle θ=2, then angle (180˚-θ)=
__________
a. 1.1416 radian
b. 1.1614 radian
c. 1.6141 radian
d. 1.4161 radian
282. The logarithm of a number to a
given base is called the ______
a. Exponent
b. Index
c. Base
d. Matrix
283. One is to fifty-two and one half as
three and one-third is to ______
a. 185
b. 175
c. 165
d. 155
284. Adjacent angles whose sum is 90
degrees are said to be _____
a. Complimentary
b. Supplementary
c. Explementary
d. Reflex angles
285. If x >y and y>z, then x _____z.
a. Less than
b. Greater than
c. Equal to
d. Less than or equal to
286. If any given triangle with sides a, b,
and c _______is equal to b(
)
a. sin A
b. sin B
c. b
d. a
287. if a>b and c>d, then (a+c) is
_______ of (b+d)
a. less than
b. greater than
c. equal to
d. less than or equal to
288. the following Fourier series equation
represents a periodic ____wave
i(x)= i + i cos x + i2 cos 2x+ i3 cos 3x
+…+i sin x + i2 sin 2x+ i3 sin 3x+…
a. cosine
b. tangent
c. cotangent
d. sine
289. a percentage is a fraction whose
denominator is ____
a. 1000
b. 100
c. 10
d. 10000
290. A swimming pool is constructed in
the shape of two partially overlapping
identical circle. Each of the circles has
a radius of 9 meters, and each circle
passes through the center of the other.
Find the area of the swimming pool.
a. 409.44 sq m
b. 309.44 sq m
c. 509.44 sq m
d. 209.44 sq m
291. The dartboard has nine numbered
blocks. Each block measuring 20cm x
20 cm. The number on each block is
the score earned when a dart hits that
block. A dart, which hits the
unnumbered portion of the dartboard,
gets a score of zero. Assuming all the
darts hit the dartboard and with two
darts, what is the probability of getting
a total score of 11?
a. 0.0128
b. 0.0328
c. 0.228
d. 0.0168
292. The dartboard has nine numbered
blocks. Each block measuring 20cm x
20 cm. The number on each block is
the score earned when a dart hits that
block. A dart, which hits the
unnumbered portion of the dartboard,
gets a score of zero. Assuming all the
darts hit the dartboard, what is the
probability of getting a score of zero
with one dart?
a. 0.64
b. 0.04
c. 0.44
d. 0.54
293. The dartboard has nine numbered
blocks. Each block measuring 20cm x
20 cm. The number on each block is
the score earned when a dart hits that
block. A dart, which hits the
unnumbered portion of the dartboard,
gets a score of zero. Assuming all the
darts hit the dartboard, what is the
probability of getting a score of seven
with one dart?
a. 0.04
b. 0.10
c. 0.07
d. 0.70
294. A rectangular metal sheet measures
22 ft long and 2R ft wide. From this
rectangular metal sheet, three identical
circles were cut, each circle measuring
R/3 ft. radius. If the area of the
remaining metal sheet is 66 sq ft, find
R.
a. 1.56 ft
b. 40.47 ft
c. 2.56 ft
d. 13.56 ft
295. If a and y are complimentary, find
the value of P if: P= cos (540˚+x)
sin(540˚+y) +cos(90˚+x)sin (90+y)
a. sin 2x
b. cos 2x
c. –cos 2x
d. –cos 2y
296. Given:
a.
b.
c.
d.
,
,
. Find a, n, and m.
2, 16, 4
16, 2, 4
4, 16, 2
2, 4, 16
297. Given: P= A sin t + B cos t, Q= A
cos t – B sin t. From the given
equations, derive another equation
showing the relationship between P,
Q, A, and B not involving any of the
trigonometric functions of angle t.
a. P2-Q2=A2+B2
b. P2+Q2=A2-B2
c. P2-Q2=A2-B2
d. P2+Q2=A2+B2
298. In a certain electronic factory, the
ratio of the number of male to female
workers is 2:3. If 100 new female
workers are hired, the number of
female workers will increase to 65%
of the total number of workers. Find
the original number of workers in the
factory.
a. 420
b. 450
c. 480
d. 490
299. During installation, a section of an
antenna was lifted to a height of 5
meters with a force of 400 kg moving
by the use of a pulley mounted on a
frame. If the efficiency of the input
multiplied by 100%, what is the
efficiency of the pulley? The tower
section weighs 1000 kg
a. 62.5%
b. 52.5%
c. 72.5%
d. 82.5%
300. An elevator can lift a load of 5000
Newtons from ground level to a height
of 20.0 meters in 10 seconds. What
horsepower, hp can the elevator
develop?
a. 12.4 hp
b. 13.4 hp
c. 14.4 hp
d. 15.4 hp
301. What is the force in Newtons,
required to move a car with 1000 kg
mass with an acceleration of 12.0
meters/sec2?
a. 12 000N
b. 10 000N
c. 8 000N
d. 6 000N
302. If the same car in problem 301, with
1000 kg mass is driven around a curve
with radius of 10.0 meters at a speed
of 20 meters per second, find the
centrifugal force in Newtons.
a. 40000N
b. 30000N
c. 20000N
d. 10000N
303. Crew 1 can finish the installation of
an antenna tower in 200 hours while
crew 2 can finish the same job in 300
hours. How long will it take both
crews to finish the same job working
together?
a. 180 hours
b. 160 hours
c. 140 hours
d. 120 hours
2
304. Evaluate the limit of x +3x-4 as x
approaches the value of 4
a. 24
b. 42
c. 35
d. 12
305. log Mn is equal to
a. log nM
b. log Mn
c. n log M
d. M log n
306. The volume of a cube is reduces to
______ if all the sides are halved
a. 1/2
b. 1/4
c. 1/8
d. 1/16
307. Evaluate the value of the determinant
|
|
a.
b.
c.
d.
-101
011
-001
111
308. Give the factors of a2-x2
a. 2a-2x
b. (a+x)(a-x)
c. 2x-2a
d. (a+x)(x-a)
309. Give the area of a triangle in square
meters when the base is equal to
24.6cm and the height is equal to 50.8
cm. One of the sides is equal to 56.53
cm
a. 0.062484
b. 0.1252
c. 2877.44
d. 1252.1
310. The cost of running an electronic
shop is made up of the following:
Office rental=40% Labor=35%
Materials=20% Miscellaneous=5%. If
the office rental is increased by 24%,
labor increased by 15%, cost of
materials increased by 20%, and the
miscellaneous costs are unchanged,
find the percentage increase in the cost
of running the shop.
a. 18.85%
b. 28.85%
c. 16.85%
d. 10.85%
311. The selling price of a TV set is
double that of its net cost. If the TV
set is sold to a customer at a profit of
255 of the net cost, how much
discount was given to the customer?
a. 27.5%
b. 47.5%
c. 37.5%
d. 30.5%
312. Find the sum of the interior angles of
a pentagram
a. 180 degrees
b. 360 degrees
c. 540 degrees
d. 720 degrees
313. Find the value of P if it I equal to
sin2 1˚ + sin22˚ + sin23˚ + .. + sin2 90˚
a. Infinity
b. 0
c. 44.5
d. Indeterminate
314. Find the value of P if it is equal to
a.
b.
c.
d.
0
1
2
4
a.
b.
c.
d.
0.3
0.4
0.5
0.6
316. Find the value of
a. 4
317. Find the value of
√
√
a.
b.
c.
d.
√
3/2
2
3
1/2
318. Find the value of
a.
b.
c.
d.
( )
25/48
125/48
125/16
125/8
319. Find the value of
a. 2
b. 4
c. 8
d. 16
320. Simplify ( )
a. 2
b. 4
c. 8
d. 16
321.
√
315.
b. 2
c. 0
d. 1
=?
=?
a.
b.
c.
d.
tan B
sec B
cot B
csc B
322. Simplify the following:
a.
b.
c.
d.
0
1
2
cot (A+B)
323. Solve for the following:
d. 3.101 to 3.104
-7a
+7a
-7-a
+7-a
327. Round off: 6785768.342 to the
nearest one tenth
a. 6785768.34
b. 6785768.3
c. 7000000.0
d. 6785770.00
a.
b.
c.
d.
324. Simplify {
*
+}
a.
b.
328. Round off: 2.371x10-8 to two
significant figures
a. 2.3x10-8
b. 2.4x10-8
c. 2.0x10-8
d. 2.5x10-8
c.
329. Round off: 0.003086 to two
significant figures
a. 0.00308
b. 0.00310
c. 0.00300
d. 0.00311
d.
325. Simplify
(
)
(
)
a.
b.
c.
d.
326. If A was originally a range of
numbers with four significant figures
which, when rounded off to three
significant figures yielded a value of
3.10, what was the original range of
values of A?
a. 3.10 to 3.105
b. 3.101 to 3.105
c. 3.101 to 3.109
330. Round off: 0.00386 to three
significant figures
a. 0.00308
b. 0.00309
c. 0.003
d. 0.00310
331. Round off: 34.2814 to four
significant figures
a. 34.2814
b. 34.2800
c. 35.0000
d. 34.2000
332. Round off: 30 562 to three
significant figures
a. 30 500
b. 30 600
c. 30 400
d. 30 300
333. Round off: 149.691 to one decimal
place
a. 149.6
b. 149.7
c. 148.5
d. 148.4
334. Round off: 149.691 to the nearest
integer
a. 149
b. 148
c. 147
d. 150
d. 77.46 meters
339. The speed of light is closest to:
a. 30x108 m/sec
b. 300x108 m/sec
c. 3000x108 m/sec
d. 3x108 m/sec
335. Round off: 149.691 to two decimal
places
a. 149.69
b. 149.70
c. 148.69
d. 148.70
340. When a ray of light is incident from
a medium, such as air, to a denser
medium, like water, the refracted ray
lie _____ to the perpendicular than
does the incident ray.
a. Closer
b. Farther
c. Parallel
d. Perpendicular
336. Which of the following is equivalent
to the expression:
a. sin
b. cos
c. sec
d. csc
341. In nuclear energy, the splitting apart
of the heavy nuclei of uranium is
called
a. Fusion
b. Fission
c. Neutron
d. Diffusion
337. A stone is thrown outward, at an
angle of 30 with the horizontal, into
the river from a cliff, which is 120
meters above the water level at a
velocity of 36 km/hr. At what height
above the water level will the stone
start to fall?
a. 121.274 m
b. 131.274 m
c. 141.274 m
d. 161.274 m
342. A parabola which opens upward and
whose vertex is at the origin is defined
by what equation?
a.
b.
c.
d.
338. A stone is thrown outward, at an
angle of 30 with the horizontal, into
the river from a cliff, which is 120
meters above the water level at a
velocity of 36 km/hr. how far from the
cliff will the stone strike the water?
a. 57.46 meters
b. 47.46 meters
c. 67.46 meters
343. The curve traced by a point moving
in a plane is shown as the _____ of
that point.
a. Parameter
b. Pattern
c. Locus
d. Formula
344. (a-b)3 is equivalent to which of the
following?
a.
b.
d. cos(A-B)
c.
d.
345. Payment for the use of borrowed
money is called
a. Loan
b. Maturity value
c. Interest
d. Rate
346. Area of a triangle is given by the
formula
a. 1/2bh
b. bh
c. 1/4bh
d. 3/4bh
347. Evaluate ∫
a.
b.
c.
d.
dx
37.6
47.6
27.6
57.6
348. In the Cartesian coordinate, the
coordinates if the vertices of a square
are (1, 1), (0, 8), (4, 5), and (-3, 4).
What is the area of the square?
a. 25 sq units
b. 16 sq units
c. 32 sq units
d. 50 sq units
349. Given log2=0.30 and log3=0,477.
Find the value of log 48
a. 1.681
b. 1.683
c. 1.685
d. 1.687
350. sinAcosB + sinBcosA= ?
a. sin(A+B)
b. sin(A-B)
c. cos(A+B)
351. sinh2 x+tanh2 x= ?
a. cosh2x-sech2x
b. cosh2x+sech2 x
c. sech2x-cosh2x
d. sech2x+cosh2 x
352. If the freezing point of water is zero
deg Celsius or 32 Fahrenheit, and its
boiling point is 100 deg Celsius or 212
Fahrenheit, which relationship is
correct?
a. F=9/5C+32
b. F=5/9C+32
c. C=9/5F+32
d. C=5/9F+32
353. What is the probability of obtaining
either four or five heads if a fair coin
is tossed 10 times?
a. 231/512
b. 233/512
c. 221/512
d. 235/512
354. Find the volume generated by
revolving the ellipse whose equation is
a.
b.
c.
d.
about the x-axis
4/3πab2
2/3 πab2
4/3 πba2
2/3 πa2b
355. A telephone pole 3ft high is to be
guyed from its middle section with a
guy wire making an angle of 45
degrees with the ground. Find the total
length of the guy wire if an additional
three feet is to be provided for
splicing. Solve by using trigonometric
functions.
a. 24.21 ft
b. 34.21 ft
c. 44.21 ft
d. 25.21 ft
a.
356. A rubber ball is made to fall from a
height of 50 feet and is observed to
rebound 2/3 of the distance it falls.
How far will the ball travel before
coming to rest if the ball continues to
fall in this manner?
a. 200 m
b. 225 m
c. 250 m
d. 300 m
357. The slope of a family of curves at
any point (x, y) is equal to 3x4-x2.
Find the equation of the curve that is
passing through point (1, 1).
a.
(
)
( )
b.
(
)
( )
c.
(
)
( )
d.
(
)
( )
358. The slope of a family of curves at
any point (x, y) is equal to (x+1)(x+2).
Find the equation of the curve that is
passing through the point (-3, -3/2)
a.
b.
c.
d.
359. Reduce the following complex
fraction into simple functions
b.
c.
d.
360. Reduce the following complex
fraction into simple fractions
a. –
b. +
c. –
d. +
361. A missile with a mass of 2200
kilograms was fired the rocket burns
for a short period of time causing a
constant force of 100 000 N to be
exerted on the missile for 10 seconds.
After the 10 second period, what is the
final velocity, v in m/sec of the
missile?
a. 365.45 m/sec
b. 352.45 m/sec
c. 356.45 m/sec
d. 256.45 m/sec
362. A missile with a mass of 2200
kilograms was fired the rocket burns
for a short period of time causing a
constant force of 100 000 N to be
exerted on the missile for 10 seconds.
After the 10 second period, what is the
acceleration of the missile in m/s2 ?
a. 35.64
b. 33.64
c. 30.64
d. 39.64
363. A consortium of international
telecommunication companies
contracted for the purchase and
installation of a fiber optic cable
linking two major Asian cities at a
total cost of US$ 960M. This amount
includes freight and installation
charges that are estimated at 10% of
the above total price, if the cable shall
be depreciated over a period of 15
years with zero salvage value, what is
the depreciation charge during the 8th
year using the sum of the year’s digit
method?
a. $64 M
b. $74 M
c. $84 M
d. $54 M
364. A consortium of international
telecommunication companies
contracted for the purchase and
installation of a fiber optic cable
linking two major Asian cities at a
total cost of US$ 960M. This amount
includes freight and installation
charges that are estimated at 10% of
the above total price, if the cable shall
be depreciated over a period of 15
years with zero salvage value. Given
the sinking fund deposit factor of
0.0430 at 6% interest where n=15,
what is the annual depreciation
charge?
a. $43.28M
b. $42.28M
c. $44.28M
d. $41.28M
365. Find the derivative of y with respect
to x in the following equations
a.
b.
(
)
c.
d.
366. Find the value of y’ at x=1 of the
equation
a. 21
b. -21
c. 12
d. -12
367. An equipment can be purchased by
paying P100 000 down payment and
24 equal monthly installments of P10
000 with 6% interest compounded
monthly? Find the cash value of the
equipment given the following:
present value of an annuity where
n=24 at 0.5% interest, PV
factor=22.563
a. P235630
b. P352630
c. P325630
d. P253630
368. Simplify the following expression:
a.
b.
c.
d.
369. Solve for the values of a in the
equation a8-17a4+16=0
a.
b.
c.
d. All of the choices
370. Log(MN) is equal to
a. logM-N
b. log M+N
c. nlogM
signal. The signal is received at station
B, from where it is retransmitted to
station C. The probability that the
signal being sent from A is receives
correctly at B is 0.98, while the
probability that the signal being
received correctly at C is 0.965. What
is the probability that when a dot
signal is transmitted from A, a dot
signal is also received at C?(Express
your answer up o four decimal places)
a. 0.9557
b. 0.9457
c. 0.4957
d. 0.5947
d. logM+logN
e. NMlog10
371. Snell’s law on light incidence and
refraction gives us the following
equation: n1sinθ1=n2sinθ2 where n1
and n2 denote the indexes on
refraction θ1 and θ2 are the angle of
incidence and refraction, respectively
through the first and second medium.
If light beamed at an angle of 30
degrees with the vertical is made pass
from air to a transparent glass with an
index of refraction equal to 1.25, what
is the angle of refraction in the glass?
a. θ=33.6˚
b. θ=43.6˚
c. θ=53.6˚
d. θ=23.6˚
376. In the figure shown, ABCD is a
square and BEC is an equilateral
triangle. Find angle AED.
a. 75˚
b. 150˚
c. 120˚
d. 140˚
D
, y’=?
372. If
a.
b.
c. -
A
eeeee
d.
373. Sin215˚+sin275˚
a. 1
b. 2
c. 3
d. 4
374. In the ECE board examinations, the
probability that an examinee pass in
each subject is 0.8. What is the
probability that he will pass in at least
2 subjects?
a. 0.896
b. 0.986
c. 0.689
d. 0.869
375. A Morse code transmitter at station
A sending out either a dot or dash
B
B
C
377. Solve for the radius of the circle
shown. Large circle r=4m, small circle
r=radius=?
4-r
45˚
4+r
a.
b.
c.
d.
378.
0.686 m
0.688 m
0.866 m
0.868 m
Differentiate the equation
a.
b.
c.
d. 1
379.
Give the slope of the curve at
point (1, 1)
a. 1/4
b. -1/4
c. 4
d. -1/3
380.
Evaluate b in the following
equation logb 1024=5/2
a. 2560
b. 2
c. 4
d. 16
381.
Obtain the differential equation
of the family of straight lines with slope
and -intercept equal.
a.
b.
c.
d.
382.
Obtain the differential equation
of all straight lines with algebraic sum
of the intercepts fixed as .
a.
b.
c.
d.
383.
Obtain the differential equation
of all straight lines at a fixed distance
from the origin.
[
]
a.
[
]
b.
[
]
c. .
[
]
d.
384.
Determine the differential
equation of the family of lines passing
through the origin.
a.
b.
c.
d.
385.
Obtain the differential equation
of all circles with center on line
and passing through the origin.
a.
b.
c.
d.
(
(
)
)
386.
Obtain the differential equation
of all parabolas with axis parallel to the
-axis.
a.
b.
c.
d.
387.
What is the differential
equation of the family of parabolas
having their vertices at the origin and
their foci on the -axis.
a.
b.
c.
d.
Obtain the particular solution of
when
,
.
388.
/
a.
a.
b.
c.
d.
b.
c.
d.
389.
Obtain the general solution of
the differential equation
.
a.
b.
c.
d.
390.
Solve the equation
395.
a.
b.
c.
d.
Obtain the general solution of
.
(
)
a.
b.
c.
d.
Solve
396.
.
a.
b.
c.
Solve the equation
.
391.
d.
a.
b.
c.
d.
Solve the equation
.
397.
a.
b.
c.
d.
398.
a.
b.
c.
d.
Obtain the particular solution of
; when
,
.
392.
a.
b.
c.
d.
Solve the equation
.
393.
394.
Solve the equation
399.
a.
b.
c.
d.
.
a.
b.
c.
d.
Solve the equation
.
Solve the equation
.
| |
|
|
| |
|
|
Solve the equation
.
400.
a.
b.
c.
d.
<MATHEMATICS>
<DIEGO INOCENCIO TAPANG
GILLESANIA>
ENCODED BY: BORBON, MARK
ADRIAN C.
MULTIPLE CHOICE QUESTIONS IN
401.
Evaluate
A. 0
C. e
B. 1
D. infinity
406.
C. 2
402.
B. 0
.
D. 3
A. 1
Simplify the expression:
B. indefinite
.
.
C. 0
A. 1
D. 2
B. 8
407.
C. 0
Evaluate:
.
A. 0
D. 16
403.
Evaluate the limit:
B. ½
Evaluate the following limit,
C. 2
.
D. -1/2
A. 2/5
408.
Evaluate the following:
B. infinity
.
C. 0
A. infinity
D. 5/2
404.
B.
Evaluate the limit
.
/(
C. 0
D.
A. 0
409.
B. undefined
A.
C. 1/7
B.
D. infinity
405. Evaluate the limit
approaches positive infinity.
A. 1
Find
/
as x
C.
D.
/
if
.
410.
Find
/
if
√
D.
.
/
A.
√
B.
√
411.
414. If is a simple constant, what is the
derivative of
?
/√
/ √
C.
D.
/ √
√
Find
.
A.
B.
√
/
if
C.
and
D.
A.
415.
B.
Find the derivative of the function
with respect to x.
C.
A.
D.
B.
C.
412. Evaluate the first derivative of the
implicit function:
.
A.
D.
416. What is the first derivative
the expression
?
B. -
A. -
C.
C. -
/
/
D. /
/
417.
Find the derivative of
A.
A.
B.
B.
/
/
C.
C.
/
of
B. 0
D. 413. Find the derivative of
with respect to x.
/
D.
/ .
418. Given the equation:
find .
B.
,
C.
A.
D.
B.
/
423.
C.
, what is
/
?
A.
D.
B. -
419. Find the derivatives with respect to x
of the function √
.
A. -
C.
D. -
/√
424.
B. -
If
Find
/
:
.
/√
A.
C. -
/√
B.
D. -
/√
C.
420. Differentiate
power.
to the ½
D.
425.
A. -
The derivative of
C. -
D.
/
if
D.
√ .
426. A function is given below, what x
value maximizes ?
A. √ /
B. x/
C. 1/2x
A. 2.23
D. 2/x
422.
Evaluate the differential of
A.
is:
B. -
C.
Find
/
A.
B.
421.
/x
B. -1
.
C. 5
D. 1
427.
The number of newspaper copies
distributed is given by
, where is in years.
Find the minimum number of copies
distributed from 1995 to 2002.
430.
. Find
A. 0
B. -1
A. 9850
C. 1
B. 9800
D. 2
C. 10200
431.
D. 7500
428.
If
to the 3rd power the maximum value of .
Given the following profit-versusproduction function for a certain
commodity:
Divide 120 into two parts so that the
product of one and the square of the
other is maximum. Find the
numbers.
A. 60 & 60
B. 100 & 120
(
)
Where P is the profit and x is the unit
of production. Determine the
maximum profit.
429.
C. 70 & 50
D. 80 & 40
A. 190000
If the sum of two numbers is , find
the minimum value of the sum of
their squares.
B. 200000
A.
⁄
C. 250000
B.
⁄
D. 550000
C.
⁄
The cost C of a product is a function
of the quantity of the product given
by the relation:
. Find the quantity for
which the cost is a minimum.
D.
⁄
A. 3000
B. 2000
432.
433.
A certain travel agency offered a tour
that will cost each person P 1500.00
if not more than 150 persons will
join, however the cost per person
will be reduced by P 5.00 per person
in excess of 150. How many persons
will make the profit a maximum?
C. 1000
A. 75
D. 1500
B. 150
434.
C. 225
C. 5.127
D. 250
D. 6.445
Two cities and are 8 km and 12
km, respectively, north of a river
which runs due east. City being 15
km east of . A pumping station is to
be constructed (along the river) to
supply water for the two cities.
Where should the station be located
so that the amount of pipe is a
minimum?
437.
A. 3.41 m
B. 3.51 m
A. 3 km east of
C. 3.71 m
B. 4 km east of
D. 4.41 m
C. 9 km east of
438.
D. 6 km east of
435.
A boatman is at , which is 4.5 km
from the nearest point on a straight
shore
. He wishes to reach, in
minimum time, a point situated on
the shore 9 km from . How far
from should he land if he can row
at the rate of 6 kph and walk at the
rate of 7.5 kph?
An iron bar 20 m long is bent to
form a closed plane area. What is the
largest area possible?
A. 21.56 square meter
B. 25.68 square meter
C. 28.56 square meter
D. 31.83 square meter
C. 5 km
A Norman window is in the shape of
a rectangle surmounted by a semicircle. What is the ratio of the width
of the rectangle to the total height so
that it will yield a window admitting
the most light for a given perimeter?
D. 8 km
A. 1
The shortest distance from the point
(5,10) to the curve
is:
B. 2/3
A. 1 km
B. 3 km
436.
A statue 3 m high is standing on a
base 4 m high. If an observer’s eye is
1.5 m above the ground, how far
should he stand from the base in
order that the angle subtended by the
statue is a maximum?
A. 4.331
B. 3.474
439.
C. 1/3
D. ½
440.
A rectangular field is to be fenced
into four equal parts. What is the size
of the largest field that can be fenced
this way with a fencing length of
1500 feet if the division is to be
parallel to one side?
capacity of 16823cc. Find the height
of the box to use the least amount of
material.
A. 65,200
C. 18.41 cm
B. 62,500
D. 28.74 cm
C. 64,500
A. 16.14 cm
B. 32.28 cm
444.
D. 63,500
441.
Three sides of a trapezoid are each 8
cm long. How long is the 4th side,
when the area of the trapezoid has
the greatest value?
A.
⁄
A. 16 cm
C.
⁄
B. 15 cm
D. ⁄
445.
D. 10 cm
An open top rectangular tank with
square bases is to have a volume of
10 cubic meters. The material for its
bottom cost P150.00 per square
meter, and that for the sides is
P60.00 per square meter. The most
economical height is:
A. 2 meters
What is the least amount of tin in
sheet, in sq. inches, that can be made
into a closed cylindrical can having a
volume of 108 cu. inches?
A. 125 square meter
B. 137 square meter
C. 150 square meter
D. 120 square meter
C. 3 meters
The volume of the closed cylindrical
tank is 11.3 cubic meter. If the total
surface area is a minimum, what is
its base radius, in m?
D. 3.5 meters
A. 1.44
A rectangular box having a square
base and open top is to have a
B. 1.88
B. 2.5 meters
443.
⁄
B.
C. 12 cm
442.
The altitude of a cylinder of
maximum volume that can be
inscribed in a right circular cone of
radius and height is:
446.
447.
C. 1.22
C. 18.56 m
D. 1.66
D. 17.89 m
A cylindrical steam boiler is to be
constructed having a capacity of
1000 cu. m. The material for the
sides cost P 2000.00 per square
meter and for the ends P 3000.00 per
square meter. Find the radius so that
the cost is least.
As increases uniformly at the rate
of 0.002 feet per second, at what rate
is the expression (1+ ) to the 3rd
power increasing when becomes 8
feet?
A. 430 cfs
A. 3.52 m
B. 0.300 cfs
B. 4.12 m
C. 0.486 cfs
C. 4.73 m
D. 0.346 cfs
D. 5.25 m
448.
450.
451.
A box is to be constructed from a
piece of zinc 20 inches square by
cutting equal squares from each
corner and turning up the zinc to
form the side. What is the volume of
the largest box that can be so
constructed?
Integrate:
A.
B.
C.
D.
A. 599.95 cubic inches
B. 579.50 cubic inches
452.
A.
C. 592.59 cubic inches
449.
Evaluate ∫
D. 622.49 cubic inches
B.
A load of 40kN is to be raised by
means of a lever weighing 250N/m,
which is supported at one end. If the
load is placed 1 m from the support,
how long should the lever be so that
the force required be a minimum?
C.
A. 13.43 m
B. 20.19 m
D.
453.
Evaluate the integral of
A.
B.
.
458.
C.
454.
D.
A.
What is the integral of
?
B.
D.
B.
459.
C.
with respect to
C. ½
460.
Evaluate ∫
C.
A. ½
D. -
B.
Integrate
D. arctan
461.
B.
Evaluate ∫
C. ⁄
A. arcsec
D. ⁄
B. [
.
C.
C. ½
D.
.
√
]
√
D. arcsin
A.
B.
.
C. ½
.
A. ⁄
Evaluate ∫
.
D. ½
B.
457.
Evaluate ∫
B.
A.
456.
√
A.
D. The integral of
is:
.
C.
A. -
455.
;∫
Evaluate ∫
462.
Evaluate ∫
A.
B.
.
467.
C.
Evaluate ∫
A.
D.
.
√
B.
463.
Evaluate ∫
.
C.
A. ½
D.
B.
464.
C.
Integrate the square root of
.
D.
A. √
Evaluate ∫
468.
B. - √
.
A.
C. -
B.
D. - √
469. Evaluate the integral of
with limits from 0 to
.
C.
D.
465.
A. 0.143
Evaluate the integral of
.
B. 0.258
A. -
C. 0.114
D. 0.186
B. 470.
C.
D. 466.
√
Evaluate ∫
Evaluate the integral of
with limits from 5 to 6.
A. 81/182
B. 82/182
.
C. 83/182
A.
D. 84/182
B. C. -
471.
Evaluate the integral of
if it
D.
has an upper limit of 1 and a lower limit of
0.
A. 0.022
B. 0.3068
B. 0.056
C. 0.6107
C. 0.043
D. 0.4105
D. 0.031
472.
Find the integral of
if lower limit = 0 and
upper limit =
.
476.
and
Find the area under the curve
and the x-axis between
.
A. 28 sq. units
A. 0.2
B. 46 sq. units
B. 0.8
C. 36 sq. units
C. 0.6
D. 54 sq. units
D. 0.4
473. Using lower limit = 0 and upper limit
=
, what is the integral of
?
477.
Find the area bounded by
, the lines
and
and the X-axis.
A. 19.456 sq. units
A. 6.783
B. 20.567 sq. units
B. 6.857
C. 22.567 sq. units
C. 6.648
D. 21.478 sq. units
D. 6.539
474.
Evaluate the integral of
using lower limit of 0 and
upper limit =
.
478.
Find the area of the region bounded
by the curves
and
A.
B. 1.7
B.
C. 1.4
C.
D. 2.3
D.
Evaluate the integral of
using lower limit = 0 and
upper limit =
.
A. 0.5046
479.
and
, the -axis,
,
.
A. 2.0
475.
,
Find the area bounded by the -axis
.
A. 25.6
B. 28.1
D.
C. 12.8
484.
D. 56.2
480. Find the area of the region bounded
by one loop of the curve
.
A.
481.
A. 62 sq. units
sq. units
B.
sq. units
C.
sq. units
D.
sq. units
Find the curved surface (area) of the
solid generated by revolving the part
of the curve
from
to
about the -axis.
√
B. 62 /3 sq. units
C. 62 /5 sq. units
D. 5/62 sq. units
485.
Find the area bounded by the curve
Find the volume generated by
rotating the region bounded by
,
, and
, about
the -axis.
A.
A.
B.
B.
C.
C.
D.
D.
482.
What is the area within the curve
?
486.
B. 28
The area bounded by the curve
and the line
is
revolved about the line
. What
is the volume generated?
C. 30
A. 186
A. 26
B. 179
D. 32
483.
C. 181
Find the area enclosed by
D. 184
A.
B.
C.
487.
Given is the area in the first quadrant
bounded by
, the line
and the -axis. What is the volume
generated by revolving this area
about the y-axis?
A. 50.26
491.
B. 52.26
C. 53.26
The area in the first quardrant,
bounded by the curve
, the
-axis and the line
is
revolved about the line
. Find
the centroid of the solid formed.
D. 51.26
A. (2.2,6)
488.
Given is the area in the first quadrant
bounded by
, the line
and the -axis. What is
the volume generated when this area
is resolved about the line
?
A. 28.41
B. (1.6,6)
C. (1.8,6)
D. (2.0,6)
492.
C. 27.32
A solid is formed by revolving about
the -axis, the area bounded by the
curve
, the -axis, and the
line
. Find its centroid.
D. 25.83
A. (0,9.6)
Find the length of the arc of
from
- to
- , in the
second quadrant.
B. (0,12.4)
B. 26.81
489.
C. (0,8.3)
D. (0,12.8)
A. 2.24
B. 2.61
493.
C. 2.75
A solid is formed by revolving about
the -axis, the area bounded by the
curve
, the -axis, and the
line
. Find its centroid.
D. 2.07
A. (0,4.75)
490.
How far from the -axis is the
centroid of the area bounded by the
curve
, the line
, and the
-axis.
B. (0,4.5)
C. (0,5.25)
D. (0,5)
A. 1.2
B. 1.4
C. 1.6
494.
Find the moment of inertia of the
area bounded by the parabola
, -axis and the line
,
with respect to the -axis.
D. 1.8
A. 1.067
B. 1.244
C. 54,448
ft-lb
C. 0.968
D. 56,305
ft-lb
D. 0.878
495.
498.
Find the work done in stretching a
spring of natural length 8 cm from
10 cm to 13 cm. Assume a force of 6
N is needed to hold it at a length of
11 cm.
A 60-m cable that weighs 4 kg/m has
a 500-kg weight attached at the end.
How much work is done in winding
up the last 20m of the cable?
A. 9,866 kg-m
B. 10,800 kg-m
A. 21 N-m
C. 12,500 kg-m
B. 2.1 N-m
D. 15,456 kg-m
C. 0.21 N-m
499.
D. 0.021 N-m
496.
A conical tank that is 5 meters high
has a radius of 2 meters, and is filled
with a liquid that weighs 800 kg per
cubic meter. How much work is
done in discharging all the liquid at a
point 3 meters above the top of the
tank?
A. 21,256
kg-m
B. 21,896
kg-m
C. 23,457
kg-m
D. 22,667
kg-m
A uniform chain that weighs 0.50 kg
per meter has a leaky 15-liter bucket
attached to it. If the bucket is full of
liquid when 30 meters of chain is out
and half-full when no chain is out,
how much work is done in winding
the chain? Assume that the liquid
leaks out at a uniform rate and
weighs 1 kg per liter.
A. 356.2 kg-m
B. 458.2 kg-m
C. 562.5 kg-m
D. 689.3 kg-m
500.
497.
How much work is required to pump
all the water from a right circular
cylindrical tank, that is 8 feet in
diameter and 9 feet tall, if it is
emptied at a point 1 foot above the
top of the tank?
A. 49,421
ft-lb
B. 52,316
ft-lb
The velocity of a body is given by
, where the velocity
is given in meters per second and is
given in seconds. The distance
covered in meters between
and
second is close to:
A. 2
B. -5
D. a fuzzy set
C. 5
D. -2
501.
505.
If equals are added to equals, the
sum is equal.
Which of the following is not a
property of probability:
A. If events
and
are mutually
exclusive, then the probability that
both events can happen is zero.
A. theorem
B. postulate
B. The probability that an event can
happen is always positive and is less
than one or equal to one.
C. axiom
D. corollary
C. If
is an event which cannot
occur in the sample space, the
probability of
is zero.
502. Any number multiplied by ________
equally unity.
A. infinity
D. If events
&
exclusive, then
B. itself
C. its reciprocal
506.
D. zero
503.
If every element of a column (or
row) of a square matrix is multiplied
by m, the determinant of the matrix
will be:
B. obtuse angle
C. reflex angle
D. acute angle
B. multiplied by m
507.
D. none of these
504.
B. tangent
C. sector
A. a sample space
C. a set of random variables
A line segment joining two point in a
circle is called:
A. arc
In probability theory, the set of
possible outcomes of an experiment
is termed as:
B. a set of random events
An angle greater that a straight angle
and less than two straight angles is
called:
A. right angle
A. unchanged
C. it depends
are mutually
D. chord
508.
All circles having the same center
but with unequal radii are called:
A. encircle
C. pentedecagon
B. tangent circles
D. nonagon
C. concyclic
513.
D. concentric circles
509.
A. rhombus
A triangle having three sides equal is
called:
B. trapezoid
A. equilateral triangle
C. square
B. scalene triangle
D. parallelogram
C. isosceles triangle
510.
514.
In a regular polygon, the
perpendicular line drawn from the
center of the inscribed circle to any
of the sides is called:
B. altitude
C. apothem
D. perimeter
B. altitude
C. median
515.
D. apothem
A line that meets a plane but not
perpendicular to it, in relation to the
plane, is:
A quadrilateral with two and only
two sides of which are parallel, is
called:
A. parallel
A. parallelogram
C. coplanar
B. trapezoid
D. oblique
C. quadrilateral
D. rhombus
512.
The sum of the sides of a polygon is
termed as:
A. circumference
A. radius
511.
A rectangle with equal sides is
called:
B. collinear
516.
A quadrilateral whose opposite sides
are equal is generally termed as:
A. a square
A polygon with fifteen sides is
called:
B. a rectangle
A. dodecagon
C. a rhombus
B. decagon
D. a parallelogram
517.
A part of a line included between
two points on the line is called:
B. vertical angles
C. horizontal angle
A. a tangent
B. a secant
D. inscribed angle
522.
C. a sector
A. perpendicular to the plane
D. a segment
518.
B. lying on the plane
The section of the sphere cut by a
plane through its center is termed as:
C. parallel to the plane
A. small circle
D. oblique to the plane
C. big circle
The chord passing through the focus
of the parabola and perpendicular to
its axis is termed as:
D. great circle
A. directrix
Line that pass through a common
point are called:
B. translated axis
B. incircle
519.
523.
C. latus rectum
A. collinear
B. coplanar
D. axis
524.
C. concurrent
D. congruent
520.
The locus of the point which move
so the sum of its distances between
two fixed points is known as:
A. a parabola
Point which lie on the same plane,
are called:
B. a circle
A. collinear
C. an ellipse
B. coplanar
D. a hyperbola
C. concurrent
521.
A normal to a given plane is:
525.
A tangent to a conic is a line
D. congruent
A. which is parallel to the normal
In two intersecting lines, the angles
opposite to each other are termed as:
B. which touches the conic at only
one point
A. opposite angles
C. which passes inside the conic
D. all of the above
526.
The locus of a point that move so
that its distance from a fixed point
and a fixed line is always equal, is
known as:
D. axis
530.
A. quadrants
A. a parabola
B. octants
B. a circle
C. axis
C. an ellipse
D. a hyperbola
527.
D. coordinates
531.
The locus of a point, which moves so
that it is always equidistant from a
fixed point, is known as:
B. a circle
C. an ellipse
B. a circle
D. a hyperbola
C. an ellipse
528.
532.
A conic section whose eccentricity is
equal to one (1) is known as:
In polar coordinate system, the polar
angle is positive when:
A. a parabola
A. measured clockwise
B. a circle
C. an ellipse
B. measured counterclockwise
D. a hyperbola
C. measured at the terminal side of
D. none of these
529.
A conic section whose eccentricity is
less than one (1) is known as;
A. a parabola
A. a parabola
D. a hyperbola
The rectangular coordinate system in
space is divided into eight
compartments, which are known as:
The plane rectangular coordinate
system is divided into four parts
which are known as:
533.
In polar coordinate system, the
distance from a point to the pole is
known as:
A. polar angle
A. coordinates
B. -coordinate
B. octants
C. radius vector
C. quadrants
D. -coorcinate
534.
The curve represented by the
equation
is:
538.
A. a parabola
A. convex
B. a line
B. equilateral
C. an ellipse
C. isopometric
D. a circle
535.
When two lines are perpendicular,
the slope of one is:
D. congruent
539.
A. equal to the other
C. equal to the reciprocal of the other
B. all right-angled triangles are
similar
D. equal to the negative reciprocal
of the other
The axis of the hyperbola, which is
parallel to its directrices, is known
as:
C. all isosceles triangle are similar
D. all rectangles are similar
540.
C. major axis
The volume of any solid of
revolution is equal to the generating
area times the circumference of the
circle described by the centroid of
the area. This is commonly known
as:
D. minor axis
A. First proposition of Pappus
The axis of the hyperbola through
the foci is known as:
B. Second proposition of Pappus
A. conjugate axis
B. transverse axis
537.
Which of the following statements is
correct?
A. all equilateral triangles are
similar
B. equal to the negative of the other
536.
A polygon is _____ if no side, when
extended, will pass through the
interior of the polygon.
C. Cavalier’s Principle
A. conjugate axis
D. Simpson’s Rule
B. transverse axis
541.
C. major axis
D. minor axis
If the product of the slopes of any
two straight lines is negative 1, one
of these lines are said to be:
A. parallel
B. skew
542.
C. perpendicular
A. orthocenter
D. non-intersecting
B. circumcenter
When two planes intersect with each
other, the amount of divergence
between the two planes is expressed
to be measuring the:
C. centroid
D. incenter
546.
A. dihedral angle
B. plane angle
A. orthocenter
C. polyhedral angle
B. circumcenter
D. reflex angle
543.
544.
The angle which the line of sight to
the object, makes with the
horizontal, which is above the eye of
the observer is called:
C. centroid
D. incenter
547.
The arc length equal to the radius of
the circle is called:
A. angle of depression
A. 1 radian
B. angle of elevation
B. 1 quarter circle
C. acute angle
C.
D. bearing
D. 1 grad
The median of a triangle is the line
connecting a vertex and the midpoint
of the opposite side. For a given
triangle, these medians intersect at a
point which is called the:
548.
D. centroid
The altitudes of the side of a triangle
intersect at the point known as:
A five pointed star is also known as:
B. pentatron
C. pentagram
D. quintagon
B. incenter
C. circumcenter
radian
A. pentagon
A. orthocenter
545.
The angular bisector of the sides of a
triangle intersects at the point which
is known as:
549.
The area bounded by two concentric
circles is called:
A. ring
B. disk
C. annulus
D. sector
550.
554.
The line passing through the focus
and perpendicular to the directrix of
a parabola is called:
A. diagonals
B. sides
A. latus rectum
C. vertices
B. axis of parabola
C. tangent line
D. bases
555.
D. secant line
551.
The altitudes of the sides of a
triangle intersect at the point known
as:
A. tetrahedron
B. prism
B. circumcenter
C. frustum
C. centroid
D. prismatoid
556.
The length of time during which the
property may be operated at a profit
is called:
In Plain Geometry, two circular arcs
that together make up a full circle are
called:
A. coterminal arcs
A. life
B. conjugate arcs
B. length of time
C. half arcs
C. physical life
D. congruent arcs
D. economic life
553.
It is a polyhedron of which two faces
are equal polygons in parallel planes
and the other faces are
parallelograms
A. orthocenter
D. incenter
552.
Prisms are classified according to
their _____.
What is the graph of the equation
?
557.
It represents the distance of a point
from the -axis.
A. ordinate
A. circle
B. coordinate
B. ellipse
C. abscissa
C. parabola
D. polar distance
D. hyperbola
558.
A. Cissoid of circles
Polygons are classified according to
the number of:
B. Folium of Descartes
A. vertices
C. Epicycloid
B. sides
C. diagonals
D. Cardioid
563.
A. an ellipse
It is the surface generated by moving
a straight line (called the generator)
which is always parallel to a fixed
line and which always intersect a
fixed plane curve (called the
directrix) is:
B. a hyperbola
A. cylindrical surface
C. a parabola
B. locus of a point
D. a circle
C. spherical surface
The family of curves which intersect
a given family of curves at an angle
less than 90° are called:
D. paraboloid
D. angles
559.
560.
561.
562.
In a conic section, if the eccentricity
> 1, the locus is;
564.
How many faces have an
icosahedron?
A. orthogonal trajectories
A. 16
B. intersecting curves
B. 18
C. isogonal trajectories
C. 20
D. acute angle
D. 22
A line perpendicular to the -axis
has a slope of:
565.
Each of the faces of a regular
hexahedron is a:
A. zero
A. square
B. unity
B. triangle
C. infinity
C. hexagon
D. none of these
D. circle
The locus of points generated when a
circle is made to roll externally along
the circumference of another circle.
566.
An arc length, which is equal to the
radius of the circle, is called:
A. 1 degree
567.
570.
B. 2 radians
In finding the distance between two
points
and
, the
most direct procedure is to use:
C. 1 radian
A. the law of cosines
D. 1 radians
B. the slope of the line
Polygons with all interior angles less
than 180° are called:
C. the translation of axes
D. the Pythagorean Theorem
A. concave polygon
C. acute polygon
In finding the distance between two
points
and
, the
most direct procedure is to use:
D. supplemental polygon
A. the law of cosines
To cut a right circular cone in order
to reveal a parabola, it must be cut
B. the slope of the line
A. perpendicular to the axis of
symmetry
D. the Pythagorean Theorem
B. convex polygon
568.
B. at any acute angle to the axis of
symmetry
571.
C. the translation of axes
572.
A. washer
C. parallel to an element of a cone
and intersecting the axis of
symmetry
B. ring
C. annulus
D. parallel to the axis of symmetry
569.
To find the angles of a triangle,
given only the lengths of the sides,
one would use
The area of a region bounded by two
concentric circles is called:
D. circular disk
573.
A. the law of cosines
It can be defined as the set of all
points in the plane the sum of whose
distance from two fixed points is a
constant.
B. the law of tangents
A. circle
C. the law of sines
B. ellipse
D. the inverse square law
C. hyperbola
D. parabola
574.
575.
576.
If the equation is unchanged by the
substitution of – for , its curve is
symmetric with respect to the:
circular motion about an axis, while
travelling at a constant speed, ,
parallel to the axis?
A. -axis
A. helix
B. -axis
B. spiral of Archimedes
C. origin
C. hypocycloid
D. line 45° with the axis
D. cycloid
A line which is perpendicular to the
-axis has a slope equal to:
579.
A. zero
A. straight angle
B. either
B. obtuse angle
C. one
C. related angle
D. infinity
D. reflex angle
In an ellipse, a chord which contains
a focus and is in a line perpendicular
to the major axis is a:
580.
C. hexagon
D. circumference
C. focal width
581.
In general triangles the expression
/
/
/ is called:
B. circle
B. law of cosines
C. radius
C. law of sines
578.
What type of curve is generated by a
point which moves in uniform
A plane closed curve, all points of
which are the same distance from a
point within, called the center:
A. arc
A. Euler’s formula
D. Pythagorean theorem
The sum of the sides of a polygon:
B. square
B. minor
D. conjugate axis
radian but less
A. perimeter
A. latus rectum
577.
An angle more than
than
radians is:
D. chord
582.
One-fourth of a great circle:
A. cone
C. circle
The point on the curve where the first
derivative of a function is zero and the
second derivative is positive is called:
D. sphere
A. maxima
Points that lie in the same plane:
B. minima
A. coplanar
C. point of inflection
B. oblique
D. point of intersection
B. quadrant
583.
C. collinear
587.
588.
At the minimum point, the slope of the
tangent line is:
D. parallel
584.
A. negative
The study of the property of figures
of three dimensions;
B. infinity
C. positive
A. physics
D. zero
B. plane geometry
589.
At the point of inflection where
,
C. solid geometry
A.
is not equal to zero
D. trigonometry
B.
585.
The volume of a circular cylinder is
equal to the product of its base and
altitude.
A. postulate
B. theorem
C.
D.
590.
Point of the derivatives, which do not
exist ( and so equals zero) is called:
A. stationary point
C. corollary
B. maximum points
D. axiom
C. maximum and minimum point
586.
A point on the curve where the second
derivative of a function is equal to zero is called:
A. maxima
B. minima
C. point of inflection
D. point of intersection
D. minimum point
591.
If the second derivative of the equation
of a curve is equal to the negative of the
equation of that same curve, the curve
is:
A. a cissoid
B. a paraboloid
C. a sinusoid
D. an exponential
MULTIPLE CHOICE QUESTIONS IN
<PHYSICS>
<DIEGO INOCENCIO TAPANG
GILLESANIA>
ENCODED BY: BORBON, MARK
ADRIAN C.
592.
It is defined as the motion of a rigid
body in which a straight line passing
through any two of its particles
always remains parallel to its initial
position.
594.
595.
The study of motion without
reference to the forces which causes
motion is known as:
A. kinetics
A. translation
B. dynamics
B. rotation
C. statics
C. plane motion
D. kinematics
Which of the following is not a
vector quantity?
A branch of physical science that
deals with state of rest or motion of
bodies under the action of forces is
known as:
A. mass
A. mechanics
B. torque
B. kinetics
C. displacement
C. kinematics
D. velocity
D. statics
D. kinetics
593.
596.
The product of force and the time
during which it acts is known as:
597.
A. impulse
In physics, work is defined in terms
of the force acting through a
distance. The rate at which the work
is done is called:
B. momentum
A. force
C. work
B. energy
D. impact
C. power
The property of the body which
measures its resistance to changes in
motion.
D. momentum
A. acceleration
B. weight
598.
599.
The point through which the
resultant of the disturbed gravity
force passes regardless of the
orientation of the body in space is
called:
C. mass
A. center of inertia
D. rigidity
B. center of gravity
600.
C. center of attraction
B. the density of ivory soap is unity
D. moment of inertia
C. the specific gravity of ivory soap
is greater than that of water
The specific gravity of the substance
is the ratio of the density of the
substance to the density of water.
Another term for specific gravity is:
D. the specific gravity of ivory
soap is less than that of water
604.
B. unit weight
One (1) gram of ice at 0°C is placed
on a container containing 2,000,000
cu. m. of water at 0°C. Assuming no
heat loss, what will happen?
C. relative density
A. ice will become water
A. specific weight
D. density
601.
The momentum of a moving object
is the product of its mass ( ) and
velocity ( ). Newton’s Second Law
of Motion says that the rate of
change of momentum with respect to
time is:
B. some part of the ice will not
change
C. the volume of the ice will not
change
D. all of the above
605.
A. power
B. energy
C. momentum
602.
When two waves of the same
frequency, speed and amplitude
travelling in opposite directions
superimposed,
D. force
A. destructive interference always
results
The acceleration due to gravity in the
English System or ft/s2 is:
B. constructive interference always
results
A. 20.2
C. standing waves are produced
B. 32.2
D. the phase difference is always
zero
C. 15.2
606.
D. 62.4
603.
Ivory soap floats in water because:
A. all matter has mass
Any two points along a steamline in
an ideal fluid in steady flow, the sum
of the pressure, the potential energy
per unit volume, and the kinetic
energy per unit volume has the same
value. This concept is known as the:
A. Pascal’s theorem
B. Bernoulli’s energy theorem
D. maximum at the free end
610.
C. Fluid theory
A. scalar
D. Hydraulic theorem
607.
Whenever a net force acts on a body,
it produces an acceleration in the
direction of the resultant force, an
acceleration which is directly
proportional to the resultant force
and inversely proportional to the
mass of the body. This theory is
popularly known as:
B. tangent
C. tensor
D. resultant
611.
B. proportional to the depth of
submergence
B. Newton’s second law of motion
C. Faraday’s law of forces
D. Hooke’s law of equilibrium
C. equal to the weight of the fluid
displaced
Kinematic viscosity in SI derived
unit is described as:
D. independent of the volume of the
body
A. watt per meter Kelvin
612.
B. sq. m. per second
C. Pascal-second
B. air resistance
In a cantilever beam with a
concentrated load at the free end, the
moment is:
A. constant along the beam
B. maximum at the wall
C. ¼ maximum halfway out on the
beam
A leak from a faucet comes out in
separate drops. Which of the
following is the main cause of this
phenomenon?
A. gravity
D. Newton per meter
609.
The loss of weight of a body
submerged in a fluid is:
A. proportional to the weight of the
body
A. Newton’s first law of motion
608.
What is the name of the vector that
represents the sum of two vectors?
C. viscosity of the fluid
D. surface tension
613.
Inelastic collision in which the total
kinetic energy after collision is
_____ before collision.
A. equal to zero
614.
B. equal
B. one inch
C. less than
C. one meter
D. greater than
D. one foot
The property by virtue of which a
body tends to return to its original
size or shape after a deformation and
when the deforming forces have
been removed.
617.
Kinetic energy equals:
A. ½
velocity
B. mass
velocity
C. mass
acceleration
A. elasticity
D. ½ mass
B. malleability
C. ductility
618.
D. plasticity
615.
A flowerpot falls off the edge of a
fifth-floor window. Just as it passes
the third-floor window someone
accidentally drops a glass of water
from the window. Which of the
following is true?
velocity2
In an ideal gas where = pressure,
= volume, and = absolute
temperature in degrees Kelvin,
which of the following is constant?
A.
B.
C.
D.
A. The flowerpot hits the ground at
the same time as the glass.
The path of the projectile is:
B. The glass hits the ground before
the flowerpot.
A. a parabola
C. The flowerpot hits the ground
first and with a higher speed than
the glass.
C. a part of a circle
D. The flowerpot and the glass hit
the ground at the same instant.
616.
619.
One Joule of work is done by a force
of one Newton acting through a
distance of:
A. one centimeter
B. an ellipse
D. a hyperbola
620.
One mole of gas at standard
temperature and pressure (STP)
conditions occupies a volume equal
to:
A. 22.4 liters
B. 9.81 liters
C. 332 liters
D. 2274.5 liters
621.
D. ascending and descending nodes
624.
“Equal volume of all gases under the
same conditions of temperature and
pressure contain the same number of
molecules”. This hypothesis is
popularly known as:
A. toughness
B. malleability
A. Dalton’s hypothesis
C. hardness
B. Avogadro’s hypothesis
C. Debye-Sear’s hypothesis
D. ductility
625.
D. Compton’s hypothesis
622.
The reciprocal of bulk modulus of
elasticity of any fluid is called:
The ratio of the uniform triaxial
stresses, to the change in volume at
equal stress in all directions is:
A. compressibility
A. modulus of flexure
C. volume stress
B. modulus of rapture
D. shape factor
C. bulk modulus of elasticity
B. volume strain
626.
D. coefficient of restitution
623.
This implies the resistance to shock
or difficulty of breaking and express
the work per unit volume required to
fracture a material.
According to the laws of Johannes
Kepler, “The orbit of satellite is an
ellipse, the radius vector sweeps
equal areas in equal intervals of time
and the square of the periods of
revolution with respect to both the
satellite and planet is proportional to
the cubes of their mean distance
from each other.” The shape of the
ellipse depends upon its:
“The resultant of the external force
applied to an object composed of a
system of particles, is equal to the
vector summation of the effective
forces acting on all particles”. This
principle is known as:
A. Archimedes’s principle
B. Bernoulli’s principle
C. D’Alembert’s principle
D. Gauss-Jordan principle
B. lengths of latera recta
Calorie is the amount of heat
required to increase the temperature
of _____ of water by one degree
centigrade.
C. apogee and perigee
A. 1 kg
A. eccentricity
627.
B. 1 lb
628.
631.
C. 1 mg
To maximize the horizontal range of
the projectile, which of the following
applies?
D. 1 gram
A. maximize the angle of elevation
It describes the luminous flux
incidence per unit area and is
expressed in lumens per square
meter.
B. maximize velocity
A. luminous intensity
D. the tangent function of the
angle of trajectory must be equal
to one
C. maximize the angle of elevation
and velocity
B. illuminance
C. radiance
The moment of inertia of a plane
figure:
According to this law, “The force
between two charges varies directly
as the magnitude of each charge and
inversely as the square of the
distance between them.
A. is zero at the centroidal axis
A. law of universal gravitation
B. increase as the distance of the
axis moves farther from the
centroid
B. Newton’s law
C. decrease as the distance of the
axis moves farther from the centroid
D. inverse square law
632.
D. luminance
629.
C. Coulomb’s law
633.
D. is maximum at the centroidal axis
630.
The distance that the top surface is
displaced in the direction of the force
divided by the thickness of the body
is known as:
Formation of bubbles in a lowpressure area in a centrifugal pump
and later their sudden collapse, is
called:
A. compression
B. corrosion
A. longitudinal strain
C. explosion
B. shear strain
D. cavitation
C. volume strain
D. linear strain
644.
The hardness of steel may be
increased by heating to
approximatelyv1500°F and
quenching in oil or water if
A. the carbon content is above 3.0%
C. lime soda treatment
B. the carbon content is from 0.2%
to 2.0%
D. thermal treatment
648.
C. the carbon content is below 0.2%
D. the steel has been hot rolled
instead of cast
645.
A. specific speed
B. impeller type
Galvanized iron is a term referring to
iron coated with:
C. Bernoulli’s equation
A. magnesium
D. overall efficiency
C. zinc
The impulse and momentum
principle is mostly useful for
problems involving;
D. tin
A. velocity, acceleration, and time
A process of welding metals in
molten or in vaporous state without
application of mechanical pressure or
blow. Such welding may be
accomplished by the oxyacetylene or
by hydrogen flame or by electric arc.
It is called:
B. force, acceleration, and time
B. aluminum
646.
Used as a guide to selecting the most
efficient centrifugal pump:
649.
C. force, velocity, and time
D. force, velocity, and acceleration
650.
Which of the following is not true
regarding the Blasius boundary layer
solution/
A. fusion welding
647.
B. TIG welding
A. It permits one to calculate the skin
friction on a flat plate
C. MIG welding
B. It is valid for laminar flow
D. cold welding
C. It is an approximate solution
A chemical method of feed water
treatment wherein water is passed
through a bed of sodium zeolite
Nesub2Z which reacts with calcium
and magnesium salts:
D. It is valid only for potential flow
A. demineralization process
B. ion exchange treatment
651.
The greatest unit pressure the soil
can continuously withstand:
A. point of raptue
B. bearing strength
C. ultimate strength
652.
D. yield point
C. internal energy
Heat transmission carried by the
movement of heated fluids away
from a hot body, as in the heating of
water by a hot surface:
D. pressure heads
656.
A. radiation
B. convection
A. enthalpy increase of the system
C. conduction
B. specific bent ratio of the moment
D. absorption
653.
The type of cooler extensively used
for medium and large size diesel
engines:
C. entropy increase of the system
D. entropy decrease of the system
657.
A. radiation cooler
B. shell and tube cooler
B. when there is no tendency
towards spontaneous change
D. plate cooler
A closed vessel intended for use in
heating water or for application of
heat to generate steam or other vapor
to be used externally to itself is
called:
A. unfired pressure vessel
C. when the system is not
accelerating
D. when all its parts are at the same
temperature
658.
B. steam generator
B. manometer
D. boiler
C. anemometer
The sum of the three types of energy
at any point in the system is called:
A. Bernoulli’s theorem
B. enthalpy
An instrument used for measuring
high temperature gas
A. plenometer
C. boiler or steam generator
655.
The system is safe to be in
thermodynamics equilibrium:
A. if it has no tendency to undergo
further chemical reaction
C. disk cooler
654.
In energy transformation process in
which the resultant condition lacks
the driving potential needed to
reverse the process, the measure of
this loss is expressed as:
D. pyrometer
659.
The power output of the engine is
increased through:
A. turbo-charging
C. the total number of pounds of
sodium bicarbonate in the water per
million pounds of water.
B. scavenging
C. all of these
D. the total number of pounds of salt
(sodium chloride) in the water per
million pounds of water
D. super-charging
660.
The equilibrium temperature that a
regular thermometer measures if
exposed to atmospheric air is:
B. °C
C. velocity2
C. wet bulb temperature
D. ½ velocity
On the hoist or load block or some
equality visible space of every hoist
designed to lift its load vertically
shall be legibly marked:
664.
An instrument used for measuring
specific gravity of fluids:
A. hygrometer
B. flowmeter
A. its electrical voltage
C. psycrometer
B. its brand and model
D. hydrometer
D. its motor hp or kW
The hardness of water is given in
ppm (parts per million, i.e., pounds
per million pounds of water). This
hardness is
A. the total number of pounds of
dissolved solids in the water per
million pounds of water
B. the total number of pounds of
calcium and magnesium
bicarbonate in the water.
_____
A. time
B. velocity
C. its rated load capacity
662.
Momentum = Force
A. dry bulb temperature
D. dew point
661.
663.
MULTIPLE CHOICE QUESTIONS IN
<MECHANICS>
<DIEGO INOCENCIO TAPANG
GILLESANIA>
ENCODED BY: BORBON, MARK
ADRIAN C.
665. A 10-lbm object is acted upon by a
4-lb force. What is the acceleration in
ft/min2 ?
A. 8.0
10 to the 4th power ft/min2
B. 9.2
10 to the 4th power ft/min2
C. 7.8
10 to the 4th power ft/min2
D. 4.637
friction with the bed is 0.4. What is
the shortest time that the truck can be
brought to a stop such that the boxes
do not shift?
A. 4.75 sec
B. 2.35 sec
C. 5.45 sec
10 to the 4th power
ft/min2
666.
667.
D. 6.37 sec
A. 343.5 N
A 40-kg block is resting on an
inclined plane making an angle 20°
from the horizontal. If the coefficient
of friction is 0.60, determine the
force parallel to the incline that must
be applied to cause impending
motion down the plane.
B. 224.5 N
A. 77
C. 53.8 N
B. 82
D. 446.2 N
C. 72
A skier wishes to build a rope tow to
pull herself up a ski hill that is
inclined at 15° with the horizontal.
Calculate the tension needed to give
the skier’s 54-kg body an
acceleration of 1.2 m/s2. Neglect
friction.
D. 87
What horizontal force P can be
applied to a 100-kg block in a level
surface with coefficient of friction of
0.2, that will cause an acceleration of
2.50m/s2 ?
669.
670.
A 50-kilogram block of wood rest on
top of the smooth plane whose length
is 3 m, and whose altitude is 0.8 m.
How long will it take for the block to
slide to the bottom of the plane when
released?
A. 202 N
A. 1.51 seconds
B. 403 N
B. 2.41 seconds
C. 106 N
C. 2.51 seconds
D. 304 N
D. 2.14 seconds
668.
A pick-up truck is travelling forward
at 25 m/s. The truck bed is located
with boxes, whose coefficient of
671.
A body weighing 40 lbs. starts from
rest and slides down a plane at an
angle of 30° with the horizontal for
which the coefficient of friction
µ=0.3. How far will it move during
the third second?
674.
A. 19.99 ft
B. 39.63 ft
C. 18.33 ft
D. 34.81 ft
672.
A car and its load weighs 27 kN and
the center of gravity is 600 mm from
the ground and midway between the
front and rear wheel which are 3 m
apart. The car is brought to rest from
a speed of 54 kph in 5 seconds by
means of the brakes. Compute the
normal force on each of the front
wheels of the car.
A. 7.576 kN
A.
B.
C.
D.
675.
B. 9.541 kN
C. 5.478 kN
D. 6 kN
673.
An elevator weighing 2,000 lb
attains an upward velocity of 16 fps
in 4 sec with uniform acceleration.
What is the tension in the supporting
cables?
C. 2,495 lb
D. 2,250 lb
A car travels on the horizontal
unbanked circular track of radius .
Coefficient of friction between the
tires and track is 0.3. If the car’s
velocity is 10 m/s, what is the
smallest radius it may travel without
skidding?
A. 50 m
B. 60 m
C. 15 m
D. 34 m
A. 1,950 lb
B. 2,150 lb
A block weighing 200 N rests on a
plane inclined upwards to the right at
a slope of 4 vertical to 3 horizontal.
The block is connected to a cable
initially parallel to the plane, passing
through the pulley and connected to
another block weighing 100 N
moving vertically downward. The
coefficient of kinetic friction
between the 200 N block and the
inclined plane is 0.10. Which of the
following most nearly gives the
acceleration
of the system?
676.
If a car travels at 15 m/s and the
track is banked 5°, what is the
smallest radius it can travel so that
the friction will not be necessary to
resist skidding?
A. 262.16 m
C. 229.6 m
B. 651.23 m
D. 285.3 m
C. 278.14 m
680.
D. 214.74 m
677.
A vertical bar of length with a
mass of 40 kg is rotated vertically
about one end at 40 rpm. Find the
length of the bar if it makes an angle
45° with the vertical?
A. 49.4 rad/s
B. 37.2 rad/s
A. 1.58 m
C. 24.9 rad/s
B. 2.38 m
D. 58.3 rad/s
C. 3.26 m
D. 1.86 m
678.
681.
The seats of a carousel are attached
to a vertical rotating shaft by a
flexible cable 8 m long. The seats
have a mass of 75 kg. What is the
maximum angle of tilt for the seats if
the carousel operates at 12 rpm?
B. 18°
C. 3.2°
B. 35°
D. 2.5°
C. 45°
679.
A highway curve is superelevated at
7°. Find the radius at the end of the
cable that will break if there is no
lateral pressure on the wheels of a
car at a speed of 40 mph.
Traffic travels at 65 mi/hr around a
banked highway curve with a radius
of 3000 ft. What banking angle is
necessary such that friction will not
be required to resist the centrifugal
force?
A. 5.4°
A. 30°
D. 39°
A 2-N weight is swung in a vertical
circle of 1-m radius at the end of a
cable that will break if the tension
exceeds 500 N. Find the angular
velocity of the weight when the cable
breaks.
682.
A concrete highway curve with a
radius of 500 feet is banked to give a
lateral pressure equivalent to
. For what coefficient of
friction will skidding impend for a
speed of 60 mph?
A. < 0.360
A. 247.4 m
B. < 0.310
B. 265.6 m
C. > 0.310
D. > 0.360
683.
A 3500 lbf car is towing a 500 lbf
trailer. The coefficient of friction
between all tires and the road is 0.80.
How fast can the car and the trailer
travel around an unbanked curve of
radius 0.12 mile without either the
car or trailer skidding?
686.
A force of 200 lbf acts on a block at
an angle of 28° with respect to the
horizontal. The block is pushed 2
feet horizontally. What is the work
done by this force?
A. 320 J
A. 87 mph
B. 540 J
B. 72 mph
C. 480 J
C. 26 mph
D. 215 J
D. 55 mph
684.
D. 26 rpm
687.
A cast-iron governor ball 3 inches in
diameter has its center 18 inches
from the point of support. Neglecting
the weight of the arm itself, find the
tension in the arm if the angle with
the vertical axis is 60°.
A 10-kg block is raised vertically 3
meters. What is the change in
potential energy. Answer in SI units
closest to:
A. 350N-m
B. 294 J
C. 350 kg-m2/s2
A. 7.63 lb
D. 320 J
B. 6.36 lb
688.
C. 7.56 lb
D. 7.36 lb
685.
An object is placed 3 feet from the
center of a horizontally rotating
platform. The coefficient of friction
is 0.3. The object will begin to slide
off when the platform speed is
nearest to:
A. 10 fps
B. 12 fps
C. 14 fps
D. 16 fps
A. 17 rpm
B. 12 rpm
C. 22 rpm
At her highest point, a girl on the
swing is 7 feet above the ground, and
at her lowest point, she is 3 feet
above the ground. What is her
maximum velocity?
689.
An automobile has a power output of
1 hp. When it pulls a cart with a
force of 300 N, what is the cart’s
velocity?
693.
B. 24.9 m/s
A ship moving North at 10 mph. A
passenger walks Southeast across the
deck at 5 mph. In what direction and
how fast is the man moving, relative
to the earth’s surface.
C. 2.49 m/s
A. N 28°40’W; 7.37 mph
D. 0.249 m/s
B. N 61°20’E; 7.37 mph
A. 249 m/s
C. N 61°20’W; 7.37 mph
690. The weight of a mass of 10
kilograms at a location where g=9.77m/s2 is:
A. 79.7 N
D. N 28°40’E; 7.37 mph
694.
B. 77.9 N
C. 97.7 N
D. 977 N
691.
A. S 14.47°W
What is the resultant velocity of a
point of -component
,
and -component
at
time
?
B. S 75.52°W
C. S 81.36°W
D. S 84.36°W
A. 63.1326
B. 62.1326
695.
C. 64.1326
D. 74.1326
692.
A man wishes to cross due west on a
river which is flowing due north at
the rate of 3 mph. if he can row 12
mph in still water, what direction
should he take to cross the river?
A plane is headed due east with air
speed of 240 kph. If a wind of 40kph
is blowing from the north, find the
ground speed of the plane.
A. 243 kph
A boat has a speed of 8 mph in still
water attempts to go directly across a
river with a current of 3 mph. What
is the effective speed of the boat?
B. 423 kph
C. 200 kph
D. 240 kph
A. 8.35 mph
C. 7.42 mph
Three forces 20N, 30N, and 40N are
in equilibrium. Find the angle
between the 30-N and 40-N forces.
D. 6.33 mph
A. 30°15’25’’
B. 8.54 mph
696.
B. 28.96°
700.
C. 40°
D. 25.97°
697.
A 10-kg weight is suspended by a
rope from a ceiling. If a horizontal
force of 5.80 kg is applied to the
weight, the rope will make an angle
with the vertical equal to:
A. 248 m
B. 390 m
C. 408 m
A. 60°
D. 422 m
B. 30°
C. 45°
701.
D. 75°
698.
The allowable spacing of towers to
carry an aluminum cable weighing
0.03 kg per horizontal meter if the
maximum tension at the lowest point
is not to exceed 1150 kg at sag of
0.50 m is:
A 100kN block slides down a plane
inclined at an angle of 30° with the
horizontal. Neglecting friction, find
the force that causes the block to
slide.
A wooden plank meters long has
one end leaning on top of a vertical
wall 1.5 m high and the other end
resting on a horizontal ground.
Neglecting friction, find if a force
(parallel to the plank) of 100 N is
needed to pull a 400 N block up the
plank.
A. 6 m
A. 86.6 kN
B. 5 m
B. 80 kN
C. 4 m
C. 20 kN
D. 3 m
D. 50 kN
702.
699.
What tension must be applied at the
ends of a flexible wire cable
supporting a load of 0.5 kg per
horizontal meter in a span of 100 m
if the sag is to be limited to 1.25 m?
A block of wood is resting on a level
surface. If the coefficient of friction
between the block and the surface is
0.30, how much can the plane be
inclined without causing the block to
slide down?
A. 423.42 kg
A. 16.7°
B. 584.23 kg
B. 30.2°
C. 500.62 kg
C. 21.2°
D. 623.24 kg
D. 33.3°
703.
A. 795
A 500-kg block is resting on a 30°
inclined plane with a µ=0.3 Find the
required force acting horizontally
that will prevent the block from
sliding.
B. 791
C. 797
D. 793
A. 1020 N
707.
B. 1160 N
C. 4236 N
D. 5205 N
704.
With a starting speed of 30 kph at a
point , a car accelerates uniformly.
After 18 minutes, it reaches point ,
21 km from . Find the acceleration
of the car in m/s2.
A. 0.126 m/s2
A 500-kg block is resting on a 30°
inclined plane with a µ=0.3 Find the
required force acting horizontally
that will start the block to block up
the plane.
B. 0.0562 m/s2
C. 0.0206 m/s2
D. 3.42 m/s2
A. 4236 N
708.
B. 1160 N
C. 5205 N
D. 2570 N
705.
What is the acceleration of the body
that increases in velocity from 20
m/s to 40 m/s in 3 seconds? Answer
in S.I. units.
A train upon passing point at a
speed of 72 kph accelerates at 0.75
m/s2 for one minute along a straight
path then decelerates at 1.0 m/s2.
How far in kilometers from point
will it be in 2 minutes after passing
point .
A. 4.95
B. 4.75
2
A. 8 m/s
C. 4.85
2
B. 6.67 m/s
D. 4.65
2
C. 5 m/s
709.
2
D. 7 m/s
706.
From a speed of 75 kph, a car
decelerates at the rate of 500 m/min2
along a straight path. Howw far in
meters, will it travel in 45 sec?
A car starting from rest moves with a
constant acceleration of 10 km/hr2
for 1 hour, then decelerates at a
constant -5 km/hr2 until it comes to a
stop. How far has it travelled?
A. 10 km
B. 20 km
713.
C. 12 km
D. 15 km
710.
The velocity of an automobile
starting from rest is given by
/
/
ft./sec.
Determine its acceleration after an
interval of 10 seconds (in ft/sec2).
A. 12.48 m
A. 2.10
B. 6.25 m
B. 1.71
C. 10.28 m
C. 2.25
D. 8.63 m
D. 2.75
711.
714.
A train running at 60 kph decelerated
at 8 m/min2 for 14 minutes. Find the
distance traveled, in kilometers
within this period.
A man driving his car at 45 mph
suddenly sees an object in the road
60 feet ahead. What constant
deceleration is required to stop the
car in this distance?
A. -36.3 ft/s2
A. 12.2
B. -45.2 ft/s2
B. 13.2
C. -33.4 ft/s2
C. 13.8
D. -42.3 ft/s2
D. 12.8
712.
A car was travelling at a speed of 50
mph. The driver saw a road block 80
m ahead and stepped on the brake
causing the car to decelerate
uniformly at 10 m/s2. Find the
distance from the roadblock to the
point where the car stopped. Assume
perception reaction time is 2
seconds.
An automobile accelerates at a
constant rate of 15 mi/hr to 45 mi/hr
in 15 seconds, while travelling in a
straight line. What is the average
acceleration?
A. 2 ft/s2
715.
Determine the outside diameter of
hallow steel tube that will carry a
tensile load of 500 kN at a stress of
140 MPa. Assume the wall thickness
to be one-tenth of the outside
diameter.
A. 123 mm
2
B. 2.39 ft/s
B. 113 mm
C. 2.12 ft/s2
C. 103 mm
2
D. 2.93 ft/s
D. 93 mm
716.
A force of 10 Newtons is applied to
one end of a 10 inches diameter
circular rod. Calculate the stress.
safety with respect to the tensile
failure?
A. 3.15
A. 0.20 kPa
B. 3.55
B. 0.05 kPa
C. 2.15
C. 0.10 kPa
D. 0.15 kPa
717.
718.
D. 2.55
What force is required to punch a 20mm diameter hole through a 10-mm
thick plate. The ultimate strength of
the plate material is 450 MPa.
A metal specimen 36-mm in
diameter has a length of 360 mm. A
force of 300 kN elongates the length
by 1.20 mm. What is the modulus of
elasticity?
A. 241 kN
A. 88.419 GPa
B. 283 kN
B. 92.564 GPa
C. 386 kN
C. 92.658 GPa
D. 252 kN
D. 95.635 GPa
A steel pipe 1.5m in diameter is
required to carry am internal
pressure of 750 kPa. If the allowable
tensile stress of steel is 140 MPa,
determine the required thickness of
the pipe in mm.
720.
721.
A. 4.56
B. 5.12
A. 3.09 mm
C. 4.25
B. 3.56 mm
D. 4.01
719.
A spherical pressure vessel 400-mm
in diameter has a uniform thickness
of 6 mm. The vessel contains gas
under a pressure of 8,000 kPa. If the
ultimate tensile stress of the material
is 420 MPa, what is the factor of
A steel wire 5-m long hanging
vertically supports a weight of 1200
N. Determine the required wire
diameter if the stress is limited to
140 MPa and the total elongation
must not exceed 4mm. Neglect the
weight of the wire and assume
GPa.
C. 3.33 mm
D. 2.89 mm
722.
During a stress-strin test, the unit
deformation at a stress of 35 MPa
was observed to be
m/m
and at a stress of 140 MPa it was
B. 54.3 mm
m/m. If the proportional
limit was 200 MPa, what is the
modulus of elasticity. What is the
strain corresponding to the stress of
80 MPa?
C. 35.4 mm
D. 45.3 mm
725.
A.
m/m
MPa;
B.
m/m
MPa;
C.
m/m
MPa;
D.
m/m
723.
724.
A steel bar 50 mm in diameter and 2
m long is surrounded by a shell of
cast iron 5 mm thick. Compute the
load that will compress the bar a
total of 1 mm in the length of 2 m.
Use
GPa and
GPa.
A. 200 kN
MPa;
An axial load of 100 kN is applied to
a flat bar 20 mm thick, tapering in
width from 120 mm to 40 mm in a
length of 10 m. Assuming
GPa, determine the total elongation
of the bar.
B. 240 kN
C. 280 kN
D. 320 kN
A. 3.43 mm
A 20-mm diameter steel rod, 250
mm long is subjected to a tensile
force of 75 kN. If the Poisson’s ratio
µ is 0.30, determine the lateral strain
of the rod. Use
GPa.
B. 2.125 mm
A.
C. 4.33 mm
B.
D. 1.985 mm
C.
Steel bar having a rectangular crosssection 15 mm 20 mm and 150 m
long is suspended vertically from
one end. The steel has a unit mass of
7850 kg/m3 and a modulus of
elasticity of 200 GPa. If a loaf of
20 kN is suspended at the other end
of the rod, determine the total
elongation of the rod.
A. 43.5 mm
726.
D.
727.
mm/mm
mm/mm
mm/mm
mm/mm
A solid aluminum shaft of 100-mm
diameter fits concentrically in a
hollow steel tube, determine the
minimum internal diameter of the
steel tube so that no contact pressure
exists when the aluminum shaft
carries an axial compressive load of
600 kN. Assume Poisson’s ratio
C. 79,698 MPa
µ=1/3 and the modulus of elasticity
of aluminum be 70 GPa.
A. 100.0364 mm
D. 82,400 MPa
731.
B. 100.0312 mm
C. 100.0303 mm
D. 100.0414 mm
728.
The maximum allowable torque, in
kN-m, for a 50-mm diameter steel
shaft when the allowable shearing
stress is 81.5 MPa is:
A. 6.28 m
A. 3.0
D. 8.56 m
B. 1.0
B. 5.23 m
C. 6.89 m
732.
C. 4.0
D. 2.0
729.
The rotation of twist in degrees of a
shaft, 800 mm long subjected to a
torque of 80 N-m, 20 mm in
diameter and shear modulus of
80,000 MPa is:
A hollow steel shaft 2540 mm long
must transmit torque of 35 kN-m.
The total angle of twist must not
exceed 3 degrees. The maximum
shearing stress must not exceed 110
MPa. Find the inside diameter and
the outside diameter of the shaft that
meets these conditions.
A.
mm;
A. 3.03
B.
mm;
B. 4.04
C.
mm;
C. 2.92
D.
mm;
733.
mm
mm
mm
mm
Compute the value the shear
modulus of steel whose modulus
of elasticity is 200 GPa and
Poisson’s ratio µ is 0.30.
Determine the maximum shearing
stress in a helical steel spring
composed of 20 turns of 20-mm
diameter wire on a mean radius of 80
mm when the spring is supporting a
load of 2 kN.
A. 72,456 MPa
A. 110.6 MPa
B. 76,923 MPa
B. 101.1 MPa
D. 1.81
730.
Determine the length of the shortest
2-mm diameter bronze wire, which
can be twisted through two complete
turns without exceeding a stress of
70 MPa. Use
GPa.
C. 120.6 MPa
midspan. What is the maximum
moment of the beam?
D. 136.5 MPa
734.
735.
A load is supported by two springs
arranged in series. The upper spring
has 20 turns of 29-mm diameter wire
on a mean diameter of 150 mm. The
lower spring consist of 15 turns of
10-mm diameter wire on a mean
diameter of 130 mm. Determine the
value of that will cause a total
deflection of 80 mm. Assume
GPa for both springs.
B. 1050 kN-m
C. 1520 kN-m
D. 1510 kN-m
A. 223.3 N
A small square 5 cm by 5 cm is cut
out of one corner of a rectangular
cardboard 20 cm by 30 cm long.
How far, in cm from the uncut longer
side, is the centroid of the remaining
area?
B. 228.8 N
A. 9.56
C. 214.8 N
B. 9.35
D. 278.4 N
C. 9.48
A 10-meter long simply supported
beam carries a uniform load of 8
kN/m for 6 meters from the left
support and a concentrated load of
15 kN 2 meters from the right
support. Determine the maximum
shear and moment.
D. 9.67
A.
kN-m
kN;
B.
kN-m
kN;
kN;
kN-m
D.
kN-m
737.
738.
What is the inertia of a bowling ball
(mass = 0.5 kg) of radius 15 cm
rotating at an angular speed of 10
rpm for 6 seconds?
A. 0.0045 kg-m2
B. 0.001 kg-m2
C. 0.005 kg-m2
D. 0.002 kg-m2
C.
736.
A. 1250 kN-m
kN;
A simple beam, 10 m long carries a
concentrated load of 500 kN at the
739.
What is the moment of inertia of a
cylinder of radius 5 m and a mass of
5 kg?
A. 62.5 kg-m2
B. 80 kg-m2
C. 72.5 kg-m2
A. 204 kPa
D. 120 kg-m2
B. 222 kPa
740. The mass of air in a room which is
3m 5m 20m is known to be 350 kg.
Find its density.
A. 1.167 kg/m3
C. 344 kPa
D. 362 kPa
744.
B. 1.176 kg/m3
C. 1.617 kg/m3
A. 90 kPa
D. 1.716 kg/m3
741.
One hundred (100) grams of water
are mixed with 150 grams of alcohol
(
kg/ cu m). What is the
specific gravity of the resulting
mixtures, assuming that the two
fluids mix completely?
B. 80 kPa
C. 100 kPa
D. 10 kPa
745.
A. 0.96
B. 521.3 kPa
C. 0.63
C. 332.8 kPa
D. 0.86
100 g of water are mixed with 150 g
of alcohol (
kg/ cu m). What
is the specific volume of the
resulting mixtures, assuming that the
two fluids mix completely?
A. 0.88 cu cm/g
B. 1.20 cu cm/g
C. 0.82 cu cm/g
D. 0.63 cu cm/g
743. The pressure 34 meters below the
ocean is nearest to:
If the pressure at a point in the ocean
is 60 kPa, what is the pressure 27
meters below this point?
A. 256.3 kPa
B. 0.82
742.
What is the atmospheric pressure on
a planet where the absolute pressure
is 100kPa and the gage pressure is 10
kPa?
D. 185.4 kPa
746.
A pressure gage 6 m above the
bottom of the tank containing a
liquid reads 90 kPa; another gage
height 4 m reads 103 kPa. Determine
the specific weight of the liquid.
A. 6.5 kN/m3
B. 5.1 kN/m3
C. 3.2 kN/m3
D. 8.5 kN/m3
747.
The weight density of a mud is given
by
, where is in
3
kN/m and is in meters. Determine
the pressure, in kPa, at a depth of
5m.
1 meter below the water surface,
what is the total water pressure
exerted on the plane surface?
A. 43.93 kN
B. 52.46 kN
A. 89.36 kPa
C. 64.76 kN
B. 56.25 kPa
D. 78.48 kN
C. 62.5 kPa
D. 78.54 kPa
748.
751.
What is the resulting pressure when
one pound of air at 15 psia and
200°F is heated at constant volume
to 800°F?
A. 138.7 kN
B. 107.9 kN
A. 28.6 psia
C. 169.5 kN
B. 52.1 psia
D. 186.5 kN
C. 36.4 psia
D. 15 psia
749.
752.
The volume of a gas under standard
atmospheric pressure 76 cm Hg is
200 in3. What is the volume when
the pressure is 80 cm Hg, if the
temperature is unchanged?
B. 7862 m3
C. 9364 m3
B. 90 in3
D. 6325 m3
C. 110 in3
750.
A two-meter square plane surface is
immersed vertically below the water
surface. The immersion is such that
the two edges of the square are
horizontal. If the top of the square is
An iceberg having specific gravity of
0.92 is floating on salt water of sp.
gr. 1.03. If the volume of ice above
the water surface is 1000 cu. m.,
what is the total volume of the ice?
A. 8523 m3
A. 190 in3
D. 30.4 in3
Find the total water pressure on a
vertical circular gate, 2 meters in
diameter, with its top 3.5 meters
below the water surface.
753.
A block of wood requires a force of
40 N to keep it immersed in water
and a force of 100 N to keep it
immersed in glycerin (sp. gr. = 1.3).
Find the weight and sp. gr. Of the
wood.
A. 0.7
B. 0.6
D. 64 ft
757.
C. 0.9
D. 0.8
754.
Reynolds number may be calculated
from:
A. diameter, density, and absolute
viscosity
B. diameter, velocity, and surface
tension
m3/s
B.
m3/s
D.
758.
D. characteristic length, mass flow
rate per unit area, and absolute
viscosity
m3/s
m3/s
An orifice has a coefficient of
discharge of 0.62 and a coefficient of
contraction of 0.63. Determine the
coefficient of velocity for the orifice.
A. 0.98
B. 0.99
The sum of the pressure head,
elevation head, and the velocity head
remains constant, this is known as:
C. 0.97
D. 0.96
B. Boyle’s Law
The theoretical velocity of flow
through an orifice 3 m below the
surface of water in a tall tank is:
C. Archimedes’ Principle
A. 8.63 m/s
D. Torrecelli’s Theorem
B. 9.85 m/s
What is the expected head loss per
mile of closed circular pipe (17-in
inside diameter, friction factor of
0.03) when 3300 gal/min of water
flow under pressure?
C. 5.21 m/s
A. Bernoulli’s Theorem
756.
A.
C.
C. diameter, velocity, and absolute
viscosity
755.
What is the rate of flow of water
passing through a pipe with a
diameter of 20 mm and speed of 0.5
m/sec?
A. 38 ft
759.
D. 7.67 m/s
760.
Water having kinematic viscosity
m2/s flows in a 100mm diameter pipe at a velocity of 4.5
m/s. the Reynolds number is:
B. 0.007 ft
A. 346,150
C. 3580 ft
B. 258,250
C. 387,450
D. 298,750
761.
Oil having specific gravity of 0.869
and dynamic viscosity of 0.0814 Pa-s
flows through a cast iron pipe at a
velocity of 1 m/s. The pipe is 50 m
long and 150 mm in diameter. Find
the head lost due to friction.
D. 19.8 m
764.
A 20-mm diameter commercial steel
pipe, 30 m long is used to drain an
oil tank. Determine the discharge
when the oil level in the tank is 3 m
above the exit of the pipe. Neglect
minor losses and assume
.
A. 0.000256 m3/s
B. 0.000179 m3/s
A. 0.73 m
C. 0.000113 m3/s
B. 0.45 m
D. 0.000869 m3/s
C. 0.68 m
D. 1.25 m
762.
What commercial size of new cast
iron pipe shall be used to carry 4490
gpm with a lost of head of 10.56 feet
per mile? Assume
.
A. 625 mm
B. 576 mm
C. 479 mm
D. 352 mm
763.
Assume that 57 liters per second of
oil (
kg/m3) is pumped
through a 300 mm diameter pipeline
of cast iron. If each pump produces
685 kPa, how far apart can they be
placed? (Assume
)
A. 23.7 m
B. 32.2 m
C. 12.6 m
MULTIPLE CHOICE QUESTIONS IN
<ENGINEERING ECONOMICS>
<DIEGO INOCENCIO TAPANG
GILLESANIA>
ENCODED BY: BORBON, MARK
ADRIAN C.
765.
C. nominal rate
The recorded current value of an
asset is known as:
A. scrap value
D. yield
769.
B. book value
C. salvage value
D. present worth
766.
A. depreciation
The ratio of the interest payment to
the principal for a given unit of time
and is usually expressed as a
percentage of the principal is known
as:
A. investment
B. depletion
C. inflation
D. incremental cost
770.
B. nominal interest
C. interest
D. interest rate
767.
A method of depreciation whereby
the amount to recover is spread over
the estimated life of the asset in
terms of the periods or units of
output is called:
The method of depreciation where a
fixed sum of money is regularly
deposited at compound interest in a
real or imaginary fund in order to
accumulate an amount equal to the
total depreciation of an asset at the
end of the asset’s estimated life is
known as:
A. straight line method
B. SYD method
A. SOYD method
C. declining balance method
B. declining balance method
D. sinking fund method
C. straight line method
D. sinking fund method
768.
The lessening of the value of an asset
due to the decrease in the quantity
available. This refers to the natural
resources such as coal, oil, and
timber in the forest.
The interest rate at which the present
worth of cash flow on a project is
zero, or the interest earned by an
investment.
A. rate of return
B. effective rate
771.
The term used to express the series
of uniform payments occurring at
equal interval of time is:
A. compound interest
B. annuity
C. perpetuity
D. depreciation
772.
A. utilities
The profit derived from a project or
business enterprise without
consideration of obligations to
financial contributors and claims of
others based on profit is known as:
B. necessities
C. luxuries
D. producer goods and services
A. yield
776.
B. earning value
C. economic return
D. expected yield
773.
A. utilities
As applied to capitalized asset, the
distribution of the initial cost by
periodic changes to operation as in
depreciation or the reduction of the
depth by either periodic or irregular
prearranged program is called:
A. amortization
B. necessities
C. luxuries
D. producer goods and services
777.
B. annuity
C. depreciation
D. capital recovery
774.
A condition where only few
individuals produce a certain product
and that any action of one will lead
to almost the same action of the
others.
A. oligopoly
Those funds that are required to
make the enterprise or project a
going concern.
B. semi-oligopoly
A. banking
D. perfect competition
B. accumulated amount
C. working capital
D. principal or present worth
775.
These are product or services that are
required to support human life and
activities, that will be purchased in
somewhat the same quantity even
though the price varies considerably.
These are product or services that are
desired by humans and will be
purchased if money is available after
the required necessities have been
obtained.
C. monopoly
778.
This occurs in a situation where a
commodity or service is supplied by
a number of vendors and there is
nothing to prevent additional vendors
entering the market.
A. perfect competition
B. monopoly
C. oligopoly
D. elastic demand
779.
It is the amount that a willing buyer
will pay to a willing seller for a
property where each has equal
advantage and is under no
compulsion to buy or sell.
D. face value
783.
A. mutually exclusive projects
A. fair value
780.
B. use value
B. evaluation over different
periods
C. market value
C. non-conventional cash flows
D. book value
D. difference in the magnitude of the
projects
It is defined to be the capacity of a
commodity to satisfy human want.
784.
A. discount
B. corporation
C. utility
C. single proprietorship
D. necessity
A form of summary of assets,
liabilities, and net worth:
A. balance method
D. all of these
785. What must two investments with the
same present worth and unequal lives have?
A. identical salvage value
B. break-even point
B. different salvage values
C. balance sheet
D. production
782.
The worth of a property, which is
equal to the original cost less
depreciation, is known as:
A. earning value
Which of the following is a form of
business/company ownership?
A. partnership
B. luxuries
781.
When using net present worth
calculations to compare two projects,
which of the following could
invalidate the calculations?
C. identical equivalent uniform
annual cash flows
D. different equivalent annual cash
flows
786.
Find the interest on P6800.00 for 3 years
at 11% simple interest.
B. scrap value
A. P1,875.00
C. book value
B. P1,987.00
C. P2,144.00
790.
D. P2,244.00
787.
A man borrowed P10,000.00 from his
friend and agrees to pay at the end of 90
days under 8% simple interest rate.
What is the required amount?
How long must a P40,000 note bearing
4% simple interest to run to amount to
P41,350.00?
A. 340 days
B. 403 days
C. 304 days
A. P10,200.00
D. 430 days
B. P11,500.00
791.
788.
C. P9,500.00
If P16,000 earns P480 in 9 months, what
is the annual rate of interest?
D. P10,700.00
A. 1%
Annie buys a television set from a
merchant who offers P25,000.00 at the
end of 60 days. Annie wishes to pay
immediately and the merchant offers to
compute the required amount on the
assumption that the money is worth 14%
simple interest. What is the required
amount?
B. 2%
C. 3%
D. 4%
792.
A. P20,234,87
B. P19,222.67
A man lends P6000 at 6% simple
interest for 4 years. At the end of this
time he invests the entire amount
(principal plus investment) at 5%
compounded annually for 12 years. How
much will he have at the end of the 16year period?
C. P24,429.97
A. P13,361.20
D. P28,456.23
B. P13,633.20
789.
What is the principal amount if the
amount of interest at the end of 2½ year
is P4500 for a simple interest of 6% per
annum?
C. P13,333.20
D. P16,323.20
C. P40,000.00
A time deposit of P110,000 for 31 days
earns P890.39 on maturity date after
deducting the 20% withholding tax on
interest income. Find the rate of interest
per annum.
D. P45,000.00
A. 12.5%
A. P35,000.00
B. P30,000.00
793.
B. 11.95%
C. 12.25%
D. 11.75%
794.
798.
A bank charges 12% simple interest on a
P300.00 loan. How much will be repaid
if the load is paid back in one lump sum
after three years?
Accumulate P5,000.00 for 10 years at
8% compounded monthly.
A. P15,456.75
B. P11,102.61
A. P408.00
C. P14,768.34
B. P551.00
D. P12,867.34
C. P415.00
799.
Accumulate P5,000.00 for 10 years at
8% compounded annually.
D. P450.00
A. P10,794.62
795.
The tag price of a certain commodity is
for 100 days. If paid in 31 days, there is
a 3% discount. What is the simple
interest paid?
B. P8,567.98
C. P10,987.90
D. P7,876.87
A. 12.15%
C. 22.32%
How long will it take P1,000 to amount
to P1,346 if invested at 6% compounded
quarterly?
D. 16.14%
A. 3 years
Accumulate P5,000.00 for 10 years at
8% compounded quarterly.
B. 4 years
A. P12,456.20
D. 6 years
B. 6.25%
796.
B. P13,876.50
800.
C. 5 years
801.
C. P10,345.80
D. P11,040.20
797.
Accumulate P5,000.00 for 10 years at
8% compounded semi-annually.
How long will it take for an investment
to double its amount if invested at an
interest rate of 6% compounded bimonthly?
A. 10 years
B. 12 years
A. P10,955.61
C. 13 years
B. P10,233.67
D. 14 years
C. P9,455.67
802.
D. P11,876.34
If the compound interest on P3,000.00
in 2 years is P500.00, then the
compound interest on P3,000.00 in 4
years is:
806.
How long will it take for an investment
to fivefold its amount if money is worth
14% compounded semiannually?
A. P956.00
A. 11
B. P1,083.00
B. 12
C. P1,125.00
C. 13
D. P1,526.00
D. 14
803.
The salary of Mr. Cruz is increased by
30% every 2 years beginning January
1,1982. Counting from that date, at what
year will his salary just exceed twice his
original salary?
807.
An interest rate of 8% compounded
semiannually is how many percent if
compounded quarterly?
A. 7.81%
A. 1988
B. 7.85%
B. 1989
C. 7.92%
C. 1990
D. 8.01%
D. 1991
804.
805.
If you borrowed P10,000 from a bank
with 18% interest per annum, what is
the total amount to be repaid at the end
of one year?
A man is expecting to receive
P450,000.00 at the end of 7 years. If
money is worth 14% compounded
quarterly, how much is it worth at
present?
A. P11,800.00
A. P125,458.36
B. P19,000.00
B. P147,456.36
C. P28,000.00
C. P162,455.63
D. P10,180.00
D. P171,744.44
What is the effective rate for an interest
rate of 12% compounded continuously?
A. 12.01%
B. 12.89%
C. 12.42%
D. 12.75%
809.
810.
A man has a will of P650,000.00 from
his father, If his father deposited an
amount of P450,000.00 in a trust fund
earning 8% compounded annually, after
how many years will the man receive his
will?
A. 4.55 years
B. 4.77 years
C. 5.11 years
25.
D. 5.33 years
A. P178,313.69
Mr. Adam deposited P120,000.00 in a
bank who offers 8% interest
compounded quarterly. If the interest is
subject to a 14% tax, how much will he
receive after 5 years?
B. P153.349.77
C. P170,149.77
D. P175,343.77
MULTIPLE CHOICE
QUESTIONS in
ENGINEERING MATHEMATICS
By
Diego Inocencio T. Gillesania
Conversion
D. 0°
Problem 1
What is the temperature in degree Celsius of
absolute zero?
A. -32
B. 0
C. 273
D. -273
Problem 6 (ME October 1994)
How many degree Celsius is 80 degrees
Fahrenheit?
A. 26.67
B. 86.4
C. 33.33
D. 16.33
Problem 2 (ME April 1996)
How many degrees Celsius is 100 degrees
Fahrenheit?
A. 37.8 °C
B. 2.667 °C
C. 1.334 °C
D. 13.34 °C
Problem 7(EE October 1990)
What is the absolute temperature of the
freezing point of water in degree Rankine?
A. -32
B. 0
C. 428
D. 492
Problem 3 (ECE November 1997)
A comfortable room temperature is 72 °F.
What is this temperature, expressed in
degrees Kelvin?
A. 263
B. 290
C. 295
D. 275
Problem 8
The angle of inclination of the road is 32.
What is the angle of inclination in mils?
A. 456.23
B. 568.89
C. 125.36
D. 284.44
Problem 4
255 °C is equivalent to:
A. 491 °F
B. 427 °F
C. 173.67 °F
D. 109.67 °F
Problem 9
An angle measures x degrees. What is its
measure in radians?
A. 180° x / π
B. π x / 180°
C. 180° π / x
D. 180° π x
Problem 5
At what temperature will the °C and °F
readings be equal?
A. 40°
B. -40°
C. 32°
Problem 10 (ECE November 1995)
Express 45° in mils.
A. 80 mils
B. 800 mils
C. 8000 mils
D. 80000mils
Problem 11 (ME April 1997)
What is the value in degrees of π radians?
A. 90°
B. 57.3°
C. 180°
D. 45°
Problem 12(CE May 1993)
How many degrees is 3200° mils?
A. 360°
B. 270°
C. 180°
D. 90°
Problem 13 (ECE November 1995)
An angular unit equivalent to 1/400 of the
circumference of a circle is called:
A. mil
B. grad
C. radian
D. degree
Problem 14 (EE October 1994)
Carry out the following multiplication and
express your answer in cubic meters:
3cm×5mm×2m.
A. 3 x 10-3
B. 3 x 10-4
C. 8 x 10-2
D. 8 x 102
Problem 15 (ME April 1994)
Add the following and express in meters:
3m + 2cm + 70mm
A. 2.90 m
B. 3.14 m
C. 3.12 m
D. 3.09 m
Problem 16
One nautical mile is equivalent to:
A. 5280 ft.
B. 6280 ft.
C. 1.256 statute mile
D. 1.854 km
Problem 17 (ME October 1991)
How many square feet is 100 square meters?
A. 328.10
B. 956.36
C. 1075.84
D. 156
Problem 18
A tank contains 1500 gallons of water. What
is the equivalent in cubic liters?
A. 4.256
B. 5.865
C. 6
D. 5.685
Problem 19 (ME October 1994)
How many cubic feet is equivalent to 100
gallons of water?
A. 74.80
B. 1.337
C. 13.37
D. 133.7
Problem 20 (ME April 1998)
How many cubic meters is 100 gallons of
liquid?
A. 0.1638 cu. meters
B. 1.638 cu. meters
C. 0.3785 cu. meters
D. 3.7850 cu. meters
Problem 21 (ME October 1995)
The number of board feet in a plank 3 inches
thick, 1 foot wide, and 20 feet long is:
A. 30
B. 60
C. 120
D. 90
Problem 22
Which of the following is correct?
A. 1horsepower = 746kW
B. 1horsepower = 0.746watts
C. 1 horsepower = 0.746kW
D. 1 horsepower = 546 watts
Problem 23 (ME October 1996)
The acceleration due to gravity in English
unit is equivalent to?
A. 32.2 ft/sec2
B. 3.22 ft/sec2
C. 9.81 ft/sec2
D. 98.1 ft/sec2
Problem 24 (ME April 1999)
The prefix nano is opposite to:
A. mega
B. tera
C. hexa
D. giga
Problem 25(ME October 1996)
10 to the 12th power is the value of the
prefix:
A. giga
B. pico
C. tera
D. peta
Exponents and Radicals
Problem 26
Solve for x: x = -(1/-27)-2/3
A. 9
B. 1/9
C. -9
D. -1/9
Problem 27
Solve for a in the equation: a = 64x4y
A. 4x+3y
B. 43xy
C. 256xy
D. 43x+y
Problem 28
Simplify 3x - 3x-1 - 3x-2
A. 3x-2
B. 33x-3
C. 5×3x-2
D. 13×3x
Problem 29
Which of the following is true?
A. √
B.
√
√
C. √
√
√
5
5
5
5
D. 5 +5 +5 +5 +55=56
Problem 30
Solve for x: x=√
A. -2√
√
√
B. 2√
C. 4
D. 4√
Problem 31
Solve for x: √
√
A. -16/25 & 0
B. 25/16 & 0
C. -25/16 & 0
D. 16/25 & 0
√
Problem 32
Simplify √
A. 5√
√
Problem 38
B. 2√
C.
√
D.
√
Solve for U if U = √
Problem 33
Solve for x: 3x5x+1 = 6x+2
A. 2.1455
B. 2.1445
C. 2.4154
D. 2.1544
Problem 34
Simplify
√
B. 3/7
C. 7/3
D. 49/9
.
A.
B.
C.
D.
Problem 35
(3x)x is equal to:
A.
B. 3xxx
C. 3xx
D. 32x
Problem 36
Solve for x: 37x+1 = 6561.
A. 1
B. 2
C. 3
D. 4
Problem 37
If 3a = 7b, then 3a2/7b2 =
A. 1
A.
B.
C.
D.
√
√
0.723
0.618
0.852
0.453
Problem 39 (ME April 1996)
If x to the ¾ power equals 8, then x equals:
A. -9
B. 6
C. 9
D. 16
Problem 40
If 33y = 1, what is the value of y/33?
A. 0
B. 1
C. Infinity
D. 1/33
Problem 41 (ME April 1998)
Find the value of x that will satisfy the
following expression: √
√
A. x = 3/2
B. x = 18/6
C. x = 9/4
D. none of these
Problem 42 (ME April 1998)
is equal to:
A. 0.048787
B. 0.049001
C. 0.049787
D. 0.048902
Problem 43
B to the m/nth power is equal to:
A. Nth root of b to the m power
B. B to the m+n power
C. 1/n square root of b to the m power
D. B to the m power over n
Problem 44 (ECE April 1993)
Find x from the following equations:
27x = 9y
81y3-x = 243
A. 2.5
B. 2
C. 1
D. 1.5
Problem 45 (ECE April 1990)
Solve for a if (am)(an) = 100000 and
A.
B.
C.
D.
15.85
10
12
12.56
Problem 46 (ECE April 1991)
Simplify
C. -7a
D. 7a
Problem 48
Simplify
A. xy3
B.
C.
D.
Problem 49
Simplify the following:
√
√
√
√
.
A. 4+√
B. 4-√
C. 8+√
D. 8-√
Problem 50
Which of the following is equivalent to
√√
A. √
B. √
C. √
D. √
A.
B.
C.
D.
Problem 47(ECE April 1991)
Simplify the following: 7a+2 – 8(7a+1) + 5(7a)
+ 49(7a-2).
A. -5a
B. 3
FUNDAMENTALS IN ALGEBRA
Problem 51 (ME Board)
Change 0.222… common fraction.
A. 2/10
B. 2/9
C. 2/13
D. 2/7
Problem 52 (ME Board)
Change 0.2272722… to a common fraction.
A. 7/44
B. 5/48
C. 5/22
D. 9/34
Problem 53 (ME Board)
What is the value of 7! or 7 fatorial?
A. 5040
B. 2540
C. 5020
D. 2520
Problem 54 (ME October 1994)
The reciprocal of 20 is:
A. 0.50
B. 20
C. 0.20
D. 0.05
Problem 55
If p is an odd number and q is an even
number, which of the following expressions
must be even?
A. p+q
B. p-q
C. pq
D. p/q
Problem 56 (ECE March 1996)
MCMXCIV is a Roman Numeral equivalent
to:
A. 2974
B. 3974
C. 2174
D. 1994
Problem 57 (ECE April 1998)
What is the lowest common factor of 10 and
32?
A.
B.
C.
D.
320
2
180
90
Problem 58
4xy – 4x2 –y2 is equal to:
A. (2x-y)2
B. (-2x-y)2
C. (-2x+y)2
D. –(2x-y)2
Problem 59
Factor x4 – y2 + y – x2 as completely as
possible.
A. (x2 + y)(x2 + y -1)
B. (x2 + y)(x2 - y -1)
C. (x2 -y)(x2 - y -1)
D. (x2 -y)(x2 + y -1)
Problem 60 (ME April 1996)
Factor the expression x2 + 6x + 8 as
completely as possible.
A. (x+8)(x-2)
B. (x+4)(x+2)
C. (x+4)(x-2)
D. (x-8)(x-2)
Problem 61(ME October 1997)
Factor the expression x3 + 8.
A. (x-2)(x2+2x+4)
B. (x+4)(x2+2x+2)
C. (-x+2)(-x2+2x+2)
D. (x+2)(x2-2x+4)
Problem 62 (ME October 1997)
Factor the expression (x4 – y4) as completely
as possible.
A. (x+y)(x2+2xy+y)
B. (x2+y2)(x2-y2)
C. (x2+y2)(x+y)(x-y)
D. (1+x2)(1+y)(1-y2)
Problem 63(ME October 1997)
Factor the expression 3x3+3x2-18x as
completely as possible.
A. 3x(x+2)(x-3)
B. 3x(x-2)(x+3)
C. 3x(x-3)(x+6)
D. (3x2-6x)(x-1)
Problem 64 (ME April 1998)
Factor the expression 16 – 10x + x2.
A. (x+8)(x-2)
B. (x-8)(x-2)
C. (x-8)(x+2)
D. (x+8)(x+2)
Problem 65
Factor the expression x6-1 as completely as
possible.
A. (x+1)(x-1)(x4+x2-1)
B. (x+1)(x-1)(x4+2x2+1)
C. (x+1)(x-1)(x4-x2+1)
D. (x+1)(x-1)(x4+x2+1)
Problem 66
What are the roots of the equation (x4)2(x+2) = (x+2)2(x-4)?
A. 4 and -2 only
B. 1 only
C. -2 and 4 only
D. 1, -2, and 4 only
Problem 67
If f(x) = x2 + x + 1, then f(x) – f(x-1) =
A. 0
B. x
C. 2x
D. 3
Problem 68
Which of the following is not an identity?
A. (x-1)2 = x2-2x+1
B. (x+3)(2x-2) = 2(x2+2x-3)
C. x2-(x-1)2 = 2x-1
D. 2(x-1)+3(x+1) = 5x+4
Problem 69 (ME October 1997)
Solve for x: 4 +
A.
B.
C.
D.
.
-18 = -18
12 = 12 or -3 = -3
Any value
-27 = -27 or 0 = 0
Problem 70 (ME October 1997)
Solve the simultaneous equations: 3x – y =
6; 9x – y = 12.
A. x = 3; y = 1
B. x = 1; y = -3
C. x = 2; y = 2
D. x = 4; y = 2
Problem 71 (ME April 1998)
Solve algebraically: 4x2 + 7y2 = 32
11y2 – 3x2 = 41
A. y = 4, x =
and y = -4, x = ±1
B. y = +2, x =
and y = -2, x = ±1
C. x = 2, y = and x = -2, y = -3
D. x = 2, y =
and x = 2, y = -2
Problem 72 (CE May 1997)
Solve for w from the following equations:
3x – 2y + w = 11
x + 5y -2w = -9
2x + y - 3w = -6
A. 1
B. 2
C. 3
D. 4
D. 13
Problem 73
When (x+3)(x-4) + 4 is divided by x – k, the
remainder is k. Find the value of k.
A. 4 or 2
B. 2 or -4
C. 4 or -2
D. -4 or -2
Problem 74
Find k in the equation 4x2 + kx + 1 = 0 so
that it will only have one real root.
A. 1
B. 2
C. 3
D. 4
Problem 75
Find the remainder when (x12 + 2) is divided
by (x – √
A. 652
B. 731
C. 231
D. 851
Problem 76 (CE November 1997)
If 3x3 – 4x2y + 5xy2 + 6y3 is divided by (x2 –
2xy + 3y2), the remainder is
A. 0
B. 1
C. 2
D. 3
Problem 77 (CE November 1007 & May
1999)
If (4y3 + 8y + 18y2 – 4) is divided by (2y +
3), the remainder is:
A. 10
B. 11
C. 12
Problem 78 (ECE April 1999)
Given f(x) = (x+3)(x-4) + 4 when divided by
(x-k), the remainder is k. Find k.
A. 2
B. 3
C. 4
D. -3
Problem 79 (EE March 1998)
The polynomial x3 + 4x2 -3x + 8 is divided
by x-5. What is the remainder?
A. 281
B. 812
C. 218
D. 182
Problem 80
Find the quotient of 3x5 – 4x3 + 2x2 + 36x +
48 divided by x3 – 2x2 + 6.
A. -3x2 – 4x + 8
B. 3x2 + 4x + 8
C. 3x2 – 4x – 8
D. 3x2 + 6x + 8
Problem 81
If 1/x = a + b and 1/y = a-b, then x-y is equal
to:
A. 1/2a
B. 1/2b
C. 2a/(a2 – b2)
D. 2b/(a2 – b2)
Problem 82
If x-1/x = 1, find the value of x3 – 1/x3.
A. 1
B. 2
C. 3
D. 4
Problem 83
If 1/x + 1/y = 3 and 2/x – 1/y = 1. Then x is
equal to:
A. ½
B. 2/3
C. ¾
D. 4/3
Problem 84
Simplify the following expression:
.
A.
B.
C.
D.
B. x = (6 + 2√
)/5 or (6 - 2√
)/5
)/5
)/5
y = (-2 + 6√ )/5 or (-2 - 6√ )/5
C. x = (6 + 2√ )/5 or (6 - 2√ )/5
y = (-2 + 6√ )/5 or (-2 - 6√ )/5
D. x = (6 + 2√ )/5 or (6 - 2√ )/5
)/5 or (-6 + 2√
Problem 89 (CE May 1996)
Find the value of A in the equation.
Problem 85
A.
B.
C.
D.
A. x = (-6 + 2√ )/5 or (-6 - 2√
y = (2 + 6√ )/5 or (-2 + 6√
y = (-6+ 2√
2/(x-3)
(x-3)/5
(x+3)/(x-1)
4/(x+3)
If 3x = 4y then
Solve the simultaneous equations:
y - 3x + 4 = 0
y + x2/y = 24/y
is equal to:
¾
4/3
2/3
3/2
A.
B.
C.
D.
2
-2
-1/2
½
Problem 90
Find A and B such that
Problem 86
Simplify: (a+1/a)2-(a-1/a)2.
A. -4
B. 0
C. 4
D. -2/a2
A.
B.
C.
D.
A = -3; B = 2
A = -3; B = -2
A = 3; B = 2
A = 3; B = 2
Problem 91 (ME October 1996)
Resolve
Problem 87 (ECE November 1996)
The quotient of (x5 +32) by (x+2) is:
A. x4 – x3 + 8
B. x3 +2x2 – 8x + 4
C. x4 – 2x3 + 4x2 – 8x + 16
D. x4 + 2x3 + x2 + 16x + 8
Problem 88 (ME April 1996)
into partial fraction.
A.
B.
C.
D.
Problem 92 (ECE April 1998)
)/5
The arithmetic mean of 80 numbers is 55. If
two numbers namely 250 and 850 are
removed what is the arithmetic mean of the
remaining numbers?
A. 42.31
B. 57.12
C. 50
D. 38.62
Problem 93 (ECE April 1998)
The arithmetic mean of 6 numbers is 17. If
two numbers are added to the progression,
the new set of number will have an
arithmetic mean of 19. What are the two
numbers if their difference is 4?
A. 21, 29
B. 23, 27
C. 24, 26
D. 22, 28
Problem 94
If 2x – 3y = x + y, then x2 : y2 =
A. 1:4
B. 4:1
C. 1:16
D. 16:1
Problem 95
If 1/a :1/b : 1/c = 2 : 3 : 4, then (a + b + c) :
(b + c) is equal to:
A. 13:7
B. 15:6
C. 10:3
D. 7:9
D. 14
Problem 97
Find the fourth proportional of 7, 12, and 21.
A. 36
B. 34
C. 32
D. 40
Problem 98 (ECE November 1997)
If (x+3):10 = (3x – 2) : 8, find (2x -1)
A. 1
B. 2
C. 3
D. 4
Problem 99
Solve for x: -4 < 3x - 1 < 11.
A. 1 < x < -4
B. -1< x < 4
C. 1 < x < 4
D. -1 < x < -4
Problem 100
Solve for x: x2 + 4x > 12.
A. -6 > x > 2
B. 6 > x > -2
C. -6 > x > -2
D. 6 > x > 2
Logarithms, Binomial Theorem,
Quadratic Equation
Problem 101
If
Problem 96
Find the mean proportional to 5 and 20.
A. 8
B. 10
C. 12
= 2, what is the value of z?
A.
B.
C.
D.
¼
25
4
5
Problem 102 (EE October 1992)
Solve for x: log 6 + x log 4 = log 4 + log (32
+ 4x)
A. 1
B. 2
C. 3
D. 4
Problem 103
Which of the following cannot be used as a
base of a system of logarithm?
A. e
B. 10
C. 2
D. 1
Problem 104
If log5.21000 = x, what is the value of x?
A. 4.19
B. 5.23
C. 3.12
D. 4.69
Problem 105
Find the value of a in the equation loga2187
= 7/2.
A. 3
B. 6
C. 9
D. 12
Problem 106
If log 2 = x and log 3 = y, find log 1.2.
A. 2x + y
B. 2xy/10
C. 2x + y - 1
D. xy - 1
Problem 107
is equal to:
A.
B.
C.
D.
xy/yx
y log x – x log y
(y log x)/ (x log y)
1
Problem 108
If 10ax+b = P, what is the value of x?
A. (1/a)(log P-b)
B. (1/a) log ( P-b)
C. (1/a) P10-b
D. (1/a) log P10
Problem 109
Find the value of log(aa)a.
A. 2a log a
B. a2 log a
C. a log a2
D. (a log a)a
Problem 110
Solve for x: x = logb a logc d
A. logb a
B. loga c
C. logb c
D. logd a
logd c
Problem 111
Find the positive value of x if log x 36 = 2.
A. 2
B. 4
C. 6
D. 8
Problem 112
Find x if logx 27 + logx 3 = 2.
A. 9
B. 12
C. 8
D. 7
Problem 113
Find a if log2 (a+2) + log2 (a-2) = 5
A. 2
B. 4
C. 6
D. 8
Problem 114
Solve for x if log5 x = 3.
A. 115
B. 125
C. 135
D. 145
B. log7 (-2) = 1/49
C. log7 (1/49) = -2
D. log7 (1/49) = 2
Problem 119 (ME April 1996)
Log of nth root of x equals log of x to the
1/n power and also equal to:
A.
B. n log (x)
C.
D. (n-1) log (x)
Problem 115
Find log P if ln P = 8.
A. 2980.96
B. 2542.33
C. 3.47
D. 8.57
Problem 120 (ME April 1996)
What is the natural logarithm of e to the xy
power?
A. 1/xy
B. 2.718/xy
C. xy
D. 2.718xy
Problem 116
If log8 x = -n, then x is equal to:
A. 8n
B. 1/8-n
C. 1/8n
D. 81/n
Problem 121 (ME April 1997)
What expression is equivalent to log x – log
(y + z)?
A. log x + log y + log z
B. log [ x/(y + z) ]
C. log x –log y –log z
D. log y + log (x + z)
Problem 117
If 3 log10 x – log10 y = 0, find y in terms of
x.
A. y = √
B. y = √
C. y = x3
D. y = x
Problem 118
Which of the following is correct?
A. -2 log 7 = 1/49
Problem 122 (ME April 1997)
What is the value of log base 10 of 10003.3?
A. 9.9
B. 99.9
C. 10.9
D. 9.5
Problem 123
If logx 2 + log2 x = 2, then the value of x is:
A. 1
B. 2
C. 3
D. 4
Problem 124 (CE November 1997)
Log6 845 =?
A. 4.348
B. 6.348
C. 5.912
D. 3.761
Problem 125 (CE May 1998, similar with
November 1998)
The logarithms of the quotient and the
product of two numbers are 0.352182518
and 1.556302501, respectively. Find the first
number?
A. 9
B. 10
C. 11
D. 12
Problem 126
The sum of two logarithms of two numbers
is 1.748188 and the difference of their
logarithms is -0.0579919. One of the
numbers is:
A. 9
B. 6
C. 8
D. 5
Problem 127 (CE November 199)
Solve for y: y = ln
A.
B.
C.
D.
.
2
x
-2
x-2
What is the value of (log 5 to the base 2) +
(log 5 to the base 3)?
A. 3.97
B. 7.39
C. 9.37
D. 3.79
Problem 129 (ME October 1997)
The logarithm of negative number is:
A. irrational number
B. real number
C. imaginary number
D. complex number
Problem 130(ME April 1998)
38.5 to the x power = 6.5 to the x-2 power,
solve for x using logarithms.
A. 2.70
B. 2.10
C. -2.10
D. -2.02
Problem 131 (CE November 1996)
Find the 6th term of the expansion of (1/2a –
3)16.
A. B. C. D. Problem 132 (ECE April 1998)
In the expansion of (x+4y) 12, the numerical
coefficient of the 5th term is.
A. 253440
B. 126720
C. 63360
D. 506880
Problem 128 (ECE April 1998)
Problem 133
The middle term in the expansion of (x2 –
3)8 is:
A. -70x8
B. 70x8
C. -5760x8
D. 5760x8
Problem 134
The term involving x9 in the expansion of
(x2 + 2/x)12 is:
A. 25434x9
B. 52344x9
C. 25344x9
D. 23544x9
Problem 135
The constant term in the expansion of ( x +
15
) is:
A.
B.
C.
D.
3003
5005
6435
7365
Problem 136
Find the sum of the coefficients in the
expansion of (x + 2y –z) 8.
A. 256
B. 1024
C. 1
D. 6
Problem 137
Find the sum of the coefficients in the
expansion of (x + 2y + z) 4 (x + 3y) 5 is:
A. 524288
B. 65536
C. 131072
D. 262 144
Problem 138 (ECE April 1995)
What is the sum of the coefficients in the
expansion of (x + y -z) 8 is:
A. less than 2
B. above 10
C. from 2 to 5
D. from 5 to 10
Problem 139 (ECE November 1995)
What is the sum of the coefficients of the
expansion of (2x -1)20?
A. 1
B. 0
C. 215
D. 225
Problem 140
In the quadratic equation Ax2 + Bx + C = 0,
the product of the roots is:
A. C/A
B. –B/A
C. –C/A
D. B/A
Problem 141
If ¼ and -7/2 are the roots of the quadratic
equation Ax2 + Bx + C = 0, what is the
value of B?
A. -28
B. 4
C. -7
D. 26
Problem 142
In the equation 3x2 + 4x + (2h – 5) = 0, find
h if the product of the roots is 4.
A. -7/2
B. -10/2
C. 17/2
D. 7/2
D. 6
Problem 143
If the roots of ax2 + bx + c = 0, are u and v,
then the roots of cx2 + bx + a = 0 are:
A. u and v
B. –u and v
C. 1/u and 1/v
D. -1/u and -1/v
Problem 144
If the roots of the quadratic equation ax2 +
bx + c = 0 are 3 and 2 and a, b, and c are all
whole numbers, find a + b + c.
A. 12
B. -2
C. 2
D. 6
Problem 145 (ECE March 1996)
The equation whose roots are the reciprocals
of the roots of 2x2 – 3x – 5 = 0 is:
A. 5x2 + 3x – 2 = 0
B. 3x2 – 5x – 3 = 0
C. 5x2 – 2x – 3 = 0
D. 2x2 – 5x -3 = 0
Problem 146 (ECE November 1997)
The roots of a quadratic equation are 1/3 and
¼. What is the equation?
A. 12x2 + 7x + 1 = 0
B. 12x2 + 7x – 1 = 0
C. 12x2 – 7x + 1 = 0
D. 12x2 – 7x – 1 = 0
Problem 147
Find k so that the expression kx2 – 3kx + 9
is a perfect square.
A. 3
B. 4
C. 12
Problem 148 (EE October 1990)
Find k so that 4x2+kx+1=0 will only have
one real solution.
A. 1
B. 4
C. 3
D. 2
Problem 149
The only root of the equation x2 – 6x + k = 0
is:
A. 3
B. 2
C. 6
D. 1
Problem 150
Two engineering students are solving a
problem leading to a quadratic equation.
One student made a mistake in the
coefficient of the first-degree term, got roots
of 2 and -3. The other student made a
mistake in the coefficient of the constant
term got roots of -1 and 4. What is the
correct equation?
A. x2 – 6x – 3 = 0
B. x2 + 6x + 3 = 0
C. x2 + 3x + 6 = 0
D. x2 – 3x – 6 = 0
Age, Mixture, Work, Clock, Number
Problem
Problem 151
Two times the father’s age is 8 more than
six times his son’s age. Ten years ago, the
sum of their ages was 44. The age of the son
is:
A.
B.
C.
D.
49
15
20
18
Problem 152
Peter’s age 13 years ago was 1/3 of his age 7
years hence. How old is Peter?
A. 15
B. 21
C. 23
D. 27
Problem 153
A man is 41 years old and in seven years he
will be four times as old as his son is at that
time. How old is his son now?
A. 9
B. 4
C. 5
D. 8
Problem 154
A father is three times as old as his son.
Four years ago, he was four times as old as
his son was at that time. How old is his son?
A. 36 years
B. 24 years
C. 32 years
D. 12 years
Problem 155
The ages of the mother and her daughter are
45 and 5 years, respectively. How many
years will the mother be three times as old
as her daughter?
A. 5
B. 10
C. 15
D. 20
Problem 156
Mary is 24 years old. Mary is twice as old as
Ana was when Mary was as old as Ana is
now. How old is Ana? (ECE November
1995)
A. 16
B. 18
C. 19
D. 20
Problem 157
The sum of the parent’s ages is twice the
sum of their children’s ages. Five years ago,
the sum of the parent’s ages is four times the
sum of their children’s ages. In fifteen years
the sum of the parent’s ages will be equal to
the sum of their children’s ages. How many
children were in the family?
A. 2
B. 3
C. 4
D. 5
Problem 158
Two thousand kilogram of steel containing
8% of nickel is to be made by mixing stell
containing 14% nickel with another steel
containing 6% nickel. How much of the
steel containing 14% nickel is needed?
A. 1500 kg
B. 800 kg
C. 750 kg
D. 500kg
Problem 159
A 40-gram alloy containing 35% gold is to
be melted with a 20-gram alloy containing
50% gold. How much percentage of gold is
the resulting alloy?
A.
B.
C.
D.
40%
30%
45%
35%
Problem 160
In what radio must a peanut costing P240.00
per kg. be mixed with a peanut costing
P340.00 per kg so that the profit of 20% is
made by selling the mixture at 360.00 per
kg?
A. 1:2
B. 3:2
C. 2:3
D. 3:5
Problem 161
A 100-kilogram salt solution originally 4%
by weight. Salt in water is boiled to reduce
water content until the concentration is 5%
by weight salt. How much water is
evaporated?
A. 10
B. 15
C. 20
D. 25
Problem 162
A pound of alloy of lead and nickel weights
14.4 ounces in water, where lead losses 1/11
of its weight and nickel losses 1/9 of its
weight. How much of each metal is in alloy?
A. Lead = 7.2 ounces; Nickel = 8.8
ounces
B. Lead = 8.8 ounces; Nickel = 7.2
ounces
C. Lead = 6.5 ounces; Nickel = 5.4
ounces
D. Lead = 7.8 ounces; Nickel = 4.2
ounces
Problem 163
An alloy of silver and gold weighs 15 oz. in
air and 14 oz. in water. Assuming that silver
losses 1/10 of its weight in water and gold
losses 1/18 of its weight, how many oz. at
each metal are in the alloy?
A. Silver = 4.5 oz.; Gold = 10.5 oz.
B. Silver = 3.75 oz.; Gold = 11.25 oz.
C. Silver = 5 oz.; Gold = 10 oz.
D. Silver = 7.8 oz.; Gold = 4.2 oz.
Problem 164(ME April 1998)
A pump can pump out a tank in 11 hours.
Another pump can pump out the same tank
in 20 hours. How long it will take both
pumps together to pump out the tank?
A. ½ hour
B. ½ hour
C. 6 hours
D. 7 hours
Problem 165
Mr. Brown can wash his car in 15 minutes,
while his son John takes twice as long as the
same job. If they work together, how many
minutes can they do the washing?
A. 6
B. 8
C. 10
D. 12
Problem 166
One pipe can fill a tank in 5 hours and
another pipe can fill the same tank in 4
hours. A drainpipe can empty the full
content of the tank in 20 hours. With all the
three pipes open, how long will it take to fill
the tank?
A.
B.
C.
D.
2 hours
2.5 hours
1.92 hours
1.8 hours
Problem 167
A swimming pool is filled through its inlet
pipe and then emptied through its outlet pipe
in a total of 8 hours. If water enters through
its inlet and simultaneously allowed to leave
through its outlet, the pool is filled in 7 ½
hours. Find how long will it take to fill the
pool with the outlet closed.
A. 6
B. 2
C. 3
D. 5
Problem 168
Three persons can do a piece of work alone
in 3 hours, 4 hours and 6 hours respectively.
What fraction of the job can they finish in
one hour working together?
A. ¾
B. 4/3
C. ½
D. 2/3
Problem 169
A father and his son can dig a well if the
father works 6 hours and his son works 12
hours or they can do it if the father works 9
hours and son works 8 hours. How long will
it take for the son to dig the well alone?
A. 5 hours
B. 10 hours
C. 15 hours
D. 20 hours
Problem 170
Peter and Paul can do a certain job in 3
hours. On a given day, they work together
for 1 hour then Paul left and Peter finishes
the rest work in 8 more hours. How long
will it take for Peter to do the job alone?
A. 10 hours
B. 11 hours
C. 12 hours
D. 13 hours
Problem 171 (ECE November 1995)
Pedro can paint a fence 50% faster than Juan
and 20% faster than Pilar and together they
can paint a given fence in 4 hours. How long
will it take Peter to paint the same fence if
he had to work alone?
A. 10 hrs.
B. 11hrs.
C. 13hrs.
D. 15hrs.
Problem 172
Nonoy can finish a certain job in 10 days if
Imelda will help for 6 days. The same work
can be done by Imelda in 12 days if Nonoy
helps for 6 days. If they work together, how
long will it take for them to do the job?
A. 8.9
B. 8.4
C. 9.2
D. 8
Problem 173
A pipe can fill up a tank with the drain open
in three hours. If the pipe runs with the drain
open for one hour and then the drain is
closed it will take 45 more minutes for the
pipe to fill the tank. If the drain will be
closed right at the start of filling, how long
will it take for the pipe to fill the tank?
A. 1.15hrs.
B. 1.125hrs
C. 1.325hrs.
D. 1.525hrs.
Problem 174
Delia can finish a job in 8 hours. Daisy can
do it in 5 hours. If Delia worked for 3 hours
and then Daisy was asked to help her finish
it, how long will Daisy have to work with
Delia to finish the job?
A. 2/5 hours
B. 25/14 hours
C. 28 hours
D. 1.923 hours
Problem 175 (CE November 1998)
A job could be done by eleven workers in 15
days. Five workers started the job. They
were reinforced with four more workers at
the beginning of the 6th day. Find the total
number of days it took them to finish the
job.
A. 22.36
B. 21.42
C. 23.22
D. 20.56
Problem 176
On one job, two power shovels excavate
20000m3 of earth, the larger the shovel
working for 40 hours and the smaller shovel
for 35 hours. Another job, they removed
40000m3 with the larger shovel working for
70 hours and the smaller working 90 hours.
How much earth can the larger shovel move
in one hour?
A. 173.91
B. 347.83
C. 368.12
D. 162.22
Problem 177 (EE April 1996)
A and B can do a piece of work in 42 days,
B and C in 31 days, and A and C in 20 days.
Working together, how many days can all of
them finish the work?
A. 18.9
B. 19.4
C. 17.8
D. 20.9
Problem 178
Eight men can dig 150 ft of trench in 7hrs.
Three men can backfill 100ft of the trench in
4hrs. The time it will take 10 men to dig and
fill 200 ft of trench is:
A. 9.867hrs.
B. 9.687hrs.
C. 8.967hrs.
D. 8.687hrs.
Problem 179
In two hours, the minute hand of the clock
rotates through an angle of :
A. 45°
B. 90°
C. 360°
D. 720°
Problem 180
In one day (24 hours), how many times will
the hour hand and minute hand of a
continuously driven clock be together
A. 21
B. 22
C. 23
D. 24
Problem 181
How many minutes after 3:00 will the
minute hand of the clock overtakes the hour
hand?
A. 14/12 minutes
B. 16-11/12 minutes
C. 16-4/11 minutes
D. 14/11 minutes
Problem 182
How many minutes after 10:00 o’clock will
the hands of the clock be opposite of the
other for the first time?
A. 21.41
B. 22.31
C. 21.81
D. 22.61
Problem 183
What time between the hours of 12:00 noon
and 1:00 pm would the hour hand and the
minute hand of a continuously driven clock
be in straight line?
A. 12:33 pm
B. 12:30 pm
C. 12:37 pm
D. 12:287 pm
Problem 184 (GE February 1997)
At what time after 12:00 noon will the hour
hand and the minute hand of a clock first
form a n angle of 120°?
A. 21.818
B. 12:21.818
C. 21.181
D. 12:21.181
Problem 185 (GE February 1994)
From the time 6:15 PM to the time 7:45 PM
of the same day, the minute hand of a
standard clock describes an arc of:
A. 360°
B. 120°
C. 540°
D. 720°
Problem 186
It is now between 3 and 4 o’clock and in
twenty minutes the minute hand will be as
much as the hour-hand as it is now behind it.
What is the time now?
A. 3:06.06
B. 3:07.36
C. 3:09.36
D. 3:08.36
Problem 187 (EE October 1990)
A man left his home at past 3:00 o’clock PM
as indicated in his wall clock. Between two
to three hours after, he returned home and
noticed that the hands of the clock
interchanged. At what time did he left his
home?
A. 3:27.27
B. 3:31.47
C. 3:22.22
D. 3:44.44
Problem 188
The sum of the reciprocals of two numbers
is 11. Three times the reciprocal of one of
the numbers is three more than twice the
reciprocal of the other number. Find the
numbers.
A. 5 and 6
B. 7 and 4
C. 1/5 and 1/6
D. 1/7 and ¼
Problem 189
If a two digit number has x for its unit’s
digit and y for its ten’s digit, represent the
number.
A. yx
B. 10y + x
C. 10x + y
D. x + y
Problem 193
Twice the middle digit of a three-digit
number is the sum of the other two. If the
number is divided by the sum of its digit, the
answer is 56 and the remainder is 12. If the
digits are reversed, the number becomes
smaller by 594. Find the number.
A. 258
B. 567
C. 852
D. 741
Problem 190
One number if five less than the other
number. If their sum is 135, what are the
numbers?
A. 70&75
B. 60&65
C. 65&70
D. 75&80
Problem 194
The product f three consecutive integers is
9240. Find the third integer.
A. 20
B. 21
C. 22
D. 23
Problem 191
In a two-digit number, the unit’s digit is 3
greater than the ten’s digit. Find the number
if it is 4 times as large as the sum of its
digits.
A. 47
B. 58
C. 63
D. 25
Problem 195
The product if two numbers is 1400. If three
(3) is subtracted from each number, their
product becomes 1175. Find the bigger
number.
A. 28
B. 50
C. 32
D. 40
Problem 192
Find two consecutive even integers such that
the square of the larger is 44 greater than the
square of the smaller integer.
Problem 196
The sum of the digits of the three-digit
number is 14. The hundreds digit being 4
times the units digit. If 594 is subtracted
from the number, the order of the digits will
be reversed. Find the number.
A. 743
B. 563
C. 653
D. 842
A.
B.
C.
D.
10&12
12&14
8&10
14&16
Problem 197 (ECE March 1996)
The sum of two numbers is 21, and one
number is twice the other. Find the numbers.
A. 7 and 14
B. 6 and 15
C. 8 and 13
D. 9 and 12
Problem 198 (ECE March 1996)
Ten less than four times a certain number is
14. Determine the number.
A. 4
B. 5
C. 6
D. 7
Problem 199 (ECE November 1997)
The denominator of a certain fraction is
three more than twice the numerator. If 7 is
added to both terms of the fraction, the
resulting fraction is 3/5. Find the original
fraction.
A. 8/5
B. 5/13
C. 13/5
D. 3/5
Problem 200
Three times the first of the three consecutive
odd integers is three more than twice the
third. Find the third integer.
A. 9
B. 11
C. 13
D. 15
Motion Variation, Percent, Miscellaneous
Problems
Problem 201
Nonoy left Pikit to drive to Davao at 6:15
PM and arrived at 11:45 PM averaged 30
mph and stopped 1 hour for dinner, how far
is Davao from Pikit.
A. 128
B. 135
C. 160
D. 256
Problem 202
A man fires a target 420 m away hears the
bullet strikes to 2 second after he pulled the
trigger. An observer 525 m away from the
target and 455 m from the man heard the
bullet strike the target one second after he
heard the report of the rifle. Find the
velocity of the bullet.
A. 525 m/s
B. 360 m/s
C. 350 m/s
D. 336 m/s
Problem 203
A man travels in a motorized banca at rate
of 12 kph from his barrio to the poblacion
and come back to his barrio at the rate of 10
kph. If his total time of travel back and forth
is 3 hours and 10 minutes, the distance from
the barrio to the poblacion is :
A. 17.27 km
B. 17.72 km
C. 12.77 km
D. 17.32 km
Problem 204
It takes Michael 60 seconds to run around a
440-yard track. How long does it take
Jordan to run around the track if they meet
in 32 second after they start together in a
race around the track in opposite direction?
A. 58.76 seconds
B. 68.57 seconds
C. 65.87 seconds
D. 86.57 seconds
Problem 205
Juan can walk from his home to his office at
the rate of 5 mph and back at the rate 2 mph.
What is his average speed in mph?
A. 2.86
B.3.56
C.4.12
D.5.89
Problem 206
Kim and Ken traveled at the same time at
the rate of20m/min,from the same pointon a
circular track of radius 600 m. If Kim walks
along a circumference and Kim towards the
center,find their distance after 10 minutes.
A.193 m
B.202 m
C.241 m
D.258 m
Problem 207
Two ferryboats ply back and forth across a
river with constant but different speeds,
turning at the river banks without loss of
time. They leave the opposite shores at the
same instant, meet for the first time 900
meters from one shore, and meet for the
second time 500 meters from the opposite
shore. What is the width of the river?
A. 1500 m
B. 1700 m
C. 2000 m
D. 2200 m
Problem 208 (CE May 1998)
A boat takes 2/3 as much time to travel
downstream from C to D, as to return, If the
rate of the river’s current is 8 kph, what is
the speed of the boat in still water?
A. 38
B. 39
C. 40
D. 41
Problem 209 (ECE November 1998)
A man rows downstream at the rate of 5mph
and upstream at the rate of 2mph. How far
downstream should he go if he is to return in
7/4 hours after leaving?
A. 2 mi
B. 3.5 mi
C. 3 mi
D. 2.5 mi
Problem 210 (EE April 1997)
A jogger starts a course at a steady rate of
8kph. Five minutes later, a second jogger the
same course at 10 kph. How long will it take
for the second jogger to catch the first?
A. 20 min
B. 25 min
C. 30 min
D. 35 min
Problem 211 (CE May 1999)
At 2:00 pm, an airplane takes off at 340mph
on an aircraft carrier. The aircraft carrier
moves due south at 25kph in the same
direction as the plane. At 4:05 pm, the
communication between the plane and
aircraft carrier was lost. Determine the
communication range in miles between the
plane and the carrier.
A. 656 miles
B. 785 miles
C. 557 miles
D. 412 miles
Problem 212
A boat going across a lake 8km wide
proceed 2 km at a certain speed and then
completes the trip at a speed 1/2kph faster.
By doing this, the boat arrives 10 minutes
earlier than if the original speed had been
maintained. Find the original speed of the
boat.
A. 2 kph
B. 4 kph
C. 9 kph
D. 5 kph
Problem 213 (CE May 1993)
Given that w varies directly as the product
of x and y and inversely as the square of z
and that w=4 when x=2, y=6, and z=3. Find
w when x=1, y=4 and z=2.
A. 4
B. 2
C. 1
D. 3
Problem 214 (ECE November 1993)
If x varies directly as y and inversely as z,
and x=14 when y=7 and z=2, find x, when
z=4 and y=16.
A. 14
B. 4
C. 16
D. 8
Problem 215
The electrical resistance of a cable varies
directly as its length and inversely as the
square of its diameter. If a cable 600 meters
long and 25 mm in diameter has a resistance
of 0.1 ohm, find the length of the cable 75
mm in diameter with resistance of 1/6 ohm.
A. 6000 m
B. 7000 m
C. 8000 m
D. 9000 m
Problem 216
The electrical resistance offered by an
electric wire varies directly as the length and
inversely as the square of the diameter of the
wire. Compare the electrical resistance
offered by two pieces of wire of the same
material, one being 100 m long and 5 mm
diameter, and the other is 50 m long and 3
mm in diameter.
A. R1 = 0.57 R2
B. R1 = 0.72 R2
C. R1 = 0.84 R2
D. R1 = 0.95 R2
Problem 217
The time required for an elevator to lift a
weight varies directly with the weight and
the distance through which it is to be lifted
and inversely as the power of the motors. If
it takes 20 seconds for a 5-hp motor to lift
50 lbs. through 40 feet, what weight can an
80-hp motor lift through a distance of 40
feet within 30 seconds?
A. 1000 lbs.
B. 1150 lbs.
C. 1175 lbs.
D. 1200 lbs.
Problem 218 (ECE November 1995)
The time required by an elevator to lift a
weight, vary directly with the weight and the
distance through which it is to be lifted and
inversely as the power of the motor. If it
takes 30 seconds for a 10-hp motor to lift
100lbs through 50 feet, what size of motor is
requires to lift 800 lbs. in 40 seconds
through a distance of 40 feet?
A. 48 hp
B. 50 hp
C. 56 hp
D. 58 hp
Problem 219
In a certain department store, the salary of
saleslady is partly constant and varies as the
value of her sales for the month, when the
value of her sales for the month is
P10000.00, her salary for that month is
P900.00. When her sales goes up to P
2000.00 her monthly salary goes up to
P1000.00. What must be the value of her
sales for the month so that her salary for that
month will be P2000.00?
A. P25000.00
B. P28000.00
C. P32000.00
D. P36000.00
Problem 220
A man sold 100 eggs, eighty of them were
sold at gain of 30% while the twenty eggs
were sold at a loss of 40%. What is the
percentage gain or loss of the whole eggs?
A. 14%
B. 15%
C. 16%
D. 17%
Problem 221
The population of the country increases 5%
each year. Find the percentage it will
increase in three years.
A. 5%
B. 15%
C. 15.15%
D. 15.76%
Problem 222
Pedro bought two cars, one for P600000.00
and the other for P400000.00. He sold the
first at a gain of 10% and the second at a
loss of 12%. What was his total percentage
gain or loss?
A. 6% gain
B. 0% gain
C. 1.20% gain
D. 6% loss
Problem 223
A grocery owner raises the prices of his
goods by 10%. Then he starts his Christmas
sale by offering the customers a 10%
discount. How many percent of discount
does the customers actually get?
A. nothing
B. 1% discount
C. 9% discount
D. they pay 1% more
Problem 224
Kim sold a watch for P3500.00 at a loss of
30% on the cost price. Find the
corresponding loss or gain if he sold it for
P5050.00.
A. 1% loss
B. 10% loss
C. 1% gain
D. 10% gain
Problem 225
By selling balut at P5.00 each, a vendor
gains 20%. The cost price of egg rises by
12.5%. If he sells the balut at the same price
as before, find his new gain in percent.
A. 7.5%
B. 5%
C. 8%
D. 6.25%
Problem 226
The enrollment at college A and college B
both grew up by 8% from 1980 to 1985. If
the enrollment in college A grew up by 800
and the enrollment in college B grew up by
840, the enrollment at college B was how
much greater than the enrollment in college
A in 1985?
A. 650
B. 504
C. 483
D. 540
Problem 227
A group consists of n boys and n girls. If
two of the boys are replaced by two other
girls, then 49% of the group members will
be boys. Find the value of n.
A. 100
B. 49
C. 50
D. 51
Problem 228
On his Christmas Sale, a merchant marked a
pair of slipper P180.00, which is 20% off the
normal retail price. If the retail price is 50%
higher than the whole sale price, what is the
wholesale price of the slipper?
A. P18.00
B. P17.00
C. P15.00
D. P22.50
Problem 229
A certain XEROX copier produces 13
copies every 10 seconds. If the machine
operates without interruption, how many
copies will it produce in an hour?
A. 780
B. 46800
C. 1835
D. 4680
Problem 230
At a certain printing plant, each of the
machines prints 6 newspapers every second.
If all machines work together but
independently without interruption, how
many minutes will it take to print the entire
18000 newspapers? ( Hint: let x = number of
machines)
A. 50x
B. 3000/x
C. 50/x
D. 3000x
Problem 231 (ME April 1996)
A manufacturing firm maintains one product
assembly line to produce signal generators.
Weekly demand for the generators is 35
units. The line operates for 7 hours per day,
5 days per week. What is the maximum
production time per unit in hours required
for the line to meet the demand?
A. 1 hour
B. 0.75 hour
C. 3 hours
D. 2.25 hours
Problem 232
Of the 316 people watching a movie, there a
re 78 more children than women and 56
more women than men. The number of men
in the movie house is:
A. 176
B. 98
C. 42
D. 210
Problem 233
A certain department store has an inventory
of Q units of a certain product at time t=0.
The store sells the product at a steady rate of
Q/A units per week, and exhausts the
inventory in A weeks. The amount of
product in inventory at any time t is:
A. Q – (Q/A) t
B. Q + (Q/A) t
C. Qt – Q/A
D. Qt – (Q/A) t
Problem 234 (ECE March 1996)
A merchant has three items on sale: namely,
a radio for P50, a clock for P30, and a
flashlight for P1. At the end of the day, she
has sold a total of 100 of three items and has
taken exactly 1000 on the total sales. How
many radios did he sale?
A. 80
B. 4
C. 16
D. 20
Problem 235
The price of 8 calculators ranges from P200
to P1000.If their average price is P950,what
is the lowest possible price of any one of
the calculators?
A. 500
B. 550
C. 600
D. 650
Problem 236
A deck of 52 playing cards is cut into two
piles. The first pile contains 7 times as many
black cards as red cards. The second pile
contains the number of red cards that is an
exact multiple as the number of black cards.
How many cards are there in the first pile.
A. 14
B. 15
C. 16
D. 17
Problem 237 (ECE November 1997)
The population of the Philippines doubled in
the last 30 years from 1967 to
1997.Assuming that the rate of population
rate increase will remain the same in what
year wills the population triple?
A. 2030
B. 2027
C. 2021
D. 2025
Problem 238
Determine the unit digit in the expansion of
3855.
A. 3
B. 9
C. 7
D. 1
Problem 239 (ECE April 1998)
Find the 1987th digit in the decimal
equivalent of 1785/9999 starting from the
decimal point.
A. 1
B. 7
C. 8
D. 5
Problem 240
Find the sum of all positive integral factors
of 2048.
A. 4095
B. 3065
C. 4560
D. 1254
Problem 241
In how many ways can two integers be
selected from the numbers 1,2,3,…50 so that
their difference is exactly 5?
A. 50
B. 5
C. 45
D. 41
Problem 242
A box contains 8 balls, 6 black balls.a8 red
balls, and 13 yellow balls. How many balls
must be drawn to ensure that there will be
three balls of the same color?
A. 8
B. 9
C. 10
D. 11
Problem 243
A shore sells 10 different sizes of shoes,
each in both high-cut and low-cut variety,
each either rubber or leather, and each with
white or black color. How many different
kinds of shoes does he sell?
A. 64
B. 80
C. 72
D. 92
Problem 244(ME October 1999)
An engineer was told that a survey had been
made on a certain rectangular field but the
dimension had been lost .An assistant
remembered that if the field had been 100 ft
longer and 25 ft narrower, the area would
have been increased by 2500 sq. ft, and that
if it had been 100 ft shorter 50 ft wider, the
area would have been decreased 5000 sq.ft.
What was the area of the field?
A. 25.000 ft2
B. 15,000 ft2
C. 20,000 ft2
D. 22,000 ft2
Problem 245 (EE April 1994)
A 10-meter tape is 5 mm short. What is the
correct length in meters?
A. 9.995 m
B. 10.05 m
C. 9.95 m
D. 10.005 m
Problem 246 (ME OCTOBER 1997)
The distance between two points measured
with a steel tape was recorded as 916.58 ft.
later. The tape was checked and to be only
99.9 ft long. What is the true distance
between the points?
A. 035.66 ft
B. 966.15 ft
C. 955.66 ft
D. 915.66 ft
Problem 247 (ME April 1996)
A certain steel tape is known to be 100000
feet long when the temperature of 70 .
When the tape is at a temperature of 10 ,
what reading corresponds to a distance of 90
ft? Coefficient of linear expansion of the
tape is 5.833 10-6 per .
A. 85.935
B. 88.031
C. 90.031
D. 93.031
Problem 248 (ME April 1996)
A line was measured with a steel tape when
the temperature was 30 . The measured
length of the line was found to be 1,256.271
feet. The tape was afterwards tasted when
the temperature was 10 and it was found
to be 100.042 feet long. What was the true
length of the line if the coefficient of
expansion of the tape was 0.000011 per ?
A. 1,275.075 feet
B. 1,375.575 feet
C. 1,256.547 feet
D. 1,249.385 feet
Problem 249 (ME April 1997)
The standard deviation of the numbers 1, 4,
&7 is:
A. 2.3567
B. 2.4495
C. 3.2256
D. 3.8876
Problem 250
Three cities are connected by roads forming
a triangle, all of different lengths. It is 30 km
around the circuit. One of the roads is 10 km
long and the longest is 10 km longer than
the shortest. What is the length of the
longest road?
A. 5 km
B. 10 km
C. 15 km
D. 20 km
Progression, Matrix, Determinant, Venn
diagram
Problem 251 (ECE November 1996)
How many terms of the sequence -9, -6, -3
… must be taken so that the sum is 66?
A. 13
B. 12
C. 4
D. 11
Problem 252 (CE November 1997)
The sum of the progression 5, 8, 11, 14 ….
is 1025. How many terms are there?
A. 22
B. 23
C. 24
D. 25
Problem 253 (CE May 1998)
There are seven arithmetic means between 3
and 35. Find the sum of all terms.
A. 169
B. 171
C. 167
D. 173
Problem 254 (CE May 1999)
There are line (9) arithmetic means between
11 and 51. The sum of the progression is:
A.
B.
C.
D.
279
341
376
254
Problem 255
The sum of all even numbers from 0 to 420
is:
A. 43410
B. 44300
C. 44310
D. 44130
Problem 256 (CE May 1997)
Which of the following numbers should be
changed to make all the numbers form an
arithmetic progression when properly
arranged?
A. 27/14
B. 33/28
C. 45/28
D. 20/14
Problem 257
The first term of an arithmetic progression
(A.P.) is 6 and the 10th term is 3 times the
second number. What is the common
difference?
A. 1
B. 2
C. 3
D. 4
Problem 258
The sum of five arithmetic means between
34 and 42 is:
A. 150
B. 160
C. 190
D. 210
Problem 259
The positive values of a so that 4x, 5x + 4,
3x2 – 1will be in arithmetic progression is:
A. 2
B. 3
C. 4
D. 5
Problem 260
Solve for x if x + 3x + 5x + 7x + … + 49x =
625
A. ¼
B. ½
C. 1
D. 1 ¼
Problem 261
The 10th term of the series a, a-b, a-2b, … is:
A. a-6b
B. a-9b
C. 2a-b
D. a+9b
Problem 262
If the sum of the first 13 terms of two
arithmetic progressions are in the ratio 7:3,
find the ratio of their corresponding 7th term.
A. 3:7
B. 1:3
C. 7:3
D. 6:7
Problem 263
If 1/x, 1/y, 1/z are in arithmetic progression,
then y is equal to:
A. X-z
B. ½(x+2z)
C. (x+z)/2xz
D. 2xz/(x+z)
C. 1/7
D. 1/9
Problem 264 (ECE November 1997)
Find the 30th term of the A.P. 4, 7, 10 …
A. 88
B. 91
C. 75
D. 90
Problem269 (ECE November 1995)
Find the fourth term of the progression ½,
0.2, 0.125, …
A. 0.102
B. 1/10
C. 1/11
D. 0.099
Problem 265 (ECE November 1997)
Find the 100th term of the sequence 1.01,
1.00, 0.99….
A. 0.05
B. 0.04
C. 0.03
D. 0.02
Problem 270
The 10th term of the progression 6/4, 4/3,
3/2, … is:
A. 12
B. 10/3
C. 12/3
D. 13/3
Problem 266
The sum of all numbers between 0 and
10000 which is exactly divisible by 77 is:
A. 546546
B. 645568
C. 645645
D. 645722
Problem 271 (ME October 1997)
The geometric mean of 4 and 64 is:
A. 48
B. 16
C. 34
D. 24
Problem 267 (ME April 1998)
What is the sum of the following finite
sequence of terms? 18, 25, 32, 39, .., 67.
A. 234
B. 181
C. 213
D. 340
Problem 268
Find x in the series: 1, 1/3, 0.2, x.
A. 1/6
B. 1/8
Problem 272 (ME October 1997)
The geometric mean of a nd b is:
A.
B.
C.
D.
√
(a+b)/2
1/b
ab/2
Problem 273 (CE May 1998)
Determine the sum of the infinite geometric
series of 1, -1/5,+1/25, …?
A. 4/5
B. 5/7
C. 4/6
D. 5/6
D. -6
Problem 274
There are 6 geometric means between 4 and
8748. Find the sum of all terms.
A. 13120
B. 15480
C. 10250
D. 9840
Problem 275 (ECE April 1998)
Find the sum of the infinite progression 6, 2, -2/3 …
A. 5/2
B. 9/2
C. 7/2
D. 11/2
Problem 276 (ECE April 1998)
Find the sum of the first 10 terms of the
Geometric Progression 2, 4, 8, 16 …
A. 1023
B. 2046
C. 1596
D. 225
Problem 277
The 1st, 4th, 8th terms of an A.P. are
themselves geometric progression (G.P.).
What is the common ratio of the G.P.?
A. 4/3
B. 5/3
C. 2
D. 7/3
Problem 278
Determine x so that x, 2x+7, 10x-7 will
form a geometric progression.
A. -7
B. 6
C. 7
Problem 279
The fourth term of a geometric progression
is 189 and the sixth term is 1701, the 8th
term is:
A. 5103
B. 1240029
C. 45927
D. 15309
Problem 280
The sum of the numbers in arithmetical
progression is 45. If 2 is added to the first
number, 3 to the second and 7 to the third,
the new numbers will be in geometrical
progression. Find the common difference in
A.P.
A. -5
B. 10
C. 6
D. 5
Problem 281
The geometric mean and the harmonic mean
of two numbers are 12 and 36/5
respectively. What are the numbers?
A. 36 & 4
B. 72 & 8
C. 36 & 8
D. 72 & 4
Problem 282
If x, 4x+8, 30x +24 are in geometrical
progression, find the common ratio.
A. 2
B. 4
C. 6
D. 8
Problem 283 (ECE April 1995)
A besiege fortress is held by 5700 men who
have provision for 66 days. If the garrison
loses 20 men each day, for how many days
can the provision hold out?
A. 60
B. 72
C. 76
D. 82
Problem 284 (ECE April 1999)
If one third of the air in the tank is removed
by each stroke of an air pump, what
fractional part of the total air is removed in 6
strokes?
A. 0.9122
B. 0.0877
C. 0.8211
D. 0.7145
Problem 285
A rubber ball is dropped from a height of
15m. On each rebound, it rises 2/3 of the
height from which it last fell. Find the
distance traveled by the ball before it
becomes to rest.
A. 75m
B. 96m
C. 100m
D. 85m
Problem 286 (CE May 1991)
In the recent Bosnia conflict, The NATO
forces captured 6400 soldiers. The
provisions on hand will last for 216 meals
while feeding 3 meals a day. The provisions
lasted 9 more days because of daily deaths.
At an average, how many died per day?
A. 15.2
B. 17.8
C. 18.3
D. 19.4
Problem 287
To build a dam, 60 men must work 72 days.
If all 60 men are employed at the start but
the number is decreased by 5 men at the end
of each 12-day period, how long will it take
to complete the dam?
A. 108 days
B. 9 days
C. 94 days
D. 60 days
Problem 288 (CE November 1994)
In a benefit show, a number of wealthy men
agreed that the first one to arrive would pay
10 centavos to enter and each later arrival
would pay twice as much as the preceding
man. The total amount collected from all of
them was P104857.50. How may wealthy
men had paid?
A. 18
B. 19
C. 20
D. 21
Problem 289
Evaluate the following determinant: |
A.
B.
C.
D.
64
44
54
-44
Problem 290
The following equation involves two
determinants:
|
The value of x is:
|
|
|
|
A.
B.
C.
D.
1
3
4
3
Given matrix A = |
|
Problem 291 (CE November 1997)
Evaluate the following determinant:
|
A.
B.
C.
D.
|
|. Find A + 2B.
A. |
|
B. |
|
C. |
|
D. |
-24
24
-46
46
|
Problem 295 (CE May 1996)
Elements of matrix B = |
of matrix C = |
Problem 292 (CE November 1996)
Compute the value of x from the following:
|
x =|
27
-28
26
-29
Find the elements of the product of the two
matrices, matrix BC.
|
|
A.
B.
C.
D.
5
-4
4
-5
Problem 294
|
B. |
|
C. |
|
D. |
Problem 293
Evaluate the following determinant:
D=
|; Elements
|
A. |
A.
B.
C.
D.
| and matrix B =
|
Problem 296 (CE Board)
Solve for x and y from the given
relationship:
|
A.
B.
C.
D.
|| |
| |
x = -2; y = 6
x = 2; y = 6
x = -2; y = -6
x =2; y = -6
Problem 297 (EE October 1993)
In a class of 40 students, 27 students like
Calculus and 25 like Geometry. How many
students liked both Calculus and Geometry?
A.
B.
C.
D.
10
14
11
12
Problem 298
A class of 40 took examination in Algebra
and Trigonometry. If 30 passed Algebra, 36
passed Trigonometry, and 2 failed in both
subjects, the number of students who passed
the two subjects is:
A. 2
B. 8
C. 28
D. 25
Problem 299 (ECE November 1992)
The probability for the ECE board
examinees from a certain school to pass the
Mathematics subject is 3/7 and that for the
Communications subject is 5/7. If none of
the examinees failed in both subjects and
there are 4 examinees who pass both
subjects, how many examinees from the
school took the examination?
A. 28
B. 27
C. 26
D. 32
Problem 300 (EE March 1998)
In a commercial survey involving 1000
persons on brand preferences, 120 were
found to prefer brand x only, 200 persons
prefer brand y only, 150 persons prefer
brand z only, 370 prefer either brand x or y
not z, 450 prefer brand y or z but not x, and
370 prefer wither brand z or x but not y, and
none prefer all the three brands at a time.
How many persons have no brand
preference with any of the three brands?
A. 120
B. 280
C. 70
D. 320
Permutation, Combination, Probability
Problem 301
How permutation can be made out of the
letters in the world island taking four letters
at a time?
A. 360
B. 720
C. 120
D. 24
Problem 302 (CE November 1996)
How many 4 digit number can be formed
without repeating any digit, from the
following digit 1,2,3,4 and 6.
A. 150
B. 120
C. 140
D. 130
Problem 303
How many permutations can made out of
the letters of the word ENGINEERING?
A. 39,916,800
B. 277,200
C. 55,440
D. 3,326,400
Problem 304
How many ways can 3 men and 4 women be
seated on a bench if the women to be
together?
A. 720
B. 576
C. 5040
D. 1024
Problem 305
In how many ways can 5 people line up to
pay their electric bills?
A. 120
B. 1
C. 72
D. 24
Problem 306
In how many ways can 5 people line up to
pay their electric bills, if two particular
persons refuse to follow each other?
A. 120
B. 72
C. 90
D. 140
Problem 307
How many ways can 7 people be seated at a
round table?
A. 5040
B. 120
C. 720
D. 840
Problem 308
In how many relative orders can we seat 7
people at a round table with a certain people
side by side.
A. 144
B. 5040
C. 720
D. 1008
Problem 309
In how many ways can we seat 7 people in a
round table with a certain 3 people not in
consecutive order?
A. 576
B. 3960
C. 5320
D. 689
Problem 310
The captain of a baseball team assigns
himself to the 4th place in the batting order.
In how many ways can he assign the
remaining places to his eight teammates if
just three men are eligible for the first
position?
A. 2160
B. 40320
C. 5040
D. 15120
Problem 311
In how many ways can PICE chapter with
15 directors choose a president, a vicepresident, a secretary, a treasurer, and an
auditor, if no member can hold more than
one position?
A. 630630
B. 3300
C. 5040
D. 15120
Problem 312
How many ways can a committee of five be
selected from an organization with 35
members?
A.
B.
C.
D.
324632
425632
125487
326597
Problem 313
How many line segments can be formed by
13 distinct point?
A. 156
B. 36
C. 98
D. 78
Problem 314
In how many ways can a hostess select six
luncheon guests from 10 women if she is to
avoid having particular two of them together
at the luncheon?
A. 210
B. 84
C. 140
D. 168
Problem 315 (ECE April 1998)
A semiconductor company will hire 7 men
and 4 women. In how many ways can the
company choose from 9 men and 6 women
who qualified for the position?
A. 680
B. 840
C. 480
D. 540
Problem 316
How many ways can you invite one or more
of five friends to a party?
A. 25
B. 15
C. 31
D. 62
Problem 317
A bag contains 4 red balls, 3 green balls, and
5 blue balls. The probability of not getting a
red ball in the first draw is:
A. 2
B. 2/3
C. 1
D. 1/3
Problem 318
Which of the following cannot be a
probability?
A. 1
B. 0
C. 1/e
D. 0.434343
Problem 319 (CE May 1996)
A bag contains 3 white and 5 black balls. If
two balls are drawn in succession without
replacement, what is the probability that
both balls are black?
A. 5/28
B. 5/16
C. 5/32
D. 5/14
Problem 320
A bag contains 3 white and 5 red balls. If
two balls are drawn at random, find the
probability that both are white.
A. 3/28
B. 3/8
C. 2/7
D. 5/15
Problem 321
In problem 320, find the probability that one
ball is white and the other is red.
A.
B.
C.
D.
15/56
15/28
¼
225/784
Problem 322
In the problem 320, find the probability that
all are of the same color.
A. 13/30
B. 14/29
C. 13/28
D. 15/28
Problem 323
The probability that both stages of a twostage rocket to function correctly is 0.92.
The reliability of the first stage is 0.97. The
reliability of the second stage is:
A. 0.948
B. 0.958
C. 0.968
D. 0.8924
Problem 324
Ricky and George each throw dice. If Ricky
gets a sum of four what is the probability
that George will get less than of four?
A. ½
B. 5/6
C. 9/11
D. 1/12
Problem 325
Two fair dice are thrown. What is the
probability that the sum of the dice is
divisible by 5?
A. 7/36
B. 1/9
C. 1/12
D. ¼
Problem 326 (ME April 1996)
An um contains 4 black balls and 6 white
balls. What is the probability of getting one
black ball and white ball in two consecutive
draws from the urn?
A. 0.24
B. 0.27
C. 0.53
D. 0.04
Problem 327
If three balls in drawn in succession from 5
white and a second bag, find the probability
that all are of one color, if the first ball is
replaced immediately while the second is
not replaced before the third draw.
A. 10/121
B. 18/121
C. 28/121
D. 180/14641
Problem 328
A first bag contains 5 white balls and 10
black balls. The experiment consists of
selecting a bag and then drawing a ball from
the selected bag. Find the probability of
drawing a white ball.
A. 1/3
B. 1/6
C. 1/2
D. 1/8
Problem 329
In problem 328, find the probability of
drawing a white ball from the first bag.
A. 5/6
B. 1/6
C. 2/3
D. 1/3
Problem 330
If seven coins are tossed simultaneously,
find the probability that will just have three
heads.
A. 33/128
B. 35/128
C. 30/129
D. 37/129
Problem 331
If seven coins are tossed simultaneously,
find the probability that there will be at least
six tails.
A. 2/128
B. 3/128
C. 1/16
D. 2/16
Problem 332 (CE November 1998)
A face of a coin is either head or tail. If three
coins are tossed, what is are the probability
of getting three tails?
A. 1/8
B. ½
C. ¼
D. 1/6
Problem 333
The face of a coin is either head or tail. If
three coins are tossed, what is the
probability of getting three tails or three
heads?
A. 1/8
B. ½
C. ¼
D. 1/6
Problem 334
Five fair coins were tossed simultaneously.
What is the probability of getting three
heads and two tails?
A. 1/32
B. 1/16
C. 1/8
D. ¼
Problem 335
Throw a fair coin five times. What is the
probability of getting three heads and two
tails?
A. 5/32
B. 5/16
C. 1/32
D. 7/16
Problem 336 (ECE March 1996)
The probability of getting credit in an
examination is 1/3. If three students are
selected at random, what is the probability
that at least one of them got a credit?
A. 19/27
B. 8/27
C. 2/3
D. 1/3
Problem 337
There are three short questions in
mathematics test. For each question, one (1)
mark will be awarded for a correct answer
and no mark for a wrong answer. If the
probability that Mary correctly answers a
question in a test is 2/3, determine the
probability that Mary gets two marks.
A. 4/27
B. 8/27
C. 4/9
D. 2/9
Problem 338
A marksman hits 75% of all his targets.
What is the probability that he will hit
exactly 4 of his next ten shot?
A. 0.01622
B. 0.4055
C. 0.004055
D. 0.001622
Problem 339
A two-digit number is chosen randomly.
What is the probability that it is divisible by
7?
A. 7/50
B. 13/90
C. 1/7
D. 7/45
Problem 340
One box contains four cards numbered 1,
3,5,and 6. Another box contains three cards
numbered 2, 4, and 7. One card is drawn
from each bag. Find the probability that the
sum is even.
A. 5/12
B. 3/7
C. 7/12
D. 5/7
Problem 341
Two people are chosen randomly from 4
married couples. What is probability that
they are husband and wife?
A. 1/28
B. 1/14
C. 3/28
D. 1/7
One letter is taken from each of the words
PARALLEL and LEVEL at random. What
is the probability of getting the same letter?
A. 1/5
B. 1/20
C. 3/20
D. ¾
Problem 343
In a shooting game, the probability that
Botoy and Toto will hit a target is 2/3 and ¾
respectively. What is the probability that the
target is hit when both shoot at it once?
A. 13/5
B. 5/13
C. 7/12
D. 11/12
Problem 344
A standard deck of 52 playing cards is well
shuffled. The probability that the first four
cards dealt from the deck will be four aces is
closes to:
A. 4 10-6
B. 2 10-6
C. 3 10-6
D. 8 10-6
Problem 345
A card is chosen from pack of playing cards.
What is the probability that it is either red or
a picture card?
A. 8/13
B. 10/13
C. 19/26
D. 8/15
Problem 346
Problem 342
In a poker game consisting of 5 cards, what
is the probability of holding 2 aces and 2
Queens?
A. 5! /52!
B. 5/52
C. 33/54145
D. 1264/45685
Problem 347
Dennis Rodman sinks 50% of all his
attempts. What is the probability that he will
make exactly 3 of his next 10 attempts?
A. 1/256
B. 3/8
C. 30/128
D. 15/128
Problem 348
There are 10 defectives per 1000 items of a
product in long run. What is the probability
that there is one and only one defective in
random lot of 100?
A. 0.3697
B. 0.3967
C. 0.3796
D. 0.3679
Problem 349
The UN forces for Bosnia uses a type of
missile that hits the target with a probability
of 0.3. How many missiles should be fired
so that there is at least an 80% probability of
hitting the target?
A. 2
B. 4
C. 5
D. 3
Problem 350 (ME April 1997)
In a dice game, one fair is used. The player
wins P20.00 if he rolls either 1 or 6. He
losses P10.00 if he turns up any other face.
What is the expected winning for one roll of
the die?
A. P40.00
B. P0.00
C. P20.00
D. P10.00
Complex Numbers, Elements
Problem 351 (CE May 1994)
In the complex number 3 + 4i, the absolute
value is:
A. 10
B. 7.211
C. 5
D. 5.689
Problem 352
In the complex number 8-21, the amplitude
is:
A. 104.04
B.
C.
D. 165.96
Problem 353
(6 cis 120 )(4 cis 30 ) is eual to:
A. 10 cis150
B. 24 cis150
C. 10 cis90
D. 24 cis90
Problem 354
is equal to:
A. 20cis30
B. 3cis130
C. 3cis30
D. 20 cis130
Problem 355
The value of x + y in the complex equation
3 + xi = y + 2i is:
A. 5
B. 1
C. 2
D. 3
Problem 356
Multiply (3-2i)(4+3i).
A. 12+i
B. 18+i
C. 6+i
D. 20+i
Problem 357 (EE October 1997)
Divide
.
A.
B. 1+2i
C.
D. 2+2i
Problem 358
Find the value of i9.
A. i
B. –i
C. 1
D. -1
Problem 359 (ECE April 1999)
Simplify i1997+i1999, where I is an imaginary
number.
A. 1+i
B. I
C. 1-i
D. 0
Problem 360
Expand (2+√ )3
A. 46+9i
B. 46-9i
C. -46-9i
D. -46+9i
Problem 361
Write -4+3i in polar form.
A. 5
B. 5
C. 5
D. 5
Problem 362
Simplify: i30-2i25+3i17.
A. I+1
B. -1-2i
C. -1+i
D. -1+5i
Problem 363 (ME April 1997)
Evaluate the value of √
√
A. Imaginary
B. -√
C. √
.
D. √
Problem 364 (EE April 1994)
Perform the indicated operation: √
.
√
A. 21
B. 21i
C. -21i
D. -21
Problem 365 (ECE April 1999)
What is the quotient when 4+8i is divided
by i3 ?
A. 8+4i
B. -8+4i
C. 8-4i
D. -8-4i
Problem366
What is the exponential form of the complex
number 4+3i?
A.
B.
C.
D.
Problem 367
What is the algebraic form of the complex
number
?
A. 12+5i
B. 5-12i
C. 12-5i
D. 5+12i
Problem 368 (ME April 1998)
Solve for x that satisfy the equation x2+36 =
9-2x2.
A. ±6i
B. ±3i
C. 9i
D. -9i
Problem 369
Evaluate ln (5+12i).
A. 2.565+1.176i
B. 2.365-0.256i
C. 5.625+2.112i
D. 3.214-1.254i
Problem 370 (EE April 1994)
Add the given vectors: (-4, 7) + (5, -9)
A. (1, 16)
B. (1, -2)
C. (9, 2)
D. (1, 2)
Problem 371 (EE April 1994)
Find the length of vector (2, 1,1).
A. √
B. √
C. √
D. √
Problem 372 (ECE November 1997)
Find the length of the vector (2, 4, 4).
A. 8.75
B. 6.00
C. 7.00
D. 5.18
Problem 373
If a=b and b=c, then a=c. This property of
real numbers is known as:
A. Reflexive Property
B. Symmetric Property
C. Transitive Property
D. Addition Property
Problem 374
If a=b, then b=a. This property of real
numbers is known as:
A. Reflexive Property
B. Symmetric Property
C. Transitive Property
D. Multiplication Property
Problem 375
A statement the truth of which is admitted
without proof is called:
A. An axiom
B. A postulate
C. A theorem
D. A corollary
Problem 376
In a proportion of four quantities, the first
and the fourth terms are referred to:
A. means
B. denominators
C. extremes
D. numerators
Problem 377 (ECE November 1997)
Convergent series is a sequence of
decreasing numbers or when the succeeding
term is ____ than the preceding term.
A. ten times more
B. greater
C. equal
D. lesser
Problem 378 (ECE November 1997)
It is the characteristics of a population which
is measurable.
A. Frequency
B. Distribution
C. Sample
D. Parameter
Problem 379 (ECE November 1997)
The quartile deviation is a measure of:
A. Division
B. Central tendency
C. Certainty
D. Dispersion
Problem 380 (ECE November 1995, 1997)
In complex algebra, we use a diagram to
represent a complex plane commonly called:
A. De Moivre’s Diagram
B. Funicular Diagram
C. Argand Diagram
D. Venn Diagram
Problem 381
A series of numbers which are perfect
square numbers (i.e. 1, 4, 9, 16, …) is
called:
A. Fourier series
B. Fermat’s series
C. Euler’s series
D. Fibonacci numbers
Problem 382
A sequence of numbers where every term is
obtained by adding all the preceding terms
such as 1, 5, 14, 30… Is called:
A. Triangular number
B. Pyramidal number
C. Tetrahedral number
D. Euler’s number
Problem 383 (ECE November 1995)
The graphical representation of the
commulative frequency distribution in a set
of statistical data is called:
A. Ogive
B. Histogram
C. Frequency polyhedron
D. Mass diagram
Problem 384 (ECE March 1996)
A sequence of numbers where the
succeeding term is greater than the
preceding term is called:
A. Dissonant series
B. Convergent series
C. Isometric series
D. Divergent series
Problem 385 (ECE March 1996)
The number 0.123123123…. is
A. Irrational
B. Surd
C. Rational
D. Transcendental
Problem 386 (ECE November 1996)
An array of m n quantities which
represent a single number system composed
of elements in rows and columns is know as:
A. Transpose of a matrix
B. Determinant
C. Co-factor of a matrix
D. Matrix
Problem 387
If equals are added to equals, the sum is
equal.
A. theorem
B. postulate
C. axiom
D. corollary
Problem 388 (ECE November 1996)
Terms that differ only in numeric
coefficients are nown as:
A. unequal terms
B. unlike terms
C. like terms
D. equal terms
Problem 389 (ECE November 1996)
______ is a sequence of terms whose
reciprocals are in arithmetic progression.
A. Geometric progression
B. Harmonic progression
C. Algebraic Progression
D. Ratio and proportion
Problem 390 (ECE November 1996)
The logarithm of a number to the base e
(2.718281828…) is called:
A.
B.
C.
D.
Naperian logarithm
Characteristic
Mantissa
Briggsian logarithm
Problem 391 (ECE November 1996)
The ratio or product of two expressions in
direct or inverse relation of the other is
called:
A. Ratio and proportion
B. Constant variation
C. Means
D. Extremes
Problem 392 (ECE November 1996)
In any square matrix, when the elements of
any two rows are the same the determinant
is:
A. Zero
B. Positive integer
C. Negative integer
D. Unity
Problem 393 (ECE November 1996)
Two or more equations are equal if and only
if they have the same
A. Solution set
B. Degree
C. Order
D. Variable set
Problem 394
What is the possible outcome of an
experiment called?
A. a sample space
B. a random point
C. an event
D. a finite set
Problem 395
If the roots of an equation are zero, then they
are classified as:
A. Trivial solutions
B. Extraneous roots
C. Conditional solutions
D. Hypergolic Solutions
Problem 396
A complex number associated with a phaseshifted sine wave in polar form whose
magnitude is in RMS and angle is equal to
the angle of the phase-shifted sine wave is
known as:
A. Argand’s number
B. Imaginary number
C. Phasor
D. Real number
Problem 397
In raw data, the term, which occurs most
frequently, is known as:
A. Mean
B. Median
C. Mode
D. Quartile
Problem 398
Infinity minus infinity is:
A. Infinity
B. Zero
C. Indeterminate
D. None of these
Problem 399
Any number divided by infinity is equal to:
A. I
B. Infinity
C. Zero
D. Indeterminate
Problem 400
The term in between any to terms of an
arithmetic progression is called:
A. Arithmetic mean
B. Median
C. Middle terms
D. Mean
Problem 401
Any equation which, because of some
mathematical process, has acquired an extra
root is sometimes called a:
A. Redundant equation
B. Literal equation
C. Linear equation
D. Defective equation
Problem 402
A statement that one mathematical
expression is greater than or less than
another is called:
A. inequality
B. non-absolute condition
C. absolute condition
D. conditional expression
Problem 403
A relation, in which every ordered pair (x, y)
has one and only one value of y that
corresponds to the values of x, is called:
A. Function
B. Range
C. Domain
D. Coordinates
Problem 404
An equation in which a variable appears
under the radical sign is called:
A.
B.
C.
D.
Literal equation
Radical equation
Irradical equation
Irrational equation
Problem 405
The number of favorable outcomes divided
by the number of possible outcomes:
A. Permutations
B. Probability
C. Combination
D. Chance
Problem 406
Two factors are considered essentially the
same if:
A. One is merely the negative of the
other
B. One is exactly the same of the other
C. Both of them are negative
D. Both of them are positive
Problem 407
An integer is said to be prime if:
A. It is factorable by any value
B. It is an odd integer
C. It has no other integer as factor
excepts itself or 1
D. It is an even integer
Problem 408
Equations in which the members are equal
for all permissible values of integer are
called:
A. a conditional equation
B. an identity
C. a parametric equation
D. a quadratic equation
Problem 409
Equations which satisfy only for some
values of unknown are called:
A. a conditional equation
B. an identity
C. a parametric equation
D. a quadratic equation
Problem 410 (ME April 1996)
The logarithm of 1 to any base is:
A. indeterminate
B. zero
C. infinity
D. one
Part 2
Plane and Spherical Trigonometry
Problems- Angles, Trigonometric
Identities and Equations
Set 10
1. Find the supplement of an angle
whose compliment is 62°.
A. 28°
B. 118°
C. 152°
D. None of these
2. A certain angle has a supplement 5
times its compliment. Find the
angle.
A. 67.5°
B. 157.5°
C. 168.5°
D. 186°
3. The sum of the two interior angles of
the triangle is equal to the third
angle and the difference of the two
angles is equal to 2/3 of the third
angle. Find the third angle.
A. 15°
B. 75°
C. 90°
D. 120°
4. The measure 0f 1 ½ revolutions
counter-clockwise is:
A. 540°
B. 520°
C. +90°
D. -90°
5. The measure of 2.25 revolutions
counterclockwise is:
A. -835°
B. -810°
C. 805°
D. 810°
6. Solve for Ѳ:
A. 40°
B. 41°
C. 47°
D. 43°
7. What are the exact values of the
cosine and tangent trigonometric
functions of acute angle A, given
that sin A = 3/7?
√
A.
√
√
B.
√
√
C.
√
√
D.
√
8. Given three angles A, B, and C
whose sum is 180°. If the tan A +
tan B + tan C = x, find the value of
tan A x tan B x tan C.
A. 1 – x
B. √
C. x/2
D. x
9. What is the sine of 820°?
A. 0.984
B. -0.866
C. 0.866
D. -0.500
10. csc 270° = ?
A. √
B. 1
C. √
D. 1
11. If coversine Ѳ is 0.134, find the
value of Ѳ.
A. 60°
B. 45°
C. 30°
D. 20°
12. Solve for cos 72° if the given
relationship is cos 2A = 2
A–
1.
A. 0.309
B. 0.258
C. 0.268
D. 0.315
13. If sin 3A = cos 6B then:
A. A + B = 180°
B. A + 2B = 30°
C. A – 2B = 30°
D. A + B = 30°
14. Find the value of sin (arcos 15/17).
A. 8/17
B. 17/9
C. 8/21
D. 8/9
15. Find the value of cos [arcsin (1/3) +
arctan (2/√ )]
A. ( )
√
B. ( ) √
C. ( ) √
D. ( ) √
16. If sin 40° + sin 20° = sin Ѳ, find the
value of Ѳ.
A. 20°
B. 80°
C. 120°
D. 60°
17. How many different value of x from
0° to 180° for the equation (2sin x –
1)(cos x + 1) = 0?
A. 3
B. 0
C. 1
D. 2
18. For what value of Ѳ (less than 2∏)
will the following equation be
satisfied?
A. ∏
B. ∏/4
C. 3∏/2
D. ∏/2
19. Find the value of x in the equation
csc x + cot x = 3.
A. ∏/4
B. ∏/3
C. ∏/2
D. ∏/5
20. If
is 5/2, the quantity
is equivalent to:
A. 2.5
B. 0.6
C. 1.5
D. 0.4
21. Find sin x if 2 sin x + 3 cos x – 2 =
0.
A. 1 & -5/13
B. -1 & 5/13
C. 1 & 5/13
D. -1 & -5/13
22. If sin A = 4/5, A in quadrant II, sin
B = 7/25, B in quadrant I, find sin
(A + B).
A. 3/5
B. 2/5
C. 3/4
D. 4/5
23. If sin A =2.571x, cos A = 3.06, and
sin 2A = 3.939x, find the value of x.
A. 0.350
B. 0.250
C. 0.100
D. 0.150
24. If cos Ѳ = √ /2, what is the value of
x if x = 1 –
.
A. -2
B. -1/3
C. 4/3
D. 2/3
25. If sin Ѳ – cos Ѳ = -1/3, what is the
value of sin 2 Ѳ?
A. 1/3
B. 1/9
C. 8/9
D. 4/9
26. If x cos Ѳ + y sin Ѳ = 1 and x sin Ѳ
– y cos Ѳ = 3, what is the
relationship between x and y?
A.
B.
C.
D.
27. If
/
√ , then
x+1/
is equal to:
A. √
B. 1
C. 2
D. 0
28. The equation 2 sin Ѳ + 2 cos Ѳ – 1
= √ is:
A. An identity
B. A parametric equation
C. A conditional equation
D. A quadratic equation
29. If x + y = 90°, then
is
equal to:
A. tan x
B. cos x
C. cot x
D. sin x
30. if cos Ѳ = x / 2 then 1 –
is
equal to:
A.
/
B.
/
C.
/
D.
/
31. Find the value in degrees of arcos
(tan 24°).
A. 61.48
B. 62.35
C. 63.56
D. 60.84
32.
√
A. ∏/3
B. ∏/4
C. ∏/6
D. ∏/2
33. Solve for x in the equation: arctan
(2x) + arctan (x) = ∏/4
A. 0.821
B. 0.218
C. 0.281
D. 0.182
34. Solve for x from the given
trigonometric equation:
A. 4
B. 6
C. 8
D. 2
35. Solve for y if y = (1/sin x – 1/tan
x)(1 + cos x)
A. sin x
B. cos x
C. tan x
D.
36. Solve for x: x =
A. sin
B. cos
C. 1
D. 2
37. Solve for x:
A. sin
B. -2 cos
C. cos 2
D. sin 2
38. Simplify
A. 2
B. 1
C. 2
D. 2
–
39. Solve for x:
A. cos a
B. sin 2a
C. cos 2a
D. sin a
40. which of the following is different
from the others?
A. 2 cos 2x – 1
B. cos 4x – sin 4x
C. cos 3x – sin 3x
D. 1 – 2 sin 2x
41. Find the value of y: y = (1 + cos
2
.
A. cos
B. sin
C. sin 2
D. cos 2
42. The equation 2 sinh x cosh x is
equal to:
A.
B.
C.
D. Cosh 2x
43. Simplifying the equation
A. 1
B.
C.
D.
44. If tan = , which of the
following is incorrect?
/√
A.
B.
C.
√
/√
D.
/
√
45. In an isosceles right triangle, the
hypotenuse is how much longer
than its sides?
A. 2 times
B. √ times
C.
D. None of these
46. Find the angle in mils subtended by
a line 10 yards long at a distance of
5000 yards.
A. 2.5 mils
B. 2 mils
C. 4 mils
D. 1 mil
47. The angle or inclination of ascend
of a road having 8.25% grade is
_____degrees.
A. 5.12 degrees
B. 4.72 degrees
C. 1.86 degrees
D. 4.27 degrees
48. The sides of a right triangle is in
arithmetic progression whose
common difference if 6 cm. its area
is:
A.
B.
C.
D.
Problems – Triangles, Angles of Elevation
& Depression
Set 11
49. The hypotenuse of a right triangle is
34 cm. Find the length of the
shortest leg if it is 14 cm shorter
than the other leg.
A. 15 cm
B. 16 cm
C. 17 cm
D. 18 cm
50. A truck travels from point M
northward for 30 min. then eastward
for one hour, then shifted N 30° W.
if the constant speed is 40 Kph, how
far directly from M, in km. will be it
after 2 hours?
A. 43.5
B. 45.2
C. 47.9
D. 41.6
51. Two sides of a triangle measures 6
cm. and 8 cm. and their included
angle is 40°. Find the third side.
A. 5.144 cm
B. 5.263 cm
C. 4.256 cm
D. 5.645 cm
52. Given a triangle: C = 100°, a = 15,
b = 20. Find c:
A. 34
B. 27
C. 43
D. 35
53. Given angle A = 32°, angle B = 70°,
and side c = 27 units. Solve for side
a of the triangle.
A. 24 units
B. 10 units
C. 14.63 units
D. 12 units
54. In a triangle, find the side c if the
angle C = 100°, side b = 20, and
side a = 15.
A. 28
B. 27
C. 29
D. 26
55. Two sides of a triangle are 50 m.
and 60 m. long. The angle included
between these sides is 30 degrees.
What is the interior angle (in
degrees) opposite the longest side?
A. 92.74
B. 93.74
C. 94.74
D. 91.74
56. The sides of a triangle ABC are AB
= 15 cm, BC = 18 cm, and CA = 24
cm. Determine the distance from the
point of intersection of the angular
bisectors to side AB.
A. 5.21 cm
B. 3.78 cm
C. 4.73 cm
D. 6.25 cm
57. If AB = 15 m, BC = 18 m and CA =
24 m, find the point of intersection
of the angular bisector from the
vertex C.
A. 11.3
B. 12.1
C. 13.4
D. 14.3
58. In triangle ABC, angle C = 70
degrees; angle A = 45 degrees; AB
= 40 m. what is the length of the
median drawn from vertex A to side
BC?
A. 36.8 meters
B. 37.1 meters
C. 36.3 meters
D. 37.4 meters
59. The area of the triangle whose
angles are 61°9’32”, 34°14’46”, and
84°35’42” is 680.60. the length of
the longest side is:
A. 35.53
B. 54.32
C. 52.43
D. 62.54
60. Given a triangle ABC whose angles
are A = 40°, B = 95° and side b =
30 cm. find the length of the
bisector of angle C.
A. 21.74 cm
B. 22.35 cm
C. 20.45 cm
D. 20.98 cm
61. The sides of a triangular lot are 130
m, 180 m, and 190 m. the lot is to
be divided by a line bisecting the
longest side and drawn from the
opposite vertex. The length of this
dividing line is:
A. 100 meters
B. 130 meters
C. 125 meters
D. 115 meters
62. From a point outside of an
equilateral triangle, the distances to
the vertices are 10m, 10m, and 18m.
Find the dimension of the triangle.
A. 25.63
B. 45.68
C. 19.94
D. 12.25
63. Points A and B 1000m apart are
plotted on a straight highway
running East and West. From A, the
bearing of a tower C is 32 degrees
N of W and from B the bearing of C
is 26 degrees N of E. Approximate
the shortest distance of tower C to
the highway.
A. 264 meters
B. 274 meters
C. 284 meters
D. 294 meters
64. An airplane leaves an aircraft
carrier and flies South at 350 mph.
The carrier travels S 30° E at 25
mph. If the wireless communication
range of the airplane is 700 miles,
when will it lose contact with the
carrier?
A. after 4.36 hours
B. after 5.57 hours
C. after 2.13 hours
D. after 4.54 hours
65. A statue 2 meters high stands on a
column that is 3 meters high. An
observer in level with the top of the
statue observed that the column and
the statue subtend the same angle.
How far is the observer from the
statue?
A. √
B. √
C. 20 meters
D. √
66. From the top of a building 100 m
high, the angle of depression of a
point A due East of it is 30°. From a
poit B due South of the building, the
angle of elevation of the top is 60°.
Find the distance AB.
A. 100 + √
B. 200 - √
C.
√ /3
D.
√ / 30
67. An observer found the angle of
elevation of the top of the tree to be
27°. After moving 10m closer (on
the same vertical and horizontal
plane as the tree), the angle of
elevation becomes 54°. Find the
height of the tree.
A. 8.65 meters
B. 7.53 meters
C. 7.02 meters
D. 8.09 meters
68. From a point A at the foot of the
mountain, the angle of elevation of
the top B is 60°. After ascending the
mountain one (1) mile to an
inclination of 30° to the horizon,
and reaching a point C, an observer
finds that the angle ACB is 135°.
A. 14386
B. 12493
C. 11672
D. 11223
69. A vertical pole is 10 m from a
building. When the angle of
elevation of the sum is 45°, te pole
cast a shadow on the building 1 m
high. Find the height of the pole.
A. 0 meter
B. 11 meters
C. 12 meters
D. 13 meters
70. A pole cast a shadow of 15 meters
long when the angle of elevation of
the sun is 61°. If the pole has leaned
15° from the vertical directly
toward the sun, what is the length of
the pole?
A. 52.43 meters
B. 54.23 meters
C. 53.25 meters
D. 53.24 meters
71. An observer wishes to determine
the height of a tower. He takes
sights at the top of the tower from A
and B, which are 50 ft. apart, at the
same elevation on a direct line with
the tower. The vertical angle at
point A is 30° and at point B is 40°.
What is the height of the tower?
A. 85.6 feet
B. 143.97 feet
C. 110.29 feet
D. 92.54 feet
72. From the top of tower A, the angle
of elevation of the top of the tower
B is 46°. From the foot of tower B
the angle of elevation of the top of
tower A is 28°. Both towers are on a
level ground. If the height of tower
B is 120m, how high is tower A in
m?
A. 38.6
B. 42.3
C. 44.1
D. 40.7
73. Points A and B are 100 m apart and
are on the same elevation as the foot
of a building. The angles of
elevation of the top of the building
from points A and B are 21° and
32°, respectively. How far is A from
the building in m?
A. 271.6
B. 265.4
C. 259.2
D. 277.9
74. A man finds the angle of elevation
of the top of a tower to be 30
degrees. He walks 85 m. nearer the
tower and finds its angle of
elevation to be 60 degrees. What is
the height of the tower?
A. 76.31 meters
B. 73.61 meters
C. 73.31 meters
D. 73.16 meters
75. The angle of elevation of a point C
from a pint B is 29°42’; the angle of
elevation of C from another point A
31.2 m directly below B is 59°23’.
How high is C from the horizontal
line through A?
A. 47.1 meters
B. 52.3 meters
C. 35.1 meters
D. 66.9 meters
76. A rectangular piece of land 40m x
30m is to be crossed diagonally by a
10-m wide roadway. If the land cost
P1,500.00 per square meter, the cost
of the roadway is:
A. P401.10
B. P60,165.00
C. P601,650.00
D. 651,500.00
77. A man improvises a temporary
shield from the sun using a
triangular piece of wood with
dimensions of 1.4m, 1.5 m, and 1.3
m. with the longer side lying
horizontally on the ground, he props
up the other corner of the triangle
with a vertical pole 0.9m long.
What would be the area of the
shadow on the ground when the sun
is vertically overhead?
A. 0.5
B. 0.75
C. 0.84
D. 0.95
78. A rectangular piece of wood 4cm x
12cm tall is titled at an angle of 45°.
Find the vertical distance between
the lower corner and the upper
corner.
A. √
B. √
C. √
D. √
79. A clock has a dial face 12 inches in
radius. The minute hand is 9 inches
long while the hour hand is 6 inches
long. The plane of rotation of the
minute hand is 2 inches above the
plane of rotation of the hour hand.
Find the distance between the tips
of the hands at 5:40 AM.
A. 9.17 inches
B. 8.23 inches
C. 10.65 inches
D. 11.25 inches
80. If the bearing of A from B is 40° W,
then the bearing of B from A is:
A. N 40° E
B. N 40° W
C. N 50° E
D. N 50° W
81. A plane hillside is inclined at an
angle of 28° with the horizontal. A
man wearing skis can climb this
hillside by following a straight path
inclined at an angle of 12° to the
horizontal, but one without skis
must follow a path inclined at an
angle of only 5° with the horizontal.
Find the angle between the
directions of the two paths.
A. 13.21°
B. 18.74°
C. 15.56°
D. 17.22°
82. Calculate the area of a spherical
triangle whose radius is 5 m and
whose angles are 40°, 65°, and
110°.
A. 12.34 sq. m.
B. 14.89 sq. m.
C. 16.45 sq. m.
D. 15.27 sq. m.
83. A right spherical triangle has an
angle C = 90°, a = 50°, and c = 80°.
Find the side b.
A. 45.33°
B. 78.66°
C. 74.33°
D. 75.89°
84. If the time is 8:00 a.m. GMT, what
is the time in the Philippines, which
is located at 120° East longitude?
A. 6 p.m.
B. 4 am
C. 4 p.m.
D. 6 am
85. An airplane flew from Manila (14°
36’N, 121° 05’E) at a course of S
30° E maintaining a certain altitude
and following a great circle path. If
its groundspeed is 350 knots, after
how many hours will it cross the
equator?
A. 2.87 hours
B. 2.27 hours
C. 3.17 hours
D. 3.97 hours
86. Find the distance in nautical miles
between Manila and San Francisco.
Manila is located at 14° 36’N
latitude and 121° 05’ E longitude.
San Francisco is situated at 37° 48’
N latitude and 122° 24’ W
longitude.
A. 7856.2 nautical miles
B. 5896.2 nautical miles
C. 6326.2 nautical miles
D. 6046.2 nautical miles
Part 3
Plane Geometry
Problems – Triangles, Quadrilaterals,
Polygons
Set 12
87. The sides of a right triangle have
lengths (a – b), a, and (a + b). What
is the ratio of a to b if a is greater
than b and b could not be equal to
zero?
A. 1 : 4
B. 3 : 1
C. 1 : 4
D. 4 : 1
88. Two sides of a triangle measure 8
cm and 12 cm. find its area if its
perimeter is 26 cm.
A. 21.33 sq. m.
B. 32.56 sq. cm.
C. 3.306 sq. in.
D. 32.56 sq. in.
89. If three sides of a triangle of an
acute triangle is 3 cm, 4 cm, and “x”
cm, what are the possible values of
x?
A. 1 < x < 5
B. 0 < x > 5
C. 0 < x < 7
D. 1 < x > 7
90. In triangle ABC, AB = 8m and BC
= 20m. one possible dimension of
CA is:
A. 13
B. 7
C. 9
D. 11
91. In a triangle BCD, BC = 25 m. and
CD = 10 m. The perimeter of the
triangle may be.
A. 72 m.
B. 70 m.
C. 69 m.
D. 71 m.
92. The sides of a triangle ABC are AB
= 25 cm, BC = 39 cm, and AC = 40
cm. Find its area.
A. 486 sq. cm.
B. 846 sq. cm.
C. 648 sq. cm.
D. 468 sq. cm.
93. The corresponding sides of two
similar triangles are in the ratio 3:2.
What is the ratio of their areas?
A. 3
B. 2
C. 9/4
D. 3/2
94. Find the area of the triangle whose
sides are 12, 16, and 21 units.
A. 95.45 sq. units
B. 102.36 sq. units
C. 87.45 sq. units
D. 82.78 sq. units
95. The sides of a right triangle are 8,
15 and 17 units. If each side is
doubled, how many square units
will be the area of the new triangle?
A. 240
B. 300
C. 320
D. 420
96. Two triangles have equal bases. The
altitude of one triangle is 3 units
more than its base and the altitude
of the other is 3 units less than its
base. Find the altitudes, if the areas
of the triangle differ by 21 square
units.
A. 5 & 11
B. 4 & 10
C. 6 & 12
D. 3 & 9
97. A triangular piece of wood having a
dimension 130 cm, 180 cm, and 190
cm is to be divided by a line
bisecting the longest side drawn
from its opposite vertex. The area of
the part adjacent to the 180-cm side
is:
A. 5126 sq. cm.
B. 5162 sq. cm.
C. 5612 sq. cm.
D. 5216 sq. cm.
98. Find EB if the area of the inner
triangle is ¼ of the outer triangle.
A. 32.5
B. 55.7
C. 56.2
D. 57.5
99. A piece of wire is shaped to enclose
a square whose area is 169 cm2. It is
then reshaped to enclose a rectangle
whose length is 15 cm. The area of
the rectangle is:
A. 165 cm2
B. 175 cm2
C. 170 cm2
D. 156 cm2
100. The diagonal of the floor of a
rectangular room is 7.50 m. The
shorter side of the room is 4.5 m.
What is the area of the room?
A. 36 sq. m.
B. 27 sq. m.
C. 58 sq. m.
D. 24 sq. m.
101. A man measuring a rectangle “x”
meters by “y” meters, makes each
side 15% too small. By how many
percent will his estimate for the area
be too small?
A. 23.55%
B. 25.67%
C. 27.75%
D. 72.25%
102. The length of the side of a square
is increased by 100%. Its perimeter
is increased by:
A. 25%
B. 100%
C. 200%
D. 300%
103. A piece of wire of length 52 cm is
cut into two parts. Each part is then
bent to form a square. It is found
that total area of the two squares is
97 sq. cm. the dimension of the
bigger square is:
A. 4
B. 9
C. 3
D. 6
104. In the figure shown, ABCD is a
square and PDC is an equilateral
triangle. Find .
A. 5°
B. 15°
C. 10°
D. 25°
105. One side of a parallelogram is 10
m and its diagonals are 16 m and 24
m, respectively. Its area is:
A. 156.8 sq. m.
B. 185.6 sq. m.
C. 158.7 sq. m.
D. 142.3 sq. m.
106. If the sides of the parallelogram
and an included angle are 6, 10 and
100 degrees respectively, find the
length of the shorter diagonal.
A. 10.63
B. 10.37
C. 10.73
D. 10.23
107. The area of a rhombus is 132
square cm. if its shorter diagonal is
12 cm, the length of the longer
diagonal is:
A. 20 centimeter
B. 21 centimeter
C. 22 centimeter
D. 23 centimeter
108. The diagonals of a rhombus are 10
cm. and 8 cm., respectively. Its area
is:
A. 10 sq. cm.
B. 50 sq. cm.
C. 60 sq. cm.
D. 40 sq. cm.
109. Given a cyclic quadrilateral whose
sides are 4 cm, 5 cm, 8 cm, and 11
cm. Its area is:
A. 40.25 sq. cm.
B. 48.65 sq. cm.
C. 50.25 sq. cm.
D. 60.25 sq. cm
110. A rectangle ABCD which measure
18 by 24 cm is folded once,
perpendicular to diagonal AC, so
that the opposite vertices A and C
coincide. Find the length of the fold.
A. 2
B. 7/2
C. 54/2
D. 45/2
111. The sides of a quadrilateral are
10m, 8m, 16m and 20m,
respectively. Two opposite interior
angles have a sum of 225°. Find the
area of the quadrilateral in sq. m.
A. 140.33 sq. cm.
B. 145.33 sq. cm.
C. 150.33 sq. cm.
D. 155.33 sq. cm.
112. A trapezoid has an area of 36 m2
and altitude of 2 m. Its two bases in
meters have ratio of 4:5, the bases
are:
A. 12, 15
B. 7, 11
C. 16, 20
D. 8, 10
113. Determine the area of the
quadrilateral ABCD shown if OB =
80 cm, OA = 120 cm, OD = 150 cm
and = 25°.
A.
B.
C.
D.
2272 sq. cm
7222 sq. cm
2572 sq. cm
2722 sq. cm
114. A corner lot of land is 35 m on one
street and 25 m on the other street.
The angle between the two lines of
the street being 82°. The other to two
lines of the lot are respectively
perpendicular to the lines of the streets.
What is the worth of the lot if its unit
price is P2500 per square meter?
A. P1,978,456
B. P1,588,045
C. P2,234,023
D. P1,884,050
115. Determine the area of the
quadrilateral having (8, -2), (5, 6),
(4, 1), and (-7, 4) as consecutive
vertices.
A. 22 sq. units
B. 44 sq. units
C. 32 sq. units
D. 48 sq. units
116. Find the area of the shaded portion
shown if AB is parallel to CD.
A. 16 sq. m.
B. 18 sq. m.
C. 20 sq. m.
D. 22 sq. m.
117. The deflection angles of any
polygon has a sum of:
A. 360°
B. 720°
C. 180°(n – 3)
D. 180° n
118. The sum of the interior angles of a
dodecagon is:
A. 2160°
B. 1980°
C. 1800°
D. 2520°
119. Each interior angle of a regular
polygon is 165°. How many sides?
A. 23
B. 24
C. 25
D. 26
120. The sum of the interior angles of a
polygon is 540°. Find the number of
sides.
A. 4
B. 6
C. 7
D. 5
121. The sum of the interior angles of a
polygon of n sides is 1080°. Find
the value of n.
A. 5
B. 6
C. 7
D. 8
122. How many diagonals does a
pentedecagon have:
A. 60
B. 70
C. 80
D. 90
123. A polygon has 170 diagonals.
How many sides does it have?
A. 20
B. 18
C. 25
D. 26
124. A regular hexagon with an area of
93.53 square centimeters is
inscribed in a circle. The area in the
circle not covered by hexagon is:
A. 18.38 cm2
B. 16.72 cm2
C. 19.57 cm2
D. 15.68 cm2
125. The area of a regular decagon
inscribed in a circle of 15 cm
diameter is:
A. 156 sq. cm.
B. 158 sq. cm.
C. 165 sq. cm.
D. 185 sq. cm.
126. The sum of the interior angle of
polygon is 2,520°. How many are
the sides?
A. 14
B. 15
C. 16
D. 17
127. The area of a regular hexagon
inscribed in a circle of radius 1 is:
A. 2.698 sq. units
B. 2.598 sq. units
C. 3.698 sq. units
D. 3.598 sq. units
128. The corners of a 2-meter square
are cut off to form a regular
octagon. What is the length of the
sides of the resulting octagon?
A. 0.525
B. 0.626
C. 0.727
D. 0.828
129. If a regular polygon has 27
diagonals, then it is a:
A. Hexagon
B. Nonagon
C. Pentagon
D. Heptagon
130. One side of a regular octagon is 2.
Find the area of the region inside
the octagon.
A. 19.3 sq. units
B. 13.9 sq. units
C. 21.4 sq. units
D. 31 sq. units
131. A regular octagon is inscribed in a
circle of radius 10. Find the area of
the octagon.
A. 228.2 sq. units
B. 288.2 sq. units
C. 282.8 sq. units
D. 238.2 sq. units
Problems – Circles, Miscellaneous
Applications
Set 13
132. The area of a circle is 89.4 square
inches. What is the circumference?
A. 35.33 inches
B. 32.25 inches
C. 33.52 inches
D. 35.55 inches
133. A circle whose area is 452 cm
square is cut into two segment by a
chord whose distance from the
center of the circle is 6 cm. Find the
area of the larger segment in cm
square.
A. 372.5
B. 363.6
C. 368.4
D. 377.6
134. A circle is divided into two parts
by a chord, 3 cm away from the
center. Find the area of the smaller
part, in cm square, if the circles has
an area of 201 cm square.
A. 51.4
B. 57.8
C. 55.2
D. 53.7
135. A quadrilateral ABCD is inscribed
in a semi-circle with side AD
coinciding with the diameter of the
circle. If sides AB, BC, and CD are
8cm, 10cm, and 12cm long,
respectively, find the area of the
circle.
A. 317 sq. cm.
B. 356 sq. cm.
C. 456 sq. cm.
D. 486 sq. cm.
136. A semi-circle of radius 14 cm is
formed from a piece of wire. If it is
bent into a rectangle whose length is
1cm more than its width, find the
area of the rectangle.
A. 256.25 sq. cm.
B. 323.57 sq. cm.
C. 386.54 sq. cm.
D. 452.24 sq. cm
137. The angle of a sector is 30 degrees
and the radius is 15 cm. What is the
area of the sector?
A. 89.5
B. 58.9
C. 59.8
D. 85.9
138. A sector has a radius of 12 cm. if
the length of its arc is 12 cm, its
area is:
A. 66 sq. cm.
B. 82 sq. cm.
C. 144 sq. cm.
D. 72 sq. cm.
139. The perimeter of a sector is 9 cm
and its radius is 3 cm. What is the
area of the sector?
A. 4
B. 9/2
C. 11/2
D. 27/2
140. A swimming pool is to be
constructed in the space of partially
overlapping identical circles. Each
of the circles has a radius of 9 m,
and each passes through the center
of the other. Find the area of the
swimming pool.
A. 302.33
B. 362.55
C. 398.99
D. 409.44
141. Given are two concentric circles
with the outer circle having a radius
of 10 cm. If the area of the inner
circle is half of the outer circle, find
the boarder between the two circles.
A. 2.930 cm
B. 2.856 cm
C. 3.265 cm
D. 2.444 cm
142. A circle of radius 5 cm has a chord
which is 6cm long. Find the area of
the circle concentric to this circle
and tangent to the given chord.
A. 14
B. 16
C. 9
D. 4
143. A reversed curve on a railroad
track consists of two circular arcs.
The central angle of one side is 20°
with radius 2500 feet, and the
central angle of the other is 25° with
radius 3000 feet. Find the total
lengths of t he two arcs.
A. 2812 ft.
B. 2218 ft.
C. 2821 ft.
D. 2182 ft.
144. Given a triangle whose sides are
24 cm, 30 cm, and 36 cm. find the
radius of a circle which is tangent to
the shortest and longest side of the
triangle, and whose center lies on
the third side.
A. 9.111 cm
B. 11.91 cm
C. 12.31 cm
D. 18 cm
145. Find the area of the largest circle
that can be cut from a triangle
whose sides are 10 cm, 18 cm, and
20 m.
A. 11
B. 12
C. 14
D. 15
146. The diameter of the circle
circumscribed about a triangle ABC
with sides a, b , c is equal to:
A. a/sin A
B. b/sin B
C. c/sin C
D. all of the above
147. The sides of a triangle are 14 cm.,
15 cm., and 13 cm. find the area of
the circumscribing circle.
A. 207.4 sq. cm.
B. 209.6 sq. cm.
C. 215.4 sq. cm.
D. 220.5 sq. cm.
148. What is the radius of the circle
circumscribing an isosceles right
triangle having an area of 162 sq.
cm?
A. 13.52
B. 14.18
C. 12.73
D. 1564
149. If the radius of the circle is
decreased by 20%, by how much is
its area decreased?
A. 36%
B. 26%
C. 46%
D. 56%
150. The distance between the center of
the three circles which are mutually
tangent to each other externally are
10, 12 and 14 units. The area of the
of the largest circle is:
A. 72
B. 23
C. 64
D. 16
151. The sides of a cyclic quadrilateral
measures 8 cm, 9 cm, 12 cm, and 7
cm, respectively. Find the area of
the circumscribing circle.
A. 8.65
B. 186.23
C. 6.54
D. 134.37 cm
152. The wheel of a car revolves n
times, while the car travels x km.
the radius of the wheel in meter is:
A. 10,000 x/ (
B. 500 x/ (
C. 500,00 x/ (
D. 5,000 x/ (
153. If the inside wheels of a car
running a circular track are going
half as fast as the outside wheel,
determine the length of the track,
described by the outer wheels, if the
wheels are 1.5 m apart.
A. 4
B. 5
C. 6
D. 8
154. A goat is tied to a corner of a 30 ft
by 35 ft building. If the rope is 40 ft
long and the goat can reach 1 ft
farther than the rope length, what is
the maximum area the goat can
cover?
A. 5281
B. 4084
C. 3961
D. 3970
155. The interior angles of a triangle
measures 2x, x + 15, and 2x + 15.
What is the value of x?
A. 30°
B. 66°
C. 42°
D. 54°
156. Two complementary angles are in
the ratio 2:1. Find the larger angle.
A. 30°
B. 60°
C. 75°
D. 15°
157. Two transmission towers 40 feet
high is 200 feet apart. If the lowest
point of the cable is 10 feet above
the ground, the vertical distance
from the roadway to the cable 50
feet from the center is:
A. 17.25 feet
B. 17.5 feet
C. 17.75 feet
D. 18 feet
158. What is the area bounded by the
curves
and
A. 6.0
B. 7.333
C. 6.666
D. 5.333
159. What is the area between y = 0, y
= 3 , x = 0, and x = 2?
A. 8
B. 12
C. 24
D. 6
Part 4
Solid Geometry
Problems – Prisms, Pyramids, Cylinders,
Cones
Set 14
160. If the edge of a cube is doubled,
which of the following is incorrect?
A. The lateral area will be
quadrupled
B. The volume is increased 8 times
C. The diagonal is doubled
D. The weight is doubled
161. The volume of a cube is reduced
by how much if all sides are
halved?
A. 1/8
B. 5/8
C. 6/8
D. 7/8
162. Each side of a cube is increased by
1%. By what percent is the volume
of the cube increased?
A. 23.4%
B. 33.1%
C. 3%
D. 34.56%
163. If the edge of a cube is increased
by 30%, by how much is the surface
area increased?
A. 67
B. 69
C. 63
D. 65
164. Find the approximate change in
the volume of a cube of side x
inches caused by increasing its side
by 1%.
A. 0.3x3 cu. in.
B. 0.1x3 cu. in.
C. 0.02 cu. in.
D. 0.03x3 cu. in.
165. A rectangular bin 4 feet long, 3
feet wide, and 2 feet high is solidly
packed with bricks whose
dimensions are 8 in. by 4 in. by 2 in.
The number of bricks in the bin is:
A. 68
B. 386
C. 648
D. 956
166. Find the total surface area of a
cube of side 6 cm.
A. 214 sq. cm.
B. 216 sq. cm.
C. 226 sq. cm.
D. 236 sq. cm.
167. The space diagonal of a cube is
4√ m. Find its volume.
A. 16 cubic meters
B. 48 cubic meters
C. 64 cubic meters
D. 86 cubic meters
168. A reservoir is shaped like a square
prism. If the area of its base is 225
sq. cm, how many liters of water
will it hold?
A. 3.375
B. 3375
C. 33.75
D. 3375
169. Find the angle formed by the
intersection of a face diagonal t the
diagonal of a cube drawn from the
same vertex.
A. 35.26°
B. 32.56°
C. 33.69°
D. 42.23°
170. The space diagonal of a cube (the
diagonal joining two non-coplanar
vertices) is 6 m. The total surface
area of the cube is:
A. 60
B. 66
C. 72
D. 78
171. The base edge of a regular
hexagonal prism is 6 cm and its
bases are 12 cm apart. Find its
volume in cu. cm.
A. 1563.45 cm3
B. 1058.45 cm3
C. 1896.37 cm3
D. 1122.37 cm3
172. The base edge of a regular
pentagonal prism is 6 cm and its
bases are 12 cm apart. Find its
volume in cu. cm.
A. 743.22 cm3
B. 786.89 cm3
C. 567.45 cm3
D. 842.12 cm3
173. The base of a right prism is a
hexagon with one side 6 cm long. If
the volume of the prism is 450 cc,
how far apart are the bases?
A. 5.74 cm
B. 3.56 cm
C. 4.11 cm
D. 4.81 cm
174. A trough has an open top 0.30 m
by 6 m and closed vertical ends
which are equilateral triangles 30
cm on each side. It is filled with
water to half its depth. Find the
volume of the water in cubic
meters.
A. 0.058
B. 0.046
C. 0.037
D. 0.065
175. Determine the volume of a right
truncated prism with the following
dimensions: Let the corner of the
triangular base be defined by A, B,
and C. the length AB = 10 feet, BC
= 9 feet and CA = 12 feet. The sides
at A, B and C are perpendicular to
the triangular base and have the
height of 8.6 feet, 7.1 feet, and 5.5
feet, respectively.
A. 413 ft3
B. 311 ft3
C. 313 ft3
D. 391 ft3
176. The volume of a regular
tetrahedron of side 5 cm is:
A. 13.72 cu. cm
B. 14.73 cu.cm
C. 15.63 cu. cm
D. 17.82 cu. cm
177. A regular hexagonal pyramid
whose base perimeter is 60 cm has
an altitude of 30 cm, the volume of
the pyramid is:
A. 2958 cu. cm.
B. 2598 cu. cm.
C. 2859 cu. cm.
D. 2589 cu. cm.
178. A frustum of a pyramid has an
upper base 100 m by 10 m and a
lower base of 80 m by 8 m. if the
altitude of the frustum is 5 m, find
its volume.
A. 4567.67 cu. m.
B. 3873.33 cu. m.
C. 4066.67 cu. m.
D. 2345.98 cu. m.
179. The altitude of the frustum of a
regular rectangular pyramid is 5m
the volume is 140 cu. m. and the
upper base is 3m by 4m. What are
the dimensions of the lower base in
m?
A. 9 x 10
B. 6 x 8
C. 4.5 x 6
D. 7.50 x 10
180. The frustum of a regular triangular
pyramid has equilateral triangles for
its bases. The lower and upper base
edges are 9m and 3m, respectively.
If the volume is 118.2 cu. m.., how
far apart are the base?
A. 9m
B. 8m
C. 7m
D. 10m
181. A cylindrical gasoline tank, lying
horizontally, 0.90 m. in diameter
and 3 m long is filled to a depth of
0.60 m. How many gallons of
gasoline does it contain? Hint: One
cubic meter = 265 gallons
A. 250
B. 360
C. 300
D. 270
182. A closed cylindrical tank is 8 feet
long and 3 feet in diameter. When
lying in a horizontal position, the
water is 2 feet deep. If the tank is
the vertical position, the depth of
water in the tank is:
A. 5.67 m
B. 5.82 m
C. 5.82 ft
D. 5.67 ft
183. A circular cylinder is
circumscribed about a right prism
having a square base one meter on
an edge. The volume of the cylinder
is 6.283 cu. m. find its altitude in m.
meter on an edge. The volume of
the cylinder is 6.283 cu. m. Find its
altitude in m.
A. 5
B. 4.5
C. 69.08
D. 4
184. If 23 cubic meters of water are
poured into a conical vessel, it
reaches a depth of 12 cm. how
much water must be added so that
the length reaches 18 cm.?
A. 95 cubic meters
B. 100 cubic meters
C. 54.6 cubic meters
D. 76.4 cubic meters
185. The height of a right circular base
down is h. If it contains water to
depth of 2h/3 the ratio of the
volume of water to that of the cone
is:
A. 1:27
B. 2:3
C. 8:27
D. 26:27
186. A right circular cone with an
altitude of 9m is divided into two
segments; one is a smaller circular
cone having the same vertex with an
altitude of 6m. Find the ratio of the
volume of the two cones.
A. 19:27
B. 2:3
C. 1:3
D. 8:27
187. A conical vessel has a height of 24
cm. and a base diameter of 12 cm. It
holds water to a depth of 18 cm.
above its vertex. Find the volume of
its content in cc.
A. 387.4
B. 381.7
C. 383.5
D. 385.2
188. A right circular cone with an
altitude of 8 cm is divided into two
segments. One is a smaller circular
cone having the same vertex with
the volume equal to ¼ of the
original cone. Find the altitude of
the smaller cone.
A. 4.52 cm
B. 6.74 cm
C. 5.04 cm
D. 6.12 cm
189. The slant height of a right circular
cone is 5m long. The base diameter
is 6m. What is the lateral area in sq.
m?
A. 37.7
B. 47
C. 44
D. 40.8
190. A right circular cone has a volume
of 128 /3 cm3 and an altitude of 8
cm. The lateral area is:
A. 16√
B. 12 √
C. 16
D. 15
191. The volume of a right circular
cone is 36 . If its altitude is 3, find
its radius.
A. 3
B. 4
C. 5
D. 6
192. A cone and hemisphere share base
that is a semicircle with radius 3
and the cone is inscribed inside the
hemisphere. Find the volume of the
region outside the cone and inside
the hemisphere.
A. 24.874
B. 27.284
C. 28.274
D. 28.724
193. A cone was formed by rolling a
thin sheet of metal in the form of a
sector of a circle 72 cm in diameter
with a central angle of 210°. What
is the volume of the cone in cc?
A. 13,602
B. 13,504
C. 13,716
D. 13,318
194. A cone was formed by rolling a
thin sheet of metal in the form of a
sector of a circle 72 cm in diameter
with a central angle of 150°. Find
the volume of the cone in cc.
A. 7733
B. 7722
C. 7744
D. 7711
195. A chemist’s measuring glass is
conical in shape. If it is 8 cm deep
and 3 cm across the mouth, find the
distance on the slant edge between
the markings for 1 cc and 2 cc.
A. 0.82 cm
B. 0.79 cm
C. 0.74 cm
D. 0.92 cm
196. The base areas of a frustum of a
cone are 25 sq. cm. and 16 sq. cm,
respectively. If its altitude is 6 cm,
find its volume.
A. 120 cm3
B. 122 cm3
C. 129 cm3
D. 133 cm3
Problems – Spheres, Prismatoid, Solids of
Revolutions, Miscellaneous Applications
Set 15
197. What is the surface area of a
sphere whose volume is 36 cu. m?
A. 52.7 m2
B. 48.7 m2
C. 46.6 m2
D. 54.6 m2
198. If the surface area of a sphere is
increased by 21%, its volume is
increased by:
A. 13.31%
B. 33.1%
C. 21%
D. 30%
199. The surface area of the sphere is
4 2. Find the percentage increase
in its diameter when the surface
area increases by 21%.
A. 5%
B. 10%
C. 15%
D. 20%
200. Find the percentage increase in
volume of a sphere if its surface
area is increased by 21%
A. 30.2%
B. 33.1%
C. 34.5%
D. 30.9%
201. The volume of a sphere is
increased by how much if its
surface area is increased by 20%?
A. 32.6%
B. 33%
C. 44%
D. 72.8%
202. Given two spheres whose
combined volume is known to be
819 cu. m. if their radii are in the
ratio 3:4, what is the volume of the
smaller sphere?
A. 576 cu. m.
B. 243 cu. m.
C. 343 cu. m.
D. 476 cu. m.
203. How much will the surface area of
a sphere be increased if its radius is
increased by 5%?
A. 25%
B. 15.5%
C. 12.5%
D. 10.25%
204. The volume of a sphere is 904.78
cu. m. Find the volume of the
spherical segment of height 4m.
A. 234.57 cu. m.
B. 256.58 cu. m.
C. 145.69 cu. m.
D. 124.58 cu. m.
205. A sphere of radius r just fits into a
cylindrical container of radius r and
altitude 2r. Find the empty space in
the cylinder.
A. (8/9) 3
3
B. (20/27)
C. (4/5) 3
D. (2/3) 3
206. If a solid steel ball is immersed in
an eight cm. diameter cylinder, it
displaces water to a depth of 2.25
cm. the radius of the ball is:
A. 3 cm
B. 6 cm
C. 9 cm
D. 12 cm
207. The diameter of two spheres is in
the ratio 2:3. If the sum of their
volumes is 1,260 cu. m., the volume
of the larger sphere is:
A. 972 cu. m.
B. 927 cu. m.
C. 856 cu. m.
D. 865 cu. m.
208. A hemispherical bowl of radius 10
cm is filled with water to such a
depth that the water surface area is
equal to 75
The volume of
water is:
A. 625/3
B. 625
C. 625
D. 625
209. A water tank is in the form of a
spherical segment whose base radii
are 4m and 3m and whose altitude
is 6m. The capacity of the tank in
gallon is:
A. 91,011
B. 92,011
C. 95,011
D. 348.72
210. Find the volume of a spherical
sector of altitude 3 cm. and radius 5
cm.
A. 75 cu. cm.
B. 100 cu. cm.
C. 50 cu. cm.
D. 25 cu. cm.
211. How far from the center of a
sphere of a radius 10 cm should a
plane be passed so that the ratio of
the areas of two zones is 3:7?
A. 3 cm
B. 4 cm
C. 5 cm
D. 6 cm
212. A 2-m diameter spherical tank
contains1396 liter of water. How
many liters of water must be added
for the water to reach a depth of
1.75 m?
A. 2613
B. 2723
C. 2542
D. 2472
213. Find the volume of a spherical
segment of radius 10 m and the
altitude 5 m.
A. 654.5 cu. m.
B. 659.8 cu. m.
C. 675.2 cu. m.
D. 680.5 cu. m.
214. Find the volume of a spherical
wedge of radius 10 cm. and central
angle 50°.
A. 425.66 sq. m.
B. 431.25 sq. m.
C. 581.78 sq. m.
D. 444.56 sq. m.
215. Determine the area of the zone of a
sphere of radius 8 in. and altitude 12
in.
A. 192
B. 198
C. 185
D. 195
216. The corners of a cubical block
touch the closed spherical shell that
encloses it. The volume of the box
is 2744 cc. What volume in cc,
inside the shell is not occupied by
the block?
A. 1356 cm3
B. 4721 cm3
C. 3423 cm3
D. 7623 cm3
217. A cubical container that measures
2 inches on each side is tightly
packed with 8 marbles and is filled
with water. All 8 marbles are in
contact with the walls of the
container and the adjacent marbles.
All of the marbles are of the same
size. What is the volume of water in
the container?
A. 0.38 cu. in.
B. 2.5 cu. in.
C. 3.8 cu. in.
D. 4.2 cu. in.
218. The volume of the water is a
spherical tank is 1470.265 cm3.
Determine the depth of water if the
tank has a diameter of 30 cm.
A. 8
B. 6
C. 4
D. 10
219. The volume of water in a spherical
tank having a diameter of 4 m. is
5.236 m3. Determine the depth of
the water on the tank.
A. 1.0
B. 1.4
C. 1.2
D. 1.6
220. A mixture compound from equal
parts of two liquids, one white and
the other black was placed in a
hemispherical bowl. The total depth
of the two liquids is 6”. After
standing for a short time the mixture
separated the white liquid settling
below the black. If the thickness of
the segment of the black liquid is
2”, find the radius of the bowl in
inches.
A. 7.53
B. 7.33
C. 7.73
D. 7.93
221. 20.5 cubic meters of water is
inside a spherical tank whose radius
is 2m. find the height of the water
surface above the bottom of the
tank, in m.
A. 2.7
B. 2.5
C. 2.3
D. 2.1
222. The volume of the sphere is
3
The surface area of this
sphere in sq. m. is:
A. 36
B. 24
C. 18
D. 12
223. Spherical balls 1.5 cm in diameter
area packed in a box measuring 6
cm by 3 cm by 3 cm. If as many
balls as possible are packed in the
box, how much free space remains
in the box?
A. 28.41 cc
B. 20.47 cc
C. 29.87 cc
D. 25.73 cc
224. A solid has a circular base of
radius r. find the volume of the solid
if every plane perpendicular to a
given diameter is a square.
A. 16 r3/3
B. 5 r3
C. 6 r3
D. 19 r3/3
225. A solid has circular base of
diameter 20 cm. Find the volume of
the solid if every cutting plane
perpendicular to the base along a
given diameter is an equilateral
triangle.
A. 2514 cc
B. 2107 cc
C. 2309 cc
D. 2847 cc
226. The base of a certain solid is a
triangle of base b and altitude h. if
all sections perpendicular to the
altitude of the triangle are regular
hexagons, find the volume of the
solid.
A.
√
B. √
C. √
D. √
227. The volume generated by the
circle by the circle
revolved about the
line 2x – 3y – 12 = 0 is:
A. 3242 cubic units
B. 3342 cubic units
C. 3452 cubic units
D. 3422 cubic units
228. The volume generated by rotating
the curve
about
the line 4x + 3y = 20 is:
A. 4
B. 58 2
C. 42
D. 48 2
229. Find the volume generated by
revolving the area bounded by the
ellipse
about the line x
= 3.
A. 347.23 cu. units
B. 355.31 cu. units
C. 378.43 cu. units
D. 389.51 cu. units
230. The area in the second quadrant of
the circle
is revolved
about the line y + 10 = 0. What is
the volume generated?
A. 2218.6
B. 2228.8
C. 2233.4
D. 2208.5
231. A square area of edge “a” revolves
about a line through one vertex,
making an angle with an edge and
not crossing the square. Find the
volume generated.
A. 3 a3 (sin + cos )
B. a3 (sin + cos ) / 2
C.
a3 (sin + cos )
D. a3 (sin + cos )
232. Given an ellipse whose semimajor axis is 6 cm. and semi-minor
axis is 3 cm. what is the volume
generated if it is revolved about the
minor axis?
A. 36 cu. cm.
B. 72 cu. cm.
C. 96 cu. cm
D. 144 cu. cm
233. A square hole 2” x 2” is cut
through a 6-inch diameter long
along its diameter and perpendicular
to its axis. Find the volume of wood
that was removed.
A. 27.32 cu. in.
B. 23.54 cu. in.
C. 21.78 cu. in.
D. 34.62 cu. in.
Part 5
Analytical Geometry
Problems – Points, Lines, Circles
Set 16
234. State the quadrant in which the
coordinate (15, -2) lies.
A. I
B. IV
C. II
D. III
235. Of what quadrant is A, if sec A is
positive and csc A is negative?
A. III
B. I
C. IV
D. II
236. The segment from (-1, 4) to (2, -2)
is extended three times its own
length. The terminal point is
A. (11, -18)
B. (11, -24)
C. (11, -20)
D. (-11, -20)
237. The midpoint of the line segment
between P1(x, y) and P2(-2, 4) is
Pm(2, -1). Find the coordinate of P1.
A. (6, -5)
B. (5, -6)
C. (6, -6)
D. (-6, 6)
238. Find the coordinates of the point
P(2,4) with respect to the translated
axis with origin at (1,3).
A. (1, -1)
B. (1, 1)
C. (-1, -1)
D. (-1, 1)
239. Find the median through (-2, -5) of
the triangle whose vertices are (-6,
2), (2, -2), and (-2, -5).
A. 3
B. 4
C. 5
D. 6
240. Find the centroid of a triangle
whose vertices are (2, 3), (-4, 6) and
(2, -6).
A. (0, 1)
B. (0, -1)
C. (1, 0)
D. (-1, 0)
241. Find the area of triangle whose
vertices are A (-3, -1), B(5, 3) and
(2, -8)
A. 34
B. 36
C. 38
D. 32
242. Find the distance between the
points (4, -2) and (-5, 1)
A. 4.897
B. 8.947
C. 7.149
D. 9.487
243. Find the distance between A(4, -3)
and B(-2, 5).
A. 11
B. 8
C. 9
D. 10
244. If the distance between the points
(8, 7) and (3, y) is 13, what is the
value of y?
A. 5
B. -19
C. 19 or -5
D. 5 or -19
245. The distance between the points
(sin x, cos x) and (cos x, -sin x) is:
A. 1
B. √
C. 2 sin x cos x
D. 4 sin x cos x
246. Find the distance from the point
(2, 3) to the line 3x + 4y + 9 = 0.
A. 5
B. 5.4
C. 5.8
D. 6.2
247. Find the distance from the point
(5, -3) to the line 7x - 4y - 28 = 0.
A. 2.62
B. 2.36
C. 2.48
D. 2.54
248. How far is the line 3x – 4y + 15 =
0 from the origin?
A. 1
B. 2
C. 3
D. 4
249. Determine the distance from (5,
10) to the line x – y = 0
A. 3.86
B. 3.54
C. 3.68
D. 3.72
250. The two points on the lines 2x +
3y +4 = 0 which are at distance 2
from the line 3x + 4y – 6 = 0 are:
A. (-8, -8) and (-16, -16)
B. (-44, 64) and (-5, 2)
C. (-5.5, 1) and (-5, 2)
D. (64, -44) and (4, -4)
251. The intercept form for algebraic
straight-line equation is:
A.
B.
C.
D.
252. Find the slope of the line defined
by y – x = 5
A. 1
B. -1/2
C. ¼
D. 5 + x
253. The slope of the line 3x + 2y + 5 =
0 is:
A. -2/3
B. -3/2
C. 3/2
D. 2/3
254. Find the slope of the line whose
parametric equation is y = 5 – 3t
and x = 2 + t.
A. 3
B. -3
C. 2
D. -2
255. Find the slope of the curve whose
parametric equations are
x = -1 + t
y = 2t
A. 2
B. 3
C. 1
D. 4
256. Find the angle that the line 2y – 9x
– 18 = 0 makes with the x-axis.
A. 74.77°
B. 4.5°
C. 47.77°
D. 77.47°
257. Which of the following is
perpendicular to the line x/3 + y/4 =
1?
A. x – 4y – 8 = 0
B. 4x – 3y – 6 = 0
C. 3x – 4y – 5 = 0
D. 4x + 3y – 11 = 0
258. Find the equation of the bisector of
the obtuse angle between the lines
2x + y = 4 and 4x - 2y = 7
A. 4y = 1
B. 8x = 15
C. 2y = 3
D. 8x + 4y = 6
259. The equation of the line through
(1, 2) and parallel to the line 3x – 2y
+ 4 = 0 is:
A. 3x – 2y + 1 = 0
B. 3x – 2y – 1 = 0
C. 3x + 2y + 1 = 0
D. 3x + 2y – 1 = 0
260. If the points (-3, -5), (x, y), and (3,
4) lie on a straight line, which of the
following is correct?
A. 3x + 2y – 1 = 0
B. 2x + 3y + 1 = 0
C. 2x + 3y – 1 = 0
D. 3x – 2y – 1 = 0
261. One line passes through the points
(1, 9) and (2, 6), another line passes
through (3, 3) and (-1, 5). The acute
angle between the two lines is:
A. 30°
B. 45°
C. 60°
D. 135°
262. The two straight lines 4x – y + 3 =
0 and 8x – 2y + 6 = 0
A. Intersects at the origin
B. Are coincident
C. Are parallel
D. Are perpendicular
263. A line which passes through (5, 6)
and (-3. -4) has an equation of
A. 5x + 4y + 1 = 0
B. 5x - 4y - 1 = 0
C. 5x - 4y + 1 = 0
D. 5x + y - 1 = 0
264. Find the equation of the line with
slope of 2 and y-intercept of -3.
A. y = -3x + 2
B. y = 2x – 3
C. y = 2/3 x + 1
D. y = 3x – 2
265. What is the equation of the line
that passes through (4, 0) and is
parallel to the line x – y – 2 = 0?
A. y + x + 4 = 0
B. y - x + 4 = 0
C. y - x - 4 = 0
D. y + x - 4 = 0
266. Determine B such that 3x + 2y – 7
= 0 is perpendicular to 2x – By + 2
=0
A. 2
B. 3
C. 4
D. 5
267. The equation of a line that
intercepts the x-axis at x = 4 and the
y-axis at y = -6 is:
A. 2x – 3y = 12
B. 3x + 2y = 12
C. 3x – 2y = 12
D. 2x – 37 = 12
268. How far from the y-axis is the
center of the curve 2x2 + 2y2 + 10x
– 6y – 55 = 0?
A. -3.0
B. 2.75
C. -3.25
D. 2.5
269. Find the area of the circle whose
center is at (2,-5) and tangent to the
line 4x + 3y – 8 = 0.
A. 6
B. 9
C. 3
D. 12
270. Determine the area enclosed by the
curve
A. 15
B. 225
C. 12
D. 144
271. Find the shortest distance from the
point (1, 2) to appoint on the
circumference of the circle defined
by the equation
A. 5.61
B. 5.71
C. 5.81
D. 5.91
272. Determine the length of the chord
common to the circles
= 64
and
A. 13.86
B. 12.82
C. 13.25
D. 12.28
273. If (3, -2) is on a circle with center
(-1, 1), then the area of the circle is:
A. 5
B. 25
C. 4
D. 3
274. The radius of the circle
is:
A. √
B. 33/16
C. √
D. 17
275. What is the radius of the circle
with the following equation?
A. 16/9
B. 4/3
C. 4
D. 8/3
277. Find the center of the circle
.
A. (3, -2)
B. (3, 2)
C. (-3, 2)
D. (-3, -2)
278. Determine the equation of the
circle whose center is at (4, 5) and
tangent to the circle whose equation
is
.
A.
B.
C.
D.
279. The equation of the circle with
center at (-2, 3) and which is
tangent to the line 20x – 21y – 42 =
0.
A.
B.
C.
D.
280. A circle has a diameter whose
ends are at (-3, 2) and (12, -6). Its
Equation is:
A.
B.
C.
A. 3.46
B. 5
C. 7
D. 6
276. The diameter of a circle described
by
is:
D.
281. Find the equation of the circle with
center on x + y = 4 and 5x + 2y + 1
= 0 and having a radius of 3.
A.
B.
C.
D.
282. If (3, -2) lies on the circle with
center (-1, 1) then the equation of
the circle is:
A.
B.
C.
D.
283. Find the equation of k for which
the equation
represents a point circle.
A. 5
B. -5
C. 6
D. -6
Problems – Parabola, Ellipse, Hyperbola,
Polar, Space
Set 17
284. The vertex of the parabola
is at:
A. (-3, 3)
B. (3, 3)
C. (-3, 3)
D. (-3, -3)
285. The length of the latus rectum of
the parabola
is:
A. 4p
B. 2p
C. P
D. -4p
286. Given the equation of the
parabola:
The length of its latus rectum is:
A. 2
B. 4
C. 6
D. 8
287. What is the length of the latus
rectum of the curve
A. 12
B. -3
C. 3
D. -12
288. Find the equation of the directrix
of the parabola
A. x = 8
B. x = 4
C. x = -8
D. x = -4
289. The curve y = opens:
A. Upward
B. To the left
C. To the right
D. Downward
290. The parabola y = opens:
A. To the right
B. To the left
C. Upward
D. Downward
291. Find the equation of the axis of
symmetry of the function y =
2
A. 4x + 7 = 0
B. x – 2 = 0
C. 4x – 7 = 0
D. 7x + 4 = 0
292. Find the equation of the locus of
the center of the circle which moves
so that it is tangent to the y-axis and
to the circle of radius one (1) with
center at (2,0).
A.
B.
C. 2
D.
293. Find the equation of the parabola
with vertex at (4, 3) and focus at (4,
-1).
A.
B.
C.
D.
294. Find the area bounded by the
curves
,x–4=
0, the x-axis, and the y-axis.
A. 10.67 sq. units
B. 10.33 sq. units
C. 9.67 sq. units
D. 8 sq. units
295. Find the area (in sq. units)
bounded by the parabolas
and
A. 11.7
B. 10.7
C. 9.7
D. 4.7
296. The length of the latus rectum of
the curve (x – 2)2 / 4 = (y + 4)2 / 25
= 1 is:
A. 1.6
B. 2.3
C. 0.80
D. 1.52
297. Find the length of the latus rectum
of the following ellipse:
25
A. 3.4
B. 3.2
C. 3.6
D. 3.0
298. If the length of the major and
minor axes of an ellipse is 10 cm
and 8 cm, respectively, what is the
eccentricity of the ellipse?
A. 0.50
B. 0.60
C. 0.70
D. 0.80
299. The eccentricity of the ellipse
+ y2 / 16 = 1 is:
A. 0.725
B. 0.256
C. 0.689
D. 0.866
300. An ellipse has the equation
16
y2 + 32x – 128 = 0. Its
eccentricity is:
A. 0.531
B. 0.66
C. 0.824
D. 0.93
301. The center of the ellipse 4
is at:
A. (2, 3)
B. (4, -6)
C. (1, 9)
D. (-2, -5)
302. Find the ratio of the major axis to
the minor axis of the ellipse:
9
A. 0.67
B. 1.8
C. 1.5
D. 0.75
303. The area of the ellipse 9
is equal to:
A. 1
sq. units
B. 20 sq. units
C. 25 sq. units
D. 30 sq. units
304. The area of the ellipse is given as
A = 3.1416 a b. Find the area of the
ellipse 25
.
A. 86.2 square units
B. 62.8 square units
C. 68.2 square units
D. 82.6 square units
305. The semi-major axis of an ellipse
is 4 and its semi-minor axis is 3.
The distance from the center to the
directrix is:
A. 6.532
B. 6.047
C. 0.6614
D. 6.222
306. Given an ellipse x2 / 36 + y2 / 32 =
1. Determine the distance between
foci.
A. 2
B. 3
C. 4
D. 8
307. How far apart are the directrices of
the curve 25
A. 12.5
B. 14.2
C. 13.2
D. 15.2
308. The major axis of the elliptical
path in which the earth moves
around the sun is approximately
186,000,000 miles and the
eccentricity of the ellipse is 1/60.
Determine the apogee of the earth.
A. 94,550,000 miles
B. 94,335.100 miles
C. 91,450,000 miles
D. 93,000,000 miles
309. Find the equation of the ellipse
whose center is at (-3, -1), vertex at
(2, -1), and focus at (1, -1).
A.
B.
C.
D.
310. Point P(x, y) moves with a
distance from point (0, 1) one-half
of its distance from line y = 4, the
equation of its locus is
A. 4x2 + 3y2 = 12
B. 2x2 - 4y2 = 5
C. x2 + 2y2 = 4
D. 2x2 + 5y3 = 3
311. The chords of the ellipse 64^2 +
25y^2 = 1600 having equal slopes
of 1/5 are bisected by its diameter.
Determine the equation of the
diameter of the ellipse.
A. 5x – 64y = 0
B. 64x – 5y = 0
C. 5x +64y = 0
D. 64x + 5y = 0
312. Find the equation of the upward
asymptote of the hyperbola whose
equation is (x – 2)2 / 9 – (y + 4)2 /
16
A. 3x + 4y – 20 = 0
B. 4x – 3y – 20 = 0
C. 4x + 3y – 20 = 0
D. 3x – 4y – 20 = 0
313. The semi-conjugate axis of the
hyperbola
4 = 1 is:
A. 2
B. -2
C. 3
D. -3
314. What is the equation of the
asymptote of the hyperbola
A. 2x – 3y = 0
B. 3x – 2y = 0
C. 2x – y = 0
D. 2x + y = 0
315. The graph y = (x – 1) / (x + 2) is
not defined at:
A. 0
B. 2
C. -2
D. 1
316. The equation x2 + Bx + y2 + Cy +
D = 0 is:
A. Hyperbola
B. Parabola
C. Ellipse
D. Circle
317. The general second degree
equation has the form Ax2 + Bxy +
Cy2 + Dx + Ey + F = 0 and
describes an ellipse if:
A. B2 – 4AC = 0
B. B2 – 4AC > 0
C. B2 – 4AC = 1
D. B2 – 4AC < 0
318. Find the equation of the tangent to
the circle x2 + y2 – 34 = 0 through
point (3, 5).
A. 3x + 5y -34 = 0
B. 3x – 5y – 34 = 0
C. 3x + 5y + 34 = 0
D. 3x – 5y + 34 = 0
319. Find the equation of the tangent to
the curve x2 + y2 + 4x + 16y – 32 =
0 through (4, 0).
A. 3x – 4y + 12 = 0
B. 3x – 4y – 12 = 0
C. 3x + 4y + 12 = 0
D. 3x + 4y - 12 = 0
320. Find the equation of the normal to
the curve y2 + 2x + 3y = 0 though
point (-5,2)
A. 7x + 2y + 39 = 0
B. 7x - 2y + 39 = 0
C. 2x - 7y - 39 = 0
D. 2x + 7y - 39 = 0
321. Determine the equation of the line
tangent to the graph y = 2x2 + 1, at
the point (1, 3).
A. y = 4x + 1
B. y = 4x – 1
C. y = 2x – 1
D. y = 2x + 1
322. Find the equation of the tangent to
the curve x2 + y2 = 41 through (5,
4).
A. 5x + 4y = 41
B. 4x – 5y = 41
C. 4x + 5y = 41
D. 5x – 4y = 41
323. Find the equation of a line normal
to the curve x2 = 16y at (4, 1).
A. 2x – y – 9 = 0
B. 2x – y + 9 =
C. 2x + y – 9 = 0
D. 2x + y + 9 = 0
324. What is the equation of the tangent
to the curve 9x2 + 25y2 – 225 = 0 at
(0, 3)?
A. y + 3 = 0
B. x + 3 = 0
C. x – 3 = 0
D. y – 3 = 0
325. What is the equation of the normal
to the curve x2 + y2 = 25 at (4, 3)?
A. 3x – 4y = 0
B. 5x + 3y = 0
C. 5x – 3y = 0
D. 3x + 4y = 0
326. The polar form of the equation 3x
+ 4y – 2 = 0 is:
A. 3r sin + 4r cos = 2
B. 3r cos + 4r sin = -2
C. 3r cos + 4r sin = 2
D. 3r sin + 4r tan = -2
327. The polar form of the equation 3x
+ 4y – 2 = 0 is:
A. r2 = 8
B.
C.
D. r2
328. the distance between points (5,
30°) and (-8, -50°) is:
A. 9.84
B. 10.14
C. 6.13
D. 12.14
329. Convert =
to Cartesian
equation.
A. x = √
B. y = x
C. 3y = √
D. y = √
330. The point of intersection of the
planes x + 5y – 2z = 9, 3x – 2y + z
= 3, and x + y + z = 2 is:
A. (2, 1, -1)
B. (2, 0, -1)
C. (-1, 1, -1)
D. (-1, 2, 1)
331. A warehouse roof needs a
rectangular skylight with vertices
(3, 0, 0), (3, 3, 0), (0, 3, 4), and (0,
0, 4). If the units are in meter, the
area of the skylight is:
A. 12 sq. m.
B. 20 sq. m.
C. 15 sq. m.
D. 9 sq. m.
332. The distance between points in
space coordinates are (3, 4, 5) and
(4, 6, 7) is:
A. 1
B. 2
C. 3
D. 4
333. What is the radius of the sphere
with center at origin and which
passes through the point (8, 1, 6)?
A. 10
B. 9
C. √
D. 10.5
Part 6
Differential Calculus
Problems – Limits, Differentiation, Rate
of Change, Slope
Set 18
334. Evaluate
A. 0
B. 1
C. 2
D. 3
335. Simplify the expression:
(
A.
B.
C.
D.
1
8
0
16
)
336. Evaluate the following limit:
A. 2/5
B. infinity
C. 0
D. 5/2
337. Evaluate the limit ( x – 4 ) /
(
as x approaches 4.
A. 0
B. undefined
C. 1/7
D. infinity
338. Evaluate the limit (1n x ) / x as x
approaches positive infinity.
A. 1
B. 0
C. e
D. infinity
339. Evaluate the following limit:
A. 1
B. Indefinite
C. 0
D. 2
340. Evaluate:
A. 0
B. ½
C. 2
D. -1/2
341. Evaluate the following:
A. Infinity
B.
C.
D.
342. Find dy/dx if y = 52x-1
A. 52x-1 ln 5
B. 52x-1 ln 25
C. 52x-1 ln 10
D. 52x-1 ln 2
343. Find dy/dx if y =
A.
√
/ 2√
B.
C.
√
/√
√
/√
D. √ √
344. Find dy/dx if y =
A. 4t3 + 14t2
B. t3 + 4t
C. 4t3 + 14t
D. 4t3 + t
345. Evaluate the first derivative of the
implicit function: 4x2 + 2xy + y2 = 0
A.
B.
C.
D.
346. Find the derivative of (x + 5) /
(
with respect to x.
A. DF(x) = (
/
2
(
B. DF(x) = (
/
2
(
C. DF(x) = (
/
2
(
D. DF(x) = (
/
2
(
347. If a simple constant, what is the
derivative of y = xa?
A. a xa-1
B. (a – 1)x
C. xa-1
D. ax
348. Find the derivative of the function
2x2 + 8x + 9 with respect to x.
A. Df(x) = 4x – 8
B. Df(x) = 2x + 9
C. Df(x) = 2x + 8
D. Df(x) = 4x + 8
349. What is the first derivative dy/dx
of the expression (xy) x = e?
A. – y(1 + ln xy) / x
B. 0
C. – y(1 – ln xy) / x2
D. y/x
350. find the derivative of
A.
B.
C.
D.
351. Given the equation: y = (e ln x)2,
find y’.
A. ln x
B. 2 (ln x) / x
C. 2x
D. 2 e ln x
352. Find the derivatives with respect to
x of the function √
A. -2
/√
B. -3x / √
C. -2 / √
D. -3x / √
353. Differentiate ax2 + b to the ½
power.
A. -2ax
B. 2ax
C. 2ax + b
D. ax + 2b
354. Find dy/dx if y = ln √
A. √ / ln x
B. x / ln x
C. 1 / 2x
D. 2 / x
355. Evaluate the differential of tan .
A. ln sec d
B. ln cos d
C. sec tan d
D. sec2 d
356. If y = cos x, what is dy/dx?
A. sec x
B. –sec x
C. sin x
D. –sin x
357. Find dy/dx: y = sin (ln x2).
A. 2 cos (ln x2)
B. 2 cos (ln x2) / x
C. 2x cos (ln x2)
D. 2 cos (ln x2) / x2
358. The derivative of ln (cos x) is:
A. sec x
B. –sec x
C. –tan x
D. tan x
359. Find the derivative of arcos 4x
with respect to x.
A. -4 / [1 – (4x)^2]^2
B. -4 / [1 – (4x)]^0.5
C. 4 / [1 – (4x)^2]^0.5
D. -4 / [(4x)^2 - 1]^0.5
360. What is the first derivative of y =
arcsin 3x.
A.
B.
C.
D.
√
√
361. If y = x (ln x), find d2 y/dx2.
A. 1 / x2
B. -1 / x
C. 1 / x
D. -1 / x2
362. Find the second derivative of y =
x-2 at x = 2.
A. 96
B. 0.375
C. -0.25
D. -0.875
363. Given the function f(x) = x3 – 5x +
2, find the value of the first
derivative at x = 2, f’ (2).
A. 7
B. 3x2 – 5
C. 2
D. 8
364. Given the function f(x) = x to the
3rd power – 6x + 2, find the value of
the first derivative at x = 2, f’(2)
A. 6
B. 3x2 – 5
C. 7
D. 8
365. Find the partial derivatives with
respect to x of the function: xy2 – 5y
+ 6.
A. y2 – 5
B. xy – 5y
C. y2
D. 2xy
366. Find the point in the parabola y2 =
4x at which the rate of change of the
ordinate and abscissa are equal.
A. (1, 2)
B. (2, 1)
C. (4, 4)
D. (-1, 4)
367. Find the slope of the line tangent
to the curve y = x3 – 2x + 1 at x = 1.
A. 1
B. ½
C. 1/3
D. ¼
368. Determine the slope of the curve
x^2 + y^2 – 6x – 4y – 21 = 0 at (0,
7).
A. 3/5
B. -2/5
C. -3/5
D. 2/5
369. Find the slope of the tangent to a
parabola y = x2 at a point on the
curve where x = ½.
A. 0
B. 1
C. ¼
D. -1/2
370. Find the slope of the ellipse x2 +
4y2 – 10x + 16y + 5 = 0 at the point
where y = -2 + 80.5 and x = 7.
A. -0.1654
B. -0.1538
C. -0.1768
D. -0.1463
371. Find the slope of the tangent to the
curve y = x4 – 2x2 + 8 through point
(2, 16).
A. 20
B. 1/24
C. 24
D. 1/20
372. Find the slope of the tangent to the
curve y2 = 3x2 + 4 through point (-2,
4)
A. -3/2
B. 3/2
C. 2/3
D. -2/3
373. Find the slope of the line whose
parametric equations are x = 4t + 6
and y = t – 1.
A. -4
B. ¼
C. 4
D. -1/4
374. What is the slope of the curve x2 +
y2 – 6x + 10y + 5 = 0 at (1, 0).
A. 2/5
B. 5/2
C. -2/5
D. -5/2
375. Find the slope of the curve y = 6(4
+ x) ½ at (0, 12).
A. 0.67
B. 1.5
C. 1.33
D. 0.75
376. Find the acute angle that the curve
y = 1 – 3x2 cut the x-axis.
A. 77°
B. 75°
C. 79°
D. 120°
377. Find the angle that the line 2y – 9x
– 18 = 0 makes with the x-axis.
A. 74.77°
B. 4.5°
C. 47.77°
D. 77.47°
378. Find the equation of the tangent to
the curve y = x + 2x1/3 through point
(8, 12)
A. 7x – 6y + 14 = 0
B. 8x + 5y + 21 = 0
C. 5x – 6y – 15 = 0
D. 3x – 2y – 1 = 0
379. What is the radius of curvature at
point (1, 2) of the curve 4x – y2 = 0?
A. 6.21
B. 5.21
C. 5.66
D. 6.66
380. Find the radius of curvature at any
point of the curve y + ln (cos x) = 0.
A. cos x
B. 1.5707
C. sec x
D. 1
381. Determine the radius of curvature
at (4, 4) of the curve y2 – 4x = 0.
A. 24.4
B. 25.4
C. 23.4
D. 22.4
382. Find the radius of curvature of the
curve x = y3 at (1, 1)
A. 4.72
B. 3.28
C. 4.67
D. 5.27
383. The chords of the ellipse 64x^2 +
25y^2 = 1600 having equal slopes
of 1/5 are bisected by its diameter.
Determine the equation of the
diameter of the ellipse.
A. 5x – 64y = 0
B. 64x – 5y = 0
C. 5x + 64y = 0
D. 64x + 5y = 0
Problems – Maxima & Minima, Time
Rates
Set 19
384. A function is given below, what x
value maximizes y?
y2 + y + x2 – 2x = 5
A. 2.23
B. -1
C. 5
D. 1
385. The number of newspaper copies
distributed is given by C = 50 t 2 –
200 t + 10000, where t is in years.
Find the minimum number of
copies distributed from 1995 to
2002.
A. 9850
B. 9800
C. 10200
D. 7500
386. Given the following profit-versusproduction function for a certain
commodity:
P = 200000 –
x–
8
Where P is the profit and x is unit of
production. Determine the maximum
profit.
A. 190000
B. 200000
C. 250000
D. 550000
387. The cost C of a product is a
function of the quantity x of the
product is given by the relation:
C(x) = x2 – 4000x + 50. Find the
quantity for which the cost is a
minimum.
A. 3000
B. 2000
C. 1000
D. 1500
388. If y = x to the 3rd power – 3x. find
the maximum value of y.
A. 0
B. -1
C. 1
D. 2
389. Divide 120 into two parts so that
product of one and the square of the
other is maximum. Find the
numbers.
A. 60 & 60
B. 100 & 20
C. 70 & 50
D. 80 & 40
390. If the sum of two numbers is C,
find the minimum value of the sum
of their squares.
A. C2 / 2
B. C2 / 4
C. C2 / 6
D. C2 / 8
391. A certain travel agency offered a
tour that will cost each person P
1500.00 if not more than 150
persons will join, however the cost
per person will be reduced by P
5.00 per person in excess of 150.
How many persons will make the
profit a maximum?
A. 75
B. 150
C. 225
D. 250
392. Two cities A and B are 8 km and
12 km, respectively, north of a river
which runs due east. City B being
15 km east of A. a pumping station
is to be constructed (along the river)
to supply water for the two cities.
Where should the station be located
so that the amount of pipe is a
minimum?
A. 3 km east of A
B. 4 km east of A
C. 9 km east of A
D. 6 km east of A
393. A boatman is at A, which is 4.5
km from the nearest point B on a
straight shore BM. He wishes to
reach, in minimum time, a point C
situated on the shore 9 km from B.
How far from C should he land if he
can row at the rate of 6 Kph and
walk at the rate of 7.5 Kph?
A. 1 km
B. 3 km
C. 5 km
D. 8 km
394. The shortest distance from the
point (5, 10) to the curve x2 = 12y
is:
A. 4.331
B. 3.474
C. 5.127
D. 6.445
395. A statue 3 m high is standing on a
base of 4 m high. If an observer’s
eye is 1.5 m above the ground, how
far should he stand from the base in
order that the angle subtended by
the statue is a maximum?
A. 3.41 m
B. 3.51 m
C. 3.71 m
D. 4.41 m
396. An iron bar 20 m long is bent to
form a closed plane area. What is
the largest area possible?
A. 21.56 square meter
B. 25.68 square meter
C. 28.56 square meter
D. 31.83 square meter
397. A Norman window is in the shape
of a rectangle surmounted by a
semi-circle. What is the ratio of the
width of the rectangle to the total
height so that it will yield a window
admitting the most light for a given
perimeter?
A. 1
B. 2/3
C. 1/3
D. ½
398. A rectangular field is to be fenced
into four equal parts. What is the
size of the largest field that can be
fenced this way with a fencing
length of 1500 feet if the division is
to be parallel to one side?
A. 65,200
B. 62,500
C. 64,500
D. 63,500
399. Three sides of a trapezoid are each
8 cm long. How long is the 4th side,
when the area of the trapezoid has
the greatest value?
A. 16 cm
B. 15 cm
C. 12 cm
D. 10 cm
400. An open top rectangular tank with
square bases is to have a volume of
10 cubic meters. The material for its
bottom cost P 150.00 per square
meter, and that for the sides is P
60.00 per square meter. The most
economical height is:
A. 2 meters
B. 2.5 meters
C. 3 meters
D. 3.5 meters
401. A rectangular box having a square
base and open at the top is to have a
capacity of 16823 cc. Find the
height of the box to use the least
amount of material.
A. 16.14 cm
B. 32.28 cm
C. 18.41 cm
D. 28.74 cm
402. The altitude of a cylinder of
maximum volume that can be
inscribed in a right circular cone of
radius r and height h is:
A. h/3
B. 2h/3
C. 3h/2
D. h/4
Problems – Integration
Set 20
Problem 1 (ME April 1997)
Integrate:
A.
+
C.
B.
D. 2x
Problem 4 (CE May 1995)
What is the integral of
A.
C.
B.
D. –
Problem 5 (Me October 1997)
The integral of
with respect to ;
∫
A.
C.
B.
D. –
Problem 6 (ME April 1998)
Integrate
A. ¼
B.
C. ¼
+
D.
Problem 2 (CE May 1999)
Evaluate: ∫
B.
D. ¼
Problem 7
Evaluate ∫
.
A.
C. ½
A.
C.
B.
D.
B.
D.
Problem 3 (CE May 1994)
Evaluate the integral of
Problem 8
Evaluate ∫
A.
C.
B.
D.
A.
C.
Problem 9
√
Evaluate ∫
Evaluate ∫
A.
A.
C. ½
C.
B.
D. ½
B.
D.
Problem 10
Evaluate ∫
.
Problem 15
Evaluate the integral of
A. ½
A. –
C. ½
B.
C.
D.
B.
Problem 11
Evaluate ∫
D. –
.
√
Problem 16
Evaluate ∫
A.
C.
B. (2/3)
D.
√
A.
C. –
B. –
Problem 12
D.
Evaluate ∫
Problem 17
Evaluate ∫
A.
C.
√
A.
C.
√
B.
D.
Problem 13
Evaluate ∫
B.
D.
Problem 18
Integrate the square root of
A.
A. ½
C. -√
C.
B.
D.
Problem 14
B.
√
D.
Problem 19 (CE November 1999)
Evaluate the integral of
with
limits from 0 to
A. 0.143
C. 0.114
B. 0.258
D. 0.186
Problem 20 (CE May 1997)
Evaluate the integral of
limits from 5 to 6.
A. 81/182
C. 83/182
B. 82/182
D. 84/182
with
Problem 21 (CE November 1996)
Evaluate the integral of
if it has an
upper limit of 1 and a lower limit of 0.
A. 0.022
C. 0.043
B. 0.056
D. 0.031
Problem 22 (CE May 1996)
Find the integral of
lower limit = 0 and upper limit =
A. 0.2
C. 0.6
B. 0.8
D. 0.4
if
.
Problem 23 (CE November 1997, Similar
to CE November 1994)
Using lower limit = 0 and upper limit =
,
what is the integral of
A. 6.783
C. 6.648
B. 6.857
D. 6.539
Evaluate the integral of
using
lower limit of 0 and upper limit = pi/2.
A. 2.0
C. 1.4
B. 1.7
D. 2.3
Problem 25 (CE May 1998, Similar to CE
November 95 & May 96)
Evaluate the integral of
using lower limit = 0 and upper limit =
.
A. 0.5046
C. 0.6107
B. 0.3068
D. 0.4105
Problem 26 (ECE April 1998)
Evaluate the integral of
to
.
A.
C.
B.
D.
Problem 27 (CE November 1996)
Evaluate the integral of
if the interior limit has an upper limit of y
and a lower limit of 0, and whose outer limit
has an upper limit of 2 and lower limit of 0.
A. 10
C. 30
B. 40
D. 20
Problem 28 (CE May 1999)
Evaluate ∫ ∫
A. 35/2
C. 17/2
B. 19/2
D. 37/2
Problem 29 (EE April 1997)
Problem 24 (CE November 1998)
from 0
Evaluate the double integral
of
, the limit of r is from 0 to
and the limit of u is from 0 to pi.
A. -1/6
C. 1/3
B. 1/6
D. 1/2
Problem 30
Evaluate ∫ ∫ ∫
A. 1/3
C. 1/2
B. 1/4
D. 1/6
Problems – Plane Areas,
Volumes, Surfaces, Centroid,
Etc.
Set 21
Problem 1
Find the area under the curve y
and the x-axis between x = 1 and x = 3.
A. 28 sq. units
C. 36
sq. units
B. 46 sq. units
D. 54
sq. units
Problem 2 (ECE April 2005)
⁄
Find the area bounded by y
,
the lines 3x = 2 and x = 10, and the X-axis.
A. 19.456 sq. units
C.
22.567 sq. units
B. 20.567 sq. units
D.
21.478 sq. units
Problem 3
Find the area of the region bounded by the
curves
, the x-axis, x = 1, and x = 4.
A.
C.
B.
D.
Problem 4 (ECE November 1996)
Find the area bounded by the y-axis and x
⁄
A. 25.6
B. 28.1
C. 12.8
D. 56.2
Problem 5
Find the area of the region bounded by one
loop of the curve
A.
sq. units
sq. units
B.
sq. units
(
sq. units
C.
D.
Problem 6 (CE November 1996,
November 1998)
Find the area bounded by the curve
A.
B.
C.
D.
Problem 7 (CE November 1997)
What is the area within the curve
A. 26
B. 28
Problem 8 (CE May 1999)
Find the area enclosed by
A.
B.
C. 30
D. 32
C.
D.
Problem 9
Find the curved surface (area) of the solid
generated by revolving the part of the curve
y = x2 from (0, 0) to (√
about the yaxis.
A.
sq. units
C.
sq. units
B.
sq. units
D.
sq. units
Problem 10
Find the volume generated by rotating the
region bounded by y = x, x = 1, and y2 = 4x,
about the x-axis.
A.
C.
B.
D.
Problem 11 (CE November 1995)
The area bounded by the curve y2 = 12x and
the line x = 3 is revolved about the line x =
3. What is the volume generated?
A. 186
C. 181
B. 179
D. 184
Problem 12 (CE May 1995)
Given is the area in the first quadrant
bounded by x2 = 8y, the line x = 4 and the xaxis. What is the volume generated by
revolving this area about the y-axis?
A. 50.26
C.
53.26
B. 52.26
D.
51.26
Problem 13 (CE November 1994)
Given is the area in the first quadrant
bounded by x2 = 8y, the line y – 2 = 0 and
the y-axis. What is the volume generated
when this area is revolved about the line y –
2 = 0?
A. 28.41
C.
27.32
B. 26.81
D.
25.83
Problem 14 (CE May 1998)
Find the length of the arc of x2 + y2 = 64
from x = -1 to x = -3, in the second
quadrant.
A. 2.24
C. 2.75
B. 2.61
D. 2.07
Problem 15
How far from the y-axis is the centroid of
the area bounded by the curve y = x3, the
line x = 2, and the y-axis.
A. 1.2
C. 1.6
B. 1.4
D. 1.8
Problem 16 (CE May 1998)
The area in the first quadrant, bounded by
the curve y = 2x1/2, the y-axis and the line y
– 6 = 0 is revolved about the line y = 6. Find
the centroid of the solid formed.
A. (2.2,6)
C.
(1.8,6)
B. (1.6,6)
D.
(2.0,6)
Problem 17
A solid formed by revolving about the yaxis, the area bounded by the curve y = x4,
the y-axis, and the line y = 16. Find its
centroid.
A. (0, 9.6)
C. (0,
8.3)
B. (0, 12.4)
D. (0,
12.8)
Problem 18 (CE November 1998)
A solid is formed by revolving about the yaxis, the area bounded by the curve x3 = y,
the y-axis and the line y = 8. Find its
centroid.
A. (0, 4.75)
C. (0,
5.25)
B. (0, 4.5)
D. (0,
5)
Problem 19 (CE November 1995)
Find the moment of inertia of the area
bounded by the parabola y2 = 4x, x-axis and
the line x = 1, with respect to the x-axis.
A. 1.067
C.
0.968
B. 1.244
D.
0.878
Problem 20
Find the work done in stretching a spring of
natural length 8 cm, from 10 cm to 13 cm.
Assume a force of 6 N is needed to hold it at
a length of 11 cm.
A. 21 N-m
C. 0.21
N-m
B. 2.1 N-m
D.
0.021 N-m
Problem 21
A conical tank that is 5 meters high has a
radius of 2 meters, and is filled with a liquid
that weighs 800 kg per cubic meter. How
much work is done in discharging all the
liquid at a point 3 meters above the top of
the tank?
A. 21,256
23,457
B. 21,896
22,667
kg-m
kg-m
kg-m
kg-m
C.
D.
Problem 22
How much work is required to pump all the
water from a right circular cylindrical tank
that is 8 feet in diameter and 9 feet tall, if it
is emptied at a point 1 foot above the top of
the tank?
A. 49,421 kg-m
C.
54,448 kg-m
B. 52,316 kg-m
D.
56,305 kg-m
Problem 23
A 60-m cable that weighs 4 kg/m has a 500kg weight attached at the end. How much
work is done in winding-up the last 20 m of
the cable?
A. 9,866 kg-m
C.
12,500 kg-m
B. 10,800 kg-m
D.
15,456 kg-m
Problem 24
A uniform chain the weighs 0.50 kg per
meter has a leaky 15-liter bucket attached to
it. If the bucket is full of liquid when 30
meters of chain is out and half-full when no
chain is out, how much work is done in
winding the chain? Assume that the liquid
leaks out at a uniform rate and weighs 1 kg
per liter.
A. 356.2 kg-m
C.
562.5 kg-m
B. 458.2 kg-m
D.
689.3 kg-m
Problem 25 (ECE Board November 1995)
The velocity of a body is given by v(t)
, where the velocity is given in
meters per second and t is given in seconds.
The distance covered in meters between t =
¼ and ½ second is close to.
A. 0.5221 m
C.
0.2251 m
B. -0.2251 m
D. 0.5221 m
Obtain the differential equation of the family
of straight lines with slope and y-intercept
equal.
A.
C.
B.
D.
Problem 26 (ECE November 1995)
The rate of change of a function of y with
respect to x equals 2 – y, and y = 8 when x =
0. Find y when x =
A. 2
C. 5
B. -5
D. -2
Problem 2
Obtain the differential equation of all
straight lines with algebraic sum of the
intercepts fixed as k.
A.
C.
B.
D.
Problem 3
Obtain the differential equation of all
straight lines at a fixed distance p from the
origin.
A.
C.
B.
D.
Problem 4 (CE May 1997)
Determine the differential equation of the
family of lines passing through the origin.
A.
C.
B.
D.
Problems – Differential
Equations & Application
Set 22
Problem 1
Problem 5
Obtain the differential equation of all circles
with center on line y = -x and passing
through the origin.
A.
B.
C.
D.
Problem 6
Obtain the differential equation of a
parabola with axis parallel to the x-axis.
A. ( )
C.
( )
B.
D.
Problem 7
Obtain the particular solution of
when
o.
A.
o
C.
o
B.
o
D.
o
Problem 11
Obtain the particular solution of
when
A.
C.
B.
D.
Problem 12
Solve the equation
A.
C.
B.
D.
Problem 13
Solve the equation (
A.
C.
B.
Problem 8
Obtain the general solution of the
differential
A.
C.
B.
D.
Problem 9
Obtain the general solution of
A.
C.
B.
D.
Problem 10
Solve the equation
.
⁄
A.
C.
⁄
B.
D.
D.
Problem 14
Solve the equation
A.
C.
B.
D.
Problem 15
Solve
A.
C.
B.
⁄
⁄
D.
B. it is homogeneous
C. it is separable
D. it can be solved using the integrating
factor
Problem 16
Solve the equation
A.
C.
B.
D.
Problem 17
Solve the equation
A. | |
C. |
|
B. |
D. |
Problem 22
A tank contains 400 liters of brine holding
100 kg of salt in solution. Water containing
125 g of salt per liter flows into the tank at
the rate of 12 liters per minute, and the
mixture, kept uniform by stirring, flows out
at the same rate. Find the amount of salt at
the end of 90 minutes.
A. 53.36 kg
C. 53.63 kg
B. 0
D. 65.33 kg
|
|
Problem 18
Solve the equation
A.
C.
B.
D.
Problem 19
Solve the equation
A.
C.
B.
D.
Problem 20
The differential equation
can
be made exact by using the integrating
factor:
A.
C.
B.
D.
Problem 21
What is not true for the differential equation
A. it is linear
Problem 23
Under certain conditions, cane sugar in
water is converted into dextrose at a rate
proportional to the amount that is
unconverted at any time. If, of 75 kg at time
t = 0, 8kg are converted during the first 30
minutes, find the amount converted in 2
hours.
A. 72.73 kg
C. 27.23 kg
B. 23.27 kg
D. 32.72 kg
Problem 24
A thermometer reading 18 oC is brought into
a room where the temperature is 70 oC; 1
minute later the thermometer reading is 31
o
C. Determine the thermometer reading 5
minutes after it is brought into the room.
A. 62.33 oC
C. 56.55 oC
B. 58.99 oC
D. 57.66 oC
Problems – Statics,
Translation, Rotation
Set 23
Problem 25
Solve the equation
A.
C.
B.
D.
Problem 26
The equation
equation of
A.
is the general
C.
B.
D.
Problem 27
Given the following simultaneous
differential equations:
.
Solve for
A.
B.
C.
D.
Problem 1
The weight of a mass of 10 kilograms at a
location where g = 9.77 m/s2 is:
A. 79.7 N
C. 97.7
N
B. 77.9 N
D. 977
N
Problem 2 (ME April 1997)
What is the resultant velocity of a point of xcomponent x
, and y-component
at time
y
A. 63.1327
C.
64.1327
B. 62.1327
D.
74.1327
Problem 3
A boat has a speed of 8 mph in still water
attempts to go directly across a river with a
current of 3 mph. What is the effective
speed of the boat?
A. 8.35 mph
C. 7.42
mph
B. 8.54 mph
D. 6.33
mph
Problem 4
A ship moving North at 10 mph. A
passenger walks Southeast across the deck at
5 mph. In what direction and how fast is the
man moving, relative to the earth’s surface.
A. N 28o40’W; 7.37 mph
C. N
o
61 20’W; 7.37 mph
B. N 61o20’E; 7.37 mph
D. N
28o40’E; 7.37 mph
Problem 5
A man wishes to cross due west on a river
which is flowing due north at the rate of 3
mph. If he can row 12 mph in still water,
what direction should he take to cross the
river?
A. S 14.47o W
C. S
o
81.36 W
B. S 75.52 o W
D. S
84.36 o W
Problem 6
A plane is moving due east with air speed of
240 kph. If a wind of 40 kph is blowing
from the south, find the ground speed of the
plane.
A. 243 kph
C. 200
kph
B. 423 kph
D. 240
kph
Problem 7
Three forces 20N, 30N, and 40N are in
equilibrium. Find the angle between the 3040N.
A. 26.96o
C. 40o
B. 28.96o
D.
o
25.96
Problem 8
A 10-kg weight is suspended by a rope from
a ceiling. If a horizontal force of 5.80 kg is
applied to the weight, the rope will make an
angle with the vertical equal to:
A. 60o
C. 45o
B. 30o
D. 75o
Problem 9
A 100 kN block slides down a place inclined
at an angle of 30o with the horizontal.
Neglecting friction, find the force that
causes the block to slide.
A. 86.6 kN
C. 20
kN
B. 80 kN
D. 50
kN
Problem 10
What tension must be applied at the ends of
a flexible wire cable supporting a load of 0.5
kg per horizontal meter in a span of 100 m if
the sag is to be limited to 1.25 m?
A. 423.42 kg
C.
500.62 kg
B. 584.23 kg
D.
623.24 kg
Problem 11
The allowable spacing of towers to carry an
aluminum cable weighing 0.03 kg per
horizontal meter if the maximum tension at
the lowest point is not to exceed 1150 at sag
of 0.50 m is:
A. 248 m
C. 408
m
B. 392 m
D. 422
m
Problem 12
A wooden plank “x” meters long has one
end leaning on top of a vertical wall 1.5 m
high and the other end resting on a
horizontal ground. Neglecting friction, find
x if a force (parallel to the plank) of 100 N is
needed to pull a 400 N block up the plank.
A. 6 m
C. 4 m
B. 5 m
D. 3 m
Problem 13
A block of wood is resting on a level
surface. If the coefficient of friction between
the block and the surface is 0.30, how much
can the plane be inclined without causing
the block to slide down?
A. 16.7o
C.
o
21.2
B. 30.2o
D.
o
33.3
Problem 14
A 500 kg block is resting on a 30o inclined
plane with a
Find the required force
P acting horizontally that will prevent the
block from sliding.
A. 1020 N
4236 N
B. 1160 N
5205 N
C.
D.
Problem 15
A 500 kg block is resting on a 30o inclined
plane with a
Find the required force
P acting horizontally that will start the block
up the plane.
A. 4236 N
C.
5205 N
B. 1160 N
D.
2570 N
Problem 16 (ME April 1996)
What is the acceleration of the body that
increases in velocity from 20 m/s to 40 m/s
in 3 seconds? Answer in S.I. units.
A. 8 m/s2
C. 5
2
m/s
B. 6.67 m/s2
D. 7
2
m/s
Problem 17 (CE May 1996)
From a speed of 75 kph, a car decelerates at
the rate of 500 m/min2 along a straight path.
How far in meters will it travel in 45 sec?
A. 795
C. 797
B. 791
D. 793
Problem 18 (CE November 1997)
With a starting speed of 30 kph at point A, a
car accelerates uniformly. After 18 minutes,
it reaches point B, 21 km from A. Find the
acceleration of the car in m/s2.
A. 0.126 m/s2
C.
0.0206 m/s2
B. 0.0562 m/s2
D. 3.42
m/s2
Problem 19 (CE November 1996)
A train upon passing point A at a speed of
72 kph accelerates at 0.75 m/s2 for one
minute along a straight path the decelerates
at 1.0 m/s2. How far in kilometers from
point A will it be 2 minutes after passing
point A?
A. 4.95
C. 4.85
B. 4.75
D. 4.65
Problem 20
A car starting from rest moves with a
constant acceleration of 10 km/hr2 for 1
hour, then decelerates at a constant -5
km/hr2 until it comes to a stop. How far has
it traveled?
A. 10 km
C. 12
km
B. 20 km
D. 15
km
Problem 21 (ECE November 1997)
The velocity of an automobile starting from
rest is given by
=
ft/sec. Determine its acceleration after an
interval of 10 seconds (in ft/sec2).
A. 2.10
C. 2.25
B. 1.71
D. 2.75
Problem 22 (CE may 1998)
A train running at 60 kph decelerated at 2
m/min2 for 14 minutes. Find the distance
traveled, in kilometers within this period.
A. 12.2
C. 13.8
B. 13.2
D. 12.8
Problem 23 (ECE November 1997)
An automobile accelerates at a constant rate
of 15 mi/hr to 45 mi/hr in 15 seconds, while
travelling in a straight line. What is the
average acceleration?
A. 2 ft/s2
C. 2.12
ft/s2
B. 2.39 ft/s2
ft/s2
D. 2.93
Problem 24 (CE November 1998)
A car was travelling at a speed of 50 mph.
The driver saw a road block 80 m ahead and
stepped on the brake causing the car to
decelerate uniformly at 10 m/s2. Find the
distance from the roadblock to the point
where the car stopped. Assume perception
reaction time is 2 seconds.
A. 12.48 m
C.
10.28 m
B. 6.25 m
D. 8.63
m
Problem 25
A man driving his car at 45 mph suddenly
sees an object in the road 60 feet ahead.
What constant deceleration is required to
stop the car in this distance?
A. -36.3 ft/s2
C. 33.4 ft/s2
B. -45.2 ft/s2
D. 42.3 ft/s2
Problem 26 (ECE March 1996)
A mango falls from a branch 5 meters above
the ground. With what speed in meters per
second will it strike the ground? Assume g =
10 m/s2.
A. 8
C. 10
B. 12
D. 14
Problem 27
A man standing at a window 5 meters tall
watches a falling stone pass by the window
in 0.3 seconds. From how high above the top
of the window was the stone released?
A. 12.86 m
C. 9.54
m
B. 11.77 m
D.
15.21 m
Problem 28
A bullet is fired at an initial velocity of 350
m/s and an angle of 50 o with the horizontal.
Neglecting air resistance, what maximum
height could the bullet rises?
A. 3,646 m
C.
3,466 m
B. 4,366 m
D.
3,664 m
Problem 29
A bullet is fired at an initial velocity of 350
m/s and an angle of 50 o with the horizontal.
Neglecting air resistance, find its range on
the horizontal plane through the point it was
fired.
A. 12,298 m
C.
12.298 m
B. 12.298 km
D.
12,298 cm
Problem 30
A bullet is fired at an initial velocity of 350
m/s and an angle of 50 o with the horizontal.
Neglecting air resistance, how long will the
bullet travel before hitting the ground?
A. 54.66 min.
C.
54.66 sec
B. 56.42 sec
D.
56.42 min.
Problem 31 (ME Board October 1997)
The muzzle velocity of a projectile is 1,500
fps and the distance of the target is 10 miles.
What must be the angle of elevation of the
gun?
A. 25o 32’
C. 24o
32’
B. 23o 34’
D. 26o
34’
Problem 32 (ME October 1997)
A shot is fired at an angle of 45o with the
horizontal and a velocity of 300 ft per
second. Find the height and the range of
projectile.
A. 600 ft and 2500 ft
C.
1000 ft and 4800 ft
B. 700 ft and 2800 ft
D. 750
ft and 3000 ft
Problem 33 (ECE April 1998)
A baseball is thrown from a horizontal plane
following a parabolic path with an initial
velocity of 100 m/s at an angle of 30 degrees
above the horizontal. How far from the
throwing point will the ball attain its original
level?
A. 890 m
C. 883
m
B. 875 m
D. 880
m
Problem 34
What is the range of a projectile if the initial
velocity is 30 m/s at an angle of 30o with the
horizontal?
A. 100 m
C.
79.45 m
B. 92 m
D. 110
m
Problem 35
A bullet is fired at an angle of 75o with the
horizontal with an initial velocity of 420
m/s. How high can it travel after 2 seconds?
A. 840 m
C. 750
m
B. 792 m
D. 732
m
Problem 36
A coin is tossed vertically upward from the
ground at a velocity of 12 m/s. How long
will the coin touch the ground?
A. 4.35 sec.
C. 2.45
sec.
B. 3.45 sec.
sec.
D. 1.45
Problem 37 (CE May 1997)
A stone is projected from the ground with a
velocity of 15 m/s at an angle of 30o with the
horizontal ground. How high in m will it
rise? Use g = 9.817 m/s.
A. 2.865 m
C.
2.586 m
B. 2.685 m
D.
8.652 m
Problem 38 (CE November 1996)
A ball is thrown from a tower 30m. high
above the ground with a velocity of 300 m/s
directed at 20o from the horizontal. How fast
will the ball hit the ground?
A. 24.2
C. 21.2
B. 23.2
D. 22.2
Problem 39
A man in a hot air balloon dropped an apple
at a height of 150 m. If the balloon is rising
at 15 m/s, find the highest point reached by
the apple.
A. 151.5 m
C.
171.5 m
B. 161.5 m
D.
141.5 m
Problem 40
A balloon is ascending at the rate of 10 kph
and is being carried horizontally by a wind
at 20 kph. If a bomb is dropped from the
balloon such that it takes 8 sec. to reach the
ground, the balloon’s altitude when the
bomb was released is:
A. 336.14 m
C. 252
m
B. 322.13 m
D. 292
m
Problem 41
A plane is flying horizontally 350 kph at an
altitude of 420 m. At this instant, a bomb is
released. How far horizontally from this
point will the bomb hit the ground?
A. 625 m
C. 785
m
B. 577 m
D. 900
m
Problem 42
A car whose wheels are 30 cm in radius is
traveling with a velocity of 110 kph. If it is
decelerated at a constant rate of 2 m/s2, how
many complete revolutions does it make
before it comes to rest?
A. 121
C. 123
B. 122
D. 124
Problem 43
The wheel of an automobile revolves at the
rate of 700 rpm. How fast does it move, in
km per hr., if the radius of its wheel is 250
mm?
A. 73.3
C. 5.09
B. 18.33
D.
34.67
Problem 44
Using a constant angular acceleration, a
water turbine is brought to its normal
operating speed of 180 rev/min in 6 minutes.
How many complete revolutions did the
turbine make in coming to normal speed?
A. 550
C. 560
B. 530
D. 540
Problem 45
A horizontal platform 6 m in diameter
revolves so that a point on its rim moves
6.28 m/s. Find its angular speed in rpm.
A. 15
C. 25
B. 20
D. 12
Problem 46 (CE May 1998)
A horizontal platform with a diameter of 6
m revolves about its center at 20 rpm. Find
the tangential speed, in m/s of a point at the
edge of the platform.
A. 6.34
C. 6.46
B. 6.28
D. 6.12
Problem 47
A flywheel rotating at 500 rpm decelerates
uniformly at 2 rad/sec2. How many seconds
will it take for the flywheel to stop?
A. 24.5 s
C. 25.1
s
B. 28.4 s
D. 26.2
s
Problem 48
A cyclist on a circular track of radius r =
800 feet is traveling at 27 ft/sec. His speed
in the tangential direction increases at the
rate of 3 ft/s2. What is the cyclist’s total
acceleration?
A. 2.8 ft/s2
C. -5.1
2
ft/s
B. -3.12 ft/s2
D. 3.13
2
ft/s
Problem 49
The radius of the earth is 3,960 miles. The
gravitational acceleration at the earth’s
surface is 32.16 ft/s2. What is the velocity of
escape from the earth in miles/s?
A. 6.94
C. 9.36
B. 8.62
D. 7.83
Problem 50
The radius of the moon is 1080 mi. The
gravitational acceleration at the moon’s
surface is 0.165 times the gravitational
acceleration at the earth’s surface. What is
the velocity of escape from the moon in
miles/second?
A. 2.38
C. 3.52
B. 1.47
D. 4.26
Set 24
Problem 1 (ME October 1997)
A 10-lbm object is acted upon by a 40-lb
force. What is the acceleration in ft/min2 ?
A. 8.0 x 10 to the 4th power ft/min2
C. 7.8
th
2
x 10 to the 4 power ft/min
B. 9.2 x 10 to the 4th power ft/min2
D.
th
4.638 x 10 to the 4 power ft/min2
Problem 2
What horizontal force P can be applied to a
100-kg block in a level surface with
coefficient of friction of 0.2, that will cause
and acceleration of 2.50 m/s2 ?
A. 343.5 N
C. 106
N
B. 224.5 N
D.
446.2 N
Problem 3
A skier wishes to build a rope tow to pull
her up a ski hill that is inclined at 15o with
the horizontal. Calculate the tension needed
to give the skier’s 54-kg body an
acceleration of 1.2 m/s2. Neglect friction.
A. 202 N
C. 106
N
B. 403 N
D. 304
N
Problems – Kinetics, Work,
Energy, Momentum, Etc.
Problem 4 (ME April 1997)
A pick-up truck is traveling forward at 25
m/s. The truck bed is loaded with boxes,
whose coefficient of friction with the bed is
0.4. What is the shortest time that the truck
can be brought to a stop such that the boxes
do not shift?
A. 4.75 sec
sec
B. 2.35 sec
sec
C. 5.45
B. 39.63 ft
34.81 ft
D.
D. 6.37
Problem 5 (CE November 1996)
A 40-kg block is resting on an inclined plane
making an angle of 20o from the horizontal.
If the coefficient of friction is 0.60,
determine the force parallel to the incline
that must be applied to cause impending
motion down the plane. Use g = 9.81 m/s.
A. 77
C. 72
B. 82
D. 87
Problem 6 (ECE November 1997)
A 50-kilogram block of wood rest on the top
of the plane whose length is 3 m. and whose
altitude is 0.8 m. How long will it take for
the block to slide to the bottom of the plane
when released?
A. 1.51 seconds
C. 2.51
seconds
B. 2.41 seconds
D. 2.14
seconds
Problem 7 (CE May 1999)
A body weighs 40 lbs. starts from rest and
inclined on a plane at an angle of 30o from
the horizontal for which the coefficient of
friction
How long will it move
during the third second?
A. 19.99 ft
C.
18.33 ft
Problem 8
A car and its load weigh 27 kN and the
center of gravity is 600 mm from the ground
and midway between the front and rear
wheel which are 3 m apart. The car is
brought to rest from a speed of 54 kph in 5
seconds by means of the brakes. Compute
the normal force on each of the front wheels
of the car.
A. 7.576 kN
C.
5.478 kN
B. 9.541 kN
D. 6
kN
Problem 9 (ME April 1998, CE November
1999 “Structural”)
An elevator weighing 2,000 lb attains an
upward velocity of 16 fps in 4 sec with
uniform acceleration. What is the tension in
the supporting cables?
A. 1,950 lb
C.
2495 lb
B. 2,150 lb
D.
2,250 lb
Problem 10 (CE November 1997
“Structural”)
A block weighing 200 N rests on a plane
inclined upwards to the right at a slope of 4
vertical to 3 horizontal. The block is
connected to a cable initially parallel to the
plane, passing through the pulley and
connected to another block weighing 100 N
moving vertically downward. The
coefficient of kinetic friction between the
200 N block and the inclined plane is 0.10.
Which of the following most nearly gives
the acceleration of the system?
A. a = 2.93 m/sec2
C. a =
2
1.57 m/sec
B. a = 0.37 m/sec2
D. a =
2
3.74 m/sec
Problem 11 (ME October 1997)
A car travels on the horizontal unbanked
circular track of radius r. Coefficient of
friction between the tires and the track is
0.3. If the car’s velocity is 10 m/s, what is
the smallest radius it may travel without
skidding?
A. 50 m
C. 15
m
B. 60 m
D. 34
m
Problem 12
If a car travels at 15 m/s and the track is
banked 5o, what is the smallest radius it can
travel so that friction will not be necessary
to resist sliding?
A. 262.16 m
C.
278.14 m
B. 651.23 m
D.
214.74 m
Problem 13 (CE May 1999)
A vertical bar of length L with a mass of 40
kg is rotated vertically about its one end at
40 rpm. Find the length of the bar if it makes
an angle of 45o with the vertical?
A. 1.58 m
C. 3.26
m
B. 2.38 m
m
D. 1.86
Problem 14
The seats of a carousel are attached to a
vertical rotating shaft by a flexible cable 8 m
long. The seats have a mass of 75 kg. What
is the maximum angle of tilt for the seats if
the carousel operates at 12 rpm?
A. 30o
C. 45o
o
B. 35
D. 39o
Problem 15 (CE November 1998)
A highway curve is super elevated at 7o.
Find the radius of the curve if there is no
lateral pressure on the wheels of a car at a
speed of 40 mph.
A. 247.4 m
C.
229.6 m
B. 265.6 m
D.
285.3 m
Problem 16 (CE November 1997
“Structural”)
A 2-N weight is swung in a vertical circle of
1-m radius at the end of the cable that will
break if the tension exceeds 500 N. Find the
angular velocity of the weight when the
cable breaks:
A. 49.4 rad/s
C. 24.9
rad/s
B. 37.2 rad/s
D. 58.3
rad/s
Problem 17 (ME April 1998)
Traffic travels at 65 mi/hr around a banked
highway curve with a radius of 3000 ft.
What banking angle is necessary such that
friction will not be required to resist the
centrifugal force?
A. 5.4o
C. 3.2o
B. 18o
D. 2.5o
Problem 18 (ME April 1997)
A concrete highway curve with a radius of
500 feet is banked to give a lateral pressure
equivalent to f = 0.15. For what coefficient
of friction will skidding impend for a speed
of 60 mph?
A. < 0.360
C. >
0.310
B. < 0.310
D. >
0.360
Problem 19 (ME October 1997)
A 3500 lbf car is towing a 500 lbf trailer.
The coefficient of friction between all tires
and the road is 0.80. How fast can the car
and trailer travel around an unbanked curve
of radius 0.12 mile without either the car or
trailer skidding?
A. 87 mph
C. 26
mph
B. 72 mph
D. 55
mph
Problem 20 (ME October 1997)
A cast-iron governor ball 3 inches in
diameter has its center 18 inches from the
point of support. Neglecting the weight of
the arm itself, find the tension in the arm if
the angle with the vertical axes is 60o.
A. 7.63 lb
C. 7.56
lb
B. 6.36 lb
D. 7.36
lb
Problem 21
An object is placed 3 feet from the center of
a horizontally rotating platform. The
coefficient of friction is 0.3. The object will
begin to slide off when the platform speed is
nearest to:
A. 17 rpm
C. 22
rpm
B. 12 rpm
D. 27
rpm
Problem 22 (ME April 1998)
A force of 200 lbf acts on a block at an
angle of 28o with respect to horizontal. The
block is pushed 2 feet horizontally. What is
the work done by this force?
A. 320 J
C. 480
J
B. 540 J
D. 215
J
Problem 23 (ME April 1998)
A 10-kg block is raised vertically 3 meters.
What is the change in potential energy?
Answer in SI units closest to:
A. 350 N-m
C. 350
2 2
kg-m /s
B. 294 J
D. 320
J
Problem 24
At her highest point, a girl on the swing is 7
feet above the ground, and at her lowest
point, she is 3 feet above the ground. What
is her maximum velocity?
A. 10 fps
C. 14
fps
B. 12 fps
D. 16
fps
Problem 25
An automobile has a power output of 1 hp.
When it pulls a cart with a force of 300 N,
what is the cart’s velocity?
A. 249 m/s
C. 2.49
m/s
B. 24.9 m/s
D.
0.249 m/s
Problem 26
A hunter fires a 50 gram bullet at a tiger.
The bullet left the gun with a speed of 600
m/s. What is the momentum of the bullet?
A. 15 kg-m/s
C. 300
kg-m/s
B. 30 kg-m/s
D. 150
kg-m/s
Problem 27
An elevator can lift a load of 5000 N from
the ground level to a height of 20 meters in
10 seconds. Find the horsepower rating of
the elevator.
A. 10000
C. 13.4
B. 13400
D.
1340
Problem 28
The average horsepower required to raise a
150-kg box to a height of 20 meters over a
period of one minute is:
A. 450 hp
C.
2960 hp
B. 0.657 hp
D. 785
hp
Problem 29
What is the force of attraction between two
90-kg bodies spaced 40 m apart? Assume
gravitational constant, G = 6.67 x 10-11 Nm2/kg2
A.
N
C.
N
B.
N
D.
N
Problem 30
What is the efficiency of the pulley system,
which lifts a 1 tonne load, a distance of 2 m
by the application of the force 150 kg for a
distance of 15 m?
A. 11%
C. 75%
B. 46%
D.
89%
Problem 31
How much mass is converted to energy per
day in a nuclear power plant operated at a
level of 100 MW?
A.
kg
C.
kg
B.
kg
D.
kg
Problem 32 (ECE April 1998)
What is the kinetic energy of a 4000-lb
automobile, which is moving at 44 ft/s?
A.
ft-lb
C.
ft-lb
B.
ft-lb
D.
ft-lb
Problem 33 (ME October 1997)
A 4000-kg elevator starts from rest
accelerates uniformly to a constant speed of
2.0 m/s and decelerates uniformly to stop 20
m above its initial position. Neglecting the
friction and other losses, what work was
done on the elevator?
A.
Joule
C.
Joule
B.
Joule
D.
Joule
Problem 34
The brakes of a 1000-kg automobile exert
3000 N. How long will it take for the car to
come to a complete stop from a velocity of
30 m/s?
A. 15 sec
C. 5
sec
B. 10 sec
D. 2
sec
Problem 35 (ME April 1997)
A car weighing 40 tons is switched to a 2%
upgrade with a velocity of 30 mph. If the
train resistance is 10 lbs/ton, how far does
the grade will it go?
A. 1124 ft
C.
1204 ft
B. 2104 ft
D.
1402 ft
Problem 36 (ME October 1997)
A body weighing 1000 lbs. fall 6 inches and
strikes a 2000 lbs (per inch) spring. What is
the deformation of the spring?
A. 3 inches
C. 100
mm
B. 6 inches
D. 2
inches
Problem 37
A 16-gram mass is moving at 30 cm/s while
a 4-gram mass is moving in an opposite
direction at 50 cm/s. They collide head on
and stick together. Their velocity after
collision is:
A. 14 cm/s
C. 13
cm/s
B. 15 cm/s
D. 18
cm/s
Problem 38
A 5-kg rifle fires a 15-g bullet at a muzzle
velocity of 600 m/s. What is the recoil
velocity of the rifle?
A. 1800 m/s
C. 18
m/s
B. 180 m/s
D. 1.80
m/s
Problem 39
A 0.50-kg ball with a speed of 20 m/s strikes
and sticks to a 70-kg block resting on a
frictionless surface. Find the block’s
velocity.
A. 142 m/s
C. 1.42
m/s
B. 14.2 m/s
D.
0.142 m/s
Problems – Stress, Strain,
Torsion, Shear & Moment, Etc.
Set 25
Problem 1
Determine the outside diameter of a hollow
steel tube that will carry a tensile load of
500 kN at a stress of 140 MPa. Assume the
wall thickness to be one-tenth of the outside
diameter.
A. 123 mm
C. 103
mm
B. 113 mm
D. 93
mm
Problem 2 (ME April 1998)
A force of 10 N is applied to one end of a 10
inches diameter circular rod. Calculate the
stress.
A. 0.20 kPa
C. 0.10
kPa
B. 0.05 kPa
D. 0.15
kPa
Problem 3
What force is required to punch a 20-mm
diameter hole through a 10-mm thick plate?
The ultimate strength of the plate material is
450 MPa.
A. 241 kN
C. 386
kN
B. 283 kN
D. 252
kN
Problem 4
A steel pipe 1.5m in diameter is required to
carry an internal pressure of 750 kPa. If the
allowable tensile stress of steel is 140 MPa,
determine the required thickness of the pipe
in mm.
A. 4.56
C. 4.25
B. 5.12
D. 4.01
Problem 5
A spherical pressure vessel 400-mm in
diameter has a uniform thickness of 6 mm.
The vessel contains gas under a pressure of
8,000 kPa. If the ultimate stress of the
material is 420 MPa, what is the factor of
safety with respect to tensile failure?
A. 3.15
C. 2.15
B. 3.55
D. 2.55
Problem 6 (CE November 1996)
A metal specimen 36-mm in diameter has a
length of 360 mm and a force of 300 kN
elongates the length bar to 1.20-mm. What
is the modulus of elasticity?
A. 88.419 GPa
C.
92.658 GPa
B. 92.564 GPa
D.
95.635 GPa
Problem 7
During a stress-strain test, the unit
deformation at a stress of 35 MPa was
observed to be
m/m and at a
stress of 140 MPa it was
m/m.
If the proportional limit was 200 MPa, what
is the modulus of elasticity? What is the
strain corresponding to stress of 80 MPa?
A. E = 210,000 MPa;
m/m
B. E = 200,000 MPa;
m/m
C. E = 211,000 MPa;
m/m
D. E = 210,000 MPa;
m/m
Problem 8
An axial load of 100 kN is applied to a flat
bar 20 mm thick, tapering in width from 120
mm to 40 mm in a length of 10 m.
Assuming E = 200 GPa, determine the total
elongation of the bar.
A. 3.43 mm
C. 4.33
mm
B. 2.125 mm
D.
1.985 mm
Problem 9
Steel bar having a rectangular cross-section
15mm 20mm and 150m long is suspended
vertically from one end. The steel has a unit
mass of 7850 kg/m3 and a modulus of
elasticity E of 200 GPa. If a loaf of 20 kN is
suspended at the other end of the rod,
determine the total elongation of the rod.
A. 43.5 mm
C. 35.4
mm
B. 54.3 mm
D. 45.3
mm
Problem 10
A steel bar 50 mm in diameter and 2 m long
is surrounded by a shell of cast iron 5 mm
thick. Compute the load that will compress
the bar a total of 1 mm in the length of 2 m.
Use Esteel = 200 GPa and Ecast-iron = 100 GPa.
A. 200 kN
C. 280
kN
B. 240 kN
D. 320
kN
Problem 11
A 20-mm diameter steel rod, 250 mm long
is subjected to a tensile force of 75 kN. If
the Poisson’s ratio is 0.30, determine the
lateral strain of the rod. Use E = 200 GPa.
A. y
mm/mm
C. y
mm/mm
B. y
mm/mm
D. y
mm/mm
Problem 12
A solid aluminum shaft of 100-mm diameter
fits concentrically in a hollow steel tube.
Determine the minimum internal diameter of
the steel tube so that no contact pressure
exists when the aluminum shaft carries an
axial compressive load of 600 kN. Assume
Poisson’s ratio = 1/3 and the modulus of
elasticity of aluminum E be 70 GPa.
A. 100.0364 mm
C.
100.0303 mm
B. 100.0312 mm
D.
100.0414 mm
Problem 13 (CE May 1996)
The maximum allowable torque, in kN-m,
for a 50-mm diameter steel shaft when the
allowable shearing stress is 81.5 MPa is:
A. 3.0
C. 4.0
B. 1.0
D. 2.0
Problem 14 (CE May 1997)
The rotation or twist in degrees of a shaft,
800 mm long subjected to a torque of 80 Nm, 20 mm in diameter and shear modulus G
of 80,000 MPa is:
A. 3.03
B. 4.04
C. 2.92
D. 1.81
Problem 15
Compute the value of the shear modulus G
of steel whose modulus of elasticity E is 200
GPa and Poisson’s ratio is 0.30.
A. 72,456 MPa
C.
79,698 MPa
B. 76,923 MPa
D.
82,400 MPa
Problem 16
Determine the length of the shortest 2-mm
diameter bronze wire, which can be twisted
through two complete turns without
exceeding a stress of 70 MPa. Use G = 35
GPa.
A. 6.28 m
C. 6.89
m
B. 5.23 m
D. 8.56
m
A. 110.6 MPa
120.6 MPa
B. 101.1 MPa
136.5 MPa
C.
D.
Problem 19
A load P is supported by two springs
arranged in series. The upper spring has 20
turns of 29-mm diameter wire on a mean
diameter of 150 mm. The lower spring
consist of 15 turns of 10-mm diameter wire
on a mean diameter of 130 mm. Determine
the value of P that will cause a total
deflection of 80 mm. Assume G = 80 GPa
for both spring.
A. 223.3 N
C.
214.8 N
B. 228.8 N
D.
278.4 N
Problem 20
A 10-meter long simply supported beam
carries a uniform load of 8 kN/m for 6
meters from the left support and a
Problem 17
concentrated load of 15 kN 2 meters from
A hollow steel shaft 2540 mm long must
the right support. Determine the maximum
transmit torque of 34 kN-m. The total angle
shear and moment.
of twist must not exceed 3 degrees. The
A. Vmax = 33.2 kN; Mmax = 85.92 KN-m
maximum shearing stress must not exceed
C. Vmax = 36.6 kN; Mmax = 83.72
110 MPa. Find the inside diameter and the
KN-m
outside diameter of the shaft the meets these
B. Vmax = 31.3 kN; Mmax = 81.74 KN-m
conditions. Use G = 83 GPa.
41.8 kN; Mmax = 92.23
A. D = 129 mm; d = 92 mm
C. D = 132 mm; D.
d =V100
mm
max =
KN-m
B. D = 125 mm; d = 65 mm
D. D = 112 mm;
d = 85 mm
Problem 18
Determine the maximum shearing stress in a
helical steel spring composed of 20 turns of
20-mm diameter wire on a mean radius of
80 mm when the spring is supporting a load
of 2 kN.
Problem 21 (ECE November 1996)
A simple beam, 10 m long carries a
concentrated load of 500 kN at the midspan.
What is the maximum moment of the beam?
A. 1250 kN-m
C.
1520 kN-m
B. 1050 kN-m
1510 kN-m
D.
Problem 22 (CE May 1997)
A small square 5cm by 5cm is cut out of one
corner of a rectangular cardboard 20cm wide
by 30cm long. How far, in cm from the
uncut longer side, is the centroid of the
remaining area?
A. 9.56
C. 9.48
B. 9.35
D. 9.67
Problem 23 (ECE April 1998)
What is the inertia of a bowling ball (mass =
0.5 kg) of radius 15 cm rotating at an
angular speed of 10 rpm for 6 seconds?
A. 0.0045 kg-m2
C.
2
0.005 kg-m
B. 0.001 kg-m2
D.
2
0.002 kg-m
Problem 24 (ECE November 1997)
What is the moment of inertia of a cylinder
of radius 5 m and mass of 5 kg?
A. 62.5 kg-m2
C. 72.5
2
kg-m
B. 80 kg-m2
D. 120
2
kg-m
Problems – Pressure,
Buoyancy, Fluid Flow, Pipes
Set 26
Problem 1
The mass of air in a room which is 3m
5m 20m is known to be 350 kg. Find its
density.
A. 1.167 kg/m3
C.
3
1.617 kg/m
B. 1.176 kg/m3
D.
3
1.716 kg/m
Problem 2 (ME October 1997)
One hundred (100) grams of water are
mixed with 150 grams of alcohol (
kg/cu m). What is the specific gravity of the
resulting mixtures, assuming that the two
fluids mix completely?
A. 0.96
C. 0.63
B. 0.82
D. 0.86
Problem 3 (ME April 1998)
One hundred grams of water are mixed with
150 grams of alcohol (
kg/cu m).
What is the specific volume of the resulting
mixtures, assuming that the fluids mix
completely?
A. 0.88 cu cm/g
C. 0.82
cu cm/g
B. 1.20 cu cm/g
D. 0.63
cu cm/g
Problem 4
The pressure 34 meters below the ocean is
nearest to:
A. 204 kPa
C. 344
kPa
B. 222 kPa
D. 362
kPa
Problem 5 (ME April 1997)
What is the atmospheric pressure on a planet
where the absolute pressure is 100 kPa and
the gage pressure is 10 kPa?
A. 90 kPa
C. 100
kPa
B. 80 kPa
D. 10
kPa
Problem 6
A pressure gage 6 m above the bottom of the
tank containing a liquid reads 90 kPa;
another gage height 4 m reads 103 kPa.
Determine the specific weight of the liquid.
A. 6.5 kN/m3
C. 3.2
3
kN/m
B. 5.1 kN/m3
D. 8.5
3
kN/m
Problem 7
The weight density of a mud is given by w =
10 + 0.5h, where w is in kN/m3 and h is in
meters. Determine the pressure, in kPa, at a
depth of 5 m.
A. 89.36 kPa
C. 62.5
kPa
B. 56.25 kPa
D.
78.54 kPa
Problem 8 (ME April 1997)
What is the resulting pressure when one
pound of air at 15 psia and 200oF is heated
at constant volume to 800oF?
A. 28.6 psia
C. 36.4
psia
B. 52.1 psia
D. 15
psia
Problem 9 (ECE November 1997)
The volume of a gas under standard
atmospheric pressure 76 cm Hg is 200 in3.
What is the volume when the pressure is 80
cm Hg, if the temperature is unchanged?
A. 190 in3
C. 110
3
in
B. 90 in3
D. 30.4
3
in
Problem 10
A two-meter square plane surface is
immersed vertically below the water surface.
The immersion is such that the two edges of
the square are horizontal. If the top of the
square is 1 meter below the water surface,
what is the total water pressure exerted on
the plane surface?
A. 43.93 kN
C.
64.76 kN
B. 52.46 kN
D.
78.48 kN
Problem 11
Find the total water pressure on a vertical
circular gate, 2 meters in diameter, with its
top 3.5 meters below the water surface.
A. 138.7 kN
C.
169.5 kN
B. 107.9 kN
D.
186.5 kN
Problem 12 (CE Board)
An iceberg having specific gravity of 0.92 is
floating on salt water of specific gravity of
1.03. If the volume of ice above the water
surface is 1000 cu. m., what is the total
volume of the ice?
A. 8523 m3
C.
3
9364 m
B. 7862 m3
D.
3
6325 m
Problem 13
A block of wood requires a force of 40 N to
keep it immersed in water and a force of 100
N to keep it immersed in glycerin (sp. gr. =
1.3). Find the weight and specific gravity of
the wood.
A. 0.7
C. 0.9
B. 0.6
D. 0.8
Problem 14 (ME April 1998)
Reynolds number may be calculated from:
A. diameter, density, and absolute
viscosity
B. diameter, velocity, and surface
tension
C. diameter, velocity, and absolute
viscosity
D. characteristic length, mass flow
rate per unit area, and absolute
viscosity
Problem 15 (ME April 1998)
The sum of the pressure load, elevation
head, and the velocity head remains
constant, this is known as:
A. Bernoulli’s Theorem
C.
Archimedes’ Principle
B. Boyle’s Law
D.
Torricelli’s Theorem
Problem 16 (ME October 1997)
What is the expected head loss per mile of
closed circular pipe (17-in inside diameter,
friction factor of 0.03) when 3300 gal/min of
water flows under pressure?
A. 38 ft
C.
3580 ft
B. 0.007 ft
D. 64
ft
Problem 17
What is the rate of flow of water passing
through a pipe with a diameter of 20 mm
and speed of 0.5 m/sec?
A. 1.24
B. 2.51
1.87
m3/s
m3/s
m3/s
m3/s
C. 1.57
D.
Problem 18
An orifice has a coefficient of discharge of
0.62 and a coefficient of contraction of 0.63.
Determine the coefficient of velocity for the
orifice.
A. 0.98
C. 0.97
B. 0.99
D. 0.96
Problem 19
The theoretical velocity of flow through the
orifice 3m above the surface of water in a
tall tank is:
A. 8.63 m/s
C. 6.38
m/s
B. 9.85 m/s
D. 7.67
m/s
Problem 20
Oil having specific gravity of 0.869 and
dynamic viscosity of 0.0814 Pa-s flows
through a cast iron pipe at a velocity of 1
m/s. The pipe is 50 m long and 150 mm in
diameter. Find the head lost due to friction.
A. 0.73 m
C. 0.68
m
B. 0.45 m
D. 1.25
m
Problem 21
What commercial size of new cast iron pipe
shall be used to carry 4490 gpm with a lost
of head of 10.56 feet per smile? Assume f =
0.019.
A. 625 mm
C. 479
mm
B. 576 mm
D. 352
mm
Problem 22
Assume that 57 liters per second of oil
(
kg/m3) is pumped through a 300
mm diameter pipeline of cast iron. If each
pump produces 685 kPa, how far apart can
they be placed? (Assume f = 0.031)
A. 23.7 m
C. 12.6
m
B. 32.2 m
D. 19.8
m
Problem 23
A 20-mm diameter commercial steel pipe,
30 m long is used to drain an oil tank.
Determine the discharge when the oil level
in the tank is 3 m above the exit of the pipe.
Neglect minor losses and assume f = 0.12.
A. 0.000256 m3/s
C.
0.000113 m3/s
B. 0.000179 m3/s
D.
3
0.000869 m /s
Problems – Simple Interest,
Compound Interest
Set 27
Problem 1
Find the interest on P6800.00 for 3 years at
11% simple interest.
A. P1,875.00
C.
P2,144.00
B. P1,987.00
D.
P2,244.00
Problem 2
A man borrowed P10,000.00 from his friend
and agrees to pay at the end of 90 days
under 8% simple interest rate. What is the
required amount?
A. P10,200.00
C.
P9,500.00
B. P11,500.00
D.
P10,700.00
Problem 3 (EE Board)
Annie buys a television set from a merchant
who offers P 25,000.00 at the end of 60
days. Annie wishes to pay immediately and
the merchant offers to compute the required
amount on the assumption that money is
worth 14% simple interest. What is the
required amount?
A. P20,234.87
C.
P24,429.97
B. P19,222.67
D.
P28,456.23
Problem 4
What is the principal amount if the amount
of interest at the end of 2½ year is P4500 for
a simple interest of 6% per annum?
A. P35,000.00
C.
P40,000.00
B. P30,000.00
D.
P45,000.00
Problem 5
How long must a P40,000.00 not bearing
4% simple interest run to amount to
P41,350.00?
A. 340 days
C. 304
days
B. 403 days
D. 430
days
Problem 6
If P16,000 earns P480 in 9 months, what is
the annual rate of interest?
A. 1%
C. 3%
B. 2%
D. 4%
Problem 7 (CE May 1997)
A time deposit of P110,000 for 31 days
earns P890.39 on maturity date after
deducting the 20% withholding tax on
interest income. Find the rate of interest per
annum.
A. 12.5%
C.
12.25%
B. 11.95%
D.
11.75%
Problem 8 (ME April 1998)
A bank charges 12% simple interest on a
P300.00 loan. How much will be repaid if
the load is paid back in one lump sum after
three years?
A. P408.00
C.
P415.00
B. P551.00
D.
P450.00
Problem 9 (CE May 1999)
The tag price of a certain commodity is for
100 days. If paid in 31 days, there is a 3%
discount. What is the simple interest paid?
A. 12.15%
C.
22.32%
B. 6.25%
D.
16.14%
Problem 10
Accumulate P5,000.00 for 10 years at 8%
compounded quarterly.
A. P12,456.20
C.
P10,345.80
B. P13,876.50
D.
P11,040.20
Problem 11
Accumulate P5,000.00 for 10 years at 8%
compounded semi-annually.
A. P10,955.62
C.
P9,455.67
B. P10,233.67
D.
P11,876.34
Problem 12
Accumulate P5,000.00 for 10 years at 8%
compounded monthly.
A. P15,456.75
C.
P14,768.34
B. P11,102.61
D.
P12,867.34
Problem 13
Accumulate P5,000.00 for 10 years at 8%
compounded annually.
A. P10,794.62
C.
P10,987.90
B. P8,567.98
D.
P7,876.87
Problem 14
How long will it take P1,000 to amount to
P1,346 if invested at 6% compounded
quarterly?
A. 3 years
C. 5
years
B. 4 years
D. 6
years
Problem 15
How long will it take for an investment to
double its amount if invested at an interest
rate of 6% compounded bi-monthly?
A. 10 years
C. 13
years
B. 12 years
D. 14
years
Problem 16
If the compound interest on P3,000.00 in 2
years is P500.00, then the compound interest
on P3,000.00 in 4 years is:
A. P956.00
C.
P1,125.00
B. P1,083.00
D.
P1,526.00
Problem 17
The salary of Mr. Cruz is increased by 30%
every 2 years beginning January 1, 1982.
Counting from that date, at what year will
his salary just exceed twice his original
salary?
A. 1988
C.
1990
B. 1989
D.
1991
Problem 18
If you borrowed P10,000.00 from a bank
with 18% interest per annum, what is the
total amount to be repaid at the end of one
year?
A. P11,800.00
C.
P28,000.00
B. P19,000.00
D.
P10,180.00
Problem 19
What is the effective rate for an interest rate
of 12% compounded continuously?
A. 12.01%
C.
12.42%
B. 12.89%
D.
12.75%
A. P125,458.36
P162,455.63
B. P147,456.36
P171,744.44
C.
D.
Problem 23
A man has a will of P650,000.00 from his
father. If his father deposited an account of
P450,000.00 in a trust fund earning 8%
compounded annually, after how many years
will the man receive his will?
A. 4.55 years
C. 5.11
years
B. 4.77 years
D. 5.33
years
Problem 20
How long will it take for an investment to
fivefold its amount if money is worth 14%
compounded semi-annually?
A. 11
C. 13
B. 12
D. 14
Problem 24
Mr. Adam deposited P120,000.00 in a bank
who offers 8% interest compounded
quarterly. If the interest is subject to a 14%
tax, how much will he receive after 5 years?
A. P178,313.69
C.
P170,149.77
B. P153,349.77
D.
P175,343.77
Problem 21
An interest rate of 8% compounded
semiannually is how many percent if
compounded quarterly?
A. 7.81%
C.
7.92%
B. 7.85%
D.
8.01%
Problem 25
What interest compounded monthly is
equivalent to an interest rate of 14%
compounded quarterly?
A. 1.15%
C.
10.03%
B. 13.84%
D.
11.52%
Problem 22
A man is expecting to receive P450,000.00
at the end of 7 years. If money is worth 14%
compounded quarterly, how much is it
worth at present?
Problem 26 (ME April 1996)
What is the present worth of two P100.00
payments at the end of the third and the
fourth year? The annual interest rate is 8%.
A. P152.87
P187.98
B. P112.34
P176.67
C.
D.
Problem 27 (ME April 1996)
Consider a deposit of P600.00, to be paid up
in one year by P700.00. What are the
conditions on the rate of interest, i% per
year compounded annually, such that the net
present worth of the investment is positive?
Assume i 0.
A.
i 16.7%
C.
12.5% i 14.3%
B.
i 14.3%
D.
16.7% i 100%
Problem 28 (ME April 1996)
A firm borrows P2000.00 for 6 years at 8%.
At the end of 6 years, it renews the loan for
the amount due plus P2000 more for 2 years
at 8%. What is the lump sum due?
A. P5,679.67
C.
P6,034.66
B. P6,789.98
D.
P5,888.77
Problem 29
At an annual rate of return of 8%, what is
the future worth of P1000 at the end of 4
years?
A. P1388.90
C.
P1765.56
B. P1234.56
D.
P1360.50
Problem 30 (ME October 1997)
A student has money given by his
grandfather in the amount of P20,000.00.
How much money in the form of interest
will he get if the money is put in a bank that
offers 8% rate compounded annually, at the
end of 7 years?
A. P34,276.48
C.
P36,279.40
B. P34,270.00
D.
P34,266.68
Problem 31 (ME October 1997)
If the interest rate on an account is 11.5%
compounded yearly, approximately how
many years will it take to triple the amount?
A. 11 years
C. 9
years
B. 10 years
D. 12
years
Problem 32 (ME October 1997)
The nominal interest rate is 4%. How much
is my P10,000.00 worth in 10 years in a
continuously compounded account?
A. P13,620.10
C.
P14,918.25
B. P13,650.20
C.
P13,498.60
Problem 33 (ME October 1997)
How much must be invested on January 1,
year 1, in order to accumulate P2,000.00 on
January 1, year 6 at 6%.
A. P1,295.00
C.
P1,495.00
B. P1,695.00
P1,595.00
D.
Problem 34 (ME April 1998)
If P5000.00 shall accumulate for 10 years at
8% compounded quarterly. Find the
compounded interest at the end of 10 years.
A. P6,005.30
C.
P6,000.00
B. P6,040.20
D.
P6,010.20
Problem 35 (ME April 1998)
A sum of P1,000.00 is invested now and left
for eight years, at which time the principal is
withdrawn. The interest that has accrued is
left for another eight years. If the effective
annual interest rate is 5%, what will be the
withdrawal amount at the end of the 16th
year?
A. P706.00
C.
P500.00
B. P774.00
D.
P799.00
Problem 36 (ME April 1998)
It is the practice of almost all banks in the
Philippines that when they grant a loan, the
interest for one year is automatically
deducted from the principal amount upon
release of money to a borrower. Let us
therefore assume that you applied for a loan
with a bank and the P80,000.00 was
approved at an interest rate of 14% of which
P11,200.00 was deducted and you were
given a check of P68,800.00. Since you have
to pay the amount of P80,000.00 one year
after, what then will be the effective interest
rate?
A. 15.90%
16.28%
B. 16.30%
16.20%
C.
D.
Problem 37 (ME April 1998)
The amount of P1,500.00 was deposited in a
bank account offers a future worth
P3,000.00. Interest is paid semi-annually.
Determine the interest rate paid on this
account.
A. 3.5%
C. 2.9
B. 2.5%
D. 4%
Problem 38 (ME April 1998)
A merchant puts in his P2,000.00 to a small
business for a period of six years. With a
given interest rate on the investment of 15%
per year, compounded annually, how much
will he collect at the end of the sixth year?
A. P4,400.00
C.
P4,390.00
B. P4,200.00
D.
P4,626.00
Problem 39
A person invests P4500 to be collected in 8
years. Given that the interest rate on the
investment is 14.5% per year compounded
annually, what sum will be collected in 8
years?
A. P13,678.04
C.
P14,888.05
B. P13,294.02
D.
P14,234.03
Problem 40
The following schedule of funds is available
to form a sinking fund.
current year (n)
50,000.00
n+1
40,000.00
n+2
30,000.00
n+3
20,000.00
At the end of the fourth year, equipment
costing P250,000.00 will have to be
purchased as a replacement for old
equipment. Money is valued at 20% by the
company. At the time of purchase, how
much money will be needed to supplement
the sinking fund?
A. P12,000.00
C.
P10,000.00
B. P11,000.00
D.
P9,000.00
Problem 41 (ME October 1995)
In year zero, you invest P10,000.00 in a
15% security for 5 years. During that time,
the average annual inflation is 6%. How
much, in terms of year zero pesos, will be in
the account at the maturity?
A. P15,386.00
C.
P13,382.00
B. P15,030.00
D.
P6,653.00
Problem 42 (ME October 1995)
A company invests P10,000 today to be
repaid in five years in one lump sum at 12%
compounded annually. If the rate of inflation
is 3% compounded annually, how much
profit in present day pesos is realized over
five years?
A. P5,626.00
C.
P3,202.00
B. P7,623.00
D.
P5,202.00
Problem 43
Compute the effective rate for an interest
rate of 16% compounded annually.
A. 16%
C.
16.98%
B. 16.64%
D.
17.03%
Problem 44
Compute the effective rate for an interest
rate of 16% compounded quarterly.
A. 16%
C.
16.98%
B. 16.64%
D.
17.03%
Problem 45
Convert 12% compounded semi-annually to
x% compounded quarterly.
A. 11.83%
C.
11.23%
B. 11.71%
D.
11.12%
Problem 46
Convert 12% compounded semi-annually to
x% compounded monthly.
A. 11.83%
C.
11.23%
B. 11.71%
D.
11.12%
Problem 47 (ME October 1995)
A bank is advertising 9.5% accounts that
yield 9.84% annually. How often is the
interest compounded?
A. daily
C. bimonthly
B. monthly
D.
quarterly
Problem 48 (ECE November 1995,
November 1998)
By the conditions of a will, the sum of
P25,000 is left to a girl to be held in a trust
fund by her guardian until it amount to
P45,000. When will the girl receive the
money if the fund is invested at 8%
compounded quarterly?
A. 7.42 years
C. 7
years
B. 7.25 years
D. 6.8
years
Problem 49 (ECE April 1995)
A man expects to receive P20,000 in 10
years. How much is that money worth now
considering interest at 6% compounded
quarterly?
A. P11,025.25
C.
P15,678.45
B. P17,567.95
D.
P12,698.65
Problem 50 (ECE March 1996)
What is the effective rate corresponding to
16% compounded daily? Take 1 year = 360
days.
A. 16.5%
C.
17.35%
B. 16.78%
17.84%
D.
Problem 51
What amount will be accumulated by a
present investment of P17,200 in 6 years at
2% compounded quarterly?
A. P19,387.15
C.
P19,856.40
B. P20,456.30
D.
P19,232.30
Problem 52
What rate of interest compounded annually
must be received if an investment of
P54,000 made now with result in a receipt of
P72,000 5 years hence?
A. 5.12%
C.
5.92%
B. 5.65%
D.
5.34%
Problem 53
With interest at 6% compounded annually,
how much is required 7 years hence to repay
an P8 M loan made today?
A. P12,456,789
C.
P12,029,042
B. P12,345,046
D.
P12,567,000
Problem 54
If money is worth 6% compounded
annually, what payment 12 years from now
is equivalent to a payment of P7000 9 years
from now?
A. P8765.10
C.
P8337.10
B. P8945.20
D.
P8234.60
Problem 55
If money is worth 6% compounded
annually, how much can be loaned now if
P6000 will be repaid at the end of 8 years?
A. P3567.30
C.
P3456.34
B. P3444.44
D.
P3764.50
annually. How much would the sum on Jan.
1, 1993?
A. P421,170
C.
P401,170
B. P521,170
D.
P621,170
Problem 56
A person invests P4500 to be collected in 8
years. Given that the interest rate on the
investment is 14.5% per year, compounded
annually, what sum, in pesos, will be
collected eight years hence?
A. P4504
C.
P13294
B. P9720
D.
P10140
Problem 57 (CE November 1996)
If P500,000 is deposited at a rate of 11.25%
compounded monthly, determine the
compounded interest after 7 years and 9
months.
A. P660,592
C.
P680,686
B. P670,258
D.
P690,849
Problem 58 (CE May 1996)
P200,000 was deposited on Jan. 1, 1988 at
an interest rate of 24% compounded semi-
Problems – Compound
Interest, Annuity
Set 28
Problem 1 (ECE November 1996)
Find the nominal rate that if converted
quarterly could be used instead for 25%
compounded semi-annually?
A. 14.93%
C.
15.56%
B. 14.73%
15.90%
D.
Problem 2 (CE November 1999)
Which of the following has the least
effective annual interest rate?
A. 12% compounded quarterly
C.
11.7% compounded semi-annually
B. 11.5% compounded monthly
D.
12.2% compounded annually
Problem 3 (CE November 1998)
One hundred thousand pesos was placed in a
time deposit that earns 9% compounded
quarterly, tax free. After how many years
would it be able to earn a total interest of
fifty thousand pesos?
A. 4.56
C. 4.32
B. 4.47
D. 4.63
Problem 4 (ECE November 1996)
The amount of P2,825.00 in 8 years at 5%
compounded quarterly is:
A. P4,166.77
C.
P4,188.56
B. P4,397.86
D.
P4,203.97
Problem 5
The amount of P2,825.00 in 8 years at 5%
compounded continuously is:
A. P4,166.77
C.
P4,397.86
B. P4,188.56
D.
P4,214.97
Problem 6 (CE May 1998)
What rate (%) compounded quarterly is
equivalent to 6% compounded semiannually?
A. 5.93
C. 5.96
B. 5.99
D. 5.9
Problem 7 (ECE April 1998)
The amount of P12,800 in 4 years at 5%
compounded quarterly is:
A. P15,614.59
C.
P16,311.26
B. P14,785.34
D.
P15,847.33
Problem 8
Fifteen percent (15%) when compounded
semi-annually is what effective rate?
A. 17.34%
C.
16.02%
B. 18.78%
D.
15.56%
Problem 9 (ECE November 1997)
What rate of interest compounded annually
is the same as the rate of interest of 8%
compounded quarterly?
A. 8.24%
C.
6.88%
B. 8.42%
D.
7.90%
Problem 10 (ECE November 1997)
How long will it take the money to triple
itself if invested at 10% compounded semiannually?
A. 13.3 years
C. 11.9
years
B. 11.3 years
D. 12.5
years
Problem 11 (ECE November 1997)
What is the accumulated amount after three
(3) years of P6,500.00 invested at the rate of
12% per year compounded semi-annually?
A. P9,500.00
C.
P9,221.00
B. P9,321.00
D.
P9,248.00
Problem 15 (ME October 1997)
A bank offers 1.2% effective monthly
interest. What is the effective annual rate
with monthly compounding?
A. 15.4%
C.
14.4%
B. 8.9%
D.
7.9%
Problem 12 (ECE November 1997)
What interest rate, compounded monthly is
equivalent to 10% effective rate?
A. 9.75%
C.
9.68%
B. 9.50%
D.
9.57%
Problem 16 (ME October 1997)
What is the present worth of P27,000.00 due
in 6 years if money is worth 13% and is
compounded semi-annually?
A. P12,681.00
C.
P15,250.00
B. P13,500.00
D.
P21,931.00
Problem 13 (ECE November 1997)
A man wishes his son to receive
P500,000.00 ten years from now. What
amount should he invest now if it will earn
interest of 12% compounded annually
during the first 5 years and 15%
compounded quarterly during the next 5
years?
A. P135,868.19
C.
P123,433.23
B. P134,678.90
D.
P145,345.34
Problem 14 (ME October 1997)
A savings association pays 4% interest
quarterly. What is the effective annual
interest rate?
A. 18.045%
C.
16.985%
B. 17.155%
D.
17.230%
Problem 17 (ME October 1997)
A student deposits P1,500.00 in a 9%
account today. He intends to deposit another
P3,000.00 at the end of two years. He plans
to purchase in five years his favorite shoes
worth P5,000.00. Calculate the money that
will be left in his account one year after the
purchase.
A. P1,280.00
C.
P1,300.00
B. P1,250.00
D.
P1,260.00
Problem 18
If money is worth 4% compounded monthly,
what payment at the end of each quarter will
replace payments of P500.00 monthly?
A. P1,500.00
C.
P1,505.00
B. P1,525.000
D.
P1,565.00
Problem 19
What amount would have to be invested at
the end of each year for the next 9 years at
4% compounded semi-annually in order to
have P5,000.00 at the end of the time?
A. P541.86
C.
P542.64
B. P553.82
D.
P548.23
Problem 20
A contractor bought a concrete mixer at
P120,000.00 if paid in cash. The mixer may
also be purchased by installment to be paid
within 5 years. If money is worth 8%, the
amount of each annual payment, if all
payments are made at the beginning of each
year, is:
A. P27,829.00
C.
P31,005.00
B. P29,568.00
D.
P32,555.00
Problem 21
A contract calls for semiannual payments of
P40,000.00 for the next 10 years and an
additional payment of P250,000.00 at the
end of that time. Find the equivalent cash
value of the contract at 7% compounded
semiannually?
A. P444,526.25
C.
P694,138.00
B. P598,589.00
D.
P752,777.00
Problem 22
A man is left with an inheritance from his
father. He has an option to receive P2 M at
the end of 10 years; however he wishes to
receive the money at the end of each year
for 5 years. If interest rate is 8%, how much
would he receive every year?
A. P400,000.00
C.
P232,020.00
B. P352,533.00
D.
P200,000.00
Problem 23 (CE November 1999)
To maintain its newly acquired equipment,
the company needs P40,000 per year for the
first five years and P60,000 per year for the
next five years. In addition, an amount of
P140,000 would also be needed at the end of
the fifth and the eighth years. At 6%, what is
the present worth of these costs?
A. P689,214
C.
P549,812
B. P512,453
D.
P586,425
Problem 24
A man receives P125,000.00 credits for his
old car when buying a new model costing
P375,000.00. What cash payment will be
necessary so that the balance can be
liquidated by payments of P12,500.00 at the
end of each month for 18 months when
interest is charged at the rate of 6%
compounded monthly?
A. P23,400.00
C.
P33,650.00
B. P28,750.00
D.
P35,340.00
Problem 25
Determine the present worth of an annual
payment of P2500.00 at the end of each year
for 12 years at 8% compounded annually.
A. P18,840.20
P15,000.00
B. P30,000.00
P17,546.04
C.
D.
Problem 26
A man borrowed P200,000.00 from a bank
at 12% compounded monthly, which is
payable monthly for 10 years (120
payments). If the first payment is to be made
after 3 months, how much is the monthly
payment?
A. P2,869.42
C.
P3,013.10
B. P2,927.10
D.
P3,124.12
Problem 27
What is the present worth of a P1000.00
annuity over a 10-year period, if interest rate
is 8%?
A. P7896.00
C.
P6234.80
B. P8976.00
D.
P6710.00
Problem 28 (ME October 1995)
How much money must you invest today in
order to withdraw P1000 per year for 10
years if interest rate is 12%?
A. P5650.00
C.
P5560.00
B. P6550.00
D.
P7550.00
Problem 29
A machine is under consideration for
investment. The cost of the machine is
P25,000. Each year it operates, the machine
will generate a savings of P15,000. Given an
effective annual interest of 18%, what is the
discounted payback period, in years, on the
investment in the machine?
A. 1.566
C.
2.155
B. 2.233
D.
2.677
Problem 30 (ME April 1996)
What is the present worth of a P100 annuity
starting at the end of the third year and
continuing to the end of the fourth year, if
the annual interest rate is 8%?
A. 153.44
C.
154.99
B. 152.89
D.
156.33
Problem 31
Consider a project which involves the
investment of P100,000 now and P100,000
at the end of one year. Revenues of
P150,000 will be generated at the end of
years 1 and 2. What is the net present value
of this project if the effective annual interest
rate is 10%?
A. P65,421.50
C.
P68,421.50
B. P67,421.50
D.
P69,421.50
Problem 32
An investment of x pesos is made at the end
of each year for three years, at an interest
rate of 9% per year compounded annually.
What will be the value of the investment
upon the deposit of the third payment?
A. 3.278x
C.
3.728x
B. 3.287x
D.
3.782x
Problem 33 (ME October 1995)
If P500 is invested at the end of each year
for 6 years, at an effective annual interest
rate of 7%, what is the total amount
available upon the deposit of the 6th
payment?
A. P3455.00
C.
P3577.00
B. P3544.00
D.
P3688.00
Problem 34
How much money must you deposit today to
an account earning 12% so that you can
withdraw P25,000 yearly indefinitely
starting at the end of the 10th year?
A. P125,000
C.
P73,767
B. P89,456
D.
P75,127
Problem 35 (ME April 1996)
In five years, P18,000 will be needed to pay
for a building renovation. In order to
generate this sum, a sinking fund consisting
of three annual payments is established now.
For tax purposes, no further payments will
be made after three years. What payment is
necessary if money is worth 15% per
annum?
A. P3,345.65
C.
P3,919.53
B. P3,789.34
P3,878.56
D.
Problem 36
An investment of P40,000.00 has revenue of
x pesos at the end of the first and second
year. Given a discount rate of 15%
compounded annually, find x so that the net
present worth of the investment is zero.
A. P33,789.54
C.
P24,604.65
B. P27,789.78
D.
P21,879.99
Problem 37
Mr. Jones borrowed P150,000 two years
ago. The terms of the loan are 10% interest
for 10 years with uniform payments. He just
made his second annual payment. How
much principal does he still owe?
A. P130,235.20
C.
P132,456.20
B. P134,567.30
D.
P129,456.78
Problem 38
Given that the discount rate is 15%, what is
the equivalent uniform annual cash flow of
the following stream of cash flows?
year 0
P
100,000.00
year 1
200,000.00
year 2
50,000.00
year 3
75,000.00
A. P158,124.60
C.
P157,345.98
B. P158,897.50
D.
P155,789.34
Problem 39
Mr. Bean borrowed P100,000 at 10%
effective annual interest rate. He must pay
back the loan over 30 years with uniform
monthly payments due on the first day of
each month. What does he pay each month?
A. P768.67
C.
P856.30
B. P987.34
D.
P839.20
Problem 40 (ECE November 1995)
An employee obtained a loan of P10,000 at
the rate of 6% compounded annually in
order to repair a house. How much must he
pay monthly to amortize the loan within a
period of ten years?
A. P198.20
C.
P110.22
B. P150.55
D.
P112.02
Problem 41
What is the accumulated value of a payment
of P12,500 at the end of each year for 9
years with interest at 5% compounded
annually?
A. P138,738.05
C.
P178,338.50
B. P137,832.05
D.
P187,833.50
Problem 42
What is the accumulated value of a payment
of P6,000 every six months for 16 years
with interest at 7% compounded
semiannually?
A. P312,345.00
C.
P347,898.00
B. P345,678.00
P344,007.00
D.
Problem 43
A mining property is offered for sale for
P5.7M. On the basis of estimated
production, an annual return of P800,000 is
foreseen for a period of 10 years. After 10
years, the property will be worthless. What
annual rate of return is in prospect?
A. 6.7%
C.
5.6%
B. 6.1%
D.
5.2%
Problem 44
If a down payment of P600,000 is made on a
house and P80,000 a year for the next 12
years is required, what was the price of the
house if money is worth 6% compounded
annually?
A. P1,270,707
C.
P1,345,555
B. P1,130,450
D.
P1,678,420
Problem 45
What annuity over a 10-year period at 8%
interest is equivalent to a present worth of
P100,000?
A. P14,903
C.
P13,803
B. P15,003
D.
P12,003
Problem 46 (CE May 1998)
The present value of an annuity of “R” pesos
payable annually for 8 years, with the first
payment at the end of 10 years, is
P187,481.25. Find the value of R if money
is worth 5%.
A. P45,000
C.
P42,000
B. P44,000
D.
P43,000
Problem 47 (ECE April 1998)
How much money must you invest today in
order to withdraw P2,000 annually for 10
years if the interest rate is 9%?
A. P12,385.32
C.
P12,835.32
B. P12,853.32
D.
P12,881.37
Problem 48 (ECE April 1998)
Money borrowed today is to be paid in 6
equal payments at the end of each of 6
quarters. If the interest is 12% compounded
quarterly, how much was initially borrowed
if quarterly payment is P2000.00?
A. P10,382.90
C.
P10,834.38
B. P10,200.56
D.
P10,586.99
Problem 49 (ME October 1997)
A car was bought on installment basis with a
monthly installment of P10,000.00 for 60
months. If interest is 12% compounded
annually, calculate the cash price of the car.
A. P455,875.00
C.
P678,519.75
B. P567,539.75
P345,539.75
D.
Problem 50 (ME October 1997)
A steel mill estimates that one of its furnaces
will require maintenance P20,000.00 at the
end of 2 years, P40,000.00 at the end 4 years
and P80,000.00 at the end of 8 years. What
uniform semi-annual amounts could it set
aside over the next eight years at the end of
each period to meet these requirements of
maintenance cost if all the funds would earn
interest at the rate of 6% compounded semiannually?
A. P7,897.35
C.
P8,897.35
B. P9,397.35
D.
P6,897.35
Problem 51 (ME April 1998)
A house and lot can be acquired at a down
payment of P500,000.00 and a yearly
payment of P100,000.00 at the end of each
year for a period of 10 years, starting at the
end of 5 years from the date of purchase. If
money is worth 14% compounded annually,
what is the cash price of the property?
A. P810,100.00
C.
P808,836.00
B. P801,900.00
D.
P805,902.00
Problem 52 (ME April 1998)
How much must be deposited at 6% each
year beginning on January 1, year 1, in order
to accumulate P5,000.00 on the date of the
last deposit, January 1, year 6?
A. P751.00
C.
P717.00
B. P715.00
P725.00
D.
Problem 53 (ME April 1998)
A piece of machinery can be bought for
P10,000.00 cash, or for P2,000.00 down and
payments of P750.00 per year for 15 years.
What is the annual interest rate for the time
payments?
A. 4.61%
C.
3.81%
B. 5.71%
D.
11.00%
Problem 54 (ME April 1998)
An instructor plans to retire in exactly one
year and want an account that will pay him
P25,000.00 a year for the next 15 years.
Assuming a 6% annual effective interest
rate, what is the amount he would need to
deposit now? (The fund will be depleted
after 15 years.)
A. P249,000.00
C.
P242,806.00
B. 248,500.00
D.
P250,400.00
Problem 55
A man invested P1,000.00 per month on a
bank that offers 6% interest. How much can
he get after 5 years?
A. P60,000.00
C.
P72,540.00
B. P69,770.00
D.
P69,491.00
Problem 56 (CE November 1995)
Find the present value in pesos, of perpetuity
of P15,000 payable semi-annually if money
is worth 8%, compounded quarterly.
A. P371,287
C.
P392,422
B. P386,227
D.
P358,477
Problem 57 (CE May 1999, May 1995)
A man paid 10% down payment of
P200,000 for a house and lot and agreed to
pay the balance on monthly installments for
60 months at an interest rate of 15%
compounded monthly. Determine the
required monthly payment.
A. P4,282.00
C.
P58,477.00
B. P42,822.00
D.
P5,848.00
Problem 58 (CE November 1998)
A debt of x pesos, with interest rate of 7%
compounded annually will be retired at the
end of 10 years through the accumulation of
deposit in the sinking fund invested at 6%
compounded semi-annually. The deposit in
the sinking fund every end of six months is
P21,962.68. What is the value of x?
A. P300,000
C.
P350,000
B. P250,000
D.
P400,000
Problems – Depreciation,
Capitalized Cost, Bonds, Etc.
Set 29
Problem 1
What is the value of an asset after 8 years of
use if it depreciation from its original value
of P120,000.00 to its salvage value of 3% in
12 years?
A. P44,200.00
C.
P44,002.00
B. P44,020.00
D.
P42,400.00
Problem 2
A man bought an equipment which cost
P524,000.00. Freight and installation
expenses cost him P31,000.00. If the life of
the equipment is 15 years with an estimated
salvage value of P120,000.00, find its book
value after 8 years.
A. P323,000.00
C.
P259,000.00
B. P244,000.00
D.
P296,000.00
Problem 3
An equipment costing P250,000 has an
estimated life of 15 years with a book value
of P30,000 at the end of the period.
Compute the depreciation charge and its
book value after 10 years using straight line
method.
A. d = P14,666.67; BV = P103,333.30
C. d =
P13,333.33; BV = P103,333.30
B. d = P14,666.67; BV = P105,666.67
D. d =
P13,333.33; BV = P105,666.67
Problem 4
An equipment costing P250,000 has an
estimated life of 15 years with a book value
of P30,000 at the end of the period.
Compute the depreciation charge and its
book value after 10 years using sinking fund
method assuming i = 8%.
A. d = P8,102.50; BV = P103,333.30
C. d =
P7,567.50; BV = P138,567.60
B. d = P6,686.67; BV = P125,666.67
D. d =
P8,102.50; BV = P132,622.60
Problem 5
An equipment costing P250,000 has an
estimated life of 15 years with a book value
of P30,000 at the end of the period.
Compute the depreciation charge and its
book value after 10 years using declining
balance method.
A. d = P9,456.78; BV = P67,456.98
C. d =
P9,235.93; BV = P60,832.80
B. d = P8,987.45; BV = P60,832.80
D. d =
P9,235.93; BV = P59,987.34
Problem 6
An equipment costing P250,000 has an
estimated life of 15 years with a book value
of P30,000 at the end of the period.
Compute the depreciation charge and its
book value after 10 years using the sum of
year’s digit method.
A. d = P11,000; BV = P67,500
C. d =
P11,500; BV = P60,000
B. d = P10,500; BV = P58,000
D. d =
P11,000; BV = P57,500
Problem 7
An asset costing P50,000 has a life
expectancy of 6 years and an estimated
salvage value of P8,000. Calculate the
depreciation charge at the end of the fourth
period using fixed-percentage method.
A. P7144.20
C.
P3878.40
B. P5264.00
D.
P2857.60
Problem 8 (CE May 1996)
A machine costing P45,000 is estimated to
have a salvage value of P4,350 when retired
at the end of 6 years. Depreciation cost is
computed using a constant percentage of the
declining book value. What is the annual
rate of depreciation in %?
A. 33.25%
C.
35.25%
B. 32.25%
D.
34.25%
Problem 9 (CE November 1997,
November 1994)
An engineer bought an equipment for
P500,000. Other expenses including
installation amounted to P30,000. At the end
of its estimated useful life of 10 years, the
salvage value will be 10% of the first cost.
Using straight line method of depreciation,
what is the book value after 5 years?
A. P281,500.00
C.
P301,500.00
B. P291,500.00
D.
P271,500.00
Problem 10 (ECE November 1997)
A machine costs P8,000.00 and an estimated
life of 10 years with a salvage value of
P500.00. What is its book value after 8 years
using straight line method?
A. P2,500.00
C.
P3,000.00
B. P4,000.00
D.
P2,000.00
Problem 11 (ME October 1997)
A factory equipment has an initial cost of
P200,000.00. Its salvage value after ten
years is P20,000.00. As a percentage of the
initial cost, what is the straight-line
depreciation rate of the equipment?
A. 5%
C. 9%
B. 6%
D. 8%
Problem 12 (ME October 1997)
An asset is purchased for P120,000.00. Its
estimated economic life is 10 years, after
which it will be sold for P12,000.00. Find
the depreciation for the first year using the
sum-of-the-year’s digit, (SOYD).
A. P20,000.00
C.
P21,080.00
B. P18,400.00
D.
P19,636.00
Problem 13 (ME April 1998)
An asset is purchased for P9,000.00. Its
estimated life is 10 years, after which it will
be sold for P1,000.00. Find the book value
during the third year if sum-of-the-year’s
digit (SOYD) depreciation is used.
A. P6,100.00
C.
P4,500.00
B. P5,072.00
D.
P4,800.00
Problem 14 (ME April 1998)
An asset is purchased for P500,000.00. The
salvage value in 25 years is P100,000.00.
What is the total depreciation in the first
three years using straight line method?
A. P48,000.00
C.
P24,000.00
B. P32,000.00
D.
P16,000.00
Problem 15 (ME April 1998)
A machine has an initial cost of P50,000.00
and a salvage value of P10,000.00 after 10
years. What is the book value after five
years using straight-line depreciation?
A. P35,000.00
C.
P25,000.00
B. P15,500.00
D.
P30,000.00
Problem 16 (ME April 1998)
A company purchased an asset for
P10,000.00 and plans to keep it for 20 years.
If the salvage value is zero at the end of the
20th year, what is the depreciation in the
third year? Use sum-of-the-years digits
depreciation.
A. P1000.00
C.
P857.00
B. P937.00
D.
P747.00
Problem 17 (ME April 1998)
An asset is purchased for P9,000.00. Its
estimated life is 10 years, after which it will
be sold for P1,000.00. Find the book value
during the first year if sum-of-the-year’s
digit (SOYD) depreciation is used.
A. P8,000.00
C.
P6,500.00
B. P7,545.00
D.
P6,000.00
Problem 18 (CE November 1998)
A machine having a first cost of P60,000.00
will be retired at the end of 8 years.
Depreciation cost is computed using a
constant percentage of the declining book
value. What is the total cost of depreciation,
in pesos, up to the time the machine is
retired if the annual rate of depreciation is
28.72%?
A. 56,000
C.
58,000
B. 57,000
D.
59,000
Problem 19 (ECE November 1998)
XYZ Corporation makes it a policy that for
any new equipment purchased; the annual
depreciation cost should not exceed 20% of
the first cost at any time with no salvage
value. Determine the length of service life
necessary if the depreciation used is the
sum-of-the-year’s digit (SOYD) method.
A. 7 years
C. 9
years
B. 8 years
D. 6
years
Problem 20
Determine the capitalized cost of an
equipment costing P 2M with and annual
maintenance of P200,000.00 if money is
worth 20% per annum.
A. P 2.5M
C. P
3M
B. P 2.75M
D. P
3.5M
Problem 21 (CE November 1996)
At 6%, find the capitalized cost of a bridge
whose cost is P250M and life is 20 years, if
the bridge must be partially rebuilt at a cost
of P100M at the end of each 20 years.
A. 245.3
C. 210
B. 215
D. 220
Problem 22 (ME October 1997)
An item is purchased for P100,000.00.
Annual costs are P18,000.00. Using 8%,
what is the capitalized cost of perpetual
service?
A. P350,000.00
C.
P320,000.00
B. P335,000.00
D.
P325,000.00
Problem 23 (CE May 1997)
A company uses a type of truck which costs
P2M, with life of 3 years and a final salvage
value of P320,000. How much could the
company afford to pay for another type of
truck for the same purpose, whose life is 4
years with a final salvage value of P400,000,
if money is worth 4%?
A. P2,565,964.73
C.
P2,585,964.73
B. P2,855,964.73
D.
P2,585,864.73
Problem 24
A P100,000, 6% bond, pays dividend semiannually and will be redeemed at 110% on
July 1, 1999. Find its price if bought on July
1, 1996, to yield an investor 4%,
compounded semi-annually.
A. P100,000.00
C.
P113,456.98
B. P112,786.65
D.
P114,481.14
Problem 25
A community wishes to purchase an existing
utility valued at P500,000 by selling 5%
bonds that will mature in 30 years. The
money to retire the bond will be raised by
paying equal annual amounts into a sinking
fund that will earn 4%. What will be the
total annual cost of the bonds until they
mature?
A. P44,667.98
C.
P34,515.05
B. P37,345.78
D.
P33,915.05
Problem 26
A man paid P110,000 for a P100,000 bond
that pays P4000 per year. In 20 years, the
bond will be redeemed for P105,000. What
net rate of interest will the man obtain on his
investment?
A. 3.37%
C.
3.56%
B. 3.47%
D.
3.40%
Problem 27 (ECE November 1996)
A man wants to make 14% nominal interest
compounded semi-annually on a bond
investment. How much should the man be
willing to pay now for a 12%, P10,000 bond
that will mature in 10 years and pays interest
semi-annually?
A. P2,584.19
P8,940.50
B. P3,118.05
P867.82
C.
D.
Problem 28
It is estimated that a timber tract will yield
an annual profit of P100,000 for 6 years, at
the end of which time the timber will be
exhausted. The land itself will then have an
anticipated value of P40,000. If a
prospective purchaser desires a return of 8%
on his investment and can deposit money in
a sinking fund at 4%, what is the maximum
price he should pay for the tract?
A. P459,480.00
C.
P578,987.00
B. P467,456.00
D.
P589,908.00
Problem 29
A mine is purchased for P1,000,000.00 and
it is anticipated that it will be exhausted at
the end of 20 years. If the sinking-fund rate
is 4%, what must be the annual return from
the mine to realize a return of 7% on the
investment?
A. P108,350
C.
P130,850
B. P150,832
D.
P103,582
Problem 30
A syndicate wishes to purchase an oil well
which, estimates indicate, will produce a net
income of P2M per year for 30 years. What
should the syndicate pay for the well if, out
of this net income, a return of 10% of the
investment is desired and a sinking fund is
to be established at 3% interest to recover
this investment?
A. P16,526,295
C.
P12,566,295
B. P15,626,245
D.
P16,652,245
Problem 31 (CE May 1995)
An investor pays P1,100,000 for a mine
which will yield a net income of P200,000 at
the end of each year for 10 years and then
will become useless. He accumulates a
replacement fund to recover his capital by
annual investments at 4.5%. At what rate
(%) does he receive interest on his
investment at the end of each year?
A. 10.04
C. 11.5
B. 8.5
D. 17.5
Problem 32 (CE May 1997)
Machine cost = $15,000; Life = 8 years;
Salvage Value = $3,000. What minimum
cash return would the investor demand
annually from the operation of this machine
if he desires interest annually at the rate of
8% on his investment and accumulates a
capital replacement fund by investing annual
deposits at 5%?
A. $5246.66
C.
$2456.66
B. $2546.66
D.
$4256.66
B.
√
D.
Problems – Recent Board
Exams, Selected Problems
Set 32
Problem 1 (CE November 2000)
A line in a map was drawn at a scale of
1:25000. An error of 0.02 mm in the
drawing is equivalent to how many meters
in actual?
A. 5 m
C. 0.05 m
B. 0.5 m
D. 50 m
Problem 2 (ME October 2000)
One day a Celsius thermometer and a
Fahrenheit thermometer registered exactly
the same numerical value for the
temperature. What was the temperature that
day?
A. -20
C. 40
B. 20
D. -40
Problem 3 (CE May 2000)
Convert 405° to mils.
A. 2,800 mils
C. 7,200 mils
B. 10,200 mils
D. 6,200 mils
Problem 4 (CE May 2000)
Rationalize the following:
A.
√
√
√
√
Problem 5 (CE November 2000)
Solve for B in the given partial fraction:
A. -3
C. -4
B. 3
D. 2
Problem 6 (ME October 2000)
Solve for the given equation,
.
A. 0.7432
C. 0.7243
B. 0.7342
D. 0.4732
Problem 7 (CE May 2000)
Log8 975 = x. Find x.
A. 3.31
C. 5.17
B. 4.12
D. 2.87
Problem 8
There are 9 arithmetic means between 6 and
18. What is the common difference?
A. 1.2
C. 5.17
B. 1
D .0.8
√
C.
√
Problem 9 (CE May 2000)
There are four geometric means between 3
and 729. Find the fourth term.
A. 81
C. 243
B. 27
D. 9
Problem 10 (CE November 2000)
The geometric mean of two numbers is 8
and their arithmetic mean is 17. What is the
first number?
A. 45
C. 32
B. 36
D. 48
Problem 11 (CE November 2000)
Twenty-eight persons can do a job in 60
days. They all start complete. Five persons
quitted the job at the beginning of the 6th
day. They were reinforced with 10 persons
at the beginning of the 45th day. How many
days was the job delayed?
A. 5.75 days
C. 1.97 days
B. 1.14 days
D. 2.45 days
Problem 12
Twenty men can finish a job in 20 days.
Twenty-five men started the job. If ten men
quitted the job after 18 days, find the total
number of days to finish the job.
A. 27
C. 26
B. 28
D. 29
Problem 13
Twelve workers could do a job in 20 days.
Six workers started the job. How many
workers should be reinforced at the
beginning of the 7th day to finish the job for
a total of 18 days from the start?
A. 10
C. 9
B. 13
D. 11
Problem 14 (ME October 2000)
Box A has 4 white balls, 3 balls, and 3
orange balls. Box B has 2 white balls, 4 blue
balls, and 4 orange balls. If one ball is drawn
from each box, what is the probability that
one of the two balls will be orange?
⁄
A.
C. ⁄
⁄
B.
D. ⁄
Problem 15
Twelve books consisting of six mathematics
books, 2 hydraulics books and four
structural books are arranged on a shelf at
random. Determine the probability that
books of the same kind are all together.
A. 1/2310
C. 1/3810
B. 1/5620
D. 1/1860
Problem 16 (ME October 16 2000)
What is the angle between two vectors A
and B?
A. 175.4°
C. 84.3°
B. -84.9°
D. 86.3°
Problem 17 (ME October 2000)
The expression [
simplifies to:
A.
C.
B.
D.
Problem 18 (CE November 2000)
Given that
, what is the value
of
?
A. 0.579
C. 0.654
B. 0.752
D. 0.925
Problem 19 (CE November 2000)
A flagpole 3 m high stands at the top of a
pedestal 2 m high located at one side of a
pathway. At the opposite side of the
pathway directly facing the flagpole, the
flagpole subtends the same angle as the
pedestal. What is the width of the pathway?
A. 4.47 m
C. 6.28 m
B. 3.21 m
D. 8.1 m
Problem 20 (CE May 2000)
Find the area in sq. m. of a spherical triangle
of whose angles are 123°, 84°, and 73°. The
radius of the sphere is 30 m.
A. 1863.3
C. 1958.6
B. 1570.8
D. 1480.2
Problem 21 (CE May 2000)
Two sides of a triangle measure 18 cm and 6
cm. The third side may be:
A. 12
C. 10
B. 13
D. 11
Problem 22 (CE May 2000)
The perimeter of an ellipse is 28.448 units.
If the major axis is 5 units, what is the
length of the minor axis?
A. 9
C. 8
B. 7
D. 6
Problem 23 (CE November 2000)
A right regular hexagonal prism is inscribed
in a right circular cylinder whose height is
20 cm. The difference between the
circumference of the circle and the perimeter
of the hexagon is 4 cm. Determine the
volume of the prism.
A. 9756 cc
C. 10857 cc
B. 114752 cc
D. 10367 cc
Problem 24 (ME October 2000)
Find the area bounded by the x-axis, the line
and the parabola
.
A. 64/2
C. 32/4
B. 32/3
D. 32/2
Problem 25 (CE November 2000)
What is the area bounded by the curves
and
A. 6.0
C. 6.666
B. 7.333
D. 5.333
Problem 26 (CE November 2000)
Given a regular hexagonal with consecutive
corners ABCDEF. If the bearing of
side AB is N
25° E, what is the bearing of side FA?
A. N 15° W
C. N 35° W
B. N 45° W
D. N 5° W
Problem 27 (CE November 2000)
The perimeter of a triangle is 58 cm and its
area is 144 sq. Cm. What is the radius of the
inscribed circle?
A. 4.97 cm
C. 5.52 cm
B. 9.65 cm
D.3.12 cm
Problem 28 (ME October 2000)
What is the area bounded by the curves
and
A. 22.4
C. 44.7
B. 26.8
D. 29.8
Problem 29
A solid sphere of radius 20 cm was placed
on top of hallow circular cylinder of radius
10 cm. What volume of the sphere was
inside the cylinder?
A. 431 cc
C. 325 cc
B. 568 cc
D. 542 cc
Problem 30
A trough is formed by nailing together, edge
two boards 130 cm in length, so that the
right section is a right triangle. If 3500 cc of
water is poured into the trough and if the
trough is held so that right section of the
water is an isosceles right triangle, how deep
is the water?
A. 6.32 cm
C. 4.21 cm
B. 5.19 cm
D. 6.93 cm
Problem 31 (CE May 2000)
The lateral area of a right circular cone of
radius 4 cm is 100.53 sq. cm. Determine the
slant height.
A. 8 cm
C. 6 cm
B. 9 cm
D. 10 cm
Problem 32 (CE May 2000)
The frustum of a regular triangular pyramid
has equilateral triangles for its bases and has
an altitude of 8 m. The lower base edge is 9
m. If the volume is 135 cu. m., what is the
upper base edge?
A. 2 m
C. 4 m
B. 5 m
D. 3 m
Problem 33 (CE May 2000)
A cylinder of radius 6 m has its axis along
the x-axis. A second cylinder of the same
radius has its axis along the y-axis. Find the
volume, in the first octant, common to the
two cylinders.
A.
C.
B.
D.
Problem 34 (CE May 2000)
Find the volume of a right circular cylinder
whose lateral area is 25.918
and base
area of 7.068
A. 19.44
C. 20.53
B. 15.69
D. 18.12
Problem 35 (CE November 2000)
A solid has a circular base of base radius 20
cm. find the volume of the solid if every
plane section perpendicular to a certain
diameter is an isosceles right triangle with
one leg in the plane of the base.
A. 21333 cc
C. 18667 cc
B. 24155 cc
D. 20433 cc
Problem 36 (CE November 2000)
The base diameter of a cone is 18 cm and its
axis is inclined 60° with the base. If the axis
is 20 cm long, what is the volume of the
cone?
A. 1524 cc
C. 1245 cc
B. 1469 cc
D. 1689 cc
Problem 37 (ME October 2000)
The equation
describes:
A. a circle
C. a hyperbola
B. a parabola
D. an ellipse
Problem 38 (CE May 2000)
Two vertices of a triangle are (6, -1) and (7,
-3). Find the ordinate of the vertex such that
the centroid of the triangle will be (0, 0).
A. -13
C. 13
B. 4
D. -4
Problem 39 (CE May 2000)
Determine the equation of the directrix of
the curve
A.
C.
B.
D.
Problem 40 (CE November 2000)
Find the area of the curve
A. 125 sq. units
C. 92 sq. units
B. 113 sq. units
D. 138 sq. units
√
C. 9
B. 0
A.
Problem 41 (CE November 2000)
Find the distance between the foci of the
curve
.
A. 7
C. 8
B. 6
D. 12
Problem 42 (CE November 2000)
What is the equivalent rectangular
coordinate of a point whose coordinate is (7,
38°).
A. (3.56, 4.31)
C. (5.52, 4.31)
B. (4.31, 5.52)
D. (4.31, 3.56)
Problem 43
The chords of the parabola
having
equal slope of 2 is bisected by its diameter.
What is the equation of the diameter?
A.
C.
B.
D.
Problem 44
Find the slope of the line whose parametric
equation is
and
A.
C. 3
B.
D.
Problem 45 (ME October 2000)
The first derivative with respect to y of the
function
√ is:
D. 3√
Problem 46 (ME October 2000)
Find the derivative of
to the 3rd
rd
power –
to the 3 power] to the 3rd
power?
A.
B.
to the 3rd power –
to the 3rd
power]2
C.
to the 3rd power –
to the 3rd power]
D.
to the 3rd power –
to the 3rd power]
Problem 47
The derivative of
to is:
A.
with respect
C.
B.
D.
Problem 48
What is the second derivative of
at
A. 8
C. 1
B. 0
D. Not defined
Problem 49 (CE May 2000)
At what value of x will the slope of the
curve
be 18?
A. 2
C. 5
B. 4
D. 3
Problem 50 (CE November 2000)
The slope of the curve at any point is given
as
and the curve passes through (5,
3). Determine the equation of the curve.
A.
C.
B.
D.
Problem 51 (CE May 2000)
The total surface area of a closed cylindrical
tank is 153.94 sq. m. If the volume is to be
maximum, what is its height in meters?
A. 6.8 m
C. 3.6 m
B. 5.7 m
D. 4.5 m
Problem 52 (CE November 2000)
A closed cylindrical tank having a volume
of 71.57
is to be constructed. If the
surface area is to e minimum, what is the
required diameter of the tank?
A. 4 m
C. 5 m
B. 5.5 m
D. 4.5 m
Problem 53
Two post, one 16 feet and the other 24 feet
are 30 feet apart. If the post are to be
supported y a cable running from the top of
the first post to a stake on the ground and
then back to the top of the second post, find
the distance from the lower post to the stake
to use the least amount of wire.
A. 6 feet
C. 15 feet
B. 9 feet
D. 12 feet
Problem 54
The motion of a body moved vertically
upwards is expressed as
Where h is the height in feet and t is the time
in seconds. What is the velocity of the body
when
seconds?
A. 21.7 fps
C. 24.1 fps
B. 28.7 fps
D. 35.6 fps
Problem 55 (CE May 2000)
A lighthouse is 2 km off a straight shore. A
searchlight at the lighthouse focuses to a car
moving along the shore. When the car is 1
km from the point nearest to the lighthouse,
the searchlight rotates 0.25 rev/hour. Find
the speed of the car in kph.
A. 3.93
C. 2.92
B. 2.56
D. 3.87
Problem 56 (CE May 2000)
Evaluate ∫
A. 15.421
C. 17.048
B. 19.086
D. 20.412
Problem 57
Determine the are enclosed by the curve
A.
C.
B.
D.
Problem 58 (CE May 2000)
Determine the moment of inertia about the
x-axis, of the area bounded by the curve
4y, the line
and the x-axis.
A. 9.85
C. 10.17
B. 13.24
D. 12.19
Problem 59 (CE November 2000)
Evaluate the integral of
with limits
from 0 to 1.
A. 0.322
C. 0.203
B. 0.018
D. 0.247
Problem 60 (CE November 2000)
The area bounded by the curve
from
to
is revolved about
the x-axis. What is the volume generated?
A. 2.145 cu. units
C. 3.452 cu. units
B. 4.935 cu. units
D. 5.214 cu.
units
Problem 61 (ME October 2000)
If you borrow money from your friend with
simple interest of 12%, find the present
worth of 20,000 which is due at the end of
nine months.
A. P18,688.20
C. P18,518.50
B. P18,691.50
D. P18,348.60
Problem 62 (ME October 2000)
Business needs to have P100, 000 in five
years. How much must he put into his 10%
account in the bank now?
A. P72,085.6
C. P70,654.1
B. P62,092.1
D. P60,345.2
Problem 63 (ME October 2000)
What is the present worth of P54, 000.00
due in five years if money is worth 11% and
is compounded semi-annually?
A. P30,367.12
C. P31,613.25
B. P28,654.11
D. P34,984.32
Problem 64 (CE May 2000)
How long will it take for money to
quadruple itself if invested at 20%
compounded quarterly?
A. 10.7 years
C. 9.5 years
B. 6.3 years
D. 7.1 years
Problem 65 (ME October 2000)
The interest on an account is 13%
compounded annually. How many years
approximately will take to triple the
amount?
A. 8 years
C. 9.5 years
B. 8.5 years
D. 9 years
Problem 66 (ME October 2000)
When will an investment of P4000 double if
the effective rate is 8% per annum?
A. 8.4
C. 9.01
B. 8.3
D. 10.2
Problem 67 (ME October 2000)
A savings association pays 1.5% interest
quarterly. What is the effective annual
interest rate?
A. 6.14%
C. 7.32%
B. 8.54%
D. 6.45%
Problem 68 (ME October 2000)
A bank offers 0.5% effective monthly
interest. What is the effective annual rate
with monthly compounding?
A. 6.2%
C. 7.2%
B. 6%
D. 7%
Problem 69 (ME October 2000)
What nominal rate converted quarterly could
be used instead of 12% compounded semiannually?
A. 10.76%
C. 11.82%
B. 11.43%
D. 11.97%
Problem 70 (CE November 2000)
P1, 000,000 was invested to an account
earning 8% compounded continuously.
What is the amount after 20 years?
A. P4,452,796.32
C. P5,356,147.25
B. P4,953,032.42
D. P3,456,254.14
Problem 71 (ME October 2000)
A sum of money is deposited now in a
savings account. The effective annual
interest rate is 12%. How much money must
be deposited to yield P500.00 at the end of
11 months?
A. P153.00
C. P446.00
B. P144.00
D. P451.00
MULTIPLE CHOICE
QUESTION in
Guidebook in Mathematics
by
Francis Jay B. Jumawa and Adrian S. Paala
1. Find the harmonic mean between the
numbers 3/8 and 4.
a.
b.
c.
d.
35/34
24/35
42/35
35/24
2. The terms of a sum may be grouped
in any manner without affecting the
result. this is law known as:
a.
b.
c.
d.
Commutative Law
Distributive Law
Associative Law
Reflexive Law
3. A number is divided into two parts
such that when the greater part is
divided by the smaller part, the
quotient is 3, and the remainder is 5.
Find the smaller number if the sum
of the two numbers is 37.
a.
b.
c.
d.
8
29
22
16
4. Mary was four times as old as Lea
ten years ago. If she is now twice as
old as Lea, how old is Mary.
a.
b.
c.
d.
25
40
30
15
5. The sum of three succeeding odd
integers is 75. The largest integer is
a.
b.
c.
d.
25
29
27
31
6. A ship propelled to move at 25 mi/
hr in still water, travels 4.2 miles
upstream in the same time that it can
travel 5.8 miles downstream. Find
the speed of the stream.
a. 4
b. 6
c. 8
d. 10
7. Jose’s rate of doing work three times
as fast as Bong. On given day Jose
and Bong work together for 4 hours
then Bong was called away and Jose
finishes the rest of the job in 2 hours.
How long would it take Bong to do
the complete job alone?
a.
b.
c.
d.
18 hrs.
22 hrs.
16 hrs.
31 hrs.
8. The length of a rectangle is 3 times
its width. If the width of the
rectangle is 5 inches, what is the
rectangle's area, in square inches?
a.
b.
c.
d.
15
20
30
75
9. For all x >2, (2x2 + 2x - 12) / (x - 2)
simplifies to
a.
b.
c.
d.
2(x + 3)
2(x - 2)
x+3
2(x + 3)(x - 2)
10. If the hypotenuse of a right triangle
is 10 inches long and one of its legs
is 5 inches long, how long is the
other leg?
a. 5
b. 5
c. 5
d. 7.5
11. In the standard (x,y) coordinate
plane, the graph of (x + 3)2 + (y + 5)2
= 16 is a circle. What is the
circumference of the circle,
expressed in coordinate units?
a.
b.
c.
d.
4π
5π
3π
8π
12. How many solutions are there to the
equation x2 - 7 = 0?
a.
b.
c.
d.
1
2
4
7
13. A circle with center (4,-5) is tangent
to the y-axis in the standard (x,y)
coordinate plane. What is the radius
of this circle?
a.
b.
c.
d.
4
5
16
25
14. Angle A is an acute angle and sin(A)
= 11/14. What is the value of
cos(A)?
a.
b.
c.
d.
√3 / 14
5√3 / 14
√(3/14)
5/14
15. What are the values of a and b, if
any, where - a|b + 4| > 0?
a. a > 0 and b ≠ 4
b. a < 0 and b ≥ -4
c. a < 0 and b ≠ -4
d. a < 0 and b ≤ -4
16. In a shipment of televisions, 1/50 of
the televisions are defective. What is
the ratio of defective to non defective
televisions?
a.
b.
c.
d.
1/50
1/49
49/1
50/1
17. Which of the following is divisible
(with no remainder) by 4?
a.
b.
c.
d.
214133
510056
322569
952217
18. A particle travels 1 x 106 meters per
second in a straight line for 5 x 10-6
seconds. How many meters has it
traveled?
a.
b.
c.
d.
4
5
6
7
19. The length of sides AB and AC in
the triangle below are equal. What is
the measure of angle ∠ A if angle ∠
C is 70°?
a.
b.
c.
d.
70°
55°
40°
110°
20. ABC is an equilateral triangle. AH is
perpendicular to BC and has a length
of 2√3 inches. What is the area, in
square inches, of triangle Δ ABC.
a.
b.
c.
d.
3√3
2√3
4√3
8√3
21. Find the value of k in the quadratic
equation (2k + 2) x2 + (4 – 4k) x + k
– 2 = 0 so that the roots are
reciprocal of each other.
a.
b.
c.
d.
4
2
-4
-2
22. Solve for x if 8y = 3x - 11
a.
b.
c.
d.
8/3) y + 11
(8/3) y - 11
(8y - 11)/3
(8y + 11)/3
23. When graphed in the (x, y)
coordinate plane, at what point do
the lines 2x + 3y = 5 and x = -2
intersect?
a.
b.
c.
d.
(-2, 0)
(-2, 5)
(0, 5)
(-2, 3)
24. The area of a trapezoid is 0.5h(b1 +
b2), where h is the altitude, and b1
and b2 are the lengths of the parallel
bases. If a trapezoid has an alitude of
15 inches, an area of 105 square
inches, and one of the bases 22
inches, what is the perimeter, in
inches, of the trapezoid?
a. 8
b. 45
c. 60
d. 30
25. If you drove at average speed of 66
miles per hour, what distance, in
miles, did you drive in 99 minutes?
a.
b.
c.
d.
65.34
108.9
150
90.45
26. If x and y are any real numbers such
that 0 < x < 2 < y , which of these
must be true?
a.
b.
c.
d.
x < (xy)/2 < y
0 < xy < 2x
x < xy < 2
0 < xy < 2
27. In the right triangle ABC below,
what is the cosine of angle A if the
opposite side is 3 and the adjacent
side is 4.
a.
b.
c.
d.
5/3
5/4
3/5
4/5
28. In a triangle ABC with segment BD,
B is on AD, ∠ BAC and ∠ ACB
measure 26° and 131° respectively.
What is the measure of ∠CBD?
a.
b.
c.
d.
26°
157°
23°
154°
29. The total surface area of all six
sides of the rectangular box
below is equal to 128 square
inches. What is x in inches?
a.
b.
c.
d.
6
4
2
8
30. ABC is a right triangle. ABDE is a
square of area 200 square inches and
BCGF is a square of 100 square
inches. What is the length, in inches,
of AC?
a.
b.
c.
d.
10√3
10√2
10√1
10
31. What is the slope of the line: 4x = 3y + 8
a.
b.
c.
d.
4
-3/4
-4/3
3
32. Which of the following is equal to
√45
a.
b.
c.
d.
5√3
9√5
3√5
3
33. What is the smallest value of x that
satisfies the equation: x(x + 4) = -3
a. 1
b. -1
c. 3
d. -3
34. A group of 7 friends are having
lunch together. Each person eats at
least 3/4 of a pizza. What is the
smallest number of whole pizzas
needed for lunch?
a. 7
b. 5
c. 6
d. 8
35. There are n students in a school. If
r% among the students are 12 years
or younger, which of the following
expressions represents the number of
students who are older than 12?
a.
b.
c.
d.
n(1 - r)
100(1 - r)n
n(1 - r) / 100
n(100 - r) / 100
36. The measures of angles A, B and C
of a triangle are in the ratio 3:4:5.
What is the measure, in degrees, of
the largest angle?
a.
b.
c.
d.
75°
15°
12°
90°
37. If x + 4y = 5 and 5x + 6y = 7, then
3x + 5y = ?
a.
b.
c.
d.
12
6
4
2
38. For all real numbers x, the minimum
value of 1 + 2cos(4x) is
a.
b.
c.
d.
0
-1
-2
-4
39. What is the largest possible product
for 2 odd integers whose sum is
equal to 32?
a. 64
b. 255
c. 256
d. 1024
40. If (a + b)2 = 25 and (a - b) 2 = 45,
then a2 + b2 = ?
a.
b.
c.
d.
35
70
140
280
41. If a = 3, then 2 / (1/7 + 1/a) = ?
a.
b.
c.
d.
21 / 10
21 / 5
21 / 15
21 / 3
42. A company makes a profit equal to
25% of its sales. The profit is shared
equally among the 4 owners of the
company. If the company generates
sales of $5,000,000, how much
money does each one of the owners
get?
a.
b.
c.
d.
312,500
500,000
1,250,000
12,500,000
43. If the expression x3 + 2hx - 2 is equal
to 6 when x = -2, what is the value of
h?
a.
b.
c.
d.
-2
-4
4
6
44. If -3/(a - 3) = 3/(a + 2), then a = ?
a. 1/2
b. 1/4
c. 1
d. 2
45. Which integer is nearest to √2100 /
√7
a.
b.
c.
d.
17
18
19
16
46. The two legs of a right triangle
measure 6 and 8 inches respectively.
What is the area of the circle that
contains all 3 vertices of the triangle?
a.
b.
c.
d.
24Pi
25Pi
35Pi
34Pi
47. X and Y are acute angles such that
tany = cotx. What is the sum, in
degrees, of the measures of the
angles X and Y?
a.
b.
c.
d.
90°
45°
60°
30°
48. What is the value of the adjacent side
if the opposite side is 1 inch and the
2 other angles of the right triangle is
30° and 60°
a.
b.
c.
d.
1/√3
√2
√3
6/√3
49. Which of the lines below is not
parallel to the line 6x - 2y = 10?
a. 3x - y = 7
b. -6x + 2y = 20
c. 3x + y = 7
d. 6x - 2y = 5
50. For what value of k the equation
below has no value of x: 2x + 3 = x 2kx – 5
a.
b.
c.
d.
1
-1
0.5
-0.5
51. Find the value of x which will satisfy
the equation
a.
b.
c.
d.
/
=1
-1, -4
1, 4
4
0
52. Find the geometric mean between
the terms -4 and -9
a.
b.
c.
d.
6
7
-6
36
53. What is the average value of 7/8 and
3/4?
a.
b.
c.
d.
5/4
5/8
5/16
13/16
54. A solution is made of water and pure
acid. If 75% of the solution is water,
how many litters of pure acid are in
20 liters of this solution?
a.
b.
c.
d.
10
5
25
15
55. The diagonal of a square has a
measure of 12 inches. What is the
perimeter, in inches, of this square.
a.
b.
c.
d.
6√2
72
24√2
48
56. In the right triangle ABC, C is a right
angle and the measure of angle B is
60°. If BC is 20 inches long, then
how long is AC?
a.
b.
c.
d.
20√3
20
√3
20/√3
57. If x = 2.0001, which of the following
expressions has the largest value?
a.
b.
c.
d.
2 / (x + 2)
2 / (x - 2)
(x + 2) / 2
2/x
58. In the rectangle ABCD, the
measure of the length AD is 3
times the measure of the width
AB. What is the slope of the line
segment BD?
a.
b.
c.
d.
3
1/3
-1/3
-3
59. What is the product of the two real
solutions of the equation: 2x = 3 - x2
a.
b.
c.
d.
2
-2
6
-3
60. The ratio of the circumference of any
circle to the diameter of the circle is:
a.
b.
c.
d.
An integer
An irrational number
A rational number
A whole number
61. Find the sum and product of roots of
the equation x3 + 2x2 – 23x – 60 = 0.
a.
b.
c.
d.
-2, 60
2, 17
17, -60
2, -60
62. The ratio of three numbers is 2:5:7.
If 7 is subtracted from the second
number, the resulting numbers form
an arithmetic progression. Determine
the smallest of the three numbers.
a.
b.
c.
d.
28
15
21
70
63. Determine the sum of the first 4
terms of the sequence whose general
term is given by 3n – 2.
a.
b.
c.
d.
121
89
98
112
64. Find the sum of all positive integers
between 84 and 719 which are
exactly divisible by 5.
a.
b.
c.
d.
23,750
45,680
50,800
38,460
65. If 3logx – logy = 0, express y in
terms of x.
a.
b.
c.
d.
y = x3
y = x2
y=x
y = 3x
66. In a certain A.P. the first, fourth and
eight terms are themselves form a
geometric progression. What is the
common ratio of the G.P.?
a.
b.
c.
d.
4/3
5/4
4/5
3/4
67. Three men A, B, and C can do a
piece of work in t hours working
together. Working alone, A can do
the work in 6 hours more, B in 1
hour more, and C in twice the time if
all working together. How long
would it take to finish the work if all
working together?
a.
b.
c.
d.
20 mins.
30 mins.
40 mins.
50 mins.
68. Solve the z if the equation is 4 x 10-5
=z
a.
b.
c.
d.
– 40,000
– 200
0.0004
0.00004
69. Two balls are drawn one at a time
from a basket containing 4 black
balls and 5 white balls. If the first
ball is returned before the second
ball is drawn, find the probability
that both balls are black.
a.
b.
c.
d.
0.198
0.898
0.167
0.264
70. There are 15 balls in a box: 8 balls
are green, 4 are blue and 3 are white.
Then 1 green and 1 blue balls are
taken from the box and put away.
What is the probability that a blue
ball is selected at random from the
box?
a.
b.
c.
d.
3/13
4/15
3/15
4/13
71. Which of the following is equivalent
to (x)(x)(x)(x3), for all x?
a.
b.
c.
d.
6x
x6
4x6
4x4
72. A number between 1 and 10000 is
randomly selected. What is the
probability that it will be divisible by
4 and 5?
a.
b.
c.
d.
0.03
0.04
0.05
0.06
73. What time after 2 o’clock will the
hands of the clock extend in opposite
directions for the first time?
a.
b.
c.
d.
2:43.64
2:43.46
2:34.64
2:34.46
74. What is the sum of the geometric
progression if there are 4 geometric
means between 3 and 729?
a.
b.
c.
d.
1212
1092
1908
1209
75. A boy on his bicycle to arrive at a
certain time to a market that is 30 km
from his school. After riding 10 km,
he rested for half an hour, and as a
result he was obliged to ride the rest
of the trip 2 km/hr faster. What was
his original speed?
a.
b.
c.
d.
7 km/hr
9 km/hr
10 km/hr
8 km/hr
76. Find the equation whose roots are
two times the roots of the equation x3
– 6x2 + 11x – 6 = 0.
a.
b.
c.
d.
x3 – 12x2 + 44x – 48 = 0
x3 – 12x2 – 44x – 48 = 0
x3 + 12x2 + 44x – 48 = 0
x3 – 12x2 + 44x + 48 = 0.
77. How many 4-digits even numbers
can be formed from the digits 0, 1, 2,
3, 4, 5, 6, 7, 8, and 9 if each digit is
to be used only once in each
number?
a.
b.
c.
d.
5,000
3,256
2,520
5,986
78. Rukia has nickels, dimes, and
quarters amounting to $1.85. If he
has twice as many dimes as quarters,
and the number of nickels is two less
than twice the number of dimes, how
many quarters does he have?
a.
b.
c.
d.
3
8
6
10
79. A club has 25 members, 4 of whom
are ECE’s. In how many ways can a
committee of 3 be formed so as to
include at least one ECE?
a.
b.
c.
d.
543
126
970
314
80. If (x -3) is a factor of the polynomial
x4 – 4x3 – 7x2 + kx + 24, what is the
value of k?
a.
b.
c.
d.
11
17
22
34
81. A guy has 8 flowers of different
variety. In how many ways can he
select 2 or more flowers to form a
bouquet?
a.
b.
c.
d.
128
247
110
540
82. At a conference, after everyone had
shaken hands with everyone else, it
was found that 45 handshakes were
exchanged. How many were at the
conference?
d. 40
83. A bag contains 4 white balls and 3
black balls. Another bag contains 3
white balls and black balls. If one
ball is drawn from each bag,
determine the probability that the
balls drawn will be 1 white and 1
black.
a.
b.
c.
d.
27/58
39/56
29/56
5/14
84. If the sides of a right triangle are in
A.P., then what is the ratio of its
sides?
a.
b.
c.
d.
3:4:5
1:2:3
4:5:6
2:3:4
85. If x: y: z = 4: -3: 2 and 2x + 4y – 3z
= 20, find x, y, z.
a.
b.
c.
d.
4, -5, 2
-8, 6, -4
5, -6, 8
2, -7, 4
86. How many numbers between 3000
and 5000 can be formed from the
digits 0, 1, 2, 3, 4, 5, 6 if repetition is
not allowed?
a.
b.
c.
d.
96
128
240
144
87. Find the mean proportional between
a. 10
b. 30
c. 20
a. 3
b.
c. 6
d. 2
88. How many liters of a 25% acid
solution must be added to 80 liters of
a 40% acid solution to have a
solution that is 30% acid?
a.
b.
c.
d.
160L
190L
150L
120L
89. A yacht can travel 10 miles
downstream in the same amount of
time as it goes 6 miles upstream. If
the velocity of the river current is
3MPH, find the speed of the yacht in
still water.
a.
b.
c.
d.
12 MPH
16MPH
15MPH
18MPH
90. Determine the 5th term of the
sequence whose sum of n terms is
given by 2n+3 – 5.
a.
b.
c.
d.
258
218
128
15
91. Find the sum of the first five terms of
the geometric progression if the third
term is 144 and the sixth term is 486.
a.
b.
c.
d.
844
972
746
548
92. A and B working together can finish
a job in 5 days, B and c together can
finish the same job in 4 days, and A
and C in 2.5 days. In how days can
all of them do the job working
together?
a.
b.
c.
d.
1.06 days
2.4 days
3.2 days
2.03 days
93. If Chicago is 10% taller than Ishida
and Ishida is 10% taller than Chad,
then Ichigo is taller than Chad by
how many percent?
a.
b.
c.
d.
31%
41%
21%
11%
94. After the price of petroleum oil went
up by 10%, a buyer reduced his oil
consumption by the same percent.
By what percent would his
petroleum bill changed?
a.
b.
c.
d.
1%
11%
10%
0.1%
95. Find the mean, median and mode
respectively of the following
numbers: 13, 13, 14, 12, 11, 10, 9,
11, 8, 11, 5, and 15.
a.
b.
c.
d.
10, 10, 10
10, 11, 10
10, 11, 11
11, 11, 11
96. There are 4 white balls and 6 red
balls in a sack. If the balls are taken
out successively (the first ball is not
replaced), what is the probability that
the balls drawn are of different
colors.
a.
b.
c.
d.
23/90
8/15
24/103
7/15
97. Solve for x in the following
equation: x + 3x + 5x + 7x + … +
49x = 625
a.
b.
c.
d.
2
1
1/2
1/3
98. An organization consists of n
engineers and n nurses. If two of the
engineers are replaced by two other
nurses, then 51% of the group
members will be nurses. Find the
value of n
a.
b.
c.
d.
70
110
50
100
99. In a certain family, the sum of the
parents’ ages is twice the sum of
their children’s ages. Five years ago,
the sum of the parents’ ages was four
times the sum of the children’s ages
during that time. In fifteen years, the
sum of the parents’ ages will be
equal to the sum of their children’s
ages. How many children are there in
the family?
a.
b.
c.
d.
5
7
6
8
100. z varies directly as x and inversely
as y2. If x = 1 and y = 2, then z = 2.
Find z when x = 3 and y = 4.
a.
b.
c.
d.
1.5
0.5
2.5
3.5
101. When two dice are thrown, what is
the probability that the sum of the
two faces shown is 6?
a.
b.
c.
d.
1/36
1/6
1/9
5/36
102. An ECE class of 40 students took
examinations in Electronics and
Communications. If 30 passed in
Electronics, 36 passed in
Communication and 2 failed in both
subjects, how many students passed
in both subjects?
a.
b.
c.
d.
28
30
26
32
103. The excess of the sum of the fourth
and fifth parts over the difference of
the half and third parts of a number
is 119. Find the number.
a.
b.
c.
d.
240
320
420
230
104. What is the area, in square feet, of
the triangle whose sides have lengths
equal to 10, 6 and 8 feet?
a. 24
b. 48
c. 30
d. 40
105. Solve for x if the equation is 3102 +
9*3100 + 3103/3 = x
a.
b.
c.
d.
3101
3102
3103
3104
106. Of the 80 students in class, 25 are
studying German, 15 French and 13
Spanish. 3 are studying German and
French; 4 are studying French and
Spanish; 2 are studying German and
Spanish; and none is studying all 3
languages at the same time. How
many students are not studying any
of the three languages?
a.
b.
c.
d.
18
53
62
36
107. There were 2 small circles C1 and
C2 inside a large circle AB. AB is a
diameter of the large circle. The
centers C1 and C2 of the smaller
circles are on AB. The two small
circles are congruent and tangent to
each other and to the larger circle.
The circumference of circle C1 is
8Pi. What is the area of the large
circle?
a.
b.
c.
d.
64Pi
32Pi
156Pi
128Pi
108. DE is parallel to CB and (length of
AE / length of EB) is 4. If the area of
triangle AED is 20 square inches,
what is the area, in square inches, of
triangle ABC?
a.
b.
c.
d.
31.25
80
320
1,600
109. Round (202)2 to the nearest
hundred.
a.
b.
c.
d.
48,000
40,800
42,000
44,000
110. If w workers, working at equal
rates, can produce x toys in n days,
how many days it takes c workers,
working at same equal rates, to
produce y toys?
a.
b.
c.
d.
y*w*c/(w*n)
y*w/(w*n*c)
y*w*n / x
y*w*n / (x*c)
111. A number of the form 213ab,
where a and b are digits, has a
reminder less than 10 when divided
by 100. The sum of all the digits in
the above number is equal to 13.
Find the digit b.
a.
b.
c.
d.
5
7
6
8
112. Find a negative value of x that
satisfies the equation: [(x+1)2 - (2x +
1)]1/2 + 2|x| - 6 = 0
a. -4
b. -3
c. -2
d. -1
113. If thrice the smaller number exceeds
the larger by 12. Find the larger
number if the two numbers are
consecutive odd integers.
a.
b.
c.
d.
7
9
10
8
114. Determine how much water should
be evaporated from 50kg of 30% salt
solution to produce a 60% salt
solution. All percentages are by
weight.
a.
b.
c.
d.
25 kg
35 kg
15 kg
20 kg
115. A runs around a circular track in 60
seconds, and in 50 seconds. Five
seconds after A starts, B starts from
the same point in the same direction.
When will they be together for the
first time, assuming they run around
the track continuously?
a.
b.
c.
d.
3.5 mins
6.5 mins
5.5 mins
7.5 mins
116. An antelope is now 50 of her leaps
ahead of a cheetah which is pursuing
her. How many more leaps will the
antelope take before it is overtaken if
she takes 5 leaps while the cheetah
takes 4 leaps, but 2 of the cheetah’s
leaps are equivalent to 3 of the
antelope’s leaps?
a. 350
b. 325
c. 420
d. 250
117. Line L passes through the points (2, 0) and (0, a). Line LL passes
through the points (4, 0) and (6, 2).
What value of a makes the two lines
parallel?
a. 1/2
b. -2
c. 2
d. -1/2
118. Solve for x if the equation is 104(54
- 24) / 21 = x
a.
b.
c.
d.
209,000
289,000
290,000
208,000
119. Two dice are tossed. What is the
probability that the sum of the two
dice is greater than 3?
a.
b.
c.
d.
3/4
5/6
11/12
1/4
120. If L is a line through the points
(2,5) and (4,6), what is the value of k
so that the point of coordinates (7,k)
is on the line L?
a.
b.
c.
d.
5
6
15/2
11/2
121. Find a negative value of k so that
the graph of y = x2 - 2x + 7 and the
graph of y = kx + 5 are tangent?
a.
b.
c.
d.
- 4√2
- 2 - 2√2
-2
- √2
122. The circle of equation (x - 3)2 + (y
- 2)2 = 1 has center O. Point M(4,2)
is on the circle. N is another point on
the circle so that angle MON has a
size of 30°. Find the coordinates of
point N.
a.
b.
c.
d.
(3 + √3/2 , 5/2)
(5/2 , 3 + √3/2)
(3 - √3/2 , 3/2)
(3/2 , 3 - √3/2)
123. Vectors u and v are given by u = (2
, 0) and v = (-3 , 1). What is the
length of vector w given by w = -u 2v?
a.
b.
c.
d.
6
√26
2√5
2
124. What is the smallest distance
between the point(-2,-2) and a point
on the circumference of the circle
given by (x - 1)2 + (y - 2) 2 = 4?
a.
b.
c.
d.
3
4
5
6
125. What is the equation of the
horizontal asymptote of function:
f(x) = 2/(x + 2) - (x + 3)/(x + 4)?
a.
b.
c.
d.
-4
-2
-1
1
126. The lines with equations x + 3y = 2
and -2x + ky = 5 are perpendicular
for k = ?
a.
b.
c.
d.
1/3
2/3
2/4
1/4
127. If f(x) = (x - 1)2 and g(x) = √x,
then (g o f)(x) = ?
a.
b.
c.
d.
|x - 1|
x-1
1-x
|1 - x|
128. The domain of f(x) = √(4 - x2) /
√(x2 - 1) is given by the interval
a.
b.
c.
d.
(-2 , 2) U (-1 , 2)
(-2 , -1) U (1 , 2)
(-2 , 2) U (-1 , 1)
(-2 , -1) U (1 , 2)
129. The area of the circle x2 + y2 - 8y 48 = 0 is
a.
b.
c.
d.
96Pi
64Pi
48Pi
20Pi
130. An investor has P100,000, part of
which he invested at 12% interest
and the rest at 18%. He received a
total annual interest of P15,300. How
much did he invest at 18% interest
rate?
a.
b.
c.
d.
65,000
60,000
55,000
75,000
131. For what value of k will the two
equations 2x + 4 = 4(x - 2) and -x +
k = 2x - 1 have the same solution?
a.
b.
c.
d.
6
2
17
20
132. An object travels at fifteen feet per
minute. How many feet does it travel
in 24 minutes and 40 seconds?
a.
b.
c.
d.
360
370
365
375
133. Solve for x if the equation is 4 /
(√20 - √12) = x
a.
b.
c.
d.
1/2
4 / √8
√5 - √3
√5 + √3
134. DE is parallel to CB and (length of
AE / length of EB) is 4. If the area of
triangle AED is 20 square inches,
what is the area, in square inches, of
triangle ABC?
a.
b.
c.
d.
31.25
80
320
1,600
135. If a and b are both even numbers,
which of the following could be and
odd integer?
a.
b.
c.
d.
a2 + b2
(a + 1)2 + (b + 1)2
(a + 1)*(b + 1) - 1
(a + 1) / (b + 1)
136. If n is a positive integer such that n!
/ (n - 2)! = 342, find n.
a.
b.
c.
d.
19
17
18
16
137. What is the sum of the reciprocals
of the solutions to the equation: x2 (3/5)x = -11/3
a.
b.
c.
d.
5/3
9/55
-11/3
94/65
138. A number is given as 987562153ab
where a and b are digits. Which
values of a and b, such that a + b =
11 and a < b, would result in
987562153ab being divisible by 4?
a.
b.
c.
d.
a=3,b=8
a=7,b=8
a=5,b=6
a=3,b=4
139. AC is parallel to DE. AE, FG and
CD intersect at the point B. FG is
perpendicular to AC and DE. The
length of DE is 5 inches, the length
of BG is 8 inches and the length of
AC is 6 inches. What is the area, in
square inches, of triangle ABC?
a.
b.
c.
d.
28.8
20
24
22
140. Points A, B and C are defined by
their coordinates in a standard
rectangular system of axes. What
positive value of b makes triangle
ABC a right triangle with AC its
hypotenuse?
a.
b.
c.
d.
6
√6
1 + √6
1 + 2√3
141. A vendor goes to market to buy
fruits for resale at his store. He
spends half his money for mangoes,
and one-third of what remains for
bananas. He spends 150 for other
fruits and still has 200 left from the
amount he originally had. How much
money did he have at the start?
a.
b.
c.
d.
1050
5100
1500
1250
142. Seven carpenters and 5 masons earn
a total of 2,300 per day. At the same
rate of pay 3 carpenters and 8
masons earn 2,040. What are the
wages per day of the carpenter and a
mason?
a.
b.
c.
d.
200 & 180
300& 210
210& 170
270 &150
143. A man and a boy can do 15 days a
piece of work which would be done
by 7 men and 9 boys in 2 days. How
long would it take one man do it
alone?
a.
b.
c.
d.
20 days
30 days
15 days
40 days
144. A certain two-digit numbers is 1
less than five times the sum of its
digits. If 9 were added to the
number, its digits would be reversed.
Find the number.
a.
b.
c.
d.
34
36
43
63
145. If one root of 9x^2 – 6x + k = 0
exceed the other by 2, find the value
of k.
a.
b.
c.
d.
8
6
-8
-6
146. Solve:
a.
b.
c.
d.
.
3
18
9
27
147. A speed boat going across a lake 8
km wide proceeds 2 km at a certain
speed and then completes the trip at
a speed ½ km/hr faster. By doing
this, the speed arrives 10 minutes
earlier than if the original speed had
been maintained. Find the original
speed of the speed boat.
a.
b.
c.
d.
5 km/hr
4 km/hr
7 km/hr
6km/hr
148. An audience of 540 people is
seated in rows having the same
number of persons in each row. If 3
more persons seat in each row, it
would require 2 rows less to seat the
audience. How many persons were in
each row originally?
a.
b.
c.
d.
17
30
27
31
149. Find the third proportional to 4 and
12.
a.
b.
c.
d.
48
20
36
16
150. How many terms of the
progression 4, 7, 10, 13, … must be
taken so that the sum will be 69.
a.
b.
c.
d.
6
9
8
12
151. Determine x so that 2x + 1, x2 + x
+ 1, 3x2 – 3x + 3 are consecutive
terms of an arithmetic progression.
a.
b.
c.
d.
3
2
5
4
152. An equipment costs P50,000.00
and depreciates 20% of the original
costs during the first year, 16%
during the second year, 12% during
the third year, and so on, for 5 years.
What is the value at the end of 5
years?
a.
b.
c.
d.
15,000
25,000
30,000
20,000
153. Find the sum of the first 100
positive integers that is exactly
divisible by 7.
a.
b.
c.
d.
35,350
25,053
53,350
25,536
154. Find the 50th term of a geometric
progression if the 20th term is 1200
and the 30th term is also 1200.
a.
b.
c.
d.
1200
2400
1400
4100
155. A woman started a chain letter by
writing to four friends and requesting
each to copy the letter and send it to
four other friends. If the chain was
unbroken until the 5th set of letters
was mailed, how much was spent for
postage at P8.00 per letter?
a.
b.
c.
d.
16,219
10,912
21,835
13,291
156. A soccer ball is dropped from
height of 6 meters. On each rebound
it rises 2/3 of the height from which
it last fell. What distance has it
traveled at the instant it strikes the
ground for the 7th time?
a.
b.
c.
d.
27.89 m
19.86 m
20.87 m
24.27 m
157. The arithmetic mean of two
numbers is 4, and their harmonic
mean is 15/4. Find the numbers.
a.
b.
c.
d.
3&5
1&7
2&6
0&8
158. Find the real values of x and y
satisfying the given equation: (2x +
3y) + i(3x – 5y) = 8 – i7.
a.
b.
c.
d.
x = 1, y = -2
x = -2, y = -1
x = 2, y = 1
x = 1, y = 2
159. From the equation 12x3 – 8x2 + kx
+ 18 = 0, find the value of k if one
root is the negative of the other.
a.
b.
c.
d.
-17
-12
-27
-36
160. In how many ways can a group of
6 people be seated on a row of 6
seats if a certain 2 refuse to sit next
to each other?
a.
b.
c.
d.
240 ways
480 ways
180 ways
320 ways
161. How many different 8-digit
numbers can be formed from the
digits 2, 2, 2, 5, 5, 7, 7, 7?
a.
b.
c.
d.
320
560
520
480
162. In how many ways can 10 different
magazines be divide among A, B,
and C so that A gets 5 magazines, B
3 magazines and C 2 magazines?
a.
b.
c.
d.
2,520
2,250
2,050
2,052
163. What is the probability of drawing 6
white balls from a jar containing 9
white, 4 red, and 3 blue balls?
a.
b.
c.
d.
0.01
0.02
0.10
0.03
164. Ten books consisting of 5
mathematics books, 3 physics books,
and 2 chemistry books are placed in
a bookcase at random. What is the
probability that the same books are
all together?
a. 1/420
b. 3/520
c. 2/241
d. 5/2463
165. In a racing contest, there are 240
vehicles which will have provisions
that will last for 15 hours. Assuming
constant hourly consumption for
each vehicle, how long will the fuels
provisions last if 8 vehicles withdraw
from the race every hour after the
first?
a.
b.
c.
d.
63
18
20
25
166. A clerk submitted the following
reports. The average rate of
production of radios is 1.5 units for
every 1.5 hrs. work by 1.5 workers.
How many radios were produce in
one month by 30 men working 200
hrs during the month?
a.
b.
c.
d.
4000
3800
5000
4200
167. A piece of rod of length 52 cm. is
cut into two unequal parts. Each part
is then bent to form a square. It is
found that the total area of the two
squares is 97cm2. Find the difference
between the sides of each square.
a.
b.
c.
d.
3
5
4
6
168. Solve the trigonometric equation:
3sec2x – 4 = 0
a. Pi / 3 + 2n*Pi , 5Pi / 3 +
2n*Pi
b. Pi / 6 + 2n*Pi , 11 Pi / 6 +
2n*Pi
c. Pi / 3 + n*Pi , 5Pi / 3 + n*Pi
d. Pi / 6 + n*Pi , 11 Pi / 6 +
n*Pi
169. In what quadrant will the angle Ө
terminate, if sin Ө is positive and sec
Ө is negative?
a.
b.
c.
d.
I
III
II
IV
170. If sec (2x – 3) =
,
determine the value x in degrees.
a.
b.
c.
d.
14.57°
16.36°
18.65°
14.61°
171. What is the maximum value of 3 –
2 cos Ө?
a.
b.
c.
d.
2
3
4
5
172. Solve the trigonometric equation:
2cosx + 1 = 0
a. Pi / 3 + 2n*Pi , 5Pi / 3 +
2n*Pi
b. -1/2
c. 2Pi / 3 + 2n*Pi , 4Pi / 3 +
2n*Pi
d. Pi / 2 + n*Pi
173. If log x + log5 = log (x + 5), what
is the value of x?
a.
b.
c.
d.
0
1.25
1.5
2
174. If the angles of the triangle are 2x,
x + 15, and 2x + 15, find the smallest
of the angle in mills.
a.
b.
c.
d.
500 mils
600 mils
800 mils
900 mils
175. If (log10x)2 = 3 – log10x2. Which of
the following choices can be a value
of x?
a.
b.
c.
d.
10-3
102
x10
10x
176. Find the value of x in the equation
= 5.
a.
b.
c.
d.
0°
45°
30°
60°
177. If ax = by and bp = aq , then
a.
b.
c.
d.
px = qy
xy = pq
xp = yq
qx = py
178. Solve the trigonometric equation:
(3cosx + 7) (-2sinx – 1) = 0
a. 7Pi / 6 + 2n*Pi , 11Pi / 6 +
2n*Pi
b. Pi / 3 + 2n*Pi , 2Pi / 3 +
2n*Pi
c. 7Pi / 6 + n*Pi , 11Pi / 6 +
n*Pi
d. -7 / 3 , -1 / 2
179. If the bearing of point A from B is S
40° W, then the bearing of B from A
is:
a.
b.
c.
d.
N40° E
S40° W
N50° W
N50° E
180. A clock has a dial face of 12 in.
radius. The minute hand is 9 inches
while the hour hand is 6 inches. The
plane of rotation of the hour hand is
2 inches above the plane of rotation
of the minute hand. Find the distance
between the tips of the minute and
hour hand at 5:40 a.m.
a.
b.
c.
d.
7.48 in
6.48 in
9.17 in
10.16 in
181. Two towers are 60 m apart from
each other. From the top of the
shorter tower, the angle of elevation
of the top of the taller tower is 40°.
How high is the taller tower if the
height of the smaller tower is 40m?
a.
b.
c.
d.
90m
100m
80 m
70 m
182. Considering the earth to be a
sphere of radius 6400 km, find the
radius of the 60th parallel of latitude.
a.
b.
c.
d.
3,200 km
1,300 km
2,300 km
3,100 km
183.
Solve the trigonometric
equation: (6tan2x – 2) (2tan2x – 6) =
0
a. Pi / 6 + n*Pi , 5Pi / 6 +
2n*Pi , Pi / 3 + n*Pi , 2Pi / 3
+ n*Pi
b. Pi / 6 , 5Pi / 6
c. sqrt(3) , sqrt(3)
d. Pi / 3 + n*Pi , 2Pi / 3 + n*Pi
184. From a point on a level ground, the
angles of elevation of the top and
bottom of the ABS-CBN tower
situated on the top of the hill are
measured as 48° and 40°,
respectively. Find the height of the
hill if the height of the tower is 116
feet.
a.
b.
c.
d.
348.56 m
368.36 m
258.96 m
358.49 m
185. A ladder, with its foot in the street,
makes an angle of 30° with the
street when its top rests on a
building on one side of the street
and makes an angle of 40°with the
street when its top rests on a
building on the other side of the
street. If the ladder is 50 ft. long,
how wide is the street?
a.
b.
c.
d.
96.2 ft.
81.6 ft.
78.5 ft.
64.3 ft.
186. A wall is 15 ft high and 10 ft from
a building. Find the length of the
shortest ladder which will just
touch the top of the wall and reach
a window 20.5 ft above.
a.
b.
c.
d.
42.54 m
35.54 m
54.45 m
47.45 m
187. A poll tilts toward the sun at an
angle 10° from the vertical casts a
shadow 9 meters long. If the angle of
elevation from the tip of the shadow
to the top of the pole is 43°, how tall
is the pole?
a.
b.
c.
d.
10.2
7.54
10.45
8.25
188. If cos ϴ =
a.
b.
c.
d.
/ 2, find 1 – tan2Ө..
-1
-1/2
2/3
2
189. Solve the trigonometric equation “2sec2x + 4 = -2secx” in the interval
[0, 2Pi].
a.
b.
c.
d.
Pi / 3 , 5Pi / 3 , Pi
Pi
-1 , 2
Pi / 6 , 5Pi / 6 , Pi
190. Solve the trigonometric equation
2sinx cos(-x) = 2sin(-x)sin(x)” in the
interval [0, 2Pi].
a.
b.
c.
d.
0 , Pi , 3Pi / 4 , 7Pi / 4
3Pi / 4 , 7Pi / 4
0 , Pi / 2
Pi / 6 , 4Pi / 3
191. From a helicopter flying at 30,000
feet, the angles of depression of
two cities are 28° and 55°. How far
apart are the two cities?
a.
b.
c.
d.
35,415.56 ft
23,587.67 ft
53,452.67 ft
43,254.76 ft
192. Two angles are adjacent and form
an angle of 120°. If the larger angle
is 20° less than three times the
smaller angle, find the larger angle.
a.
b.
c.
d.
75°
30°
85°
65°
193. A pine tree broken over by the
wind forms a right triangle with the
ground. If the broken part makes
an angle of 50° with the ground
and the top of the tree is now 20 ft
from its base, how tall was the pine
tree?
a.
b.
c.
d.
55 ft
65 ft
45 ft
35 ft
194. A ball, 5 ft in diameter, rolls up an
incline of 18°20’. What is the
height of the center of the ball
above the base of the incline when
the ball has rolled up 5 ft up the
incline?
a.
b.
c.
d.
3 ft
5 ft
4 ft
6 ft
195. If coversed Sin Ө = 0.134, find the
value of versed Sin Ө.
a.
b.
c.
d.
0.8
0.3
0.5
0.2
196. A vertical pole consists of two
parts, each one half of the whole
pole. At a point in the horizontal
plane which passes through the
foot of the pole and 36 m from it,
the upper half of the pole subtend
an angle whose tangent is 1/3. How
high is the pole?
a.
b.
c.
d.
72
25
46
66
197. Solve the trigonometric equation
“sin2x = -sin(-x)” in the interval [0,
2Pi].
a.
b.
c.
d.
0 , 2Pi
0 , Pi / 3 , Pi , 5Pi / 3
0 , Pi
Pi / 3 , Pi
198. If the sides of the triangle are
2x+3, x2+3x+3, and x2+2x, find the
greatest angle.
a.
b.
c.
d.
100 deg.
130 deg.
120 deg.
110 deg.
199. ABDE is a square section and
BDC is an equilateral triangle with
C outside the square. Compute the
value of angle ACE.
a.
b.
c.
d.
30 deg.
60 deg.
50 deg.
20 deg.
200. The angle of elevation of the top of
a tower from a point A is 23°30’.
From another point B, the angle of
elevation of the top of the tower is
55°30’. The point A and B are
217.45 m apart and on the same
horizontal plane as the foot of the
tower. The horizontal angle
subtended by A and B at the foot
of the tower is 90 degrees. Find the
height of the tower.
a.
b.
c.
d.
90.6 m
86.7 m
89.5 m
55.9 m
201. The y coordinates of all the points
of intersection of the parabola y2 = x
+ 2 and the circle x2 + y2 = 4 are
given by
a.
b.
c.
d.
2 , -2
0 , √3 , - √3
1 , 2 , -1
1 , -2 , 1
202. What is the smallest positive zero
of function f(x) = 1/2 - sin(3x +
Pi/3)?
a.
b.
c.
d.
Pi/3
Pi/6
Pi/18
Pi/36
203. A cylinder of radius 5 cm is
inserted within a cylinder of radius
10 cm. The two cylinders have the
same height of 20 cm. What is the
volume of the region between the
two cylinders?
a.
b.
c.
d.
500Pi
1000Pi
1500Pi
2000Pi
204. A data set has a standard deviation
equal to 1. If each data value in the
data set is multiplied by 4, then the
value of the standard deviation of the
new data set is equal to
a.
b.
c.
d.
3
1
2
4
205. A cone made of cardboard has a
vertical height of 8 cm and a radius
of 6 cm. If this cone is cut along the
slanted height to make a sector, what
is the central angle, in degrees, of the
sector?
a.
b.
c.
d.
216
180
90
36
206. If sin(x) = -1/3 and Pi ≤ x ≤ 3Pi/2,
then cot(2x) = ?
a.
b.
c.
d.
4√2
2√2
√2
7/(4√2)
207. If in a triangle ABC, sin(A) = 1/5,
cos(B) = 2/7, then cos(C) = ?
a.
b.
c.
d.
(√45 - 2√24)/35
(√45 + 2√24)/35
(7√24 + 10)/35
0.85
208. What value of x makes the three
terms x, x/(x + 1) and 3x/[(x + 1)(x +
2)] those of a geometric sequence?
a.
b.
c.
d.
1
1/2
1/4
-1/2
209. The sum of the sides of a triangle
is equal to 100 cm. If the angles of
the triangle are in the continued
proportions of 1:2:4. Compute the
shortest side of the triangle.
a.
b.
c.
d.
17.545
19.806
18.525
14.507
210. The sides of the triangular field
which contains an area of 2400 sq.
cm. are in continued proportion of
3:5:7. Find the smallest side of the
triangle.
a.
b.
c.
d.
45.74
63.62
95.43
57.67
211. In triangle ABC, angle A=80 deg.
And point D is inside the triangle. If
BD and CD are bisectors of angle B
and C, solve for the angle BDC.
a.
b.
c.
d.
100 deg.
130 deg.
120 deg.
140 deg.
212. Simplify the equation Sin2x
(1+cot2x).
a.
b.
c.
d.
0
cos2x
1
sec2xsin2x
213. Assuming the earth to be a sphere
of radius 3960 mi, find the distance
of point 36° N latitude from the
equator.
a.
b.
c.
d.
2844 mi
2488 mi.
2484 mi.
4288 mi.
214. If sinxcosx+sin2x=1, what are the
values of x in degrees?
a.
b.
c.
d.
32.2, 69.3
-32.2, 69.3
20.9, 69.1
20.9, -69.1
215. If sin3x = cos6y then:
a.
b.
c.
d.
x - 2y=30
x + y=180
x + 2y=30
x + y=90
216. Evaluate cot-1 [2cos (sin-10.5)].
a.
b.
c.
d.
20°
45°
30°
60°
217. An airplane can fly at airspeed of
300 mph. if there is a wind blowing
towards the east at 50 mph, what
should be the planes compass
heading in order for its course to be
30 degrees. What will be the planes
groundspeed if it flies at this course?
a.
b.
c.
d.
21.7°, 321.86 mph
31.6°, 351.68 mph
51.7°, 121.86 mph
12.7°, 331.86 mph
218. From the given parts of a spherical
triangle ABC, compute for angle A.
(a=120°, b=73°15’, c=62°45’)
a.
b.
c.
d.
127°45’
115°26’
185°15’
137°56’
219. The diagonals of a parallelogram
are 18 cm and 30 cm respectively.
One side of a parallelogram is 12 cm.
Find the area of the parallelogram.
a. 214
b. 216
c. 361
d. 108
220. A quadrilateral has sides equal to
12 cm, 20 cm, 8 cm, and 17 cm
respectively. If the sum of the two
opposite angles is 225°, find the area
of the parallelogram.
a.
b.
c.
d.
168.18
78.31
70.73
186.71
221. The sides of the cyclic
quadrilateral are a=3 cm, b=3 cm,
c=4 cm and d=4 cm. Find the radius
of the circle that can be inscribed in
it.
a.
b.
c.
d.
2.71 cm
3.1 cm
1.51 cm
1.71 cm
222. How many diagonals can be drawn
from a 12 sided polygon?
a.
b.
c.
d.
66
48
54
36
223. Find the area of a regular polygon
whose side is 25m and apothem is
17.2m.
a.
b.
c.
d.
1075
925
1175
1275
224. Find the area of a pentagon which
is circumscribing a circle having an
area of 420.60 sq. cm.
a.
b.
c.
d.
386.57
450.54
486.29
260.24
225. As x increases from Pi/4 to 3Pi/4,
|sin(2x)|
a.
b.
c.
d.
always increases
always decreases
increases then decreases
decreases then increases
226. If ax3 + bx2 + cx + d is divided by
x - 2, then the reminder is equal to
a.
b.
c.
d.
a-b+c-d
8a + 4b + 2c + d
-8a + 4b -2c + d
a+b+c+d
227. A committee of 6 teachers is to be
formed from 5 male teachers and 8
female teachers. If the committee is
selected at random, what is the
probability that it has an equal
number of male and female teachers?
a.
b.
c.
d.
140/429
150/429
160/429
170/429
228. The range of the function f(x) = -|x
- 2| - 3 is
a.
b.
c.
d.
y≥2
y ≤ -3
y ≥ -3
y ≤ -2
229. What is the period of the function
f(x) = 3sin2(2x + Pi/4)?
a.
b.
c.
d.
3Pi
2Pi
Pi/2
Pi/3
230. It is known that 3 out of 10
television sets are defective. If 2
television sets are selected at random
from the 10, what is the probability
that 1 of them is defective?
a.
b.
c.
d.
1/15
1/10
1/2
1/3
231. In a triangle ABC, angle B has a
size of 50o, angle A has a size of 32o
and the length of side BC is 150
units. The length of side AB is
a.
b.
c.
d.
232
280
260
270
232. For the remainder of the division
of x3 - 2x2 + 3kx + 18 by x - 6 to be
equal to zero, k must be equal to
a.
b.
c.
d.
1
5
-9
-10
233. It takes pump (A) 4 hours to empty
a swimming pool. It takes pump (B)
6 hours to empty the same swimming
pool. If the two pumps are started
together, at what time will the two
pumps have emptied 50% of the
water in the swimming pool?
a.
b.
c.
d.
1 hour 12 minutes
1 hour 20 minutes
2 hours 30 minutes
3 hours
234. The graph of r = 10 cos(Θ) , where
r and Θ are the polar coordinates, is
a. a circle
b. an ellipse
c. a horizontal line
d. a hyperbola
235. If (2 - i)*(a - bi) = 2 + 9i, where i is
the imaginary unit and a and b are
real numbers, then a equals
a.
b.
c.
d.
2
1
0
-1
236. Lines L1 and L2 are perpendicular
that intersect at the point (2, 3). If L1
passes through the point (0, 2), then
line L2 must pass through the point
a.
b.
c.
d.
(0 , 3)
(1 , 1)
(3 , 1)
(5 , 0)
237. In a plane there are 6 points such
that no three points are collinear.
How many triangles do these points
determine?
a.
b.
c.
d.
8
10
18
12
238. In a circle with a diameter of 10
meters, a regular five pointed star
touching its circumference is
inscribed. What is the area of the part
not covered by the star?
a.
b.
c.
d.
60.42
40.58
40.68
50.47
239. Find the area of a hexagon with a
square having an area of 72 sq. cm.
inscribed in a circle which is
inscribed in a hexagon.
a.
b.
c.
d.
124.71 sq. cm.
150.26 sq. cm.
150.35 sq. cm.
130.77 sq. cm.
240. The tangent and a secant are to a
circle from the same external point.
If the tangent is 6 inches and the
external segment of the secant is 3
inches, compute the length of the
secant.
a.
b.
c.
d.
10
13
12
14
241. Two circles with radii 8 and 3 m
are tangent to each other externally.
What is the distance between the
points of tangency of one of their
common external tangencies?
a.
b.
c.
d.
7.8 m
9.8 m
10.7 m
6.7 m
242. The diameters of the two circles
that are tangent internally are 18 and
8, respectively. What is the length of
the tangent segment from the center
of the larger circle to the smaller
circle?
a.
b.
c.
d.
2
4
3
5
243. Three identical circles are tangent
to each other externally. If the area
of the curvilinear triangle enclosed
between the points of tangency of the
3 circles is 16.13 cm2, compute the
radius of each circle.
a.
b.
c.
d.
10 cm
13 cm
9 cm
15cm
244. A semi – circle of radius 14 cm is
bent to form a rectangle whose
length is 1 cm more than its width.
Find the area of the rectangle.
a.
b.
c.
d.
323.75 cm2
322.32 cm2
233.57 cm2
233.75 cm2
245. A swimming pool is constructed in
the shape of two partially
overlapping circles, each of radius 9
m. If the center each circle lies on
the circumference of the other, find
the perimeter of the swimming pool.
a.
b.
c.
d.
85.7 m
75.4 m
56.5 m
96.8 m
246. The length of the side of a rhombus
is 5 cm. If the shorter diagonal is of
length 6 cm. What is the area of the
rhombus?
a.
b.
c.
d.
24 cm2
14 cm2
18 cm2
25 cm2
247. Two squares each of 12 cm sides
overlap each other such that the
overlapping region is a regular
polygon. Determine the area of the
overlapping region thus formed.
a.
b.
c.
d.
110.9 cm2
119.3 cm2
121.5 cm2
117.4 cm2
248. The side of a regular pentagon is
25 cm. If the radius of its inscribed
circle is 15 cm, find the area of the
pentagon.
a.
b.
c.
d.
937.5 cm2
784.6 cm2
825.75 cm2
857.65 cm2
249. The capacities of two
hemispherical tanks are in the ratio
64:125. If 4.8 kg of paint is required
to paint the outer surface of the
smaller tank, then how many
kilograms of paint would be needed
to paint the outer surface of the
larger tank?
a.
b.
c.
d.
8.5 kg
6.7 kg
7.5 kg
9.4 kg
250. A wooden cone of altitude 10 cm is
to be cut into two parts of equal
weight. How far from the vertex
should the cut parallel to the base be
made?
a.
b.
c.
d.
6.65 cm
3.83 cm
7.94 cm
8.83 cm
251. A sphere of radius 5 cm and a right
circular cone of base radius 5 cm and
a height 10 cm stand on a plane. Find
the position of a plane that cuts the
two solids in equal circular sections.
a.
b.
c.
d.
2 cm
2.5 cm
1.5 cm
3.2 cm
252. A regular triangular pyramid has
an altitude of 9 m and a volume of
187.06 cu.m. What is the base edge
in meters?
a.
b.
c.
d.
10
11
12
13
253. Two cylinders of equal radius 3m
have their axes at right angles. Find
the volume of the common part.
a.
b.
c.
d.
122 cu.cm.
144 cu.cm.
154 cu.cm.
134 cu.cm.
254. A solid has a circular base of
radius 20 cm. find the volume of the
solid if every plane section
perpendicular to a certain diameter is
an equilateral triangle.
a.
b.
c.
d.
18,475.21 cm3
20,475.31 cm3
12,775.21 cm3
21,475.21 cm3
255. If the edge of a cube is increased
by 30%, by how much is the surface
increased?
a.
b.
c.
d.
30%
21%
69%
33%
256. If the edge of a cube decreases by
x%, its volume decrease by 48.8%.
Find the value of x.
a.
b.
c.
d.
10%
20%
16%
25%
257. Find the acute angles between the
two planes 2x – y + z = 8 and x + y +
2z – 11 = 0.
a.
b.
c.
d.
30°
60°
45°
40°
258. Find the volume of the solid
bounded by the plane x + y + z = 1
and the coordinate planes.
a.
b.
c.
d.
1/3
1/4
1/5
1/6
259. Compute the volume of a regular
icosahedron with sides equal to 6
cm.
a.
b.
c.
d.
470.88 cm3
520.78 cm3
340.89 cm3
250.56 cm3
260. Compute the volume (in cm3) of a
sphere inscribe in an octahedron
having sides equal to 18 cm.
a.
b.
c.
d.
1622.33
1875.45
1663.22
1892.63
261. Find the volume of a spherical
cone in a sphere of radius 17 cm if
the radius of its zone is 8 cm.
a.
b.
c.
d.
2120.35
1426.34
1210.56
2316.75
262. A spherical wooden ball 15 cm in
diameter sinks to depth 12 cm in a
certain liquid. Calculate the area
exposed above the liquid in cm2.
a.
b.
c.
d.
45 pi.
20 pi.
15 pi.
10 pi.
263. Given a solid right circular cone
having a height of 8 cm. has a
volume equal to 4 times the volume
of the smaller cone that could be cut
from the same cone having the same
axis. Compute the height of the
smaller cone.
a.
b.
c.
d.
5.04 cm
3.25 cm
4.45 cm
2.32 cm
264. The diameter of a sphere and the
base of a cone are equal. What
percentage of that diameter must the
cones height be so that both volumes
are equal.
a.
b.
c.
d.
100%
200%
50%
400%
265. The volume a regular pyramid
whose base is a regular hexagon is
156 m3. If the altitude of the pyramid
is 5 m., find the sides of the base.
a.
b.
c.
d.
4m
8m
6m
3m
266. The base of a cylinder is a hexagon
inscribed in a circle. If the difference
in the circumference of the circle and
the perimeter of the hexagon is 4
cm., find the volume of the prism if
it has an altitude of 20 cm.
a.
b.
c.
d.
10,367 cm3
12,239 cm3
10,123 cm3
11,231 cm3
267. The volume of a truncated prism
with an equilateral triangle as its
horizontal base is equal to 3600 cm3.
The vertical edges at each corner are
4, 6, and 8 cm., respectively. Find
one side of the base.
a.
b.
c.
d.
22.37
25.43
37.22
17.89
268. Aluminum and lead have specific
gravities of 2.5 and 16.48
respectively. If a cubical aluminum
has edge of 0.30 m., find the edge of
a cubical block of lead having the
same weight as the aluminum.
a.
b.
c.
d.
10 cm
14 cm
13 cm
16 cm
269. Find the area of a pentagonal
spherical pyramid the angles of
whose base are 105°, 126°, 134°,
146° and 158° on the sphere of
radius 12 m.
a.
b.
c.
d.
324.21
343.56
222.34
433.67
270. If the surface areas of two spheres
are 24 cm2 and 96 cm2 respectively.
Find the ratio of their volume.
a.
b.
c.
d.
1/4
5
1/8
3/5
271. Considering the earth as a sphere
of radius 6400 km, find the radius of
the 60th parallel of latitude.
a.
b.
c.
d.
3200 km
1300 km
2300 km
3100 km
272. A conical vessel has a height of 24
cm. and a base diameter of 12 cm. It
holds water to a depth of 18 cm
above its vertex. Find the volume of
its content.
a.
b.
c.
d.
381.7 cm2
281.6 cm2
451.2 cm2
367.4 cm2
273. A sphere is dropped in a can
partially filled with water. What is
the rise in height of the water if they
have equal diameters?
a. 0.75d
b. 0.67d
c. 1.33d
d. 1.5d
274. A wooden cone is to be cut into
two parts of equal volume by a plane
parallel to its base. Find the ratio of
the heights of the two parts.
a.
b.
c.
d.
2.35
3.85
1.26
1.86
275. The ratio of the area of regular
polygon circumscribed in a circle to
the area of inscribed regular polygon
of the same number of sides is 4:3.
Find the number of sides.
a.
b.
c.
d.
4
6
8
10
276. A rectangle ABCD which
measures 18 by 24 cm is folded
once, perpendicular to diagonal AC,
so that the opposite vertices A and C
coincide. Find the length of the fold.
a.
b.
c.
d.
18.5 cm
22.5 cm
21.5 cm
19.5 cm
277. If the straight lines ax + by +c = 0
and bx + cy + a = 0 are parallel, then
which of the following is correct?
a.
b.
c.
d.
2
b = 4ac
b2 = ac
b2 + ac = 0
a2 =bc
278. Find the equation of the
perpendicular bisector of the
segment joining the points (2, 6) and
(-4, 3).
a.
b.
c.
d.
2x - 4y + 5 = 0
2x + 4y + 5 = 0
4x + 2y – 5 = 0
5x – 2y + 4 = 0
279. The vertices of the base of an
isosceles triangle are (-1, -2) and (1,
4). If the third vertex lies on the line
4x + 3y = 12, find the area of the
triangle.
a.
b.
c.
d.
8
10
9
11
280. The coordinates of the two vertices
of a triangle are (6, -1) and (-3, 7).
Find the coordinates of the third
vertex so that the centroid of the
triangle will be at the origin.
a.
b.
c.
d.
(-3, -6)
(-5, -5)
(4, -6)
(6, -4)
281. Compute the angle between the
line 2y-9x-18=0 and the x-axis.
a.
b.
c.
d.
64.54°
45°
77.47°
87.65°
282. Find the value of k if the yintercept of the line 3x-4y-8k=0 is
equal to 2.
a.
b.
c.
d.
1
2
-1
3
283. Find the area of the polygon whose
vertices are (2, -6), (4, 0), (2, 4), (-3,
2), (-3, 3).
a.
b.
c.
d.
32.5
23.5
47.5
35.5
284. In the triangle ABC having vertices
at A(-2, 5), B(6, 1) and C(-2, -3),
find the length of the median from
vertex B to side AC.
a.
b.
c.
d.
5
7
6
8
285. A line has an equation of 3x-ky8=0. Find the value of k if this line
makes an angle of 45° with the line
2x+5y-17=0.
a.
b.
c.
d.
5
7
8
6
286. The points (1, 3) and (5, 5) are two
opposite vertices of a rectangle. The
other two vertices lie on the line 2xy+k=0. Find the value of k.
a.
b.
c.
d.
-2
2
-3
4
287. Let m1 and m2 be the respective
slopes of two perpendicular lines.
Then
a. m1 + m2 = -1
b. m1 x m2 = -1
c. m1 = m2
d. m1 x m2 =0
288. The abscissa of a point is 3. If its
distance from a point (8, 7) is 13,
find its ordinate.
a.
b.
c.
d.
-5 or 19
3 or 5
5 or 19
-3 or 7
289. If the points (-3, -5), (p, q) and (3,
4) lie on a straight line, then which
of the following is correct?
a.
b.
c.
d.
2p – 3q =1
p + q = -3
3p – 2q =1
2p – q =3
290. Find the equation of the line
parallel to 7x + 2y – 4 = 0 and
passing through (-3, -5).
a.
b.
c.
d.
7x + 2y + 31 = 0
2x – 4y -7 = 0
3x – 4y + 7 = 0
2x – 7y + 31 = 0
291. Find the area of a triangle whose
vertices are (1, 1), (3,-3), and (5,-3).
a.
b.
c.
d.
4
7
10
12
292. Determine the x – intercept of the
line passing through (4, 1) and (1, 4).
a.
b.
c.
d.
3
5
4
6
293. Find the slope of the line having a
parametric equations of x=2+t and
y=5-3t.
a.
b.
c.
d.
1
1/3
-3
-1
294. The midpoint of the line segment
joining a moving point to (6, 0) is on
the line y=x. Find the equation of its
locus.
a.
b.
c.
d.
x–y+6=0
x – 2y + 6 = 0
2x – y -3 = 0
2x + 3y – 5 = 0
295. The base of an isosceles triangle is
the line from (4,-3) to (-4, 5). Find
the locus of the third vertex.
a.
b.
c.
d.
x–y+1=0
x+y+1=0
x–y–2=0
x+y–3=0
296. What is the new equation of the
line 5x + 4y + 3 = 0 if the origin is
translated to the point (1, 2)?
a.
b.
c.
d.
4x’ + 3y’ + 16 = 0
5x’ + 4y’ + 16 = 0
5x’ – 4y’ – 16 = 0
6x’ + 6y’ – 16 = 0
297. One end of the diameter of the
circle (x – 4)2 + y2 = 25 is the point
(1, 4). Find the coordinates of the
other end of this diameter.
a.
b.
c.
d.
(7, -4)
(3, 4)
(-4, 7)
(-7, 4)
298. Determine the area bounded by the
curve x2 + y2 – 6y = 0
a.
b.
c.
d.
27.28 sq. units
72.28 sq. units
28.27 sq. units
18.27 sq. units
299. How far is the center of the circle
x2 + y2 – 10x – 24y + 25 = 0 from
the line y = 2?
a.
b.
c.
d.
10
14
12
16
300. Find the equation of the circle
tangent to the y-axis and the center is
at (5, 3).
a.
b.
c.
d.
(x+5)2 + (y-3)2 = 25
(x-5)2 + (y+3)2 = 25
(x-5)2 + (y-3)2 = 25
(x-5)2 + (y-3)2 = 50
301. Find the equation of the circle
circumscribing a triangle whose
vertices are (0, 0), (0, 5) and (3, 3).
a.
b.
c.
d.
x2 + y2 – x - 5y = 0
x2 + y2 - 2x – y = 0
x2 + y2 -5x -5y + 8 = 0
x2 + y2 – x - 5y + 6 = 0
302. A parabola having its axis along
the x-axis passes through (-3, 6).
Compute the length of the latus
rectum if the vertex is at the origin.
a.
b.
c.
d.
4
8
6
12
303. A hut has a parabolic cross-section
whose height is 30m. and whose
base is 60m. wide. If the ceiling 40
m. is to be placed inside the hut, how
high will it be above the base?
a.
b.
c.
d.
16.67 m
15.48 m
14.47 m
19.25 m
304. Find the coordinates of the focus of
the parabola x2=4y-8.
a.
b.
c.
d.
(0, -3)
(0, 3)
(2, 0)
(0, -2)
305. An ellipse has an eccentricity of
1/3. Compute the distance between
directrices if the distance between
foci is 4.
a.
b.
c.
d.
18
36
32
38
306. An ellipse has a length of semimajor axis of 300 m. compute the
second eccentricity of the eclipse.
a.
b.
c.
d.
1.223
1.222
1.333
1.233
307. Compute the circumference of an
ellipse whose diameters are 14 and
10 meters.
a.
b.
c.
d.
28.33 m
38.22 m
18.75 m
23.14 m
308. Find the eccentricity of a hyperbola
having distance between foci equal
to 18 and the distance between
directrices equal to 2.
a.
b.
c.
d.
2
3
2.8
3.7
309. Find the length of the tangent from
point (7, 8) to the circle x2 + y2 – 9 =
0
a.
b.
c.
d.
10.2
14.7
11.3
13.6
310. What is the equation of the
equation of the directrix of the
parabola y2 = 16x?
a.
b.
c.
d.
x=4
y=4
x = -4
y = -4
311. Find the radius of the circle 2x2 +
2y2 – 3x + 4y – 1 = 0
a.
/4
b.
/4
c.
/4
d.
/4
312. Find an equation for the hyperbola
with foci at (1, 3) and (9, 3), and
eccentricity 2.
a. x2 – 3y2 – 30x + 6y + 54 = 0
b. 3x2 – y2 – 30x + 6y + 54 = 0
c. x2 – y2 – 30 x + 6y + 54 = 0
d. 3x2 – y2 – 6x + 30y = 54 = 0
313. Find the equation of the locus of a
point which moves so that its
distance from (1, -7) is always 5.
a.
b.
c.
d.
x2 + y2 – 2x + 14y + 25 = 0
x2 + y2 – 2x – 14y + 25 = 0
x2 + y2 + 2x + 14y + 25 = 0
x2 + y2 – 2x + 14y + 25 = 0
314. The difference of the distances of a
moving point from (1, 0) and (-1, 0)
is 1. Find the equation of its locus.
a.
b.
c.
d.
4x2 - 12y2 = 3
3x2 - 4y2 = 12
12x2 - 4y2 = 3
4x2 - 9y2 = 3
315. A circle has its center on the line
2y=3x and tangent to the x-axis at (4,
0). Find the radius.
a.
b.
c.
d.
6
7
5
8
316. Find the shortest distance from (3,
8) to the curve x2+y2+4x-6y=12.
a.
b.
c.
d.
1.21
2.07
4.09
3.73
317. The focus of the parabola y2=4x is
at:
a.
b.
c.
d.
(4, 0)
(1, 0)
(0, 4)
(0, 1)
318. An arc in the form of a parabola is
60 m across the bottom. The highest
point is 16 m above the horizontal
base. What is the length of the beam
placed horizontally across the arc 3
m below the top.
a.
b.
c.
d.
19.36
24.86
25.98
27.34
319. A curve has an equation of x2 = cy
+ d. the length of latus rectum is 4
and the vertex is at (0, 2). Compute
the value of C and d.
a.
b.
c.
d.
4, -8
6, -2
2, -5
3, -7
320. What conic section is 2x2 - 8xy +
4x = 12?
a.
b.
c.
d.
Parabola
Ellipse
Hyperbola
Circle
321. What conic section is described by
the equation r = 6 / (4 – 3cosӨ)?
a.
b.
c.
d.
Circle
Ellipse
Hyperbola
Parabola
322. An ellipse has its center at (0, 0)
with its axis horizontal. The distance
between the vertices is 8 and its
eccentricity is 0.5. Compute the
length of the longest focal radius
from point (2, 3) on the curve.
a. 3
b. 5
c. 4
d. 6
323. Determine the equation of the
common tangents to the circles
x2+y2+2x+4y-3=0 and x2+y2-8x6y+7=0.
a.
b.
c.
d.
x+y–1=0
2x + y – 1 = 0
x–y–1=0
x -2x + 1 = 0
324. An arc in the form of a parabolic
curve is 40 m across the bottom. A
flat horizontal beam 26 m long is
placed 12 m above the base. Find the
height of the arc.
a.
b.
c.
d.
20.78 m
18.67 m
25.68 m
15.87 m
325. Evaluate: Lim (tan33x) / x3 as x
approaches 0.
a.
b.
c.
d.
0
31
27
Infinity
d. 0
328. A snowball is being made so that
its volume is increasing at the rate of
8 ft3/min. Find the rate at which the
radius is increasing when the
snowball is 4 ft in diameter.
a.
b.
c.
d.
0.159 ft/min
0.015 ft/min
0.259 ft/min
0.325 ft/min
329. A stone is dropped into a still
pond. Concentric circular ripples
spread out, and the radius of the
disturbed region increases at the rate
of 16 cm/s. At what rate does the
area of the disturbed increase when
its radius is 4 cm?
a.
b.
c.
d.
304.12 cm2/s
503.33 cm2/s
402.12 cm2/s
413.13 cm2/s
330. Find the limit (x+2)/(x-3) as x
approaches 3.
a.
b.
c.
d.
0
infinity
indeterminate
3
326. Evaluate the integral xcosxdx.
a.
b.
c.
d.
xsinx + cosx + C
x2sinx + C
xcosx + sinx + C
2xsinx + cosx + C
327. Find the limit: sin2x/sin3x as x
approaches to 0.
a. 1/3
b. 3/4
c. 2/3
331. A man 1.8 m. tall is walking at the
rate of 1.2 m/s away from a lamp
post 6.7 m high. At what rate is the
tip of his shadow receding from the
lamp post?
a.
b.
c.
d.
2.16 m/s
1.64 m/s
1.83 m/s
1.78 m/s
332. A man on a wharf is pulling a rope
tied to a raft at a rate of 0.6 m/s. If
the hands of the man pulling the rope
are 3.66 m above the water, how fast
is the raft approaching the wharf
when there are 6.1 m of rope out?
a.
b.
c.
d.
-1.75 m/s
-0.25 m/s
-0.75 m/s
-0.54 m/s
333. Evaluate the limit: tanx / x as x
approaches 0.
a.
b.
c.
d.
0
undefined
1
infinity
334. A man is riding his car at the rate
of 30 km/hr toward the foot of a pole
10 m high. At what rate is he
approaching the top when he is 40 m
from the foot of the pole?
a.
b.
c.
d.
-5.60 m/s
-6.78 m/s
-8.08 m/s
-4.86 m/s
335. Find the point on the curve y = x3
at which the tangent line is
perpendicular to the line 3x + 9y = 4.
a.
b.
c.
d.
(1, 1)
(1, -1)
(-1, 2)
(-2, -1)
336. A boy wishes to use 100 feet of
fencing to enclose a rectangular
garden. Determine the maximum
possible area of his garden.
a. 625 ft2
b. 524 ft2
c. 345 ft2
d. 725 ft2
337. Find the equation of the tangent
line to the curve x3 + y3 = 9 at the
given point (1, 2).
a.
b.
c.
d.
x + 4y = 9
2x + 4y = 5
4x – y = 9
4x – 2y = 10
338. Find the area of the largest
rectangle whose base is on the x axis
and whose upper two vertices lie on
the curve y = 12 – x2.
a.
b.
c.
d.
24
32
16
36
339. Find the radius of the largest right
circular cylinder inscribed in a
sphere of radius 5.
a.
b.
c.
d.
4.08 units
1.25 units
5.14 units
8.12 units
340. A rectangular box open at the top
is to be constructed from a 12x12inch piece of cardboard by cutting
away equal squares from the four
corners and folding up the sides.
Determine the size of the cutout that
maximizes the volume of the box.
a.
b.
c.
d.
6 inches
1.5 inches
2 inches
3 inches
341. Find dy / dx if y = 5^(2x + 1).
a.
b.
c.
d.
(5^(2x + 1))ln25
(5^(2x + 1))ln(2x + 1)
(5^(2x + 1))ln5
(5^(2x + 1)ln15
342. An athlete at point A on the shore
of a circular lake of radius 1 km
wants to reach point B on the shore
diametrically opposite A. If he can
row a boat 3 km/hr and jog 6 km/hr,
at what angle with the diameter
should he row in order to reach B in
the shortest possible time?
a.
b.
c.
d.
30°
50°
45°
60°
343. Find the area of the region above
the x axis bounded by the curve y = x2 + 4x – 3.
a.
b.
c.
d.
1.333 square units
3.243 square units
2.122 square units
1.544 square units
344. Find the volume of the solid of
revolution formed by rotating the
region bounded by the parabola y =
x2 and the lines y = 0 and x = 2 about
the x axis.
a.
b.
c.
d.
25.01 cu. units
15.50 cu. units
20.11 cu. units
30.14 cu. units
345. A publisher estimates that in t
months after he introduces a new
magazine, the circulation will be C(t)
= 150t2 + 400t + 7000 copies. If this
prediction is correct, how fast will
the circulation increase 6 months
after the magazine is introduced?
a.
b.
c.
d.
1200 copies/month
2202 copies/month
2000 copies/month
2200 copies/month
346. What is the order and degree of the
differential equation y’’’ + xy’’ +
2y(y’)2 +xy = 0.
a.
b.
c.
d.
first order, second degree
second order, third degree
third order, first degree
third order, second degree
347. A curve is defined by the condition
that at each of its points (x, y), its
slope is equal to twice the sum of the
coordinates of the point. Express the
condition by means of a differential
equation.
a.
b.
c.
d.
dy / dx = 2x + 2y
dy / dx = 2x + 2ydy
dy = 2xdx + 2y
x + y = 2y’
348. Find the first derivative of
ln(cosx).
a.
b.
c.
d.
cscx
–tanx
secx
cotx
349. Find the number of equal parts into
which a given number N must be
divided as that their product will be a
maximum.
a.
b.
c.
d.
N/2e
N/e
2N/e2
2N/e
350. An object moves along the x – axis
so that its x-coordinate obeys the law
x = 3t2 + 8t + 1. Find the time when
its velocity and acceleration are the
same.
a.
b.
c.
d.
2/3
3/5
3/4
4/5
351. Assuming that the earth is a perfect
sphere, with radius 4000 miles. The
volume of ice at the north and south
poles is estimated to be 8,000,000
cubic miles. If this ice were melted
and if the resulting water were
distributed uniformly over the globe,
approximately what should be the
depth of the added water at any point
on the earth?
a.
b.
c.
d.
120 ft.
320 ft.
210 ft.
230 ft.
352. Find the equation of the curve
passing through the point (3, 2) and
having slope 2x2 – 5 at any point (x,
y).
a.
b.
c.
d.
2x3 – 15x – 3y + 2 = 0
3x3 – 5x – 2y – 1 = 0
2x3 + 5x – 3y – 21 = 0
5x3 – 3x – 3y + 1 = 0
353. Find the centroid of the region
bounded by y = x2, y = 0, and x = 1.
a.
b.
c.
d.
(1/4, 2/3)
(2/3, 5/4)
(3/4, 3/10)
(3/5, 5/10)
354. Find the point of inflection of the
curve x3 – 3x2 – x + 7.
a.
b.
c.
d.
2, 3
2, 6
1, 5
1, 4
355. Find two numbers whose sum is 36
if the product of one by the square of
the other is a maximum.
a.
b.
c.
d.
12, 23
25, 11
16, 20
12, 24
356. Find the minimum distance from
the curve y = 2 square root of 2x to
the
a.
b.
c.
d.
3.56
4.66
5.66
2.66
357. Divide 60 into 3 parts so that the
product of the three parts will be the
maximum. Find the product.
a.
b.
c.
d.
6,000
8,000
4,000
12,000
358. A particle moves along a path
whose parametric equations are x =
t3 and y = 2t2. What is the
acceleration of that particle when t =
5 seconds?
a.
b.
c.
d.
30.26 m/s2
18.56 m/s2
21.62 m/s2
23.37 m/s2
359. Find the area bounded by the curve
5y2 = 16x and the curve y2 = 8x – 24.
a.
b.
c.
d.
36
25
16
14
360. Find the area in the first quadrant
bounded by the parabola y2=4x and
the line x=3 and x=1
a.
b.
c.
d.
5.595
4.254
6.567
7.667
361. Find the area enclosed by the curve
x2+8y+16=0, the line x=4 and the
coordinate axes.
a.
b.
c.
d.
8.97
10.67
9.10
12.72
362. Find the volume of the solid
formed by rotating the curve 4x2 +
9y2 = 36 about the line 4x + 3y – 20
=0
a.
b.
c.
d.
356.79
138.54
473.74
228.56
363. Determine the moment of inertia of
a rectangle 100cm by 300cm with
respect to a line through its center of
gravity and parallel to the shorter
side.
a.
b.
c.
d.
225x106 cm4
125x106 cm4
325x106 cm4
235x106 cm4
364. Find the area of the region
bounded by y2=8x and y=2x.
a.
b.
c.
d.
3/4
5/4
4/3
5/6
365. Two posts, one 8 ft. high and the
other 12 ft. high, stand 15 ft. apart
from each other. They are to be
stayed by wires attached to a single
stake at ground level, the wires
running to the tops of the posts. How
far from the shorter post should the
stake be placed to use the least
amount of wire?
a.
b.
c.
d.
6 ft.
5 ft.
9 ft.
8 ft.
366. At the maximum point, the second
derivative of the curve is
a.
b.
c.
d.
0
Negative
Undefined
Positive
367. Determine the curvature of the
curve y2=16x at the point (4, 8).
a.
b.
c.
d.
-0.0442
-0.1043
-0.0544
-0.0254
368. Determine the value of the integral
of sin53xdx from 0 to pi over 6.
a. 0.457
b. 1.053
c. 0.0178
d. 0.178
369. A body moves such that its
acceleration as a function of time is
a=2+12t, where “a” is in m/s2. If its
velocity after 1 s is 11 m/s. find the
distance traveled after 5 seconds.
a.
b.
c.
d.
256 m
340 m
290 m
420 m
370. A runner and his coach are
standing together on a circular track
of radius 100 meters. When the
coach gives a signal, the runner starts
to run around the track at a speed of
10 m/s. How fast is the distance
between the runners has run ¼ of the
way around the track?
a.
b.
c.
d.
5.04 m/s
6.78 m/s
5.67 m/s
7.07 m/s
371. A telephone company has to run a
line from a point A on one side of a
river to another point B that is on the
other side, 30 km down from the
point opposite A. the river is
uniformly 10 km wide. The company
can run the line along the shoreline
to a point C then run the line under
the river to b. the cost of laying the
line along the shore is P5000 per km,
and the cost of laying it under water
is P12, 000 per km. Where the point
C should be located to minimize the
cost?
a.
b.
c.
d.
5.167 km
6.435 km
4.583 km
3.567 km
372. The height of a projectile thrown
vertically at any given time is define
by the equation h(t) = -16t2 + 256t.
What is the maximum height reach
by the projectile?
a.
b.
c.
d.
1567 ft
1920 ft
1247 ft
1024 ft
373. The density of the rod is the rate of
change of its mass with respect to its
given length. A certain rod has
length of 9 feet and a total mass of
24 slugs. If the mass of a section of
the rod of length x from its left end is
proportional to the square root of this
length, calculate the density of the
rod 4 ft from its left end.
a.
b.
c.
d.
1 slug/ft
2 slugs/ft
3 slugs/ft
4 slugs/ft
374. It costs 0.05 x2 + 6x + 100 dollars
to produce x pounds of soap.
Because of quantity discounts, each
pound sells for 12 – 0.15x dollars.
Calculate the marginal profit when
10 pounds of soap is produced.
a.
b.
c.
d.
$9
$2
$ 12
$7
375. Find the area of the region
bounded by y = x2 – 5x + 6, the axis,
and the vertical lines x = 0 and x = 4.
a. 5/7
b. 19/4
c. 17/3
d. 9/2
376. A police car is 20 ft away from a
long straight wall. Its beacon,
rotating 1 revolution per second,
shines a beam of light on the wall.
How fast is the beam moving when it
is nearest to the police car?
a.
b.
c.
d.
10pi
20pi
30pi
40pi
377. Find area of the largest rectangle
that can be inscribed in an equilateral
triangle of side 20.
a.
b.
c.
d.
46.83
59.23
91.23
62.73
379. Find the maximum area of a
rectangle circumscribed about a
fixed rectangle of length 8 and width
4
a.
b.
c.
d.
a.
b.
c.
d.
13/25
1/46
5/21
11/14
381. Determine the area of the region
bounded by the curve y = x3 – 4x2 +
3x and the axis, from x = 0 to x =3.
a.
b.
c.
d.
28/13
13/58
29/11
37/12
382. Find the volume of a solid formed
by rotating the area bounded by y =
x2, y = 8 – x2 and the y axis about the
x axis.
378. A hole of 2 radius is drilled
through the axis of a sphere of radius
3. Compute the volume of the
remaining part.
a.
b.
c.
d.
isosceles trapezoid 30 cm wide of 50
cm. If the through leaks water at the
rate of 2000 cm3/min, how fast is the
water level decreasing when the
water is 20 cm deep.
67
38
72
81
380. A trough filled with liquid is 2 m
long and has a cross section of an
a.
b.
c.
d.
268.1
287.5
372.9
332.4
383. The price p of beans, in dollars per
basket, and the daily supply x, in
thousands of basket, are related by
the equation px + 6x + 7p = 5950. If
the supply is decreasing at the rate of
2000 baskets per day, what is the rate
of change of daily basket price of
beans when 100,000 baskets are
available?
a.
b.
c.
d.
2.35
1.05
3.15
4.95
384. A flying kite is 100 m above the
ground, moving in a horizontal
direction at a rate of 10 m/s. How
fast is the angle between the string
and the horizontal changing when
there is 300 m of string out?
a.
b.
c.
d.
1/90 rad/sec
1/30 rad/sec
1/65 rad/sec
1/72 rad/sec
385. If functions f and g have domains
Df and Dg respectively, then the
domain of f / g is given by
a. the union of Df and Dg
b. the intersection of Df and Dg
c. the intersection of Df and
Dg without the zeros of
function g
d. None of the above
386. Let the closed interval [a, b] be the
domain of function f. The domain of
f(x - 3) is given by
a. the open interval (a , b)
b. the closed interval [a , b]
c. the closed interval [a - 3 , b 3]
d. the closed interval [a + 3 , b
+ 3]
387. Let the interval (a , +infinity) be
the range of function f. The range of
f(x) - 4 is given by
a. the interval (a - 4 ,
+infinity)
b. the interval (a + 4, +infinity)
c. the interval (a, +infinity)
d. None of the above
388. If functions f(x) and g(x) are
continuous everywhere then
a. (f / g)(x) is also continuous
everywhere.
b. (f / g)(x) is also continuous
everywhere except at the
zeros of g(x).
c. (f / g)(x) is also continuous
everywhere except at the
zeros of f(x).
d. more information is needed
to answer this question
389. If functions f(x) and g(x) are
continuous everywhere and f(1) = 2,
f(3) = -4, f(4) = 8, g(0) = 4, g(3) = -6
and g(7) = 0 then lim (f + g)(x) as x
approaches 3 is equal to
a.
b.
c.
d.
-9
-10
-11
-12
390. If f(x) and g(x) are such that lim
f(x) as x --> a = + infinity and lim
g(x) as x --> a = 0, then
a. lim [ f(x) . g(x) ] as x --> a is
always equal to 0
b. lim [ f(x) . g(x) ] as x --> a is
never equal to 0
c. lim [ f(x) . g(x) ] as x --> a
may be +infinity or -infinity
d. None of the above
391. A critical number c of a function f
is a number in the domain of f such
that
a.
b.
c.
d.
f '(c) = 0
f '(c) is undefined
(A) or (B) above
None of the above
392. The values of parameter a for
which function f defined by f(x) = x3
+ ax2 + 3x has two distinct critical
numbers are in the interval
a. (-infinity , + infinity)
b. (-infinity , -3] U [3 ,
+infinity)
c. (0 , + infinty)
d. None of the above
393. If f(x) has one critical point at x =
c, then
a. function f(x - a) has one
critical point at x = c + a
b. function - f(x) has a critical
point at x = - c
c. f(k x) has a critical point at x
=c/k
d. (A) and (C) only
394. The values of parameter a for
which function f defined by f(x) =
3x3 + ax2 + 3 has two distinct critical
numbers are in the interval
a. (-infinity , + infinity)
b. (-infinity , -3] U [3 ,
+infinity)
c. (0 , + infinty)
d. None of the above
395. If f(x) = x3 -3x2 + x and g is the
inverse of f, then g '(3) is equal to
a.
b.
c.
d.
10
1 / 10
1
None of the above
396. If functions f and g are such that
f(x) = g(x) + k where k is a constant,
then
a.
b.
c.
d.
f '(x) = g '(x) + k
f '(x) = g '(x)
Both (A) and (B)
None of the above
397. If f(x) = g(u) and u = u(x) then
a.
b.
c.
d.
f '(x) = g '(u)
f '(x) = g '(u) . u '(x)
f '(x) = u '(x)
None of the above
398. lim [ex -1] / x as x approaches 0 is
equal to
a. 1
b. 0
c. is of the form 0 / 0 and
cannot be calculated.
d. None of the above
399. If f(x) is a differentiable function
such that f '(0) = 2, f '(2) = -3 and f
'(5) = 7 then the limit lim [f(x) - f(4)]
/ (x - 4) as x approaches 4 is equal to
a.
b.
c.
d.
2
-3
7
4
400. If f(x) and g(x) are differentiable
functions such that f '(x) = 3x and
g'(x) = 2x2 then the limit lim [(f(x) +
g(x)) - (f(1) + g(1))] / (x - 1) as x
approaches 1 is equal to
a.
b.
c.
d.
5
10
20
15
Multiple Choice
Question
In
Engineering Mathematics
By JAS Tordillo
1. A man sold a book by mistake at 120% of
the marked price instead of discounting the
marked price by 20%. If he sold the book for
P14.40, what was the price for which he
have sold the book?
a) P8.00
b) P8.50
c) P9.00
d) P9.60
2. In how many ways can 9 books be
arranged on a shelf so that 5 of the books are
always together?
a) 30,200
b) 25,400
c) 15,500
d) 14,400
3. If one third of the air tank is removed by
each stroke of an air pump, what fractional
part of the total air is removed in 6 strokes?
a) 0.7122
b) 0.6122
c) 0.8122
d) 0.9122
4. If 3^x = 9^y and 27^y = 81^z, find x/z?
a) 3/5
b) 4/3
c) 3/8
d) 8/3
5. Determine x, so that x, 2x+7, 10x-7 will
be geometric progression.
a) 7,-5/6
b) 7, -14/5
c) 7, -7/12
d) 7, -7/6
6. A man invested part of P20,000 at 18%
and the rest at 16%. The annual income
from 16% investment was P620 less than
three times the annual income from 18%
investment. How much did he invest at
18%?
a) P5,457.20
b) P6,457.20
c) P7,457.20
d) P8,457.20
7. The sum of four positive integers is 32.
Find the greatest possible product of these
four numbers.
a) 5013
b) 645
c) 4069
d) 4913
8. A piece of paper is 0.05 in thick. Each
time the paper is folded into half, the
thickness is doubled. If the paper was folded
12 times, how much thick in feet the folded
paper be?
a) 10.1 ft
b) 12.1 ft
c) 15.1 ft
d) 17.1 ft
9. A seating section in a certain athletic
stadium has 30 seats in the first row, 32
seats in the second row, 34 seats in the third
row, and so on, until the tenth row is
reached, after which there are ten rows each
containing 50 seats. Find the total number of
seats in the section.
a) 1200
b) 980
c) 890
d) 750
10. One pipe can fill a tank in 6 hours and
another pipe can fill the same tank in 3
hours. A drain pipe can empty the tank in 24
hours. With all three pipes open, how long
will it take to fill in the tank?
a) 5.18 hours
b) 4.18 hours
c) 3.18 hours
d) 2.18 hours
11. The ten’s digit of a certain two digit
number exceeds the unit’s digit by four and
is one less than twice the unit’s digit. Find
the number.
a) 65
b) 75
c) 85
d) 95
12. The sum of two numbers is 35 and their
product is 15. Find the sum of there
reciprocal.
a) 2/7
b) 7/3
c) 2/3
d) 5/2
13. The smallest natural number for which 2
natural numbers are factors.
a) Least common divisor
b) Least common denominator
c) Least common factor
d) Least common multiple
14. Ana is 5 years older than Beth. In 5
years, the product of their ages is 1.5 times
the product of their present ages. How old is
Beth now?
a) 30
b) 25
c) 20
d) 15
15. The time required for the examinees to
solve the same problem differ by two
minutes. Together they can solve 32
problems in one hour. How long will it take
for the slower problem solver to solve a
problem?
a) 2 minutes
b) 3 minutes
c) 4 minutes
d) 5 minutes
16. Find the value of m that will make 4x^2
– 4mx + 4m ) 5 a perfect square trinomial.
a) 3
b) -2
c) 4
d) 5
17. How many liters of water must be added
to 35 liters of 89% hydrochloric acid
solution to reduce its strength to 75%?
a) 3.53
b) 4.53
c) 5.53
d) 6.53
18. A purse contains $11.65 in quarters and
dimes. If the total number of coins is 70,
find how many dimes are there.
a) 31
b) 35
c) 39
d) 42
19. Equations relating x and y that cannot
readily be solved explicitly for y as a
function of x or for x as a function of y.
Such equations may nonetheless determine y
as a function of x or vice versa, such
function called _________.
a) logarithmic function
b) implicit function
c) explicit function
d) continuous function
20. A piece of wire of length 50 m is cut into
two parts. Each part is then bent to form a
square. It is found that the total area of the
square is 100 sq. m. Find the difference in
length of the two squares.
a) 6.62
b) 7.62
c) 8.62
d) 9.62
21. A tank is filled with an intake pipe that
fills it in 2 hours and an outlet pipe that
empty in 6 hours. If both pipes are left open,
how long will it take to fill in the empty
tank?
a) 1.5 hrs
b) 2.0 hrs
c) 2.8 hrs
d) 3 hrs
22. Maria sold a drafting pen for P612 at a
loss of 25% on her buying price. Find the
corresponding loss or gain in percent if she
had sold it for P635?
a) 20.18%
b) 11.18%
c) 22.18%
d) 28.18%
23. Divide 1/8 by 8.
a) 1/64
b) 18
c) 1
d) 64
24. Given 2 x 2 matrix [
], find its
determinant.
a) 31
b) 44
c) -20
d) 20
25. If the sum is 220 and the first term is 10,
find the common difference if the last term
is 30.
a) 2
b) 5
c) 3
d) 2/3
26. Find the sum of the sequence 25, 30,
35, .....
a) (2/5)(n^2 + 9n)
b) (5/2)(n^2 + 9n)
c) (9/2)(n^2 + 9n)
d) (9/2)(n^2 – 9n)
27. Solve for x: √
.
a) 4, -5
b) -4, -5
c) -4, 5
d) no solution
28. Solve for x: 10x^2 + 10x + 1 =0.
a) -0.113, -0.887
b) -0.331, -0.788
c) -0.113, -0.788
d) -0.311, -0.887
29. The number x, 2x + 7, 10x – 7 form a
Geometric Progression. Find the value of x.
a) 5
b) 6
c) 7
d) 8
30. Find the 30th term of A.P. 4,7,10,...
a) 91
b) 90
c) 88
d) 75
31. Find the sum of the first 10 terms of the
geometric progression 2, 4, 8, 16,...
a) 1023
b) 2046
c) 225
d) 1596
32. Find the sum of the infinite geometric
progression 6, -2, 2/3,...
a) 9/2
b) 5/2
c) 11/2
d) 7/2
33. Find the ratio of an infinite geometric
series if the sum is 2 and the first term is ½.
a) 1/3
b)1/2
c) 3/4
d) 1/4
34. Find the 1987th digit in the decimal
equivalent to 1785/9999 starting from the
decimal point.
a) 8
b) 1
c) 7
d) 5
35. What is the lowest common factor of 10
and 32.
a) 320
b) 2
c) 180
d) 90
36. Ten less than four times a certain
number is 14. Determine the number.
a) 6
b) 7
c) 8
d) 9
37. Jolo bought a second hand betamax
VCR and sold it to Rudy at a profit of 40%.
Rudy then sold the VCR to Noel at a profit
of 20%. If Noel paid P2856 more than it cost
to Jolo, how much did Jolo paid the unit?
a) P4000
b) 4100
c) 4200
d) P4300
38. A club of 40 executives, 33 likes to
smoke Malboro, and 20 likes to smoke
Philip Morris. How many like both?
a) 13
b) 10
c) 11
d) 12
39. A merchant has three items on sale,
namely a radio for P50, a clock for P30 and
a flashlight for P1.00. At the end of the day,
he has sold a total of 100 of the three items
and has taken exactly P1000 on the total
sales. How many radios did he sale?
a) 16
b) 20
c) 18
d) 24
40. What is the sum of the coefficients of the
expansion of (2x – 1)^20?
a) 0
b) 1
c) 2
d) 3
41. Find the ratio of the infinite geometric
series if the sum is 2 and the first term is 1/2.
a) 1/3
b) 1/2
c) 3/4
d) 1/4
42. A stack of bricks has 61 bricks in the
bottom layer, 58 bricks in the second layer,
55 bricks in the third layer and sol until
there are 10 bricks in the last layer. How
many bricks are there together?
a) 638
b) 637
c) 640
d) 639
43. Once a month a man put some money
into the cookie jar. Each month he put 50
centavos more into the jar than the month
before. After 12 years he counted his
money; he had P5436. How much did he put
in the jar in the last month?
a) 73.5
b) P75.50
c) P74.50
d) P72.50
44. The seventh term is 56 and the 12th term
is -1792 of the geometric progression. Find
the ratio and the first term. Assume the
ratios are equal.
a) -2, 7/8
b) -1. 5/8
c) -1, 7/8
d) -2, 5/8
45. Find the value of x in the equation 24x^2
+ 5x -1 = 0.
a) (1/6, 1)
b) (1/6, 1/5)
c) (1/2, 1/5)
d) (1/8, -1/3)
46. The polynomial x^3 + 4x^2 -3x +8 is
divided by x – 5, then the remainder is:
a) 175
b) 140
c) 218
d) 200
47. Find the rational number equivalent to
repeating decimal 2.3524242424...
a) 23273/9900
b) 23261/990
c) 23289/9900
d) 23264/9900
48. The sum of Kim’s and Kevin’s ages is
18. In three years, Kim will be twice as old
as Kevin. What are their ages now?
a) 4, 14
b) 5, 13
c) 7, 11
d) 6, 12
49. Ten liters of 25% salt solution and
15%liters of 35% solution are poured into a
drum originally containing 30 liters of 10%
salt solution. What is the percent
concentration in the mixture?
a) 19.55%
b) 22.15%
c) 27.05
d) 26.72%
50. Determine the sum of the infinite series:
S = 1/3 + 1/9 + 1/27 + .... (1/3)^n.
a) 4/5
b) 3/4
c) 2/3
d) 1/2
51. Determine the sum of the positive valued
solution to the simultaneous equations: xy =
15, yz = 35, zx = 21.
a) 15
b) 13
c) 17
d) 19
52. The areas of two squares differ by 7 sq.
ft. and their perimeters differ by 4 ft.
Determine the sum of their areas.
a) 25 ft^2
b) 27 ft^2
c) 28 ft^2
d) 22 ft^2
53. A bookstore purchased a bestselling
book at P200 per copy. At what price should
this book be sold so that, giving a 20%
discount, the profit is 30%?
a) P450
b) P500
c) P375
d) P400
54. In a certain community of 1,200 people,
60% are literate. Of the males, 50% are
literate and of the females 70% are literate.
What is the female population?
a) 850
b) 500
c) 550
d) 600
55. Gravity causes a body to fall 16.1 ft. in
the 1st second, 48.3 ft. in the 2nd second,
80.5 ft. in the 3rd second, and so on. How
far did the body fall during the 10th second?
a) 248.7 ft
b) 308.1 ft
c) 241.5 ft
d) 305.9 ft
56. In a commercial survey involving 1,000
persons on brand reference, 120 were found
to prefer brand x only, 200 prefer brand y
only, 150 prefer brand z only. 370 prefer
either x or y but not z, 450 prefer brand y or
z but not x, and 420 prefer either brand z or
x but not y. How many persons have no
brand preference, satisfied with any of the 3
brands?
a) 280
b) 230
c) 180
d) 130
57. The electric power which a transmission
line can transmit is proportional to the total
product of its design voltage and current
capacity, and inversely to the transmission
distance. A 115 kilovolt line rated at 1000
amperes can transmit 150 Megawatts over
150 km. How much power, in Megawatts,
can a 230 kilovolt line rated 1500 amperes
transmit over 100km?
a) 785
b) 485
c) 675
d) 595
58. Find the geometric mean of 64 and 4.
a) 16
b) 34
c) 32
d) 28
59) Factor the expression x^2 + 6x + 8 as
completely as possible.
a) (x + 8)(x – 2)
b) (x + 4)(x – 2)
c) (x + 4)(x + 2)
d) (x – 4)(x – 2)
60. A batch of concrete consisted of 200 lbs.
Fine aggregate, 350 lbs coarse aggregate, 94
lbs cement, and 5 gallons water. The
specific gravity of the sand and gravel may
be taken as 2.65 and that of the cement as
3.10. What was the weight of concrete in
place per cubic foot?
a) 172 lb
b) 236 lb
c) 162 lb
d) 153 lb
61. Dalisay’s Corporation gross margin is
45% sales. Operating expenses such as sales
and administration are 15% of sales. Dalisay
is in 40% tax bracket. What percent of sales
is their profit after taxes?
a) 18%
b) 5%
c) 24%
d) 50%
62. A and B working together can finish
painting a home in 6 days. A working alone,
can finish it in five days less than B. How
long will it take each of them to finish the
work alone?
a) 10, 15
b) 15, 20
c) 20, 25
d) 5, 10
63. Determine the sum of the progression if
there are 7 arithmetic mean between 3 and
35.
a) 171
b) 182
c) 232
d) 216
64. Find the sum of 1, -1/5, 1/25,...
a) 5/6
b) 2/3
c) 0.84
d) 0.72
65. Find the remainder if we divide 4y^3 +
18y^2 + 8y -4 by (2y + 3).
a) 10
b) 11
c) 15
d) 13
66. What time after 3 o’clock will the hands
of the clock be together for the first time?
a) 3:16.36
b) 3:14.32
c) 3:12.30
d) 3:13.37
67. The difference of the squares of the
digits of a two digit positive number is 27. If
the digits are reversed in order and the
resulting number subtracted from the
original number, the difference is also 27.
What is the original number?
a) 63
b) 54
c) 48
d) 73
68. The boat travels downstream in 2/3 of
the time as it does going upstream. If the
velocity of the river current is 8 kph,
determine the velocity of the boat in still
water.
a) 40 kph
b) 50 kph
c) 30 kph
d) 60 kph
69. Given that w varies directly as the
product of x and y and inversely as the
square of z, and that w = 4, when x = 2, y =
6, and z = 3. Find the value of ―w‖ when x =
1, y = 4, and z = 2.
a) 2
b) 3
c) 4
d) 5
70. The third term of a harmonic progression
is 15 and 9th term is 6. Find the eleventh
term?
a) 4
b) 5
c) 6
d) 7
71. Solve for x for the given equation, 7.4 x
10^-4 = e^-9.7x.
a) 0.7621
b) 0.7432
c) 0.7243
d) 0.7331
72. Find the 10th term of the geometric
progression: 3, 6, 12, 24,....
a) 1536
b) 1653
c) 1635
d) 3156
73. Find the sum of odd integers from 1 to
31.
a) 256
b) 526
c) 265
d) 625
74. Box A has 4 white balls, 3 blue balls,
and 3 orange balls. Box B has 2 white balls,
4 blue balls, and 4 orange balls. If one ball is
drawn from each box, what is the probability
that one of the two balls will be orange?
a) 27/50
b) 9/50
c) 23/50
d) 7/25
75. Solve: x^2 + y^2 = 5z and x^2 – y^2 =
3z. How many and what numerical values
for x, y, and z will satisfy these
simultaneous equations?
a) if z = 3^2, then x = 6 and y = 3
b) if z = 2^2, then x =4 and y =2
c) if z = 1^2, then x =2 and y = 1
d) There are an infinite no. of values that
will satisfy
76. Two people driving towards each other
between two towns 160 km apart. The first
man drives at the rate of 45 kph and the
other drives at 35 kph. From their starting
point, how long would it take that they
would meet?
a) 3 hr
b) 4 hr
c) 2 hr
d) 1 hr
77. Solve x for the equation 6x – 4 = 2x + 6.
a) 10
b) 5/2
c) 5
d) 2.5
78. The man has a total of 33 goats and
chickens. If the total of their feet is 900, find
the number of goats and chickens.
a) 12 goats and 21 chickens
b) 9 goats and 27 chickens
c) 6 cats and 5 dogs
d) 13 goats and 20 chickens
79. Express 5y – [3x – (5y + 4)] into
polynomial.
a) 10y – 3x +4
b) 5y + 5x – 4
c) 5y + 5x + 4
d) 5y – 5x +4
80. What is the exponential form of the
complex number 3 + 4i?
a) e^i53.1°
b) 5e^i53.1°
c) 5e^i126.9°
d) 7e^i53.1°
81. Simplify the complex numbers: (3 + 4i)
– (7 – 2i)
a) -4 + 6i
b) 10 + 2i
c) 4 – 2i
d) 5 – 4i
82. Solve for x: x^2 + x -12 = 0
a) x = 6, x = -2
b) x = 1, x = 12
c) x = 3, x = -4
d) x = 4, x = -3
83. √
√ =
a) 0
b) √
c) √
d) 10
84. What us the value of x in the expression:
x – 1/x = 0?
a) x = -1
b) x = 1, 1/2
c) x = 1
d) x = 1, -1
85. What is the value of A: A^-6/8 = 0.001?
a) 10
b) 100
c) 0
d) 10000
86. Find the value of x: ax – b = cx + d
a) x = (a – b)/(c + d)
b) x = (b + d)/(a – c)
c) x = (a – d)/(c – b)
d) x = (c + d)/(a – c)
87. Divide: 15x^4 +6x^3 + 15x + 6 by 3x^3
+ 3.
a) 5x + 2
b) 5x^2 + 2
c) 5x^2
d) 5x – 4
88. Simplify: √
√
a)
√
b) √
c) √
d) √
89. Find the value of x in the equation: csc x
+ cot x = 3
a) π/5
b) π/4
c) π/3
d) π/2
90. If A is in the III quadrant and cos A = 15/17, find the value of cos (1/2)A.
a) –(8/17)^1/2
b) –(5/17)^1/2
c) –(3/17)^1/2
d) –(1/17)^1/2
91. Simplify the expression: (sin B + cos B
tan B)/cos B
a) 2 tan B
b) tan B + tan B
c) tan B cos B
d) 2 sin B cos B
92. If cot 2A cot 68° = 1, then tan A is equal
to ________.
a) 0.194
b) 0.419
c) 0.491
d) 0.914
93. A ladder 5 m long leans against the wall
of an apartment house forming an angle of
50 degrees, 32 minutes with ground. How
high up the wall does it reach?
a) 12.7 m
b) 10.5 m
c) 3.86 m
d) 1.55 m
94. The measure of 2.25 revolutions
counterclockwise is:
a) -810 deg
b) -805 deg
c) 810 deg
d) 805 deg
95. If sin A = 2.5 x and cos A = 5.5x, find
the value of A in degrees.
a) 14.5 deg
b) 24.5 deg
c) 34.5 deg
d) 44.5 deg
96. Solve angle A of an oblique triangle wit
vertices ABC, if a = 25, b = 16 and C = 94
degrees and 6 minutes.
a) 50 deg and 40 min
b) 45 deg and 35 min
c) 55 deg and 32 min
d) 54 deg and 30 min
97. Given: x = (cos B tan B – sin B)/cos B.
Solve for x if B = 30 degrees.
a) 0.577
b) 0
c) 0.500
d) 0.866
98. (cos A)^4 – (sin A)^4 is equal to
_________.
a) cos 2A
b) sin 2A
c) 2tan A
d) sec A
99. 174 degrees is equivalent to _________
mils.
a) 3094
b) 2084
c) 3421
d) 2800
100. What is the resultant of a displacement
6 miles North and 9 miles East?
a) 11 miles, N 56° E
b) 11 miles, N 54° E
c) 10 miles, N 56° E
d) 10 miles, N 54° E
101. Which is identically equal to (sec A +
tan A)?
a) 1/(sec A + tan A)
b) csc A – 1
c) 2/(1 – tan A)
d) csc A + 1
102. Determine the simplified form of (cos
2A – cos A)/(sin A).
a) cos 2A
b) –sin A
c) cos A
d) sin 2A
103. Ifsec 2A = 1/sin 13A, determine the
angle A in degrees.
a) 5 deg
b) 6 deg
c) 3 deg
d) 7 deg
104. Solve for x in the equation: arctan (x +
1) + arctan (x – 1) = arctan (12).
a) 1.50
b) 1.34
c) 1.20
d) 1.25
105. Solve for x if tan 3x = 5tan x.
a) 20.705 deg
b) 30.705 deg
c) 15.705 deg
d) 35.705 deg
106. If sin A = 2.511x, cos A = 3.06x and
sin 2A = 3.939x, find the value of x.
a) 0.265
b) 0.256
c) 0.562
d) 0.625
107. The angle of inclination of ascend of a
road having 8.25% grade is ______.
a) 4.72
b) 4.27
c) 5.12
d) 1.86
108. A man finds the angle of elevation of
the top of a tower to be 30 degrees. He
walks 85 m nearer the tower and finds its
angle of elevation to be 60 degrees. What is
the height of the tower?
a) 76. 31 m
b) 73.31 m
c) 73.16 m
d) 73.61 m
109. If the sides of a parallelogram and an
included angle are 6, 10, and 100 degrees
respectively, find the length of the shorter
diagonal.
a) 10.63
b) 10.37
c) 10.73
d) 10.23
110. What is the value of log2 5 + log3 5?
a) 7.39
b) 3.79
c) 3.97
d) 9.37
111. Points A and B 1000 m apart are
plotted on a straight highway running east
and west. From A, the bearing of a tower C
is 32 degrees W of N and from B the bearing
of C is 26 degrees N of E. Approximate the
shortest distance of tower C to the highway.
a) 364 m
b) 374 m
c) 394 m
d) 384 m
112. If log of 2 to base 2 plus log of x to the
base of 2 is equal to 2, then the value of x is:
a) 4
b) -2
c) 2
d) -1
113. Arctan [2cos (arcsin √ /2)] is equal to:
a) π/3
b) π/4
c) π/6
d) π/2
114. Solve A for the given equations cos^2
A = 1 – cos^2 A.
a) 45, 125, 225, 335 degrees
b) 45, 125, 225, 315 degrees
c) 45, 135, 115, 315 degrees
d) 45, 150, 220, 315 degrees
115. If sin A = 2/5, what is the value of 1 –
cos A?
a) 0.083
b) 0.916
c) 0.400
d) 0.614
116. Sin A cos B – cos A sin B is equivalent
to:
a) cos (A – B)
b) sin (A – B)
c) tan (A – B)
d) cos (A –B)
117. How many degrees is 4800 mils?
a) 270 deg
b) 90 deg
c) 180 deg
d) 215 deg
118. ln 7.18^xy equals
a) 1.97xy
b) 0.86xy
c) xy
d) 7.18xy
119. The log10 (8)(6) equal to:
a) log10 8 + log10 6
b) log10 8 - log10 6
c) log10 8 log10 6
d) log10 8 / log10 6
120. 38.5 to the x power = 6.5 to the x – 2
power, solve for x using logarithms.
a) 2.70
b) -2.10
c) 2.10
d) -2.02
121. Given the triangle ABC in which A =
30°30’, b = 100 m and c = 200 m. Find the
length of the side a.
a) 124.64 m
b) 142.24 m
c) 130.5 m
d) 103.00 m
122. An observer wishes to determine the
height of the tower. He takes sight at the top
of the tower from A and B, which are 50 ft
apart at the same elevation on a direct line
with the tower. The vertical angle at point A
is 30 deg and at point B is 40 deg. What is
the height of the tower?
a) 85.60 ft
b) 110.29
c) 143.97
d) 92.54 ft
123. What is the value of log to the base of
1000^3.3?
a) 9.9
b) 99.9
c) 10.9
d) 9.5
124. In a triangle, find the side c if angle C =
100 deg, side b = 20, and side a = 15.
a) 28
b) 29
c) 27
d) 26
125. Given a triangle with an angle C = 28.7
deg, side a = 132 units and side b = 224
units. Solve for the side c.
a) 95 units
b) 110 units
c) 125.4 units
d) 90 units
126. A PLDT tower and a monument stand
on a level plane. The angles of depression of
the top and bottom of the monument viewed
from the top of the PLDT tower are 13 deg
and 35 deg respectively. The height of the
tower is 50 m. Find the height of the
monument.
a) 33.51 m
b) 47.3 m
c) 7.48 m
d) 30.57 m
127. Find the value of x if log12 x = 2.
a) 144
b) 414
c) 524
d) 425
128. If tan x = 1/2, tan y = 1/3. What is the
value of tan (x + y)?
a) 1
b) 2
c) 3
d) 4
129. The logarithm of the quotient M/N and
the logarithm of the product MN is equal to
1.55630251 and 0.352182518 respectively.
Find the value of M.
a) 6
b) 7
c) 8
d) 9
130. The angle of elevation of the top tower
B from the top of the tower A is 28 deg and
the angle of elevation of the top tower A
from the base of the tower B is 46 deg. The
two towers lie in the same horizontal plane.
If the height of the tower B is 120 m, find
the height of tower A.
a) 87.2 m
b) 90.7 m
c) 79.3 m
d) 66.3 m
131. Evaluate the log6 845 = x.
a) 3.76
b) 5.84
c) 4.48
d) 2.98
132. Find the value of log8 48.
a) 1.86
b) 6.81
c) 8.61
d) 1.68
133. Find the value of sin 920 deg.
a) 0.243
b) -0.243
c) 0.342
d) -0.342
134. Log (x)^n =
a) log x
b) n log x
c) 1/n log x
d) n
135. Sin 2θ is equal to:
a) 2 sin θ cos θ
b) 1/2 sin θ
c) sin θ cos θ
d) 1 – sin^2 θ
136. What is the interior angle (in radian) of
an octagon?
a) 2.26 rad
b) 2.36 rad
c) 2.8 rad
d) 2.75 rad
137. The trigonometric function (1 + tan^2
θ) is also equal to:
a) sec^2 θ
b) cos^2 θ
c) csc^2 θ
d) sin θ
138. Derive the formula of each interior
angle (in degrees).
a) (no. of sides – 2)180
b) [(no. of sides – 2)180/no. of sides]
c) [(no. of sides – 1)180/no. of sides]
d) [no. of sides – 2]/180
139. What is the Cartesian logarithm of
402.9?
a) 2.605
b) 2.066
c) 3.05
d) 3.60
140. What is the value of the following
[
]
limit?
a) 3
b) 6
c) 9
d) 0
141. Given the three sides of a triangle: 2, 3,
4. What is the angle in radians opposite the
side with length 3?
a) 0.11
b) 0.41
c) 0.55
d) 0.81
142. Find the area of the geometric figure
whose vertices are at (3, 0, 0), (3, 3, 0), (0, 0,
4) and (0, 3, 4).
a) 12 sq. units
b) 14 sq. units
c) 15 sq. units
d) 24 sq. units
143. A central angle of 45 degrees subtends
an arc of 12 cm. What is the radius of the
circle?
a) 15.28 cm
b) 18.28 cm
c) 20.28 cm
d) 30.28 cm
144. It is a part of circle bounded by a chord
and an arc.
a) slab
b) segment
c) section
d) sector
145. What is the area (in sq. inches) of a
parabola with a base of 15 cm and a height
of 20 cm?
a) 87
b) 55
c) 31
d) 11
146. Triangle ABC is a right triangle with
right angle at C. CD is perpendicular to AB.
BC = 4 and CD = 1. Find the area of the
triangle ABC.
a) 2.95
b) 2.55
c) 2.07
d) 1.58
147. The tangent and a secant are drawn to a
circle from the same external point. If the
tangent is 6 inches and the external segment
of the secant is 3 inches, the length of the
secant is ________ inches.
a) 15
b) 14
c) 13
d) 12
148. If a regular polygon has 27 diagonals,
then it is a,
a) nonagon
b) pentagon
c) hexagon
d) heptagon
149. A regular dodecagon is inscribed in a
circle of radius 24. Find the perimeter of the
dodecagon.
a) 125
b) 135
c) 149
d) 169
150. An annulus is a plane figure, which is
composed of two concentric circles. The
area of the annulus can be calculated by
getting the difference between the area of
the larger circle and the area of the smaller
circle. Also, it can be calculated by
removing the hole. The method is called:
a) Law of Extremities
b) Law of Reduction
c) Law of Deduction
d) Sharp Theorem
151. The sides of a triangle are 195, 157,
and 210 respectively. What is the area of the
triangle?
a) 73250 sq. units
b) 14586 sq. units
c) 10250 sq. units
d) 11260 sq. units
152. Given a triangle of sides 10 cm and 15
cm an included angle of 60 degrees. Find the
area of the triangle.
a) 70
b) 80
c) 72
d) 65
153. The sides of a triangle are 8 cm, 10 cm,
and 14 cm. Determine the radius of the
inscribed and circumscribed circle.
a) 3.45, 7.14
b) 2.45, 7.14
c) 2.45, 8.14
d) 3.45, 8.14
154. The sides of a cyclic quadrilateral are a
= 3m, b = 3m, c = 4m and d = 4m. Find the
radius of the inscribed and circumscribed
circle.
a) 1.71, 2.50
b) 1.91, 2.52
c) 2.63, 4.18
d) 2.63, 3.88
155. From the point inside a square the
distance to three corners are 4, 5 and 6 m
respectively. Find the length of the sides of a
square.
a) 7.53
b) 8.91
c) 6.45
d) 9.31
156. A regular pentagon has sides 20 cm. An
inner pentagon with sides of 10 cm is inside
and concentric to the larger pentagon.
Determine the area inside and concentric to
the larger pentagon but outside of the
smaller pentagon.
a) 430.70 cm^2
b) 573.26 cm^2
c) 473.77 cm^2
d) 516.14 cm^2
157. A rhombus has diagonals of 32 and 20
inches. Determine its area.
a) 360 in^2
b) 280 in^2
c) 320 in^2
d) 400 in^2
158. In a circle with a diameter of 10 m, a
regular five pointed star touching its
circumference is inscribed. What is the area
of the part not covered by the star?
a) 60.2 m^2
b) 50.48 m^2
c) 45.24 m^2
d) 71.28^m
159. Find the area of a regular octagon
inscribed in a circle of radius 10 cm.
a) 186.48 cm^2
b) 148.91 cm^2
c) 282.24 cm^2
d) 166.24 cm^2
160. Find the area of a regular pentagon
whose side is 25 m and apothem is 17.2 m.
a) 846 m^2
b) 1090 m^2
c) 1075 m^2
d) 988 m^2
161. The area of a circle circumscribing a
hexagon is 144π m^2. Find the area of the
hexagon.
a) 374.12 m^2
b) 275.36 m^2
c) 415.26 m^2
d) 225.22 m^2
162. Determine the area of a regular 6-star
polygon if the inner regular hexagon has 10
cm sides.
a) 441.66 cm^2
b) 467.64 cm^2
c) 519.60 cm^2
d) 493.62 cm^2
163. Find each interior angle of a hexagon.
a) 90 deg
b) 120 deg
c) 150 deg
d) 180 deg
164. Find the length of the side of pentagon
if the line perpendicular to its side is 12
units from the center.
a) 8.71
b) 17.44
c) 36.93
d) 18.47
165. How many sides are in a polygon if
each interior angle is 165 degrees.
a) 12 sides
b) 24 sides
c) 20 sides
d) 48 sides
166. Find the area of triangle whose sides
are: 25, 39 and 40.
a) 468
b) 684
c) 486
d) 864
167. Find the area of a regular hexagon
inscribed in a circle of radius 1.
a) 2.698
b) 2.598
c) 3.698
d) 3.598
168. A goat is tied to a corner of a 30 ft by
35 ft building. If the rope is 40 ft long and
the goat can reach 1 ft farther than the rope
length. What is the maximum area the goat
can cover.
a) 4840
b) 4804
c) 8044
d) 4084
169. In triangle BCD, BC = 25 m, and CD =
10 m. The perimeter of the triangle maybe:
a) 79 m
b) 70 m
c) 71 m
d) 72 m
170. A quadrilateral have sides equal to 12
m, 20 m, 8 m and 16.97 m respectively. If
the sum of the two opposite angles is equal
to 225, find the area of the quadrilateral.
a) 168
b) 100
c) 124
d) 158
171. The area of a circle inscribed in a
hexagon is 144π m^2. Find the area of the
hexagon.
a) 498.83 m^2
b) 489.83 m^2
c) 439.88 m^2
d) 349.88 m^2
172. Each angle of the regular dodecagon is
equal to _________ degrees.
a) 135
b) 150
c) 125
d) 105
173. If an equilateral triangle is circumscribe
about a circle of radius 10 cm, determine the
side of the triangle.
a) 34.64 cm
b) 64.12 cm
c) 36.44 cm
d) 32.10 cm
174. The angle of a sector is 30 degrees and
the radius is 15 cm. What is the area of the
sector.
a) 59.8 cm^2
b) 58.9 cm^2
c) 89.5 cm^2
d) 85.9 cm^2
175. The distance between the center of the
three circles which are mutually tangent to
each other externally are 10, 12 and 14 units.
Find the area of the largest circle.
a) 72π
b) 64π
c) 23 π
d) 16 π
176. Two triangles have equal bases. The
altitude of one triangle is 3 units more than
its base and the altitude of the other is 3
units less than its base. Find the altitude, if
the areas of the triangles differ by 21 square
units.
a) 6 & 12
b) 5 &11
c) 3 & 9
d) 4 & 10
177. If the sides of a parallelogram and an
included angle are 6, 10 and 100 degreess
respectively, find the length of the shorter
diagonal.
a) 10.63
b) 10.73
c) 10.23
d) 10.37
178. In triangle ABC, angle C = 34 degrees,
side a = 29 cm, b = 40 cm. Solve the area of
the triangle.
a) 324 cm^2
b) 342 cm^2
c) 448 cm^2
d) 484 cm^2
179. An oblique equilateral parallelogram.
a) square
b) rectangle
c) rhombus
d) recession
180. What is the interior angle (in radian) of
an octagon
a) 2.26 rad
b) 2.36 rad
c) 2.8 rad
d) 2.75 rad
181. The circumference of a great circle of a
sphere is 18π. Find the volume of the sphere.
a) 3053.6
b) 4053.6
c) 5053.6
d) 6053.6
182. A pyramid whose altitude is 5 ft weighs
800 lbs. At what distance from its vertex
must it be cut by a plane parallel to its base
so that the two solids of equal weight will be
formed?
a) 3.97 ft
b) 2.87 ft
c) 4.97 ft
d) 5.97 ft
183. Find the increase in volume of a
spherical balloon when its radius is
increased from 2 to 3 inches.
a) 75. 99 cu. in.
b) 74.59 cu. in.
c) 74.12 cu. in.
d) 79.59 cu. in.
184. If the lateral area of a right cylinder is
88 and its volume is 220, find its radius.
a) 2 cm
b) 3 cm
c) 4 cm
d) 5 cm
185. It is desired that the volume of the
sphere be tripled. By how many times will
the radius be increased?
a) 2^1/2
b) 3^1/3
c) 3^1/2
d) 3^3
186. A cone and a cylinder have the same
height and the same volume. Find the ratio
of the radius of the cone to the radius of the
cylinder.
a) 0.577
b) 0.866
c) 1.732
d) 2.222
187. Compute the surface area of the cone
having a slant height of 5 cm and a diameter
of 6 cm.
a) 47.12 cm^2
b) 25.64 cm^2
c) 38.86 cm^2
d) 30.24 cm^2
188. The ratio of the volume of the lateral
area of a right circular cone is 2:1. If the
altitude is 15 cm, what is the ratio of the
slant height to the radius?
a) 5:2
b) 5:3
c) 4:3
d) 4:2
189. A conical vessel has a height of 24 cm
and a base diameter of 12 cm. It holds water
to a depth of 18 cm above its vertex. Find
the volume of its contents in cubic
centimeter.
a) 387.4
b) 381.7
c) 383.5
d) 385.2
190. A circular cylinder is circumscribed
about a right prism having a square base one
meter on an edge. The volume of the
cylinder is 6.283 m^3. Find its altitude in m.
a) 4.5
b) 5.5
c) 4
d) 5
191. The volume of water in a spherical tank
having diameter of 4 m is 5.236 m^3.
Determine the depth of the water in the tank.
a) 1.6
b) 1.4
c) 1.2
d) 1.0
192. The corners of a cubical block touched
the closest spherical shell that encloses it.
The volume of the box is 2744 cm^3. What
volume in cm^3 inside the shell is not
occupied by the block?
a) 4713.56
b) 3360.14
c) 4133.25
d) 5346.42
193. A circular cone having an altitude of 9
m is divided into 2 segments having the
same vertex. If the smaller altitude is 6m,
find the ratio of the volume of the small
cone to the big cone.
a) 0.296
b) 0.396
c) 0.186
d) 0.486
194. A frustum of a regular pyramid has an
upper base of 8 m x 80 m and a lower base
of 10 m x 100 m and an altitude of 5 m. Find
the volume of the pyramid.
a) 4066.67 m^3
b) 5066.67 m^3
c) 6066.67 m^3
d) 7066.67 m^3
195. The bases of a right prism is a hexagon
with one each side equal to 6 cm. The bases
are 12 cm apart. What is the volume of a
right prism?
a) 1211.6 cm^3
b) 2211.7 cm^3
c) 1212.5 cm^3
d) 1122.4 cm^3
196. The volume of the water in hemisphere
having a radius of 2 m is 2.05 m^3. Find the
height of the water.
a) 0.602
b) 0.498
c) 0.782
d) 0.865
197. Find the volume of a cone to be
constructed from a sector having a diameter
of 72 cm and a central angle of 150 deg.
a) 7711.82 cm^3
b) 6622.44 cm^3
c) 5533.32 cm^3
d) 8866.44 cm^3
198. A cubical container that measures 2 in
on a side is tightly packed with marbles and
is filled with water. All the 8 marbles are in
contact with the walls of the container and
the adjacent marbles are the same size. What
is the volume of water in the container?
a) 0.38 in^3
b) 2.5 in^3
c) 3.8 in^3
d) 4.2 in^3
199. If one edge of a cube measures12 cm,
calculate for the surface area of the cube and
the volume of the cube.
a) 864 cm^2; 1728 cm^3
b) 468 cm^2; 1728 cm^3
c) 863 cm^2; 8721 cm^3
d) 468 cm^2; 8721 cm^3
200. A pyramid with a square base has an
altitude of 25 cm. If the edge of the base is
15 cm. Calculate the volume of the pyramid.
a) 1785 cm^3
b) 1875 cm^3
c) 5178 cm^3
d) 5871 cm^3
201. If a right cone has a base radius of 35
cm and an altitude of 45 cm. Solve for the
total surface area and the volume of the cone.
a) 10,116.89 cm^2 and 57,726.76 cm^3
b) 9,116.89 cm^2 and 57,726.76 cm^3
c) 10,116.89 cm^2 and 67,726.76 cm^3
d) 9,116.89 cm^2 and 67,726.76 cm^3
202. If the volume of a sphere is 345 cm^3.
Solve for its diameter.
a) 8.70 cm
b) 7.70 cm
c) 6.70 cm
d) 9.70 cm
203. A group of children playing with
marbles placed 50 pieces of the marbles
inside a cylindrical container with water
filled to a height of 20 cm. If the diameter of
each marble is 1.5 cm and that of the
cylindrical container 6 cm. What would be
the new height of water inside the
cylindrical container after the marbles were
placed inside?
a) 23.125 cm
b) 24.125 cm
c) 22.125 cm
d) 25.125 cm
204. A pipe lining material silicon carbide
used in a conveyance of pulverized coal to
fuel a boiler, has a thickness of 2 cm and
inside diameter of 10 cm. Find the volume
of the material with pipe length of 6 meters.
a) 45,239 cm^3
b) 42,539 cm^3
c) 49,532 cm^3
d) 43,932 cm^3
205. Given of diameter x and altitude h.
What percent is the volume of the largest
cylinder which can be inscribed in the cone
to the volume of the cone?
a) 44%
b) 56%
c) 46%
d) 65%
206. Each side of a cube is increased by 1%.
By what percent is the volume of the cube
increased?
a) 23.4%
b) 30.3%
c) 34.56%
d) 3.03%
207. Two vertical conical tanks are joined at
the vertices by a pipe. Initially the bigger
tank is full of water. The pipe valve is open
to allow the water to flow to the smaller tank
until it is full. At this moment, how deep is
the water in the bigger tank? The bigger tank
has a diameter of 6 ft and a height of 10 ft,
the smaller tank has a diameter of 6 ft and a
height of 8 ft. Neglect the volume of water
in the pipeline.
a) √
b) √
c) √
d) √
208. A pyramid has a square base of 8 m on
a side and an altitude of 10 m. How many
liters of water will it hold when full and
inverted?
a) 223,330
b) 203,330
c) 213,330
d) 233,330
209. What solid figure that has many faces?
a) octagon
b) decagon
c) polygon
d) polyhedron
210. If the length of the latus rectum of an
ellipse is three-fourth of the length of its
minor axis, find its eccentricity.
a) 0.15
b) 0.33
c) 0.55
d) 0.66
211. Find the equation of a line where xintercept is 2 and y-intercept is -2.
a) 2x + 2y +2 = 0
b) x – y – 2 = 0
c) -2x + 2y = -2
d) x – y – 1 = 0
212. A point (x, 2) is equidistant from the
points (-2, 9) and (4, -7). The value of x is:
a) 11/3
b) 20/3
c) 19/3
d) 3
213. A parabola y = -x^2 – 6x – 9 opens
______________.
a) to the right
b) upward
c) to the left
d) downward
214. A line with a curve approaches
indefinitely near as its tracing point passes
off infinitely is called the:
a) tangent
b) asymptote
c) directly
d) latus rectum
215. Find the eccentricity of an ellipse when
the length of the latus rectum is 2/3 of the
length of the major axis.
a) 0.58
b) 0.68
c) 0.78
d) 0.98
216. The directrix of a parabola is the line y
= 5 and its focus is at the point (4, -3).
a) 20
b) 18
c) 16
d) 12
217. The radius of a sphere is r inches at
time t seconds. Find the radius when the
rates of increase of the surface area and the
radius are numerically equal.
a) 1/(8π) in
b) 1/(4π) in
c) 2π in
d) π^2 in
218. In general quadratic equation, if the
discriminant is zero, the curve is a figure
that represents ________.
a) hyperbola
b) circle
c) parabola
d) ellipse
219. The equation of the tangent to the curve
y = x + 5/x at point P(1, 3) is:
a) 4x – y + 7 = 0
b) x + 4y – 7 = 0
c) 4x + y -7 = 0
d) x – 4y + 7 = 0
220. A line 4x + 2y – 2 = 0 is coincident
with the line:
a) 4x + 4y – 2 = 0
b) 4x + 3y + 33 = 0
c) 8x + 4y – 2 = 0
d) 8x + 4y – 4 = 0
221. A locus of a point which moves so that
it is always equidistant from a fixed point
(focus) to a fixed line (directrix) is a
_____________.
a) circle
b) ellipse
c) parabola
d) hyperbola
222. Find the equation of the line passing
through (7, -3) and (-3, -5).
a) x + 5y + 22 = 0
b) x + 5y – 22 = 0
c) x – 5y + 22 = 0
d) x – 5y – 22 = 0
223. Find the vertex of the parabola, x^2 =
8y
a) (0, 0)
b) (0, 4)
c) (4, 0)
d) (0, 8)
224. What type of conics is x^2 – 4y + 3x +
5 = 0.
a) parabola
b) ellipse
c) hyperbola
d) circle
225. Determine the coordinates of the point
which is three-fifths of the way from the
point (2, -5) to the point (-3, 5).
a) (-1, 1)
b) (-2, -1)
c) (-1, -2)
d) (1, 1)
226. A line passing through a point (2, 2).
Find the equation of the line if the length of
the segment intercepted by the coordinate’s
axes is equal to the square root of 5.
a) 2x – y – 2 = 0
b) 2x + y + 2 = 0
c) 2x – y + 2 = 0
d) 2x + y – 2 = 0
227. Point P(x, y) moves with a distance
from point (0, 1) one half of its distance
from line y = 4, the equation of its locus is:
a) 2x^2 – 4y^2 = 5
b) 4x^2 + 3y^2 = 12
c) 2x^2 + 5y^2 = 3
d) x^2 + 2y^2 = 4
228. The major axis of the elliptical path in
which the earth moves around the sun is
approximately 186,000,000 miles and the
eccentricity of the ellipse is 1/60. Determine
the apogee of the earth.
a) 93,000,000 miles
b) 94,335,000 miles
c) 91, 450,000 miles
d) 94,550,000 miles
229. What is the equation of the asymptote
of the hyperbola (x^2)/9 – (y^2)/4 = 1.
a) 2x – 3y = 0
b) 3x – 2y = 0
c) 2x – y = 0
d) 2x + y = 0
230. Compute the focal length and the
length of the latus rectum of the parabola
y^2 + 8x – 6y + 25 = 0.
a) 2, 8
b) 4, 16
c) 16, 64
d) 1, 4
231. Find the equation of the axis of
symmetry of the function y = 2x^2 – 7x + 5.
a) 7x + 4 = 0
b) 4x + 7 = 0
c) 4x – 7 = 0
d) x – 2 = 0
232. Find the value of k for which the
equation x^2 + y^2 + 4x – 2y – k = 0,
represents a point circle.
a) 5
b) 6
c) -6
d) -5
233. Find the equation of the circle whose
center is at (3, -5) and whose radius is 4.
a) x^2 + y^2 – 6x + 10y + 18 = 0
b) x^2 + y^2 + 6x + 10y + 18 = 0
c) x^2 + y^2 – 6x – 10y + 18 = 0
d) x^2 + y^2 + 6x – 10y + 18 = 0
234. Determine B such that 3x + 2y – 7 = 0
is perpendicular to 2x – By + 2 = 0.
a) 5
b) 4
c) 3
d) 2
235. In a Cartesian coordinates, the
coordinates of a square are (1, 1), (0, 8), (4,
5), and (-3, 4). What is the area?
a) 25
b) 20
c) 18
d) 14
236. The segment from (-1, 4) to (2, -2) is
extended three times its own length. Find the
terminal point.
a) (11, -24)
b) (-11, -20)
c) (11, -18)
d) (11, -20)
237. Find the distance between A(4,-3) and
B(-2, 5).
a) 10
b) 8
c) 9
d) 11
238. Given three vertices of a triangle whose
coordinates are A(1, 1), B(3, -3) and C(5, -3).
Find the area of the triangle.
a) 3
b) 4
c) 5
d) 6
239. The line segment connecting (x, 6) and
(9, y) is bisected by the point (7, 3). Find the
values of x and y.
a) 33, 12
b) 5, 0
c) 6, 9
d) 14, 6
240. A line passes through (1, -3) and (-4, 2). Write the equation of the line in slopeintercept form.
a) y – 4 = x
b) y = -x – 2
c) y = x – 4
d) y – 2 = x
241. What is the x-intercept of the line
passing through (1, 4) and (4, 1).
a) 4.5
b) 5
c) 6
d) 4
242. Find the distance between the lines, 3x
+ y – 12 = 0 and 3x + y – 4 = 0.
a) 16/√
b) 12/√
c) 4/√
d) 8/√
243. Find the area of the circle whose
equation is x^2 + y^2 = 6x – 8y.
a) 25π
b) 5π
c) 15π
d) 20π
244. Find the major axis of the ellipse x^2 +
4y^2 – 2x – 8y + 1 = 0.
a) 2
b) 10
c) 4
d) 6
245. An arch 18 m high has the form of
parabola with a vertical axis. The length of a
horizontal beam placed across the arch 8 m
from the top is 64 m. Find the width of the
arch at the bottom.
a) 86 m
b) 96 m
c) 106 m
d) 76 m
246. Find the equation of the hyperbola
whose asymptotes are y = 2x and which
passes through (5/2, 3).
a) 4x^2 – y^2 – 16 = 0
b) 2x^2 – y^2 – 4 = 0
c) 3x^2 – y^2 – 9 = 0
d) 5x^2 – y^2 – 25 = 0
247. Find the eccentricity of the curve 9x^2
– 4y^2 – 36x + 8y = 4.
a) 1.80
b) 1.90
c) 1.70
d) 1.60
248. The equation of a line that intercepts
the x-axis at x = 4 and the y-axis at y = - 6
is:
a) 3x + 2y = 12
b) 2x – 3y = 12
c) 3x – 2y = 12
d) 2x – 3y = -12
249. What is the radius of a circle defined by
the equation x^2 – 6x + y^2 – 4y – 12 = 0.
a) 3.46
b) 7
c) 5
d) 6
250. Find the slope of the line defined by y –
x = -5.
a) 1
b) 1/4
c) -1/2
d) 5 + x
251. What conic section is represented by
4x^2 – y^2 + 8x + 4y = 15.
a) parabola
b) ellipse
c) hyperbola
d) circle
252. What conic section is represented by
x^2 + y^2 – 4x + 2y – 20 = 0
a) circle
b) parabola
c) ellipse
d) hyperbola
253. Find the equation of the straight line
with a slope of 3 and a y-intercept of 1.
a) 3x – y + 1 = 0
b) 3x + y + 1 = 0
c) 3x – y – 1 = 0
d) 3x + y – 1 = 0
254. What is the equation of the line that
passes through (4, 0) and is parallel to the
line x – y – 2 = 0?
a) y + x + 4 = 0
b) y – x – 4 = 0
c) x – y – 4 = 0
d) x + y – 4 = 0
255. Find the distance from the line 4x – 3y
+ 5 = 0 to the point (2, 1).
a) 1
b) 2
c) 3
d) 4
256. What is the center of the curve x^2 +
y^2 – 2x – 4y – 31 = 0.
a) (-1, -2)
b) (1, -2)
c) (-1, 2)
d) (1, 2)
257. Determine the equation of the curve
such that the sum of the distances of any
point on the curve from two points whose
coordinates are (-3, 0) and (3, 0) is always
equal to 8.
a) 7x^2 + 16y^2 – 112 = 0
b) 16x^2 + 7y^2 – 112 = 0
c) 7x^2 + 16y^2 + 112 = 0
d) 16x^2 + 7y^2 + 112 = 0
258. The equation 9x^2 + 16y^2 + 54x 64y = -1 describes:
a) a hyperbola
b) a sphere
c) a circle
d) an ellipse
259. The sum of the distances from the two
foci to any point in a/an ______________ is
a constant.
a) a parabola
b) any conic
c) hyperbola
d) ellipse
260. Determine the curve: 9x^2 + 6y^2 + 2x
+ 3y + 9 = 0.
a) ellipse
b) hyperbola
c) parabola
d) circle
261. Locus of points on a side which rolls
along a fixed line:
a) cardoid
b) epicycloid
c) cycloid
d) hypocycloid
262. What is the radius of a circle with the
following equation? x^2 – 6x + y^2 – 12 = 0
a) 2
b) 5
c) 7
d) 25
253. Find the slope of the line passing to the
point (-3, -4) and (2, 4).
a) 0
b) 5
c) 10
d) 1.6
254. What is the slope of the line
perpendicular to y = (1/4)x + 6?
a) 4
b) 1
c) -4
d) -1
255. Given the polar coordinates (4, 20°).
Find the rectangular coordinates.
a) -2, 3.46
b) -3.46, -2
c) 2, -3.46
d) -3.46, 4
256. Find the equation of the line which
passes through the point (2, 1) and
perpendicular to the line whose equation is y
= 4x + 3.
a) x – 4y + 6 = 0
b) y – 4x + 6 = 0
c) x + 4y – 6 = 0
d) y – 4x + 6 = 0
257.What is the second derivative of a
function y = 5x^3 + 2x + 1?
a) 25x
b) 30x
c) 18
d) 30
258. Find the height of a circular cylinder of
a maximum volume, which can be inscribed
in a sphere of radius 10 cm.
a) 11.55 cm
b) 12.55 cm
c) 14.55 cm
d) 15.55 cm
259. Find the maximum point of y = x + 1/x.
a) (2, 5/2)
b) (1, 2)
c) (-1, -2)
d) (2, 3)
260. Simplify the expression Lim(x^2 –
16)/(x – 4) as x approaches 2.
a) 8
b) 6
c) 4
d) 2
261. Evaluate the Lim (x^2 + 3x – 4) as x
approaches 3.
a) 18
b) 12
c) 4
d) 2
262. The distance a body travels is a
function of time t and is defined by: x(t) =
18t + 9t^2. What is its velocity at t = 3?
a) 36
b) 45
c) 72
d) 92
263. Water running out a conical funnel at
the rate of 1 cu. in per second. If the radius
of the base of the funnel is 4 in and the
altitude is 8 in, find the rate at which the
water level is dropping when it is 2 in from
the top.
a) -1/9 π in/sec
b) -3/2 π in/sec
c) -8/9 π in/sec
d) -4/9 π in/sec
264. ________ is the concept of finding the
derivative of composite functions.
a) Logarithmic differentiation
b) Chain rule
c) Trigonometric differentiation
d) Implicit differentiation
265. The volume of the sphere is increasing
at the rate of 6 cm^3/hr. At what rate is its
surface area increasing (in cm^2/hr) when
the radius is 50 cm?
a) 0.54
b) 0.44
c) 0.34
d) 0.24
266. A man on a wharf 3.6 m above sea
level is pulling a rope tied to a raft at 0.60 m
per second. How fast is the raft approaching
the wharf when there are 6 m of rope out?
a) -0.95 m/s
b) -0.85 m/s
c) -0.75 m/s
d) -0.65 m/s
267. If the distance x from the point of
departure at time t is defined by the equation
x = -16t^2 + 5000t + 5000, what is the initial
velocity?
a) 2000
b) 0
c) 5000
d) 3000
268. Using two existing corner sides of an
existing wall, what is the maximum
rectangular area that can be fenced by a
fencing material 30 ft long?
a) 225 sq. ft
b) 240 sq. ft
c) 270 sq. ft
d) 335 sq. ft
269. The radius of a sphere is r inches at
time t seconds. Find the radius when the
rates of increase of the surface area and the
radius are numerically equal.
a) 1/(8π) in
b) 1/(4π) in
c) 2π in
d) π^2 in
270. Three sides of a trapezoid are each 8
cm long. How long is the fourth side when
the area of the trapezoid has the greatest
value?
a) 8 cm
b) 12 cm
c) 16 cm
d) 20 cm
271. Find the change in y = 2x – 3 if x
changes from 3.3 to 3.5.
a) 0.1
b) 0.2
c) 0.3
d) 0.4
272. If y = arctan(ln x), find dy/dx at x = 1/e.
a) e
b) e/2
c) e/3
d) e^2
273. Evaluate the limit (ln x)/x as x
approaches positive infinity.
a) 1
b) 0
c) infinity
d) -1
274. lim[(x^3 – 27)/(x – 3)] as x approaches
3.
a) 0
b) infinity
c) 9
d) 27
275. A box is to be constructed from a piece
of zinc 20 in square by cutting equal squares
from each corner and turning up zinc to
form the side. What is the volume of the box
that can so constructed?
a) 599.95 in^3
b) 592.59 in^3
c) 579.50 in^3
d) 622.49 in^3
276. Given the function f(x) = x to the 3rd
power – 6x + 2, find the value of the first
derivative at x = 2, f(2).
a) 6
b) 7
c) 3x^2 – 5
d) 8
277. Water is pouring into a swimming pool.
After t hours there are t + √ gallons in the
pool. At what rate is the water pouring into
the pool when t = 9 hours?
a) 7/6 gph
b) 1/6 gph
c) 2/3 gph
d) 1/2 gph
278. Evaluate Lim [(x^2 – 16)/(x – 4)] as x
approaches 4.
a) 1
b) 8
c) 0
d) 16
279. Evaluate Lim [(x - 4)/(x^2 – x – 12)]
as x approaches 4.
a) undefined
b) 0
c) infinity
d) 1/7
280. Evaluate Lim [(x^3 – 2x + 9)/(2x^3 –
8)] as x approaches infinity.
a) 0
b) 2
c) 1/2
d) 1/4
281. If y = 1/(t + 1) and x = t/(t + 1), find
dy/dx or y’.
a) 1
b) -1
c) t
d) –t
282. Differentiate: y = [(sin x)/(1 – 2cos x)].
a) (cos x – 1)/(1 – 2cos x)^2
b) (cos x – 2)/(1 – 2cos x)^2
c) (cos x)/(1 – 2cos x)^2
d) (-2)/(1 – 2cos x)^2
283. Given the curve y = 12 – 12x + x^3,
determine its maximum, minimum and
inflection points.
a) (-2, 28), (2, -4), & (0, 12)
b) (2, -28), (2, 4), & (0, 2)
c) (-2, -28), (-2 -4) & (2, 12)
d) (-2, 28), (-2, 4) & (1, 12)
284. Given the curve y^2 = 5x – 1 at point
(1, -2), find the equation of tangent and
normal to the curve.
a) 5x + 4y + 3 = 0 & 4x – 5y – 14 = 0
b) 5x + 4y – 3 = 0 & 4x + 5y – 14 = 0
c) 5x – 4y + 3 = 0 & 4x + 5y + 14 = 0
d) 5x – 4y – 3 = 0 & 4x + 5y – 14 = 0
285. Find the radius of the curvature at any
point on the curve, y + ln cos x = 0
a) cos x
b) 1.5707
c) sec x
d) 1
286. Find the minimum volume of a right
circular cylinder that can be inscribed in a
sphere having a radius r.
a) 1/√ volume of sphere
b) √ volume of sphere
c) 2/√ volume of sphere
d) √
volume of sphere
287. Find the point in the parabola y^2 = 4x
at which rate change of the ordinate and
abscissa are equal.
a) (1, 2)
b) (-1, 4)
c) (2, 1)
d) (4, 4)
288. What is the allowable error in
measuring the edge of cube that is intended
to hold 8 m^3, if the error of the computed
volume is not to exceed 0.03 m.
a) 0.002
b) 0.003
c) 0.0025
d) 0.001
289. Find the slope of x^2 y = 8 at point (2,
2)
a) 2
b) -1
c) -2
d) 1/2
290. Water is flowing into a conical vessel
15 cm deep and having a radius of 3.75 cm
across the top. If the rate at which the water
rises is 2 cm/sec, how fast is the water
flowing into the conical vessel when the
water is 4 cm deep?
a) 6.28 m^3/s
b) 2.37 m^3/s
c) 4.57 m^3/s
d) 5.73 m^3/s
291. Find the slope of the line having a
parametric equation y = 4t + 6 and x = t + 1.
a) 1
b) 2
c) 3
d) 4
292. Determine the diameter of a closed
cylindrical tank having a volume of 11.3
m^3 to obtain a minimum surface area.
a) 1.44
b) 2.44
c) 3.44
d) 4.44
293. Determine the velocity of progress with
the given equation, D = 20t + 5/(t + 1) when
t = 4 sec.
a) 16.8 m/s
b) 17.8 m/s
c) 18.8 m/s
d) 19.8 m/s
294. Find the slope of the curve x^2 + y^2 –
6x + 10y + 5 = 0 at point (1, 0).
a) 1/3
b) 3/4
c) 2/5
d) 1/5
295. Two posts 10 m high and the other is
15 m high stands 30 m apart. They are to be
stayed by transmission wires attached to a
single stake at ground level, the wires
running to the top of the posts. Where
should the stake be placed to use the least
amount of wire?
a) 12 m
b) 14 m
c) 18 m
d) 16 m
296. Find the slope of the line having the
parametric equations x = t – 1 and y = 2t.
a) 1
b) 3
c) 2
d) 4
297. Find the second derivative of y with
respect to x for: 4x^2 + 8y^2 = 36.
a) 9/4y^3
b) 4y^3
c) -9/4y^3
d) -4y^3
298. Find the derivative of h with respect to
u; for h = π^2u.
a) π^2x
b) 2u ln π
c) 2π^2u ln π
d) 2π^2u
299. Find y’ if y = x ln x – x.
a) ln x
b) x ln x
c) (ln x)/x
d) x/ln x
300. Differentiate, y = sec x^2.
a) 2x sec x^2
b) 2sec x^2
c) 2xtan x^2
d) 2xsec x^2 tan x^2
301. What is the derivative of the function
with respect to x of (x + 1)^3 – x^3?
a) 3x + 3
b) 3x – 3
c) 6x – 3
d) 6x + 3
302. Evaluate the Lim [(x^2 – 1)/(x^2 + 3x –
4)] as x approaches 1.
a) 3/5
b) 2/5
c) 4/5
d) 1/5
303. Evaluate: Lim [(1 – cos x)/x^2] as x
approaches 0
a) 0
b) 1/2
c) 2
d) -1/2
304. Evaluate: Lim [(3x^4 – 2x^2 +
7)/(5x63 + x – 3)] as x approaches infinity.
a) undefined
b) 3/5
c) infinity
d) 0
305. Differentiate: (x^2 + 2)^1/2
a) [(x^2 + 1)^1/2]/2
b) x/(x^2 + 2)^1/2
c) 2x/(x + 2)^1/2
d) (x^2 + 2)^2
306. Differentiate y = e^x cos x^2
a) –e^x sin x^2
b) e^x (cos x^2 – 2xsin x^2)
c) e^x cos x^2 – 2xsin x^2
d) -2xe^x sin x
307. Differentiate: y = log (x^2 + 1)^ 2
a) log e (x)(x^2 + 1)^2
b) 4x(x^2 + 1)
c) (4xlog e)/(x^2 +1)
d) 2x(x + 1)
308. If y = 4cos x + sin 2x, what is the slope
of the curve then x = 2.
a) -2.21
b) -4.94
c) -3.25
d) -2.22
309. Find y’ = arcsin cos x.
a) -1
b) -2
c) 1
d) 2
310. A poster is to contain 300 m^2 of
printed matter with margins of 10 cm at the
top and bottom and 5 cm at each side. Find
the overall dimensions, if the total area of
the poster is a minimum.
a) 27.76 cm, 47.8 cm
b) 20.45 cm, 35.6 cm
c) 22.24 cm, 44.5 cm
d) 25.55 cm, 46.7 cm
311. Water is flowing into a conical cistern
at the rate of 8 m^3/min. If the height of the
inverted cone is 12 m and the radius of its
circular opening is 6 m. How fast is the
water level rising when the water is 4 m
deep?
a) 0.74 m/min
b) 0.64 m/min
c) 0.54 m/mid
d) 0.84 m/min
312. An isosceles triangle with equal sides
of 20 cm has these sides at variable equal
angles with the base. Determine the
maximum area attainable by the triangle.
a) 250 cm^2
b) 200 cm^2
c) 180 cm^2
d) 300 cm^2
313. A triangle has variable sides x, y, z
subject to the constraint such that the
perimeter P is fixed to 18 cm. What is the
maximum possible area for the triangle?
a) 15.59 cm^2
b) 18.71 cm^2
c) 14.03 cm^2
d) 17.15 cm^2
314. What is the limit value of y = (x^3 +
x)/(x^2 + x) as x approaches zero?
a) 1
b) indeterminate
c) 0
d) 3
315. A fencing is limited to 20 ft high. What
is the maximum rectangular area that can be
fenced in using two perpendicular corner
sides of an existing wall?
a) 120
b) 100
c) 140
d) 190
316. Find the point on the curve x^2 = 2y
which is nearest to the point (4, 1).
a) (2, 4)
b) (4, 2)
c) (2, 2)
d) (2, 3)
317. Find the largest area of a rectangle
which can be inscribed in the ellipse, 4x^2 +
9y^2 = 36.
a) 12
b) 24
c) 6
d) 48
318. The derivative with respect ot v of the
function f(y) = √ is:
a) (y^-2/3)/3
b) 3y^2/3
c) 3y^-2/3
d) (y^2/3)/3
319. If a is the simple constant, what is the
derivative of y = x^a?
a) ax – x
b) ax
c) ax to the a - 1 power
d) x to the a – 1 power
320. The first derivative with respect to y of
the function d(y) = 3√ is _____.
a) 3(9/2)
b) 3(9) to the 1/2 power
c) 0
d) 9
321. Find the derivative of f(x) = [x to the
3rd power – (x – 1) to the 3rd power] to the
3rd power?
a) 3x – 3 (x – 1)
b) 3[x to the 3rd power – x – 1] to the 3rd
power
c) 9[x to the 3rd power – (x – 1) to the 3rd
power]^2 [x –(x – 1)]^2
d) 9[x to the 3rd power – (x – 1) to the 3rd
power]^2 [x^2 – (x – 1)^2]
322. Water from the filtering facility is
pouring into a swimming pool. After n hours,
there are n + √ gallons in the pool. At what
rate is the water pouring into the pool when
n = 16 hrs?
a) 1/2 gph
b) 9/8 gph
c) 1 gph
d) 7/6 gph
323. Find the slope of the equation y = x^2
when x = 2.
a) 2
b) 6
c) 4
d) 1
324. What is the value of the following
limit? Lim (x^2 – 9)/(x – 3) as x approaches
3.
a) 3
b) 6
c) 9
d) 0
325. The position of an object as a function
of time is describe by x = 4t^3 + 2t^2 – t + 3.
What is the distance traveled by an object at
t = -2 and t = 2?
a) 44
b) 63
c) 78
d) 108
326. Lim (x^2 0 4)/(x – 2) as x approaches 2,
compute the indicated limit.
a) 4
b) 8
c) 6
d) 10
327. Evaluate the integral of [(3^x)
/(e^x)]dx from 0 to 1.
a) 1.510
b) 1.051
c) 1.105
d) 1.510
328. Evaluate the integral of tan^2 x dx.
a) tan x – x + c
b) sec^2 x + x + c
c) 2sec x – x + c
d) (tan^2 x)/s + x + c
329. Evaluate the integral of sqrt(3t – 1) dt.
a) (2/9)(3t – 1)^5/2 + c
b) (2/9)(3t – 1)^3/2 + c
c) (1/2)(3t – 1)^5/2 + c
d) (1/2)(3t – 1)^3/2 + c
330. Evaluate the integral of (3t – 1)^3 dt.
a) (1/12)(3t – 1)^4 + c
b) (1/4)(3t – 1)^4 + c
c) (1/3)(3t – 1)^4 + c
d) (1/12)(3t – 1)^3 + c
331. Integrate the square root of (1 – cos x)
dx.
a) -2 sqrt(2) cos (x/2) + c
b) -2sqrt(2) cos x + c
c) 2sqrt(2) cos (x/2) + c
d) -2sqrt(2) cos x+ c
332. Find the area bounded by the parabolas
x^2 – 2y = 0 and x^2 + 2y – 8 = 0.
a) 32/2
b) 20/3
c) 16/3
d) 64/3
333. Evaluate: integral of cos^8 3A dA from
0 to π/6.
a) 35π/768
b) 45π/768
c) 125π/768
d) 5π/768
334. Evaluate: integral of 1/(4 + x^2)^3/2 dx.
a) x/(4sqrt(x^2 + 4)) + c
b) -1/(4sqrt(x^2 + 4)) + c
c) - x/(4sqrt(x^2 + 4)) + c
d) 1/(4sqrt(x^2 + 4)) + c
335. Evaluate: integral of (e^x)/(e^x + 1) dx
a) ln(e^x + 1) + c
b) ln(e^-x + 1) + c
c) ln^2 (e^x + 1) + c
d) ln^2 (e^x + 1) + c
336. Evaluate: integral of (e^x – 1)/(e^x + 1)
a) ln (e^x -1)^2 + x + c
b) ln (e^x + 1) + x + c
c) ln (e^x + 1)^2 –x + c
d) ln (e^x + 1)^2 –x + c
337. Evaluate integral of ln x dx from 1 to 0.
a) infinity
b) 1
c) 0
d) e
338. Find the area bounded by the line x –
2y + 10 = 0, the x-axis, the y-axis and x = 10.
a) 75
b) 45
c) 18
d) 36
339. Find the area bounded by the curves
x^2 + y^2 = 9 and 4x^2 + 9y^2 = 36, on the
first quadrant.
a) 2/3π
b) 3/4π
c) 1/2π
d) 3/2π
340. Determine the integral of z sin z with
respect to z, then r from r = 0 to r = 1 and
from z = 0 to z = π/2.
a) 1/2
b) 4/5
c) 1/4
d) 2/3
341. Integrate 1/(3x + 4) with respect to x
and evaluate the result from x = 0 to x = 2.
a) 0.278
b) 0.336
c) 0.252
d) 0.305
342. An area in the xy plane is bounded by
the following lines: x = 0 (y-axis), y = 0 (xaxis), x + 4y = 20, and 4x + y = 20. The
linear function z = 5x + 5y attains its
maximum value within the bounded area
only at one of the vertices (intersections of
the above lines). Determine the maximum
value of z.
a) 40
b) 25
c) 50
d) 45
343. Find the area bounded by the parabola
x^2 = 4y and y = 4.
a) 21.33
b) 33.21
c) 31.32
d) 13.23
344. Find the area in the first quadrant
bounded by the parabola y^2 = 4x, x = 1 ad
x = 3.
a) 9.555
b) 5.955
c) 5.595
d) 9.955
345. Evaluate integral of 12 sin^5 x cos^5 x
dx from 0 to π/2.
a) 0.20
b) 0.50
c) 0.25
d) 0.35
346. Evaluate integral of x(x – 5)^12 dx
from 5 to 6.
a) 0.456
b) 0.587
c) 0.708
d) 0.672
347. What is the area bounded by the curve
y^2 = x and the line x – 4 = 0.
a) 32/3
b) 34/7
c) 64/3
d) 16/3
348. Find the area bounded by the curve r =
8 cos 2θ.
a) 16π
b) 32π
c) 12π
d) 8π
349. The area bounded by the curve y =
2x^1/2, the line y = 6 and the y-axis is to be
resolved at y = 6. Determine the centroid of
the volume generated.
a) 0.56
b) 1.80
c) 1.0
d) 1.24
350. Find the area of the region bounded by
the polar curve r^2 = a^2 cos 2θ.
a) 2a^2
b) 4a^2
c) 3a^2
d) a^2
351. The area bounded by the curve y^2 =
12x and the line x = 3 is resolved about the
line x = 3. What is the volume generated?
a) 185
b) 187
c) 181
d) 183
352. Find the moment of inertia with respect
to the x-axis of the area bounded by the
parabola y^2 = 4x and the line x = 1.
a) 2.35
b) 2.68
c) 2.13
d) 2.56
353. Given the area in the first quadrant
bounded by x^2 = 8y, the line y – 2 = 0 and
the y-axis. What is the volume generated
when the area is resolved about the line y –
2 = 0?
a) 28.41
b) 27.32
c) 26.81
d) 25.83
354. Find the area of the horizontal
differential rectangle xdy by the x-axis and
the line y = 4. The parabola y = 4x.
Rectangle area = (4 – x)dy.
a) 64/2
b) 32/3
c) 32/4
d) 32/2
355. What is the approximate area bounded
by the curves y = 8 – x^2 and y = -2 + x^2?
a) 22.4
b) 29.8
c) 44.7
d) 26.8
356. What retarding force is required to stop
a 0.45 caliber bullet of mass 20 grams and
speed of 200 m/s as it penetrates a wooden
block to a depth of 2 inches?
a) 17,716 N
b) 19,645 N
c) 15,500 N
d) 12,500 N
357. A freely falling body is a body in
rectilinear motion and with constant
________.
a) velocity
b) speed
c) deceleration
d) acceleration
358. A ball is thrown upward with an initial
velocity of 50 ft/s. How high does it go?
a) 39 ft
b) 30 ft
c) 20 ft
d) 45
359. It takes an airplane one hour and fortyfive minutes to travel 500 miles against the
wind and covers the same distance in one
hour and fifteen minutes with the win. What
is the speed of the airplane?
a) 342 mph
b) 375 mph
c) 450 mph
d) 525 mph
360 When the total kinetic energy of a
system is the same as before and after the
collision of two bodies, it is called:
a) static collision
b) elastic collision
c) inelastic collision
d) plastic collision
361. An airplane travels from points A to B
with a distance of 1500 km and a wind along
its flight. If it takes the airplane 2 hours from
A to B with the tailwind and 2.5 hours from
B to A with the headwind, what is the
velocity?
a) 700 kph
b) 675 kph
c) 450 kph
d) 750 kph
362. The periodic oscillations either up or
down or back and forth motion in a straight
line is known as ________.
a) transverse harmonic motion
b) resonance
c) rotational harmonic motion
d) translational harmonic motion
363. A flywheel of radius 14 inches is
rotating at the rate of 1000 rpm. How fast
does a poin on the rim travel in ft/sec?
a) 122
b) 1456
c) 100
d) 39
364. Pedro started running at a speed of 10
kph. Five minutes later, Mario started
running in the same direction and catches up
with Pedro in 20 minutes. What is the speed
of Mario?
a) 12.5 kph
b) 15.0 kph
c) 17.5 jph
d) 20.0 kph
365. A flywheel accelerates uniformly from
rest to a speed of 200 rpm in one-half
second. It then rotates at the same speed for
2 seconds before decelerating to rest in onethird second. Determine the total number of
revolutions of the flywheel during the entire
time interval?
a) 8.06 rev
b) 9.12 rev
c) 6.90 rev
d) 3.05
366. A ball is thrown upward with an initial
velocity of 60 ft/s. Determine the velocity at
the maximum height.
a) 6.12 ft/s
b) 2.61 ft/s
c) 2.12 ft/s
d) 0 ft/s
367. A bullet if fired vertically upward with
a mass of 3 grams. If it reaches an altitude of
100 m, what is its initial velocity?
a) 54.2 m/s
b) 47.4 m/s
c) 52.1 m/s
d) 44.2 m/s
368. What is the acceleration of a point on a
rim of a flywheel 0.8 m in diameter turning
at the rate of 1400 rad/min?
a) 214.77 m/s
b) 217.77 m/s
c) 220.77 m/s
d) 227.77 m/s
369. Impulse causes ______________.
a) the object’s momentum to change
b) the object’s momentum to decrease
c) the object’s momentum to increase
d) the object’s momentum to remain
constant or to be conserve
370. A DC-9 jet with a takeoff mass of 120
tons has two engines producing average
force of 80,000 N during takeoff. Determine
the plane’s acceleration down the runway if
the takeoff time is 10 seconds.
a) 1.52 m/s^2
b) 1.33 m/s^2
c) 3.52 m/s^2
d) 2.45 m/s^2
371. In a hydraulic press, the small cylinder
has a diameter of 8 cm, while the larger
piston has a diameter of 2 cm. If the force of
600 N is applied to the small piston, what is
the force of the large piston, neglecting
friction?
a) 3895 N
b) 4125 N
c) 4538 N
d) 5395 N
372. A car accelerates uniformly from
standstill to 80 mi/hr in 5 seconds. What is
its acceleration?
a) 23.47 ft/sec^2
b) 33.47 ft/sec^2
c) 43.47 ft/sec^2
d) 53.47 ft/sec^2
373. A stone is thrown vertically upward at
the rate of 20m/s. It will return to the ground
after how many seconds?
a) 3.67 sec
b) 5.02 sec
c) 4.08 sec
d) 2.04 sec
374. A plane is headed due east with
airspeed of 240 mph. If a wind at 40 mph is
blowing from the north, find the ground
speed of the plane.
a) 190 mph
b) 210 mph
c) 243 mph
d) 423 mph
375. The study of motion without reference
to the force that causes the motion is known
as __________.
a) statics
b) dynamics
c) kinetics
d) kinematics
376. A car accelerates from rest and reached
a speed of 90 kph in 2- seconds. What is the
acceleration in meter per second?
a) 0.667
b) 0.707
c) 0.833
d) 0.866
377. Momentum is a property related to the
object’s __________.
a) motion and mass
b) mass and acceleration
c) motion and weight
d) weight and velocity
378. A gulf weighs 1.6 ounce. If its velocity
immediately after being driven is 225 fps,
what is the impulse of the bow in slugft/sec?
a) 0.855
b) 0.812
c) 0.758
d) 0.699
379. A missile is fired with a speed of 100
fps in a direction 30 degrees above the
horizontal. Determine the maximum height
to which it rises?
a) 60 ft
b) 52 ft
c) 45 ft
d) 39 ft
380. When the total kinetic energy of a
system is the same as before and after
collision of two bodies, it is called:
a) plastic collision
b) inelastic collision
c) elastic collision
d) static collision
381. A man travels in a motorized banca at
the rate of 15 kph from his barrio to the
poblacion and come back to his barrio at the
rate of 12 kph. If his total time of travel back
and forth is 3 hours, the distance from the
barrio to the poblacion is:
a) 10 km
b) 15 km
c) 20 km
d) 25 km
382. A 50,000 N car travelling with a speed
of 150 km/hr rounds a curve whose radius is
150 m. Find the centripetal force.
a) 70 kN
b) 25 kN
c) 65 kN
d) 59 kN
383. A ball is dropped from a building 100
m high. If the mass of the ball is 10 grams,
after what time will the ball strikes the
earth?
a) 5.61 s
b) 2.45 s
c) 4.52 s
d) 4.42 s
384. A 900 N weight hangs on a vertical
plane. A man pushes this weight
horizontally until the rope makes an angle of
40° with the vertical. What is the tension in
the rope?
a) 1286 N
b) 1175 N
c) 918 N
d) 825 N
385. A plane dropped a bomb at an elevation
1000 meters from the ground intended to hit
a target which is 200 m from the ground. If
the plane was flying at a velocity of 300 kph,
at what distance from the target must the
bomb be dropped to hit the target? Wind
velocity and atmospheric pressure to be
disregarded.
a) 1864.71 m
b) 2053.20 m
c) 1574.37 m
d) 1064.20 m
386. What is the minimum distance can a
truck slide on a horizontal asphalt road if it
is travelling at 25 m/s? The coefficient of
sliding friction between the asphalt and
rubber tire is at 0.60. The weight of the truck
is 8500 kg.
a) 44.9
b) 58.5
c) 53.2
d) 63.8
387. A concrete highway curve with a radius
of 500 ft is banked to give lateral pressure
equivalent to f = 0.15. For what coefficient
of friction will skidding impend for a speed
of 60 mph.
a) µ > 0.360
b) µ < 0.310
c) µ > 0.310
d) µ < 0.360
388. A circle has a diameter of 20 cm.
Determine the moment of inertia if the
circular area relative to the axis
perpendicular to the area through the center
of the circle in cm^4.
a) 14,280
b) 15,708
c) 17,279
d) 19,007
389. An isosceles triangle has a 10 cm base
and a 10 cm altitude. Determine the moment
of inertia of the triangle area relative to a
line parallel to the base and through the
upper vertex in cm^4.
a) 2,750
b) 3,025
c) 2,500
d) 2,273
390. Two electrons have speeds of 0.7c and
x respectively. If their relative velocity is
0.65c, find x.
a) 0.02c
b) 0.12c
c) 0.09c
d) 0.25c
391. A baseball is thrown from a horizontal
plane following a parabolic path with an
initial velocity of 100 m/s at an angle of 30°
above the horizontal. How far from the
throwing point will the ball attain its original
level?
a) 890 m
b) 883 m
c) 878 m
d) 875 m
392. What is the speed of a synchronous
earth’s satellite situated 4.5 x 10^7 m from
the earth?
a) 11,070 kph
b) 12,000 kph
c) 11,777.4 kph
d) 12,070.2 kph
393. What is the inertia of a bowling ball
(mass 0.50 kg) of radius 15 cm rotating at an
angular speed of 10 rpm for 6 seconds.
a) 0.001 kg-m^2
b) 0.002 kg-m^2
c) 0.0045 kg-m^2
d) 0.005 kg-m^2
394. The angle or inclination of ascend of a
road having 8.25% grade is ____________
degrees.
a) 4.72
b) 4.27
c) 5.12
d) 1.86
395. A highway curve has a super elevation
of 7 degrees. What is the radius of the curve
such that there will be no lateral pressure
between the tires and the roadway at a speed
of 40 mph?
a) 265.71 m
b) 438.34 m
c) 345.34 m
d) 330.78 m
396. A shot is fired at an angle of 30 degrees
with the horizontal and a velocity of 120 m/s.
Calculate the range of the projectile.
a) 12.71 km
b) 387.57 ft
c) 0.789 mile
d) 423.74 yd
397. A stone dropped from the top of a
building 55 yd elevation will hit the ground
with a velocity of:
a) 37 ft/sec
b) 33 ft/sec
c) 105 ft/sec
d) 103 ft/sec
398. What is the kinetic energy of a 4000 lb
automobile which is moving at 44 ft/sec?
a) 1.21 x 10^5 ft-lb
b) 2.10 x 10^5 ft-lb
c) 1.80 x 10^5 ft-lb
d) 1.12 x 10^5 ft-lb
399. Find the rate of increase of velocity if a
body increases its velocity from 50 m/sec to
130 m/sec in 16 sec.
a) -4.0 m/sec^2
b) 80 m/sec^2
c) -80 m/sec^2
d) 5.0 m/sec^2
400. A 20 kg sack is raised vertically 5
meters in 0.50 sec. What is the change in
Potential Energy?
a) 98.1 J
b) 981 J
c) 200 J
d) 490.5 J
401. A 350 lbf acts on a block at an angle of
15 degrees with the horizontal. What is the
work done by this force if it is pushed 5 feet
horizontally?
a) 1350.3 ft-lb
b) 1690 ft-lb
c) 1980 ft-lb
d) 2002 ft-lb
402. A 20 kg object moving at 10 m/sec
strikes an unstretched spring to a vertical
wall having a spring constant of 40 kN/m.
Find the deflection of the spring.
a) 111.8 mm
b) 223.6 mm
c) 70.7 mm
d) 50.0 mm
403. A 300 kg box impends to slide down a
ramp inclined at an angle of 25 degrees with
the horizontal. What is the frictional
resistance?
a) 1243.76 N
b) 9951.50 N
c) 1468.9 N
d) 3359.7 N
404. A marksman fires a rifle horizontally at
a target. How much does the bullet drop in
flight if the target is 150 m away and the
bullet has a muzzle velocity of 500 m/sec?
a) 0.34 m
b) 0.44 m
c) 0.64 m
d) 0.54 m
405. A ball is thrown from a building at an
angle of 60 degrees with the horizontal at an
initial velocity of 30 m/sec. After hiting
level ground at the base of the building, it
has covered a total distance of 150 m. How
tall is the building?
a) 230.7 m
b) 756.7 m
c) 692.5 m
d) 1089 m
406. A highway curve with radius 800 ft is
to be banked so that a car travelling 55 mph
will not skid sideways even in the absence
of friction. At what angle should the curve
be banked?
a) 0.159 deg
b) 75 deg
c) 6.411 deg
d) 14.2 deg
407. An airplane flying horizontally at a
speed of 200 m/sec drops a bomb from an
elevation of 2415 meters. Determine the
time required for the bomb to reach the earth.
a) 11.09 sec
b) 22.18 sec
c) 44.37 sec
d) 8.20 sec
408. Find the banking angle of a highway
curve of 100 m radius designed for cars
travelling at 180 kph, if the coefficient of
friction between the tires and the road is
0.58.
a) 19.23 deg
b) 38.5 deg
c) 76.9 deg
d) 45 deg
409. A pulley has a tangential speed of
14m/sec and an angular velocity of 6/5
rad/sec. What is the normal acceleration of
the pulley?
a) 91 m/sec^2
b) 99 m/sec^2
c) 105 m/sec^2
d) 265 m/sec^2
410. An elevator weighing 4000 kb attains
an upward velocity of 4 m/sec in 3 sec with
uniform acceleration. Find the apparent
weight of a 40 kg man standing inside the
elevator during its ascent.
a) 339 N
b) 245 N
c) 446 N
d) 795 N
411. A stone is dropped from a cliff and 2
sec later another stone is thrown downward
with a speed of 22 m/sec. How far below the
top of the cliff will the second stone
overtake the first?
a) 375 m
b) 507 m
c) 795 m
d) 994 m
412. How much horizontal force is needed
to produce an acceleration of 8 m/sec^2 on a
75 kg box?
a) 600 N
b) 500 N
c) 400 N
d) 200 N
413. An elevator with a mass of 1500 kg
descends with a acceleration of 2.85
m/sec^2. What is the tension in the
supporting cable?
a) 10,440 N
b) 12,220 N
c) 15,550 N
d) 20,220 N
414. A dictionary is pulled to the right at a
constant velocity by a 25 N force pulling
upward at 60 degrees above the horizontal.
What is the weight of the dictionary if the
coefficient of kinetic friction is 0.30?
a) 31 N
b) 21 N
c) 20 N
d) 63 N
415. The breaking strength of a string is 500
N. Find the maximum speed that it can
attain if a 1.5 kg ball is attached at one end
while the other end is held stationary and is
whirled in a circle. The string is 0.65 m long.
a) 15.4 m/sec
b) 55.2 m/sec
c) 24.4 m/sec
d) 14.7 m/sec
416. The position of a body weighing 72.6
kg is given by the expression S = 5t^2 + 3t +
4, where S is in meters and t is in seconds.
What force is required for this motion?
a) 625 N
b) 695 N
c) 726 N
d) 985 N
417. Assuming a shaft output of 3,000 kW
and a fuel rate of (JP-4) 34.2 lbs/min. What
is the overall thermal efficiency of the
machine? (HHV of JP-4 is 18,000 Btu/lb)
a) 24.2%
b) 28.3%
c) 27.7%
d) 29.1%
418. g = 32.2 ft/sec^2. How is it expressed
in SI?
a) 9.81 m/sec^2
b) 9.86 m/sec^2
c) 9.08 m/sec^2
d) 9.91 m/sec^2
419. A winch lifted a mass of 1600 kg
through a height of 25 m in 30 sec. If the
efficiency of the winch is 60%, calculate the
energy consumed in kWh.
a) 0.1718 kWh
b) 0.1881 kWh
c) 0.1817 kWh
d) 0.218 kWh
420. Cast iron weighs 640 pounds per cubic
foot. The weight of a cast iron block 14‖ x
12‖ x 18‖ is:
a) 1120 lbs
b) 1000 lbs
c) 1200 lbs
d) 1088 lbs
421. A solid disk flywheel (l = 2—kg-,^2) is
rotating with a speed of 900 rpm. What is its
rotational kinetic energy?
a) 730 x 10 to the 3rd power J
b) 680 x 10 to the 3rd power J
c) 1100 x 10 to the 3rd power J
d) 888 x 10 to the 3rd power J
422. The path of a projectile is a:
a) ellipse
b) parabola
c) part of a circle
d) hyperbola
423. What is the name for a vector that
represent the sum of two vectors?
a) moment
b) torque
c) scalar
d) resultant
424. Determine the super elevation of the
outer rail of a 4-ft wide railroad track on a
10 degrees curve. (A 10 degrees curve is one
which a chord 100 ft long subtends an angle
of 10 degrees at the center). Assumed
velocity of 45 mph.
a) 0.90 ft
b) 2.80 ft
c) 2.50 ft
d) 1.15 ft
425. A 10‖ diameter helical gear carries a
torque of 4000 in-lb. It has a 20 degree
involute stub teeth and a helix angle of 30
degree. Determine the axial component of
the load on the teeth.
a) 451.4 lb
b) 218 lb
c) 471.5 lb
d) 461.6 lb
426. A winch lifted a mass of 1600 kg
through a height of 25 m in 30 sec. Calculate
the input power in kW if the efficiency of
the winch is 60%.
a) 18.1 kW
b) 21.8 kW
c) 28.1 kW
d) 13.08 kW
427. A diagram which shows only the forces
acting on the body:
a) free body diagram
b) cash flow
c) forces flow diagram
d) motion diagram
428. One horse power is equivalent to:
a) 746 watts
b) 7460 watts
c) 74.6 watts
d) 7.46 watts
429. Which is a true statement about the
vector? V1 = i + 2j + k and v2 = i + 3j – 7k
a) the vectors coincide
b) the angle between them is 17.4 degree
c) the vectors are parallel
d) the vectors are orthogonal
430. In a lifting machine, a load of 50 kN is
moved by a distance of 10 cm using an
effort of 10 kN which moves through a
distance of 1 m, the efficiency of the
machine is:
a) 20%
b) 50%
c) 10%
d) 40%
431. What is the angle between two vectors
A and B? A = (3, 2, 1) and B = (2, 3, 2)
a) 24.8 deg
b) 36.7 deg
c) 42.5 deg
d) 77.5 deg
432. What is the equivalent of one
horsepower?
a) 746 W
b) 3141 kW
c) 33,000 ft-lb/min
d) 2545 Btu/lb
433. Two people are driving towards each
other between two towns 160 km apart. The
first man drives at the rate of 45 kph and the
other drives at 35 kph. From their starting
point how long would it take that they will
meet.
a) 3 hr
b) 4 hr
c) 2 hr
d) 1 hr
434. Resistance to motion, caused by one
surface rubbing against another.
a) inertia
b) resistance
c) gravity
d) friction
435. What happens to the acceleration if the
mass is tripled and the force remains the
same?
a) it will be tripled
b) it will be 1/3 of the original
c) it will remain the same
d) it will be 3 times the original
436. Which number has five significant
digits?
a)0.01410
b)0.00101
c)1.0140
d)0.01414
437. The prefix of a no. 10 raise ot the
power minus 6 is:
a) tera
b)deci
c) centi
d) micro
438. The length of a bar is one million of a
meter is called:
a) omicron
b) micron
c) one bar
d)one milli
439. 120 Giga Newton is how many Mega
Newton?
a) 12,000
b) 120
c) 1,200
d) 120,000
440. Factor the expression ( 289x^3 204x^2 + 36x )
a)4x( 17/2 x – 3)( 17/2 x – 3 )
b) 4x(17x-3)(17x-3)
c) 4x(4x-3)(4x+3)
d)4x(17x-3)(17x+3)
441. Factor the expression as completely as
possible: (2x^3 -7x^2 +6x)
a) x(x-2)(x-3)
b) x(x-2)(x+3)
c) x(x-2)(2x+3)
d) x(x-2)(2x-3)
442. ( (xyz)^(1/n) )^n is equal to:
a) (xyz)^(1/n)
b) (xyz)^n
c) xyz
d) (xyz)^(n-1)
443. If x raise to the one half of one equals 4,
x equal to:
a) 24
b) 8
c) 12
d) 16
444. If the numbers one and above divided
by zero the answer is:
a) zero
b) infinity
c) indeterminate
d) absurd
445. Solve for x and y: 4x + 3y = 11 and
8x^2 – 9y^2 = -7.
a) x = 5/3 and y = 3/2
b) x = 3/2 and y = 3/2
c) x = 3/5 and y = 5/3
d) x = 3/2 and y = 5/3
446. If A can do the work in a days and B in
b days, how long will it take to do the job
working together?
a) ( a + b ) / ab days
b) ( a + b ) / 2 days
c) ab / ( a + b ) days
d) a + b days
447. Five hundred kg of steel containing 8%
nickel to be made by mixing a steel
containing 14% nickel with another
containing 6% nickel. How much of each is
needed?
a) 125 kg and 375 kg
b) 150 kg and 350 kg
c) 200 kg and 300 kg
d) 250 kg and 250 kg
448. Logarithm of 10th root of, x raise to 10
equals to:
a) log x
b) ( log x^(1/10) ) / 10
c) 10 log x
d) log x^10
449. What is the natural logarithm of e to the
a plus b power?
a) ab
b) log ab
c) a + b
d) 2.718 ( a + b)
450. What is the logarithm of negative one
hundred?
a) No logarithm
b) Zero
c) Positive log
d) Negative log
451. The logarithm of 1 to base e is:
a) One
b) 2.718
c) Infinity
d) Zero
452. What is the value of (0.101)^(5/6)?
a) antilog [ log 0.101/(5/6) ]
b) antilog [ 6/5 log 0.101 ]
c) 6/5 antilog [ log 0.101 ]
d) antilog [ 5/6 log 0.101]
453. A box contains 8 black and 12 white
balls. What is the probability of getting 1
black and 1 white ball in two consecutive
draws from the box?
a) 0.53
b) 0.45
c) 0.50
d) 0.55
454. What is the sum of the following finite
sequence of terms? 28, 35, 42, ..., 84.
a) 504
b) 525
c) 540
d) 580
455. Solve for x that satisfy the equation,
x^2 + 36 = 9 – 2x^2
a) ±6i
b) +9i
c) ±3i
d) -9i
456. 35.2 to the x power = 7.5 to the x-2
power, solve for x using logarithms.
a) -2.06
b) -2.10
c) -2.60
d) +2.60
457. Solve algebraically: 4x^2 + 7y^2 = 32
and 11y^2 – 3x^2 = 41.
a) y = 4, x = ±1 and y = -4, x = ±1
b) y = +2, x = ±1 and y = -2 , x = ±1
c) x = 2, y = 3 and x = -2, y = -3
d) x = 2, y = -2 and x = 2, y = -2
458. Factor the expression 16 – 10x + x^2.
a) (x+8)(x-2)
b) (x-8)(x+2)
c) (x-8)(x-2)
d) (x+8)(x+2)
459. What is the value of e^-4 =
_____________.
a) 0
b) 0.183156
c) 0.1381560
d) 0.0183156
460. A pump can pump out a tank in 15 hrs.
Another pump can pump out the same tank
in 20 hrs. How long will it take both pumps
together to pump out the tank?
a) 8.57 hrs
b) 7.85 hrs
c) 6.58 hrs
d) 5.50 hrs
461. A tank can be filled by one pipe in 9
hrs and another pipe in 12 hrs. Starting
empty, how long will it take to fill the tank
if water is being taken out by a third pipe at
a rate per hour equal to one-sixth the
capacity of the tank?
a) 36 hrs
b) 25 hrs
c) 30 hrs
d) 6 hrs
462. A rubber ball was dropped from a
height of 42 m and each time it strikes the
ground it rebounds to a height of 2/3 of the
distance from which it fell. Find the total
distance travelled by the ball before it comes
to rest.
a) 180 m
b) 190 m
c) 210 m
d) 220 m
463. From a box containing 8 red balls, 8
white balls and 12 blue balls, one ball is
drawn at random. Determine the probability
that it is red or white:
a) 0.571
b) 0.651
c) 0.751
d) 0.0571
464. If 1/x, 1/y, 1/z are in A.P., then y is
equal to:
a) x-z
b) ½(x+2z)
c) (x+z)/2xz
d) 2xz/(x+z)
465. A class of 40 took examination in
Algebra and Trigonometry. If 30 passed
algebra, 36 passed Trigonmetry, and 2 failed
in both subjects, the number of students who
passed the two subjects is:
a) 22
b) 28
c) 30
d) 60
466. Simplify: ( ab / (ab)^(1/3) )^(1/2)
a) (ab)^(1/3)
b) ab
c) (ab)^(1/2)
d) (ab)^(1/5)
467. Combine into a single fraction: (3x1)/(x^2-1) – (x+3)/(x^2+3x+2) – 1/(x+2)
a) x-1
b) x+1
c) 1/(x+1)
d) 1/(x-1)
468. Two cars start at the same time from
nearby towns 200 km apart and travel
toward each other. One travel at 60 kph and
the other at 40 kph. After how many hours
will they meet on the road?
a) 1 hour
b) 2 hrs
c) 3 hrs
d) 2.5 hrs
469. A single engine airplane has an
airspeed of 125 kph. A west wind of 25 kph
is blowing. The plane is to patrol due to east
and then return toa is base. How far east can
it go if the round trip is to consume 4 hrs?
a) 240 km
b) 180 km
c) 200 km
d) 150 km
470. A car travels from A to B, a distance of
100 km, at an average speed of 30 kph. At
what average speed must it travel back from
B to A in order to average 45 kph for the
round trip of 200 km?
a) 70 kph
b) 110 kph
c) 90 kph
d) 50 kph
471. Two trains A and B having average
speed of 75 mph and 90 kph respectively,
leave the same point and travel in opposite
direstions. In how many minutes would they
be 1600 miles apart?
a) 533
b) 733
c) 633
d) 833
472. It takes Butch twice as long as it takes
Dan to do a certain piece of work. Working
together, they can do the work in 6 days.
How long would it take Dan to do it alone?
a) 12 days
b) 10 days
c) 11 days
d) 9 days
473. A man leaving his office one afternoon
noticed the clock at past two o’clock.
Between two to three hours, he returned to
his office noticing the hands of the clock
interchanged. At what time did he leave the
office?
a) 2:26.01
b) 2:10.09
c) 2:30.01
d) 2:01.01
474. A company has a certain number of
machines of equal capacity that produced a
total of 180 pieces each working day. If two
machines breakdown, the work load of the
remaining machines is increased by three
pieces per day to maintain production. Find
the number of machines.
a) 12
b) 18
c) 15
d) 10
475.A rectangular field is surrounded by a
fence 548 meters long. The diagonal
distance from corner to corner is 194 meters.
Determine the area of the rectangular field.
a) 18,270 m^2
b) 18,720 m^2
c) 18,027 m^2
d) 19,702 m^2
476. Solve for x: (x+2)^(1/2) + (3x-2)^(1/2)
=4
a) x = 1
b) x = 3
c) x = 2
d) x = 4
477. Solve for x: (1/x) + (2/x^2) = (3/x^3).
a) x=1,x=-3
b) x=3,x=1
c) x=-1,x=3
d) x=2,x=3
478. Solve for x: x^(2/3) + x^(-2/3) = 17/4
a) x=-4,x=-1/4
b) x=8,x=-1/4
c) x=4,x=1/8
d) x=8,x=1/8
479. A rectangular lot has a perimeter of 120
meters and an area of 800 square meters.
Find the length and width of the lot.
a) 10m and 30m
b) 30m and 20m
c) 40m and 20m
d) 50m and 10m
480. A 24-meter pole is held by three guy
wires in its vertical position. Two of the guy
wires are of equal length. The third wire is 5
meters longer than the other two and is
attached to the ground 11 meters farther
from the foot of the pole than the other two
equal wires. Find the length of the wires.
a) 25m and 30m
b) 15m and 40m
c)20m and 35m
d) 50 and 10m
481. In a racing contest, there are 240 cars
which will have fuel provisions that will last
for 15 hours. Assuming a constant hourly
consumption for each car, how long will the
fuel provisions last if 8 cars withdraw from
race every hour after the first?
a) 20 hours
b)10 hours
c) 15 hours
d) 25 hours
482. A pile of boiler pipes contains 1275
pipes in layers so that the top layer contains
one pipe and each lower layer has one more
pipe than the layer above. How many layers
are there in the pile?
a) 50
b) 45
c) 40
d) 55
483. A production supervisor submitted the
following report on the average rate of
production of printed circuit boards(PCB) in
an assembly line: ―1.5 workers produce 12
PCB’s in 2 hours‖. How many workers are
employed in the assembly line working 40
hours each per week with a weekly
production of 8000 PCB’s/
a) 50 workers
b) 60 workers
c) 55 workers
d) 70 workers
484. A man bought 20 calculators for
P20,000.00. There are three types of
calculators bought, business type costs
P3,000 each, scientific type costs P1,500
each and basic type costs P500 each. How
many calculators of each type were
purchased?
a) 3, 6, 11
b) 2, 6, 12
c) 1, 4, 15
d) 2, 5, 13
486. A veterans organization in cebu city
consists of men who fought in World War II
and men who fought in Korea. The secretary
noted that 180 members had fought in Korea
and that 70% had taken part in World War II,
while 10% of the members had fought in
both World War II and Korea. How many
members are there together?
a) 400
b) 500
c) 450
d) 700
487. An angle greater than a straight angle
and less than two straight angles is called:
a) Right angle
b) Obtuse angle
c) Reflex angle
d) Acute angle
488. A line segment joining two points on a
circle is called:
a) Arc
b) Tangent
c) Sector
d) Chord
489. All circles having the same center but
with unequal radii are called:
a) encircle
b) tangent circles
c) concyclic
d) concentric circles
490. A triangle having three sides equal is
called:
a) equilateral triangles
b) scalene triangles
c) isosceles triangles
d) right triangles
491. In a regular polygon, the perpendicular
line drawn from the center of the inscribed
circle to any one of the sides is called:
a) radius
b) altitude
c) median
d) rhombus
492. A quadrilateral with two and only two
sides of which are parallel is called:
a) parallelogram
b) trapezoid
c) quadrilateral
d) rhombus
493. A polygon with fifteen sides is termed
as:
a) dodecagon
b) decagon
c) pentedecagon
d) nonagon
494. A statement the truth of which is
admitted without proof is called:
a) an axiom
b) a postulate
c) a theorem
d) a corollary
495. A rectangle with equal sides is termed
as:
a) rhombus
b) trapezoid
c) square
d) parallelogram
496. The sum of the sides of a polygon is
termed as:
a) circumference
b) altitude
c) apothem
d) perimeter
497. A line that meets a plane but not
perpendicular to it, in relation to the plane,
is:
a) parallel
b) collinear
c) coplanar
d) oblique
498. A quadrilateral whose opposite sides
are equal is generally termed as:
a) a square
b) a rectangle
c) a rhombus
d) a parallelogram
499. A part of a line included between two
points on the line is called:
a) a tangent
b) a secant
c) a sector
d) a segment
500. Lines which pass through a common
point are called:
a) collinear
b) coplanar
c) concurrent
d) congruent
501. Points which lie on the same plane is
called:
a) collinear
b) coplanar
c) concurrent
d) congruent
502. In two intersecting lines, the angles
opposite to each other are termed as:
a) opposite angles
b) vertical angles
c) horizontal angles
d) inscribed angles
503. A normal to a given plane is:
a) perpendicular to the plane
b) lying on the plane
c) parallel to the plane
d) oblique to the plane
504. Which of the following statements is
correct?
a) all equilateral triangles are similar
b) all right-angled triangles are similar
c) all isosceles triangles are similar
d) all rectangles are similar
505. A polygon is ________ when no side,
when extended, will pass through the
interior of the polygon.
a) equilateral
b) isoperimetric
c) congruent
d) none of the above
506. The sum of the sides of a polygon:
a) perimeter
b) hexagon
c) square
d) circumference
507. What are the exact values of the cosine
and tangent trigonometric functions of the
acute angle A, given sin A = 5/8?
a) cos A = 8 / 39^(1/2) and tan A = 39^(1/2)
/5
b) cos A = 39^(1/2) / 5 and tan A = 8 /
39^(1/2)
c) cos A = 39/8 and tan A = 5/ 39^(1/2)
d) cos A = 8/5 and tan A = 5/8
508. Given a triangle with angle C=290, side
a =132 units and side b=233.32 units. Solve
for angle B.
a) B=1200
b) B=122.50
c) B=125.20
d) B=1300
509. Simplify: cos2 θ ( 1 + tan2 θ )
a) tan 2θ
b) 1
c) sin 2θ
d) cos θ
510. What is the cosine of 1200?
a) -0.500
b) -0.450
c) -0.866
d) 0.500
511. What is the sine of 8400?
a) -0.866
b) -0.500
c) 0.866
d) 0.500
512. If the sine of angle A is given as k,
what would be then tangent of angle A?
Symbol h for hypotenuse, o for opposite and
a for adjacent.
a) hk/o
b) hk/a
c) ha/k
d) ok/a
513. Which is true regarding the signs of the
natural functions for angles between 900 and
1800?
a) The tangent is positive
b) The cotangent is positive
c) The cosine is negative
d) The sine is negative
514. What is the inverse natural function of
the cosecant?
a) secant
b) sine
c) cosine
d) tangent
515. What is the sum of the squares of the
sine and cosine of an angle?
a) 0
b) 1
c) 3^(1/2)
d) 2
516. What is an equivalent expression for
sin 2x?
a) ½ sin x cos x
b) 2 sin x cos ½ x
c) -2 sin x cos x
d) 2 sin x/sec x
517. A transit set-up 112.1 feet from the
base of a vertical chimney reads 32030’ with
the crosshairs set on top of the chimney.
With the telescope level, the vertical rod at
the base of the chimney is 5.1 feet. How tall
is the chimney?
a) 66.3 ft
b) 71.4 ft
c) 76.5 ft
d) 170.9 ft
518. If sin θ – cos θ = 1/3, what is the value
of in 2θ?
a) 1/3
b) 1/9
c) 8/9
d) 4/9
519. If cos θ = 3^(1/2)/2, then find the value
of x if x = 1 – tan2 θ:
a) -2
b) -1/3
c) 4/3
d) 2/3
520. Solve for x: x = 1-(sin θ-cos θ)^2
a) sin θcos θ
b) -2cos θ
c) cos 2 θ
d) sin 2 θ
521. A mobiline tower and a Nipa Hut stand
on a level plane. The angles of depression of
the top and bottom of the Nipa Hut viewed
from the top of the mobiline tower are 150
and 400, respectively. The height of the
tower is 100m. Find the height of the Nipa
hut.
a) 78.08 m
b) 87.08 m
c) 68.07 m
d) 77.08 m
522. Ship A started sailing N40032’E at the
rate of 3 mph. After 2 hours, ship B started
from the same port going S45018’E at the
rate of 4 mph. After how many hours will
the second ship be exactly south of ship A?
a) 2.25 hrs
b) 2.97 hrs
c) 3.73 hrs
d) 4.37 hrs
523. Solve for the value of x in the equation:
ln (2x+7) – ln (x-1) = ln 5
a) x=4
b) x=5
c) x=6
d) x=8
524. Two ships started sailing from the same
point. One travelled N200E at 30 mph while
the other travelled S500E at 20 mph. After 3
hrs, how far apart are the ships?
a) 124 miles
b) 129 miles
c) 135 miles
d) 145 miles
525. A quadrilateral ABCD is inscribed in a
semi-circle such that one of the sides
coincides with the diameter AD. AB = 10
meters, and BC = 20 meters. If the diameter
AD of the semi-circle is 40 meters, find the
area of the quadrilateral.
a) 350 m^2
b) 420 m^2
c) 470 m^2
d) 530 m^2
526. Solve for x: Arcsin 2x - Arcsin x = 150
a) 0.1482
b) 0.2428
c) 0.3548
d) 0.4282
527. Solve for x: 2^x + 4^x = 8 ^x
a) 0.694242
b) 0.692424
c) 0.964242
d) 0.742420
528. Given: Triangle ABC whose angle A is
320 and a = 75 m. The opposite side of angle
B is 100m. Find angle C.
a) 1000
b) 1030
c) 1100
d) 1150
529. Given triangle ABC with sides
AB=210 m, BC=205 m, and AC=110 m.
Find the largest angle.
a) 72.7510
b) 75.7210
c) 77.1570
d) 82.5170
530. A pole which leans 10015’ from the
vertical towards the sun casts a shadow
9.43m long on the ground when the angle of
elevation of the sun is 54050’. Find the
length of the pole.
a) 12.5m
b) 14.2m
c) 15.4m
d) 18.3m
531. Two points lie on a horizontal line
directly south of a building 35 m high. The
angles of depression to the points are 29010’
and 43050’, respectively. Determine the
distance between the points.
a) 26.3 m
b) 28.7 m
c) 30.2 m
d) 36.4 m
532. Two points lie on a horizontal line
directly south of a building 35 m high. The
angles of depression to the points are 29010’
and 43050’, respectively. Determine the
distance between the building and the
farthest point.
a) 62.7 m
b) 36.5 m
c) 26.5 m
d) 72.6 m
533. Given triangle ABC with sides
AB=210 m, BC=205 m, and AC=110 m.
Find the largest angle.
a) C = 1100
b) C = 85.20
c) C = 77.10
d) C = 43.50
534. Given triangle ABC whose angle A is
320 and opposite side of A is 75 meters. The
opposite side of angle B is 100 m. find the
opposite side of angle C.
a) c = 137.8 m
b) c = 181.2 m
c) c = 117.7 m
d) c = 127.8 m
535. A point P within an equilateral triangle
has a distance of 4m, 5m, and 6m
respectively from the vertices. Find the side
of the triangle.
a) 8.53m
b) 6.78m
c) 9.45m
d) 17.8m
536. The diagonal of the floor of a
rectangular room is 7.50 m. The shorter side
of the room is 4.5 m. What is the area of the
room?
a) 36 sq. m
b) 27 sq. m
c) 58 sq. m
d) 24 sq. m
537. A semi-circle of radius 14 cm is formed
from a piece of wire. If it is bent into a
rectangle whose length is 1 cm more than its
width, find the area of the rectangle.
a) 256.25 sq. cm
b) 323.57 sq. cm
c) 386.54 sq. cm
d) 452.24 sq. cm
538. The length of the side of’ a square is
increased by 100%. Its perimeter is
increased by:
a) 25%
b) 100%
c) 200%
d) 300%
539. A piece of wire of length 52 cm is cut
into two parts. Each part is then bent to form
a square. It is found that total area of the two
squares is 97 sq. cm. the dimension of the
bigger square is:
a) 4
b) 9
c) 3
d) 6
540. A sector has a radius of 12 cm. If the
length of its arc is 12 cm, its area is:
a) 66 sq. cm
b) 82 sq. cm
c) 144 sq. cm
d) 72 sq. cm
541. The perimeter of a sector is 9 cm and
its radius is 3 cm. What is the area of the
sector?
a) 4 sq. cm
b) 9/2 sq. cm
c) 11/2 sq. cm
d) 27/2 sq. cm
542. An iron bar 20 cm long is bent to form
a closed plane area. What is the largest area
possible?
a) 21.56 sq. m
b) 25.68 sq. m
c) 28.56 sq. m
d) 31.83 sq. m
543. A swimming pool is to be constructed
in the shape of partially-overlapping
identical circles. Each of the circles has a
radius of 9 cm, and each passes through the
center of the other. Find the area of the
swimming pool.
a) 302.33 sq. m
b) 362.55 sq. m
c) 398.99 sq. m
d) 409.44 sq. m
544. A circle of radius 5 cm has a chord
which is 6 cm long. Find the area of the
circle concentric to this circle and tangent to
the given chord.
a) 14 π
b) 16 π
c) 9 π
d) 4 π
545. The diagonals of a rhombus are 10 cm
and 8 cm, respectively. Its area is:
a) 10 sq. cm
b) 50 sq. cm
c) 60 sq. cm
d) 40 sq. cm
546. The diagonals of a parallelogram are 10
cm and 16 cm, respectively, if one of its side
measures 6 cm, what is the area?
a) 59.92 sq. cm
b) 65.87 sq. cm
d) 69.56 sq. cm
d) 78.56 sq. cm
547. Given a cyclic quadrilateral whose
sides are 4 cm, 5cm, 8cm and 11cm. its area
is:
a) 40.25 sq. cm
b) 48.65 sq. cm
c) 50.25 sq. cm
d) 60.25 sq. cm
548 How many cubic meters is 100 gallons
of liquid?
a) 1.638
b) 37.85
c) 3.7850
d) 0.37854
549. How many cubic meters is 100 cubic
feet of liquid?
a) 3.785
b) 28.31
c) 37.85
d) 2.831
550. The volume of a sphere is 904.78 m^3.
Find the volume of the spherical segment of
height 4 m.
a) 234.57 m^3
b) 256.58 m^3
c) 145.69 m^3
d) 124.58 m^3
551. A sector of radius of 6 cm and central
angle of 600 is bent to form a cross. Find the
volume of the cone.
a) (35)^(1/2) π / 3
b) π (35)^(1/2)
c) 35 π / 3^(1/2)
d) 35 π / 3
552. A spherical wedge of a sphere of radius
10 cm has an angle of 400. Its volume is:
a) 523.42 cm^3
b) 465.42 cm^3
c) 683.42 cm^3
d) 723.45 cm^3
553. If a solid steel ball is immersed in an
eight cm diameter cylinder, if displaces
water to a depth of 2.25 cm. The radius of
the ball is:
a) 3 cm
b) 6 cm
c) 9 cm
d) 12 cm
554. The volume of a cube is reduced by
how much if all sides are halved?
a) 1/8
b) 5/8
c) 6/8
d) 7/8
555. If 23 cm^3 of water are poured into a
conical vessel, it reaches a depth of 12 cm.
How much water must be added so that the
depth reaches 18 cm?
a) 95 cm^3
b) 100 cm^3
c) 54.6 cm^3
d) 76.4 cm^3
556. A cylindrical tank, lying horizontally,
0.90 m in diameter and 3 m long is filled to
a depth of 0.60 m. How many gallons of
gasoline does it contain?
a) 250
b) 360
c) 300
d) 270
557. A closed cylindrical tank is 8 ft long
and 3 ft in diameter. When lying in a
horizontal position, the water is 2 feet deep.
If the tank is in the vertical position, the
depth of the water tank is:
a) 5.67 m
b) 5.82 m
c) 5.82 ft
d) 5.67 ft
558. The surface area of a sphere is 4πr^2.
Find the percentage increase in its diameter
when the surface area increases by 21%.
a) 5%
b) 10%
c) 15%
d) 20%
559. Find the percentage increase in volume
of a sphere if its surface area is increased by
21%.
a) 30.2%
b) 33.1%
c) 34.5%
d) 30.9%
560. Determine the estimated weight of steel
plate size ¼ x 4 x 8.
a) 184.4 kg
b) 148.7 kg
c) 327 kg
d) 841 kg
561. The no. of board feet in a plank 2 in.
thick, 6 in. wide and 20 ft long is:
a) 15
b) 30
c) 20
d) 25
562. Determine the volume of a right
truncate triangle prism with the following
dimensions: Let the corners of the triangular
base be defined by A, B ad C. The length
AB=11ft, BC=10ft and CA=13ft. The sides
at A, B and C are perpendicular to the
triangular base and have the height of 8.6ft,
7.1ft and 5.5ft, respectively.
a) 377 ft^3
b) 337 ft^3
c) 358 ft^3
d) 389 ft^3
563. A right circular conical vessel is
constructed to have a volume of 100,000
liters. Find the diameter if depth is to be
1.25 times the diameter.
a) 6.736 m
b) 7.632 m
c) 8.24 m
d) 9.45 m
564. A hollow sphere with an outer radius of
32 cm is made of a metal weighing 8 grams
per cubic cm. The weight of the sphere is
150 kg so that the volume of the metal is
24,000 cubic cm. Find the inner radius.
a) 30 cm
b) 35 cm
c) 40 cm
d) 45 cm
565. A circular cylindrical tank, axis
horizontal, diameter 1 meter, and length 2
meters, is filled with water to a depth of 0.75
meters. How much water is in the tank?
a) 2.578 m^3
b) 2.125 m^3
c) 1.2638 m^3
d) 1.0136 m^3
566. A machine foundation has the shape of
a frustrum of a pyramid with lower base 6m
x 2m, upper base 5.5m x 1.8m, and altitude
of 1.5m. Find the volume of the foundation.
a) 12.5 m^3
b) 14.2 m^3
c) 15.6 m^3
d) 16.4 m^3
567. An elevated water tank is in the form a
circular cylinder with diameter of 3 m and a
hemispherical bottom. The total height of
the tank is 5 m. Water is pumped into the
tank at a rate of 30 gallons per minute. How
long will it take to fully fill the tank starting
empty?
a) 4.668 hrs
b) 5.468 hrs
c) 7.725 hrs
d) 9.245 hrs
568. The intercept form for algebraic
straight equation:
a) a/x + y/b = 1
b) y = mx + b
c) Ax + By + C = 0
d) x/a + y/b = 1
569. Find the slope of the line y-x=5.
a) 1
b) 5+x
c) -1/2
d) ¼
570. Find the equation of the line that passes
through the points (0,0) and (2,-2).
a) y=x
b) y=-2x+2
c) y=-2x
d) y=-x
571. Find the equation of the line with
slope=2 and y-intercept=-3.
a) y=-3x+2
b) y=2x-3
c) y=2/3x+1
d) y=2x+3
572. The equation y=a1+a2x is an algebraic
expression for which of the following:
a) A cosine expansion
b) projectile motion
c) a circle in polar form
d) a straight line
573. In finding the distance, d, between two
point, which equation is the appropriate one
to use?
a) d=((x1-x2)^2 + (y2-y1)^2)^(1/2)
b) d=((x1-y1)^2 + (x2-y2)^2)^(1/2)
c) d=((x1^2 – x2)^2 + (y1^2 - y2^2))^(1/2)
d) d=((x2-x1)^2 + (y2-y1)^2)^(1/2)
574. The slope of the line 3x + 2y + 5 = 0 is:
a) -2/3
b) -3/2
c) 3/2
d) 2/3
575. Find the area of the circle whose center
is at (2,-5) and tangent to the lien 4x+3y-8=0.
a) 6π
b) 3 π
c) 9 π
d) 12 π
576. Given the equation of the parabola: y^2
– 8x -4y -20 =0. The length of its latus
rectum is:
a) 2
b) 4
c) 6
d) 8
577. Find the equation of the tangent to the
circle x^2 + y^2 – 34 = 0 through point (3,5).
a) 3x+5y-34=0
b) 3x-5y-34=0
c) 3x+5y+34=0
d) 3x-5y+34=0
578. If the distance between the points (8,7)
and (3,y) is 13, what is the value of y?
a) 5
b) -19
c) 19 or -5
d) 5 or -19
579. Which of the following is
perpendicular to the line x/3 + y/4 =1?
a) x-4y-8=0
b) 4x-3y-6=0
c) 3x-4y-5=0
d) 4x+3y-11=0
580. The two straight lines 4x-y+3=0 and
8x-2y+6=0
a) intersects at the origin
b) are coincident
c) are parallel
d) are perpendicular
581. A line which passes through (5,6) and
(-3,-4) has an equation of:
a) 5x+4y+1=0
b) 5x-4y-1=0
c) 5x-4y+1=0
d) 5x+4y-1=0
582. The equation of the line through (1,2)
parallel to the line 3x-2y+4=0.
a) 3x-2y+1=0
b) 3x-2y-1=0
c) 3x+2y+1=0
d) 3x+2y-1=0
583. Find the area of the polygon which is
enclosed by the straight lines x-y=0, x+y=0,
x-y=2a and x+y=2a.
a) 2a^2
b) 4a^2
c) 2a
d) 3a^2
584. Find the equation of the circle with
center at (2, -3) and radius of 4.
a) x^2 + y^2 -6x + 4y + 3 = 0
b) x^2 + y^2 -4x + 6y - 3 = 0
c) x^2 + y^2 -6x + 4y - 3 = 0
d) x^2 + y^2 -2x + 3y - 1 = 0
585. Find the area of the curve whose
equation is : 2x^2 – 8x + 2y^2 + 12y = 1.
a) 35.4 sq. units
b) 39.2 sq. units
c) 42.4 sq. units
d) 44.2 sq. units
586. Find the area of the curve whose
equation is : 9x^2 – 36x + 25y^2 = 189.
a) 41.7 sq. units
b) 43.4 sq. units
c) 46.2 sq. units
d) 47.1 sq. units
587. Given the curve Ax^2 + By^2 + F = 0.
It passes through the points (4,0) and (0,3).
Find the value of A, B and F.
a) 9,16,144
b) 9,16,121
c) 3,4,112
d) 3,4,144
588. A straight line passes through (2,2)
such that the length of the line segment
intercepted between the coordinate axis is
equal to the square root of 5. Find the
equation of the straight line.
a) 4x-y-2=0
b) x-4y-2=0
c) 2x-y-2=0
d) 2y-x-4=0
589. Find the area of the circle whose
equation is : 2x^2 – 8x + 2y^2 + 12y = 1.
a) 24.4 sq. units
b) 34.2 sq. units
c) 42.4 sq. units
d) 54.2 sq. units
590. Find the area of the curve whose
equation is : 9x^2 – 36x + 25y^2 = 189.
a) 27.2 sq. units
b) 32.8 sq. units
c) 47.1 sq. units
d) 75.4 sq. units
591. What is the first derivative with respect
to x of the function G(x) = 4 * 9^(1/2) ?
a) 0
b) 4/9
c) 4
d) 4(9^(1/2))
592. If a is a simple constant, what is the
derivative of y = x^a?
a) ax
b) x^(a-1)
c) a x^(a-1)
d) (a-1)x
593. Find the derivative of F(x) = [x^3 – (x1)^3]^3.
a) 3x^2 – 3(x-1)^2
b) 3[x^3 – (x-1)^3]^2
c) 9[x^3 – (x-1)^3][x^2 – (x-1)^2]
d) 9[x^3 – (x-1)^3]^2 [x^2 – (x-1)^2]
594. Differentiate f(x) = [2x^2 +4x
+1]^(1/2)
a) 2x+2
b) ½[2x^2 + 4x + 1]^(1/2)
c) (2x + 2)/ [2x^2 +4x +1]^(1/2)
d) (4x + 4)/ [2x^2 +4x +1]^(1/2)
595. Find the second derivative of y = (x^2
+ x^-2)^(1/2)
a) 1 - 2x^-3
b) 1 - 6x^4
c) 3
d) 6 / x^4
596. If y=cos x, what is dy/dx?
a) sec x
b) – sec x
c) csc x
d) – sin x
597. What is the slope of the graph y = -x^2
at the point (2,3)?
a) -4
b) -2
c) 1
d) 3
598. Given the function f(x) = x^3 – 5x + 2,
find the value of the first derivative at x=2.
a) 2
b) 3x^2 – 5
c) 7
d) 8
599. Find the slope of the tangent to a
parabola y = x^2, at a point on the curve
where x=1/2.
a) 0
b) 1/2
c) -1/2
d) 1
600. What is the slope of the curve y = x^2 4x as it passes through the origin?
a) 0
b) -3
c) -4
d) 4
601. Find the slope of the line tangent to the
curve y = x^3 – 2x + 1 at the point (1,2).
a) 1/4
b) 1/3
c) 1/2
d) 1
602. Determine the equation of the line
tangent to the graph y = 2x^2 + 1, at the
point (1,3).
a) y = 2x + 1
b) y = 4x - 1
c) y = 2x - 1
d) y = 4x + 1
603. Given Y1 = 4x + 3 and Y2 = x^2 + C,
find C such that Y2 is tangent to Y1.
a) 2
b) 4
c) 5
d) 7
604. The distance of a body travels is a
function of time and is given by x(t) = 18t +
9t^2. Find its velocity at t=2.
a) 20
b) 24
c) 36
d) 54
605. If x increases uniformly at the rate of
0.001 feet per second, at what rate is the
expression (1+x)^3 increasing when x
becomes 9 feet?
a) 0.001
b) 0.003
c) 0.3
d) 1.003
606. A spherical balloon is being filled with
air at a rate of 1 cubic foot per second.
Compute the time rate of rate of the surface
area of the balloon at the instant when its
volume is 113.1 cubic feet.
a) 0.67 ft^2 / s
b) 1.73 ft^2 / s
c) 3.0 ft^2 / s
d) 3.7 ft^2 / s
607. What is the maximum of the function y
= -x^3 +3x for x=-1?
a) -2
b) -1
c) 0
d) 2
608. The cost C of a product is a function of
the quantity x, of the product: C(x) = x^2 –
4000x + 50. Find the quantity for which the
cost is minimum.
a) 1000
b) 1500
c) 2000
d) 3000
609. Compute the following limit Lim
x+2
x →∞
x-2
a) 0
b) 1
c) 2
d) ∞
610. Find the equation of the tangent to the
ellipse: 4x^2 + 9y^2 = 40 at point (1,-2).
a) 2x – 9y – 20 = 0
b) 9x + 5y + 2 = 0
c) 9x – 2y + 20 = 0
d) 2x + 9y +20 = 0
611. Find the equation of the tangents to the
graph y = x^3 + 3x^2 – 15x – 20 at the
points of the graph where the tangents to the
graph have a slope of 9.
a) 9x + y + 70 = 0
b) 9y + x + 60 = 0
c) 9x – y – 48 = 0
d) x - y - 9 = 0
612. A rectangular field to contain a given
area is to be fenced off along a straight river.
If no fencing is needed along the river, show
that the least amount of fencing will be
required when the length of the field is twice
its width.
a) L = 3W
b) L = 4W
c) L = W
d) L = 2W
613. Find the shape of the largest rectangle
that can be inscribed in a given circle.
a) Trapezoid
b) Rectangle
c) Parallelogram
d) Square
614. Divide the number 60 into two parts so
that the product P of one part and the square
of the other is a maximum.
a) 30 and 30
b) 25 and 35
c) 50 and 10
d) 40 and 20
615. What is the maximum volume of a box
that is constructed from a piece of cardboard
16 inches square by cutting equal squares
out of the corners and turning up the sides.
a) 303.4 in^3
b) 404.5 in^3
c) 202.2 in^3
d) 101.1 in^3
616. A square sheet of galvanized iron, 100
cm x 100 cm will be used in making an
open-top container by cutting a small square
from each corner and bending up the sides.
Determine how large the square should be
cut from each corner in order to obtain the
largest possible volume.
a) 16 2/3 cm x 16 2/3 cm
b) 11 ½ cm x 11 ½ cm
c) 12 1/3 cm x 12 1/3 cm
d) 14 ¼ cm x 14 ¼ cm
617. The sum of two positive numbers is 36.
What are the numbers if their product is to
be the largest possible?
a) 10 and 10
b) 15 and 15
c) 12 and 12
d) 18 and 18
618. A bus company charges P85 per
passenger from Manila to Baguio for 100 or
less passengers. For group tours, the
company allows for P0.50 discount of the
ticket price for every passenger in excess of
100. How many passengers give the
maximum income?
a) 110
b) 150
c) 120
d) 135
619. A tinsmith wishes to make a gutter of
maximum cross-section (carrying capacity)
whose bottom and sides are each 6 inches
wide and whose sides have the same slope.
What will be the width at the top?
a) 10 in
b) 12 in
c) 8 in
d) 14 in
620. A lot is in the shape of a quadrant of a
circle of radius 100 meters. Find the area of
the e largest rectangular building that can be
constructed inside the lot.
a) 2500 m^2
b) 7500 m^2
c) 5000 m^2
d) 9000 m^2
621. The cost of setting up a geothermal
power plant is P10M for the first MW,
P11M for the second MW, P12M for the
third MW, etc., the other expenses (land
rights, desing fee, etc.) amount to P50M. If
the expected annual income per MW is 2M,
find the plant capacity that will yield a
maximum rate of return of investment.
a) 8 MW
b) 10 MW
c) 9 MW
d) 14 MW
622. If the fuel cost to run a boat is
proportional to the square of her speed and
is P25 per hour for a speed of 30 kph, find
the most economical speed to run the boat,
other expenses independent from the speed
amount to P100 per hour and the distance is
200 km.
a) 60 kph
b) 100 kph
c) 70 kph
d) 30 kph
623. The strength of a rectangular beam is
proportional to the breadth and the square of
the depth. Find the dimensions of the
strongest beam that can be cut from a log 30
cm in diameter.
a) b = 17.32 cm, h = 24.49 cm
b) b = 22.45 cm, h = 31.55 cm
c) b = 12.45 cm, h = 19.85 cm
d) b = 19.65 cm, h = 28.49 cm
624. Two posts, one 8 meters high and the
other 12 meters high, stand 15 meters apart.
They are to be stayed by wires attached to a
single stake at ground level, the wires
running to the tops of the posts. How far
from the shortest post should the stake be
placed, to use the least amount of wire?
a) 6m
b) 4m
c) 8m
d) 12m
625. A cylindrical glass jar has a metal top.
If the metal costs three times as much as the
glass per unit area, find the proportions of
the least costly jar that holds a given amount.
a) H = D
b) H = ¼ D
c) H = ½ D
d) H = 2D
626. The parcel post regulations limit the
size of a package to such a size that the
length plus the girth equals 6 feet.
Determine the volume of the largest
cylindrical package that can be sent by the
parcel post.
a) 2.546 cu. ft
b) 3.846 cu. ft
c) 4.234 cu. ft
d) 6.870 cu. ft
627. A cylindrical steam boiler is to be
constructed having a capacity of 30 cu.
meters. The material for the sides costs P430
per sq. meter and for the ends P645 per sq.
meter. Find the radius when the cost is least.
a) 1m
b) 1.47m
c) 2.1m
d) 1.7m
628. A boat is being towed toward a pier
which is 20 feet above the water. The rope is
pulled in at a rate of 6 ft/sec. How fast is the
boat approaching the base of the pier when
25 feet of rope remain to be pulled in?
a) 8 ft/sec
b) 12 ft/sec
c) 10 ft/sec
d) 15 ft/sec
629. A water tank is in the form of a right
circular cone with vertex down, 12 feet deep
and 6 feet across the top. Water is being
pumped into the tank at the rate of 10 cu.
ft/min. How fast is the surface of the water
in the tank rising when the water is 5 feet
deep?
a) 8 ft/min
b) 4 ft/min
c) 6 ft/min
d) 2 ft/min
630. Water is flowing out of a conical funnel
at a rate of 1 cu. in/sec. If the radius of the
funnel is 2 inches and the altitude is 6 inches,
find the rate at which the water level is
dropping when it is 2 inches from the top.
a) 0.179 in/sec
b) 1.245 in/sec
c) 0.889 in/sec
d) 2.225 in/sec
631. A helicopter is rising vertically from
the ground at constant rate of 15 ft per
second. When it is 250 feet off the ground, a
jeep passed beneath the helicopter travelling
in a straight line at a constant speed of 50
mph. Determine how fast is the distance
between them is changing after one second.
a) 34 ft/sec
b) 45 ft/sec
c) 38 ft/sec
d) 60 ft/sec
632. A plane flying north at 640 kph passes
over a certain town at noon and a second
plane going east at 600 kph is directly over
he same town 15 minutes later. If the planes
are flying at the same altitude, how fast will
they be separating at 1:15 PM?
a) 872 kph
b) 287 kph
c) 782 kph
d) 728 kph
633. The height of a cylindrical cone is
measured to be four meters which is equal to
its radius with a possible error of 0.04.
Determine the percentage error in
computing the volume.
a) 3%
b) 10%
c) 5%
d) 1%
634. Divide 94 into three parts such that
one-half the product of one pair, plus onethird the product of another pair, plus onefourth the product of the third pair may seem
to be a maximum value.
a) 42,40,12
b) 35,40,19
c) 38,40,16
d) 30,50,14
635. Integrate (3x^4 + 2x^3 + x^2 + 1)dx
a) (3x^3)/5 + (2x^2)/4 + x + 1 + c
b) (3x^5)/5 + (x^4)/2 + (x^3)/3 + x + c
c) (5x^5)/3 + 4x^2 + x + c
d) 3x^3 + 2x^4 + x^3 + x^2 + c
636. The integral of cos x dx with respect to
x:
a) –sin x +c
b) sin x +c
c) cos x +c
d) –cos x +c
637. Find the area under the curve y = 1/x
between the limits y=2 and y=10.
a) 1.61
b) 2.39
c) 3.71
d) 3.97
638. Fill in the blank in the following
statement: The integral of a function
between certain limits divided by the
difference in abscissas between those limits
gives the ___________ of the function.
a) average
b) middle
c) intercept
d) limit
639. Find the area bounded between y = 6x1 and y = x/4 + 3 by x=0 and the intersection
point.
a) 32/529
b) 16/23
c) 32/23
d) 64/23
640. If it is known that y=1 when x=1, what
is the constant of integration for the
following integral? Y(x) = (e^(2x) 2x)dx
a) c = 2 – e^2
b) c = 3 – e^2
c) c = 4 – e^2
d) ½(4 – e^2)
641. Evaluate integral of Tan (ln x) dx
x
a) ln cos (ln x) + c
b) ln sec (ln x) + c
c) 1/2 Tan^2 (ln x) + c
d) Tan (ln x) + c
642. Evaluate integral of cos x ln sin x dx
a) sin x (1- ln sin x) + c
b) sin x (1+ ln sin x) + c
c) sin x (ln sin x - 1) + c
d) ln sin x + c
643. Evaluate ∫ _e^x_dx_
1 + e^(2x)
a) 1/2 ln (1 + e^2x) + c
b) ln (1 + e^2x) + c
c) 1/2 (1 + e^2x)^2 + c
d) Arctan (e^x) + c
644. Evaluate ∫ _______dx__________
ln x^x [(ln x)^2 -1]^(1/2)
a) Arc sec (ln x) + c
b) 2/3[(ln x)^2 -1]^(3/2) + c
c) ln (ln x)^2 – 1 + c
d) Arc sin (ln x) + c
645. Evaluate ∫
a) 2
b) -2
c) -3
d) 3
646. Evaluate ∫
a) ln (10x + 1) + c
b) 1/10 ln(10x + 1) + c
c) ln(10x) + c
d) 10x + 1 + c
647. Evaluate ∫ 8dx / x^5
a) 8x^4 + c
b) 2x^4 + c
c) -2x^-4 + c
d) 2x^-4 + c
648. Evaluate ∫ (x^2)[(8 - x^3)^(1/2)]dx
a) -2/9 (8 – x^3)^(3/2) + c
b) -8 (8 – x^3)^(3/2) + c
c) 2/9 (8 – x^3)^(3/2) + c
d) -2/3 (8 – x^3)^(3/2) + c
649. Evaluate ∫ x^2a dx
a)
+c
b)
+c
c) x^a / a + c
d) x / 2a + c
650. Find the area bounded by the parabola
y = x^2, the x-axis and the lines x=1 and
x=3.
a) 8 2/3 sq. units
b) 7 1/2 sq. units
c) 9 1/4 sq. units
d) 12 sq. units
651. An ellipsoidal tank measuring 6 ft by
12 ft has its axis vertical, the axis of rotation
being the major axis. It is filled with water
to a depth of 7 feet. Find the amount of
water in the tank.
a) 111 cu. ft
b) 121 cu. ft
c) 141 cu. ft
d) 161 cu. ft
652. Find the area enclosed by the curves:
y^2 = 8x – 24 and 5y^2 = 16x.
a) 20 sq. units
b) 16 sq. units
c) 18 sq. units
d) 22 sq. units
653. An open cylindrical tank 3 feet in
diameter and 4.5 feet high is full of water. It
is then tilted until one-half of its bottom is
exposed. How many gallons of water was
spilled out?
a) 187.4 gal
b) 148.7 gal
c) 178.4 gal
d) 147.8 gal
654. The parabolic reflector of an
automobile headlight is 12 inches in
diameter and 4 inches depth. What is the
surface area in square inches?
a) 135.9 sq. in
b) 195.3 sq. in
c) 153.9 sq. in
d) 159.3 sq. in
655. A cistern in the form of an inverted
right circular cone is 20 meters deep and 12
meters diameter at the top. If the water is 16
meters deep in the cistern, find the work in
kJ in pumping out the water to a height of 10
meters above the top of the cistern.
a) 61,817 kJ
b) 55,004 kJ
c) 64,890 kJ
d) 68,167 kJ
656. A flour bag originally weighing 60 kg
is lifted through a vertical distance of 9
meters. While the bag is being lifted, flour is
leaking from the bag at such a rate that the
weight lost is proportional to the square root
of the distance travelled. If the total loss is
12 kg, find the amount of work in kJ done in
lifting the bag?
a) 4.59 kJ
b) 9.54 kJ
c) 5.94 kJ
d) 4.95 kJ
657. What is the name for a vector that
represents the sum of two vectors?
a) scalar
b) tensor
c) resultant
d) tangent
658. What is the acceleration of a body that
increases its velocity from 60 m/s to 110
m/s?
a) 5 m/s
b) 3.0 m/s
c) 4.0 m/s
d) 5.0 m/s
659. A cyclists on a circular track of radius r
= 250 m is travelling at 9 m/s. His speed in
the tangential direction increases at a rate of
1.5 m/s^2. What is the cyclist’s total
acceleration?
a) -1.53 m/s^2
b) 1.53 m/s^2
c) 2.3 m/s^2
d) -2.3 m/s^2
660. A bus weighing 9000N is switched to a
2% upgrade with a velocity of 40 kph. If the
train resistance is 950 N, how far up the
grade will it go?
a) 50 m on slope
b) 5 m on slope
c) 500 m on slope
d) 75 m on slope
661. Moment of inertia on SI is described
as:
a) N-m
b) N/m
c) kg/m
d) Farad/m
662. A solid disks flywheel (I=200 kg-m^2)
is rotating with a speed of 900 rpm. What is
the rotational KE?
a) 730 x 10^3 J
b) 680 x 10^3 J
c) 888 x 10^3 J
d) 1100 x 10^3 J
663. The weight of a mass 10 kg at a
location where the acceleration of gravity is
9.7 m/s^2 is:
a) 79.7 N
b) 77.9 N
c) 97.7 N
d) 977 N
664. A standard acceleration due to gravity
in SI unit:
a) 32.2 ft/s^2
b) 35.5 m/s^2
c) 9.81 ft/s^2
d) 9.81 m/s^2
665. A 50 kg sack is raised vertically 5
meters. What is the change in potential
energy?
a) 2452.5 kJ
b) 2.4525 kJ
c) 2452.5 N
d) 2.4525 kN
666. A shot is fired at an angle of 300 with
the horizontal and a velocity of 90 m/s.
Calculate the range of the projectile.
a) 715 km
b) 715 cm
c) 0.444 mi
d) 250 ft
667. A ball dropped from the top of a
building 60 meters elevation will hit the
ground with a velocity of:
a) 34.31 m/s
b) 31.34 m/s
c) 43.31 m/s
d) 33.41 m/s
668. What horizontal force P can be applied
to a 100 kg block in a level surface (µ =
0.20) that will cause an acceleration of 2.50
m/s^2?
a) 343.5 N
b) 224.5 N
c) 53.8 N
d) 446.2 N
669. Which of the following is not a vector
quantity?
a) mass
b) torque
c) displacement
d) velocity
670. The product of force and the time
during which it acts is known as:
a) impulse
b) momentum
c) work
d) impact
671. The property of the body which
measures its resistance to changes in motion:
a) acceleration
b) weight
c) mass
d) rigidity
672. The study of motion without reference
to the forces which causes motion is known
as:
a) kinetics
b) dynamics
c) statics
d) kinematics
673. The branch of physical science which
deals with state of rest or motion of bodies
under the action of forces is known as:
a) mechanics
b) kinetics
c) kinematics
d) statics
674. In physics, work is defined in terms of
the force acting through a distance. The rate
at which the work is done is called:
a) force
b) energy
c) power
d) momentum
675. The point through which the resultant
of the distributed gravity force passes
regardless of the orientation of the body in
space is known as:
a) center of inertia
b) center of gravity
c) center of attraction
d) moment of inertia
676. The momentum of a moving object is
the product of its mass(m) and velocity(v).
Newton’s second law of motion says that the
rate of change of momentum with respect to
time is:
a) power
b) energy
c) momentum
d) force
677. A coin is tossed vertically upward from
ground at a velocity of 12 m/s. How long
will the coin touch the ground?
a) 4.45 asec
b) 3.45 sec
c) 2.45 sec
d) 1.45 sec
678. A bullet is fired at an angle of 750 with
the horizontal with an initial velocity of 420
m/s. How high can it travel after 2 seconds?
a) 840 m
b) 792 m
c) 750 m
d) 732 m
679. A flywheel rotates at 150 rpm slowed
down to 120 rpm during the punching
portion of the cycle. Compute the angular
acceleration of the flywheel in rad/sec^2, if
time is 1 sec.
a) 3.14 rad/sec/sec
b) -3.14 rad/sec/sec
c) 4.31 rad/sec/sec
d) -4.31 rad/sec/sec
680. A shot is fired at an angle of 300 with
the horizontal and a velocity of 400 ft per
sec. Find the height of the projectile.
a) 600 ft
b) 622 ft
c) 700 ft
d) 680 ft
681. A projectile is fired with a velocity of
1600 fps and the target distance is 50,000 ft.
Determine the angle of elevation of the
projectile.
a) 38057’
b)32017’
c) 24032’
d) 19028’
682. Given the component velocities Vsubx
and Vsuby, what is the resultant velocity at t
= 3.
a) 19
b) 23
c) 21
d) 24
683. A 500 lbf acts on a block at an angle of
300 with respect to the horizontal. The block
is pushed 5 feet horizontally. What is the
work done by this force?
a) 2.936 kJ
b) 2,936 kJ
c) 3.396 kJ
d) 3,396 kJ
684. Traffic travels at 110 mph around a
banked highway curve with a radius of 2000
ft and f = 0.3. What banking angle to resist
the centrifugal force?
a) 5.330
b) 5.990
c) 6.660
d) 7.770
685. A plane dropped a bomb at an elevation
of 1000m from the ground intending to hit a
target which elevation is 200 m from the
ground. If the plane was flying at a velocity
of 300 kph, at what distance from the target
must the bomb be dropped to hit the target?
a) 1064 m
b) 1046 m
c) 1275 m
d) 1146 m
686. A projectile is launched from a level
plane at 300 from the horizontal with an
initial velocity of 1500 ft/sec. What is the
maximum height and maximum range the
projectile can reach?
a) 2772 m ; 18,500 m
b) 2727 m ; 18,885 m
c) 2266 m ; 18,994 m
d) 2663 m ; 18,449 m
687. A flywheel stops in 10 sec from a speed
of 80 rpm. Compute the number of turns the
flywheel makes before it stops.
a) 6.56 rev
b) 6.96 rev
c) 5.56 rev
d) 6.65 rev
688. An elevator weighing 4000 lb attains an
upward velocity of 20 fps in 5 sec with
uniform acceleration. What is the tension in
the supporting cables?
a) 4947 lbs
b) 4974 lbs
c) 4749 lbs
d) 4497 lbs
689. A gun is fired horizontally at a 10 kg
block of wood suspended at the end of a
cord. The block with the bullet embedded in
it rises vertically by 10 cm. Mass of bullet is
40 grams. Find the velocity of the bullet just
before it hit the block.
a) 354.1 m/s
b) 351.4 m/s
c) 341.5 m/s
d) 315.4 m/s
690. A body weighing 100 kg is hanging at
the end of a rope 5 m long. What horizontal
force is needed to move the body a
horizontal distance of 1m.
a) F = 24.1 kg
b) F = 22.4 kg
c) F = 21.4 kg
d) F = 20.4 kg
691. A light rail transit travels between two
terminals 1 km apart in a minimum time of 1
min. If the LRT cart accelerates and
decelerates at 3.4 m/s^2, starting from rest at
the first terminal and coming to stop at the
second terminal, find the maximum speed in
km per hr.
a) 63.9 kph
b) 64.9 kph
c) 65.9 kph
d) 66.9 kph
692. A body weighing 2000 kg is suspended
by a cable 20 meters and pulled 5 meters to
one side by a horizontal force. Find the
tension in the cable.
a) 2066 kg
b) 2660 kg
c) 5166 kg
d) 3020 kg
693. A body weighing 350 kg rests on a
plane inclined 300 with the horizontal. The
angle of static friction between the body and
the plane is 15 degrees. What horizontal
force P is necessary to hold the body from
sliding down the plane?
a) 93.7 kg
b) 73.9 kg
c) 97.3 kg
d) 119 kg
694. A 200 kg crate is on a 300 ramp. The
coefficient of friction between the crate and
the ramp is 0.35. If a force is applied to the
crate horizontally, calculate the force F to
start the crate moving up the ramp.
a) 244 kg
b) 38 kg
c) 232 kg
d) 223 kg
695. A 600 N block rests on a 300 inclined
plane. The coefficient of static friction is
0.30 and the coefficient of kinetic friction is
0.20. If a force P is applied to the block
horizontally, find the value of P needed to
keep the block moving up the plane.
a) 257 N
b) 750 N
c) 275 N
d) 527 N
696. A steam pipe weighing 200 kg per
meter will cross a road by suspension on a
cable anchored between supports 6 meters
apart. The maximum allowable sag of the
cable is 50 cm, calculate the length of the
cable.
a) 2.5 m
b) 3.6 m
c) 6.1 m
d) 9.5 m
697. A parabolic cable has a span of 400 feet.
The difference in elevation of the supports is
10 feet and the lowest point of the cable is 5
feet below the lower support. If the load
supported by the cable is 12 lbs per
horizontal foot, find the maximum tension in
the cable.
a) 25,902 lbs
b) 27,857 lbs
c) 29,345 lbs
d) 34,876 lbs
698. A tripod whose legs are each 4 meters
long supports a load of 1000 kg. The feet of
the tripod are the vertices of a horizontal
equilateral triangle whose side is 3.5 m.
Determine the load on each leg.
a) 256 kg
b) 386 kg
c) 296 kg
d) 458 kg
699. Two cars A and B accelerate from a
stationary start. The acceleration of A is 4
ft/sec^2 and that of B is 5 ft/sec^2. If B was
originally 20 feet behind A , how long will it
take B to overtake A.
a) 18.6 sec
b) 10 sec
c) 12.5 sec
d) 6.32 sec
700. Two cars, A and B, are travelling at the
same speed of 80 km/hr in the same
direction on a level road, with car A 100
meters ahead of car B. Car A slows down to
make a turn decelerating at 7 ft/sec^2. In
how many seconds will B overtake A.
a) 6.96 sec
b) 5.55 sec
c) 7.85 sec
d) 9.69 sec
701. In a 25 storey office building, the
elevator starting from rest at first floor, is
accelerated at 0.8 m/sec^2 for 5 seconds
then continues at constant velocity for 10
seconds more and is stopped in 3 seconds
with constant deceleration. If the floors are 4
meters apart, at what floor does the elevator
stop?
a) 12th floor
b) 14th floor
c) 10th floor
d) 15th floor
702. A stone is dropped from a cliff into the
ocean. The sound of the impact of the stone
on the ocean surface is heard 5 seconds after
it is dropped. The velocity of sound is 1100
fps. How high is the cliff?
a) 352.5 ft
b) 255.5 ft
c) 325.5 ft
d) 335.5 ft
703. Water drips from a faucet at a rate of 8
drops per second. The faucet is 18 cm above
the sink. When one drop strikes the sink,
how far is the next drop above the sink?
a) 15.8 cm
b) 12.5 cm
c) 18.5 cm
d) 25.6 cm
704. Bombs from a plane drop at a rate of
one drop per second. Calculate the vertical
distance after two bombs after the first had
dropped for 7 seconds. Assume freely
falling body with g = 9.8 m/sec^2.
a) 37.6 m
b) 73.6 m
c) 63.7 m
d) 76.3 m
705. A weight is dropped from a helicopter
that is rising vertically with a velocity of 6
m/sec. If the weight reaches the ground in
15 seconds, how high above the ground was
the helicopter when the weight was
dropped?
a) 1100 m
b) 1013 m
c) 1580 m
d) 1130 m
706. A bomber flying at a horizontal speed
of 800 kph drops a bomb. If the bomb hits
the ground in 20 seconds, calculate the
vertical velocity of the bomb as it hit the
ground.
a) 169 m/sec
b) 196 m/sec
c) 175 m/sec
d) 260 m/sec
707. A flywheel starting from rest develops
a speed of 400 rpm in 30 seconds. How
many revolutions did the flywheel make in
30 seconds it took to attain 400 rpm.
a) 100 rev
b) 150 rev
c) 120 rev
d) 360 rev
708. A 100 kg block of ice is released at the
top of a 300 incline 10 meters above the
ground. If the slight melting of the ice
renders the surface frictionless, calculate the
velocity at the foot of the incline.
a) 30 m/sec
b) 24 m/sec
c) 14 m/sec
d) 10 m/sec
709. What drawbar pull is required to
change the speed of a 120,000 lb car from
15 mph to 30 mph on a half mile while the
car is going up a 1.5% upgrade? Car
resistance is 10 lb/ton.
a) 3425 lbs
b) 3542 lbs
c) 3245 lbs
d) 4325 lbs
710. A body weighing 200 kg is being
dragged along a rough horizontal plane by a
force of 45 kg. If the coefficient of friction is
assumed to be 1/12 and the line pull makes
an angle of 180 with the horizontal, what is
the velocity acquired from rest in the first 3
meters.
a) 2.8 m/sec
b) 3.1 m/sec
c) 3.5 m/sec
d) 4.9 m/sec
711. A 50 kN Diesel Electric Locomotive
(DEL) has its speed increased from 30 kph
to 120 kph in a distance of 1 km while
ascending a 3% grade. What constant trust
(drawbar pull) parallel to the surface of the
railway must be exerted by the wheel? The
total frictional resistance is 30 N/kN of DEL
weight.
a) 5.655 kN
b) 7.889 kN
c) 6.556 kN
d) 7.996 kN
712. Water is flowing through a cast iron
pipe at the rate 3500 GPM. The inside
diameter of pipe is 6 in. Find the flow
velocity?
a) 39.7 m/s
b) 32.5 m/s
c) 12.1 m/s
d) 17.84 m/s
713. Find the water pressure reading if
manometer is 0.45 m Hg. Mercury is 13.6
times heavier than water.
a) 60 kPa
b) 50 kPa
c) 70 kPa
d) 65 kPa
714. Determine the velocity of the fluid in a
tank at the exit, given that surface h1 = 1m
and h2 = 100 cm.
a) 3.9 m/s
b) 4.2 m/s
c) 4.8 m/s
d) 5.6 m/s
715. Water is flowing at a rate of 3500 GPM.
The inside radius is 8cm and coefficient of
friction is 0.0181. What is the pressure drop
over a length of 50 m?
a) 317 kPa
b) 301 kPa
c) 341 kPa
d) 386 kPa
716. The unit of kinematic viscosity in SI is
described as:
a) Newton per meter
b) Watt per meter
c) Pascal second
d) Sq. m per sec
717. Which of the following is not a unit of
viscosity?
a) Pa-sec
b) Poise
c) stoke
d) Dyne
718. Which of the following describes
laminar flow?
a) NR = 2180
b) NR = 1989
c) NR = 4100
d) NR = 2100
719. Water is flowing in a pipe with radius
of 30 cm at a velocity of 12 m/s. The density
and viscosity of water are: Density = 1000
kg/m^3 ; Viscosity = 1.12 Pa-s. What is the
Reynold’s number?
a) 6428
b) 6386
c) 4534
d) 2187
720. What is the density of a solid that
weights 194 N (43.9 lbf) in air and 130 N
(29.4 lbf) in water?
a) 3534.50 kg/m^3
b) 3031.25 kg/m^3
c) 2989.34 kg/m^3
d) 3235.96 kg/m^3
721. What is the buoyant force of a body
that weighs 100 kg in air and 70 kg in
water?
a) 234.17 N
b) 329.68 N
c) 285.6 N
d) 294.3 N
722. A venturi meter with a 15 cm throat is
installed in a 20 cm pipe which inclined
upward at an angle of 300 to the horizontal.
If the distance between pressure tape along
the pipe is 1 m, the differential pressure is
65 kPA. What is the discharge of water in
m^3/s? Assume coefficient of 0.995.
a) 0.109 m^3/s
b) b) 0.536 m^3/s
c) 0.233 m^3/s
d) 0.0123 m^3/s
723. What is the pressure of point A in the
tank if h = 2 feet from the water level? (g =
32.2 ft/s^2 and ρ = 1.94 slug/ft^3).
a) 75 lbf/ft^2
b) 85 lbf/ft^2
c) 100 lbf/ft^2
d) 125 lbf/ft^2
724. Steam with an enthalpy of 700 kcal/kg
enters a nozzle and leaves with an enthalpy
of 650 kcal/kg. Find the initial velocity if
steam leaves with a velocity of 700 m/s,
assuming the nozzle is horizontal and
disregarding heat losses.
a) 276 m/s
b) 296 m/s
c) 376 m/s
d) 267 m/s
725. The flow of water through a cast iron
pipe is 6000 GPM. The pipe is 1 ½ ft
nominal diameter. What is the velocity of
water?
a) 8.56 ft/sec
b) 7.56 ft/sec
c) 6.56 ft/sec
d) 5.56 ft/sec
726. A perfect venturi with throat diameter
of 2 in is placed horizontally in a pipe with a
2 inches is placed horizontally in a pipe with
a 6 inches inside diameter. What is the
difference between the pipe and venturi
throat static pressure if the mass flow rate of
water is 100 lb/sec.
a) 38.8 lb/in^2
b) 36.8 lb/in^2
c) 37.8 lb/in^2
d) 35.8 lb/in^2
727. A deposit of P1000 is made in a bank
account that pays 8% interest compounded
annually. Approximately how much money
will be in the account after 10 years?
a) P2160
b) P2345
c) P1860
d) P1925
728. You need P4000 per year for your
college four year course. Your father
invested P5000 in 7% account for your
education when you were born. If you
withdraw P4000 at the end of your 17th,
18th,19th, and 20th birthday, how much
money will be left in the account at the end
of the 21st year?
a) P2500
b) P3400
c) P1700
d) P4000
729. What is the acid test ratio?
a) The ratio of the owners equity to the total
current liabilities
b) The ratio of all assets to total liabilities
c) The ratio of gross margin to operating
sales and administrative expenses
d) The ratio of current assets (exclusive of
inventory) to total current liabilities
730. An interest rate is quoted as being 7
1/2 % compounded quarterly. What is the
effective annual interest rate?
a) 21.8 %
b) 7.71%
c) 7.22%
d) 15.78%
731. Mr. Ayala borrows P100,000.00 at 10%
effective annual interest. He must pay back
the loan over 30 years with uniform monthly
payments due on the first day of each month.
What does Mr. Ayala pay each month?
a) P870
b) P846
c) P878
d) P839
732. A steel drum manufacturer incurs a
yearly fixed operating cost of P200,000.
Each drum manufactured cost P160 to
produce and sells for P200. What is the
manufacturers break-even sales volume in
drums per year?
a) 1250
b) 2500
c) 1000
d) 5000
733. The length of time, usually in years, for
the cumulative net annual profit to equal the
initial investments is called:
a) receivable turnover
b) return on investment
c) price earning ratio
d) pay back period
734. A local firm is establishing a sinking
fund for the purpose of accumulating a
sufficient capital to retire its outstanding
bonds at maturity. The bonds are
redeemable in 10 years, and their maturity
value is P150,000. How much should be
deposited each year if the fund pays interest
at the rate of 3%?
a) P12,547.14
b) P13,084.58
c) P14,094.85
d) P16,848.87
735. What is the formula for a straight line
depreciation rate?
a) 100% - %net salvage value over
estimated life
b) 100% net salvage value over estimated
service life
c) 100% net salvage value over estimated
service life
d) average net salvage value over estimated
service life
736. A machine is under consideration for
investment. The cost of the machine is
P25,000. Each year it operates, the machine
will generate a savings of P15,000. Given an
effective annual interest rate of 18%, what is
the discounted payback period, in years, on
the investment of the machine?
a) 1.75 years
b) 3.17 years
c) 1.67 years
d) 2.16 years
737. A businessman wishes to earn 7% on
his capital after payment of taxes. If the
income from an available investment will be
taxed at an average rate of 42%, what
minimum rate of return, before payment of
taxes, must the investment offer to be
justified?
a) 12.1 %
b) 10.7%
c) 11.1 %
d) 12.7 %
738. Liquid assets such as cash and other
assets that can be converted quickly into
cash such as accounts receivable, and
merchandise is called:
a) current assets
b) fixed assets
c) total assets
d) land and buildings
739. Instead of the profits being paid out to
the stockholders or owners as dividends,
they are retained in the business and used to
finance expansion. This is called:
a) retained earnings
b) flow back
c) bonds
d) deposits
740. A term used to describe payment of an
employee for time spent on the property of
the employer though not actually working at
the job, e.g. time spent changing clothes to
get ready for work or time spent travelling
from the plant entrance to the place of work.
a) portal-to-portal pay
b) down-time pay
c) call-in pay
d) lost time pay
741. A machine has an initial cost of
P50,000 and a salvage value of P10,000
after 10 years. What is the straight-line
method depreciation rate as a percentage of
the initial cost?
a) 10%
b) 8%
c) 12%
d) 9%
742. Fifteen years ago, P1000 was deposited
in a bank account, and today it is worth
P2370. The bank pays interest semi-annually.
What was the interest rate paid on this
account?
a) 4.9%
b) 5.8%
c) 5.0%
d) 3.8%
743. Company A purchases P200,000 of
equipment in year zero. It decides to use
straight-line depreciation over the expected
20 year life of the equipment. The interest
rate is 14%. If its average tax rate is 40%,
what is the present worth of the depreciation
tax held?
a) P3,500
b) P26,500
c) P98,700
d) P4,000
744. A product has a current selling price of
P325. If its selling price is expected to
decline at the rate of 10% per annum
because of obsolescence, what will be its
selling price four years hence?
a) P213.23
b) P202.75
c) P302.75
d) P156.00
745. You borrow P3500 for one year from a
friend at an interest rate of 1.5% per month
instead of taking a loan from a bank at a rate
of 18% per year. Compare how much money
will you save or lose on the transaction.
a) You will pay P155 more than if you
borrowed from the bank
b) You will save P55 by borrowing from
your friend
c) You will pay P85 more than if you
borrowed from the bank
d) You will pay P55 less than if you
borrowed from the bank
746. Instead of paying P100,000 in an
annual rent for offices space at the
beginning of each year for the next 10 years,
an engineering has decided to take out a 10
year P1,000,000 loan for a new building at
6% interest. The firm will invest P100,000
of the rent save and earn 18% annual interest
on that amount. What will be the difference
between the firm’s annual revenue and
expenses?
a) The firm will need P17,900 extra.
b) The firm will break even.
c) The firm will have P21,500 left over.
d) The firm will need P13,000 extra.
747. The peso amount as earned from an
investment or project is called:
a) ROI
b) Interest
c) ROR
d) Surplus
748. Those funds that are required to make
the enterprise or project a going concern:
a) Working capital
b) Accumulated amount
c) Banking
d) Principal or present worth
749. You borrowed the amount of P10,000
for 120 days at 30% per annum simple
interest. How much will be due at the end of
120 days?
a) P10,100
b) P11,000
c) P11,600
d) P12,000
750. You obtain a loan of P0.5 million at the
rate of 12% compounded annually in order
to build a house. How much must you pay
monthly to amortize a loan within a period
of five years?
a) P10,968
b) P11,968
c) P12,968
d) P13,968
751. An asset is purchased for P25,000. Its
estimated life is 10 years after which it will
be sold for P500. Find the depreciation for
the first three years using the sum of the
years digit.
a) P11,000.72
b) P13,007.72
c) P12,027.27
d) P13,027.72
752. If P10,000 is invested at the end of
each year for 6 years, at an annual interest of
10%, what is the total amount available
upon the deposit of the sixth payment?
a) P77,651
b) P80,156
c) P78,156
d) P77,156
753. The original cost of an equipment is
P50,000, the salvage value after 5 years is
P8,000, and the rate of interest on the
investment is 10%. Determine the capital
recovery per year.
a) P11,879.50
b) P12,897.50
c) P10,879.50
d) P11,379.50
754. A small shop in Leyte fabricates
portable threshers for palay producers in the
locality. The shop can produce each thresher
at a labor cost of P2000. The cost of
materials for each unit is P4500. The
variable costs amount to 800 per unit, while
fixed charges incurred per annum totals to
P90,000. If the portable threshers are sold at
P14,000 per unit, how many units must be
produced and sold per annum to break even?
a) 14 units
b) 17 units
c) 19 units
d) 21 units
755. You want to save an amount of
P100,000 at the end of 10 years. You are
given 8% interest compounded quarterly.
How much would you have to save per
month in order to accumulate the sum of
P100,000 ten years from now.
a) P864.50
b) P590.00
c) P648.50
d) P548.40
756. With an interest at 10% compounded
annually, after how many years will a
deposit now of P1000 become P1331?
a) 3 years
b) 4 years
c) 5 years
d) 6 years
757. What rate (%) compounded quarterly is
equivalent to 6% compounded semiannually?
a) 5.93
b) 5.99
c) 5.96
d) 5.9
758. Determine the break-even point in
terms of number of units produced per
month using the following data:
(the costs are in pesos per unit)
Selling price per unit
= 600
Total monthly overhead expenses
= 428,000
Labor cost
= 115
Cost of materials
= 76
Other variable cost
= 2.32
a) 1036
b) 1044
c) 1053
d) 1025
759. The present value of an annuity of ―R‖
pesos payable annually for 8 years, with the
first payment at the end of 10 years, is
P187,481.25. Find the value of R if money
is worth 5%.
a) P45,000
b) P44,000
c) P42,000
d) P43,000
760. The amount of P50,000 is deposited in
a bank. How much money are you going to
withdraw after 8 years at 8% compounded
annually?
a) P83,546
b) P85,456
c) P92,546
d) P97.856
761. A machine has an initial cost of
P300,000. Its salvage value after 5 years is
P30,000. What is the straight line
depreciation rate of the machine?
a) 25%
b) 23%
c) 18%
d) 15%
762. An asset is purchased for P120,000 and
it can be sold for P12,000. Its estimated life
is 10 years. Find the depreciation for the
second year using the sum-of-the-years digit
method.
a) P17,672
b) P17,850
c) P18,276
d) P19,636
763. A bank offers 2% effective monthly
interest. What is the effective annual rate?
a) 26.82%
b) 25.28%
c) 24.65%
d) 22.45%
764. How much must you invest today in
order to accumulate P20,000 at 8% after 6
years?
a) P20,004.50
b) P18,450.80
c) P15,305.60
d) P12,603.40
765. A machine that cost P1000 will save
P0.10 per unit produced. Maintenance cost
will be P100 annually. 2000 units are
produced annually. What is the payback
period at 12% interest?
a) 8 years
b) 9 years
c) 10 years
d) 12 years
766. An item is purchased for P100,000.
Annual cost is P18,000. Using 10%, what is
the capitalized cost of the perpetual service?
a) P220,000
b) P250,000
c) P265,000
d) P280,000
767. A car was bought at P549,492.13 with
14% down payment and the remaining
balance will be paid on installment basis
with a monthly payment of P12,000 for 60
months. Determine the interest rate
compounded annually.
a) 19.56%
b) 18.25%
c) 16.45%
d) 14.35%
768. A businessman wishes to earn 9% on
his capital after payment of taxes. If the
minimum rate of return, before payment of
taxes is 12.1 %. What is the available
average taxed rate of the income from a
businessman’s investment?
a) 25.6 %
b) 24.6%
c) 22.4%
d) 20.5%
769. A debt of P1000 is to be paid in five
equal yearly payments, each payment
combining an amortization installment an
interest at 8% on the previously unpaid
balance of the debt. What should be the
amount of each payment?
a) P365.50
b) P310.20
c) P290.60
d) P250.45
770. A father wishes to develop a fund for
his new born son’s college education. The
fund is to pay P5000 on the 18th, 19th 20th
and the 21st birthdays of his son. The fund
will be built up by the deposit of a fixed sum
on the son’s first to seventeenth birthdays. If
the fund earns 4%, what should the yearly
deposit into the fund be?
a) P985.44
b) P845.66
c) P795.65
d) P765.88
771. A man owns a building on which there
is a P100,000 mortgage which earns 6% per
annum. The mortgage is being paid for in 20
equal year-end payments. After making 8
payments, the man desires to reduce his
payments by refinancing the balance of the
debt with a 30-year mortgage at 8%, and to
be retired by equal annual payments. What
would be the reduction in the yearly
payment?
a) P2,225.70
b) P2,550.80
c) P2,985.30
d) P3,120.90
772. An engineer borrows P150,000 at 12%
effective annual interest. He must pay back
the loan over 25 years with uniform monthly
payments due on the first day of each month.
What is this monthly payment?
a) P1126
b) P1265
c) P1398
d) P1498
773. Funds are deposited in a savings
account at an interest rate of 8% per annum
compounded semi-annually. What is the
initial amount that must be deposited to
yield a total of P10,000 in 10 years?
a) P1458
b) P2550
c) P3875
d) P4564
774. A machinery has an initial cost of
P40,000 and results in an increase in annual
maintenance costs of P2000. If the
machinery saves the company P10,000 per
year, in how many years will the machine
pay for itself if compounding is considered?
(i = 7%)
a) 8 years
b) 9 years
c) 7 years
d) 11 years
775. How long will it take a sum of money
to double at a 5% annual percentage rate?
a) 14.2 years
b) 15.9 years
c) 18.4 years
d) 19.3 years
776. A sum of P1000 is invested now and
left for eight years, at which time the
principal is withdrawn. The interest that has
accrued is left for another eight years. If the
effective annual interest rate is 5%, what
will be the withdrawal amount at the end of
the 16th year?
a) P980
b) P830
c) P780
d) P706
777. How many horsepower is 746 kW?
a) 1 HP
b) 100 HP
c) 74.6 HP
d) 1000 HP
778. What is the origin of the energy
conservation equation used in flow system?
a) Newton’s First Law of Motion
b) Newton’s Second Law of Motion
c) First Law of Thermodynamics
d) Second Law of Thermodynamics
779. A volume of 560 cc of air is measured
at a pressure of 10 mm Hg vacuum and a
temperature of 200C. What will be the
volume at standard pressure and 00C?
a) 6.9 cc
b) 535.5 cc
c) 437.5 cc
d) 1071 cc
780. What is the specific weight of a liquid
substance if it specific weight relative to
water is 8.77 and the specific weight of
water is 62.4 lb per cubic foot?
a) 86.03 kN/m^3
b) 82.20 kN/m^3
c) 102.56 kN/m^3
d) 89.90 kN/m^3
781. Steam at a pressure of 12.5 MPa has a
specific volume of 1160 x 10^-6 m^3 per kg
and a specific enthalpy of 2560 kJ/kg. Find
the internal energy per mass of steam.
a) 2574.5 kJ per kg
b) 2545.5 kJ per kg
c) 2634.17 kJ per kg
d) 2560.50 kJ per kg
782. A heat engine (Carnot cycle) has its
intake and exhaust temperature of 2100C and
1200C respectively. What is its efficiency?
a) 42.86%
b) 34.85%
c) 16.34%
d) 18.63%
783. One kilogram of water is heated by
2000 Btu energy. What is the change in
temperature in 0K?
a) 55.6 0K
b) 54.1 0K
c) 50.4 0K
d) 48.5 0K
784. A pressure reading of 35 psi in kPa abs
is:
a) 427.3
b) 724
c) 273.4
d) 342.72
785. What conditions exists in a adiabatic
throttling process?
a) Enthalpy is variable
b) Enthalpy is constant
c) Entropy is constant
d) Volume is constant
786. The specific gravity of a substance is
the ratio of its density to the density of:
a) mercury
b) gas
c) air
d) water
787. What do you call the weight of the
column of air above the earth’s surface?
a) air pressure
b) aerostatic pressure
c) wind pressure
d) atmospheric pressure
788. An air bubble rises from the bottom of
a well where the temperature is 200C, to the
surface where the temperature is 320C. Find
the percent increase int eh volume of the
bubble if the depth of the well is 8.5 m.
Atmospheric pressure is 101,325 Pascals.
a) 45.5%
b) 72.5%
c) 89.76%
d) 91.34%
789. Gas being heated at constant volume is
undergoing the process:
a) isentropic
b) adiabatic
c) isometric
d) isobaric
790. What is the required heating energy in
raising the temperature of a given amount of
water when the energy applied is 1000 kwhr with heat losses at 25%?
a) 1000
b) 1500
c) 1333
d) 1250
791. What is the process that has no heat
transfer?
a) reversible
b) isothermal
c) polytropic
d) adiabatic
792. Heat normally flowing from a high
temperature body to a low temperature body
where in it is impossible to convert heat
without other effects is called the:
a) First Law of Thermodynamics
b) Second Law of Thermodynamics
c) Third Law of Thermodynamics
d) Zeroth Law of Thermodynamics
793. What equation applies in the first law
of thermodynamics for an ideal gas in a
reversible open steady state system?
a) Q – W = U2 – U1
b) Q + VdP = H2 – H1
c) Q - VdP = H2 – H1
d) Q - PdV = H2 – H1
794. Form of energy associated with kinetic
energy of the random motion of large
number of molecules:
a) internal energy
b) kinetic energy
c) heat of fusion
d) heat
795. Which of the following is a set of
standard condition of atmospheric air?
a) 1 atm, 255 0K, 22 cu./kg mole
b) 101.325 kPa, 273 0K, 22.4 cu./kg mole
c) 101.325 kPa, 273 0K, 23.66 cu./kg mole
d) 1 atm, 10 0C, 22.41 cu./kg mole
796. Steam flows into a turbine at a rate of
20 kg/s and 21 kw of heat/ are lost from the
turbine. Ignoring elevation and other energy
effects, calculate the power output from the
turbine if the energy input is 2850 kJ/kg and
energy output is 2410 kJ/kg.
a) 8800 kw
b) 8821 kw
c) 8779 kw
d) 8634 kw
797. What pressure of water is a column of
100 cm high equivalent to:
a) 9807 dynes/cm^2
b) 9807 N/m^2
c) 0.1 bar
d) 100 kPa
798. An engine has an efficiency of 26%. It
uses 2 gallons of gasoline per hour. Gasoline
has heating value of 20,500 Btu/lb and a
specific gravity of 0.80. What is the power
output of the engine?
a) 41.7 kw
b) 0.33 kw
c) 26.0 kw
d) 20.8 kw
799. A thermodynamic system which
undergoes a cyclic process during a positive
amount of work done by the system:
a) reversed Rankine cycle
b) heat pump
c) reversible-irreversible process
d) heat engine
800. In a constant temperature, closed
system process, 100 Btu of heat is
transferred to the working fluid at 1000F.
What is the change in entropy of the
working fluid?
a) 0.18 kJ/0K
b) 0.57 kJ/0K
c) 0.25 kJ/0K
d) 0.34 kJ/0K
801. If an initial volume of an ideal gas is
compressed to one-half of its original
volume and to twice its original temperature,
the pressure:
a) doubles
b) quadruples
c) remains constant
d) halves
802. (u + pv) is a quantity called:
a) flow energy
b) shaft work
c) enthalpy
d) internal energy
803. What horsepower is required to
isothermally compress 800 ft^3 per minute
of air from 14.7 psia to 120 psia?
a) 13,800 HP
b) 28 HP
c) 256 HP
d) 108 HP
804. A pressure of one bar is equivalent to:
a) 110 kPa
b) 14 psi
c) 720 mm Hg
d) 1,000,000 dynes/cm^2
805. A pressure reading of 4.5 kg/cm^2 is
equal to:
a) 441.40 kPaa
b) 451.60 kPaa
c) 542.72 kPaa
d) 582.92 kPaa
806. A water temperature rise of 380F in the
condenser is equivalent to:
a) 3.33 0C
b) 33.3 0C
c) 21.1 0C
d) 38.1 0C
807. A boiler installed where the
atmospheric pressure is 752 mm Hg has a
pressure of 12 kg/cm^2. What is the
absolute pressure in MPa?
a) 1.277 MPa
b) 1.772 MPa
c) 2.177 MPa
d) 3.771 MPa
808. An oil storage tank contains oil with
specific gravity of 0.88 and depth of 20
meters. What is the absolute pressure in
kPa?
a) 274
b) 247
c) 724
d) 742
809. A pressure tank for a water pump
system contains 2/3 water by volume when
the pressure is 10 kg/cm^2 gauge. What is
the absolute pressure at the bottom of the
tank if the water is 2 meters depth?
a) 1012 kPa
b) 1201 kPa
c) 1102 kPa
d) 1080 kPa
810. Convert 360F to temperature difference
to 0C.
a) 36
b) 40
c) 20
d) 25
811. At what temperature are the two
temperatures scales 0C and 0F equal?
a) -20 0C
b) -40 0C
c) -30 0C
d) 40 0C
812. The temperature inside a furnace is 320
0
C and the temperature of the outside/ is -
100C. What is the temperature difference in
0
F?
a) 495 0F
b) 549 0F
c) 594 0F
d) 645 0F
813. Convert 60 lbs/ft^3 to kN/m^3:
a) 9.426
b) 7.356
c) 8.956
d) 5.479
814. A boiler feed pump delivers 200,000 kg
of water per hour at 10 MPa and 2300C.
What is the volume flow rate in m^3/sec?
a) 0.0666
b) 0.0888
c) 0.0777
d) 0.0999
815. The radiator of a heating system was
filled with dry and saturated steam at 0.15
MPa after which the valves on the radiator
were closed. As a result of heat transfer to
the room, the pressure drops to 0.10 MPa.
What percentage of steam has condensed?
a) 31.6%
b) 25.4%
c) 36.1%
d) 45.7%
816. A throttling calorimeter receives a
sample of steam from a steam main in which
the pressure is 1 MPa. After throttling, the
steam is at 100 kPa and 120 0C. What is the
quality of steam in the steam main?
a) 96.9 %
b) 95.5%
c) 99.6%
d) 92.4%
817. Steam at 2.5 MPa and 320 0C expands
through a nozzle to 1.5 MPa at the rate of
10,000 kg/hr. If the process occurs
isentropically and the initial velocity is low,
calculate the exit area of the nozzle.
a) 853 x 10^-6 m^2
b) 358 x 10^-6 m^2
c) 835 x 10^-6 m^2
d) 583 x 10^-6 m^2
818. Water at a pressure of 10 MPa and the
temperature of 2300C is throttled to a
pressure of 1 MPa in an adiabatic process.
What is the quality after throttling?
a) 11.3%
b) 12.5%
c) 14.5%
d) 19.3%
819. An air compressor delivers air to an air
receiver having a volume of 2 m^3. At the
start, the air in the receiver is at atmospheric
condition of 250C and 100 kPa. After 5
minutes, the pressure of the air in the tank is
1500 kPa and the temperature is 600C. What
is the capacity of the compressor in m^3/min
of free air?
a) 4.97
b) 5.55
c) 6.95
d) 8.45
820. At the suction of an air compressor, in
which the conditions are 97.9 kPa and 270C,
the air flow rate is 10.3 m^3/min. What is
the volume flow rate at the free air
conditions of 100 kPa and 200C?
a) 7.635 m^3/min
b) 6.590 m^3/min
c) 9.848 m^3/min
d) 3.568 m^3/min
821. Steam at 5 MPa and 3500C enters a
turbine and expands isentropically to 0.01
MPa. If the steam flow rate is 100,000 kg/hr,
determine the turbine power.
a) 28.5 kw
b) 22.5 kw
c) 25.8 kw
d) 33.8 kw
MULTIPLE CHOICE
QUESTIONS IN
MATHEMATICS
Fausto Uy and Jimmy Ocampo
1. A sequence of numbers such that the same quotient is obtained by dividing a term by the
preceding term is called
A. arithmetic progression
B. harmonic progression
C. infinite progression
D. geometric progression
2.If x is an irrational number not equal to zero and x2 = N, then which of the following best
describes N?
A. N is a natural number.
B. N is any rational number.
C. N is a positive rational number.
D. N is a positive integral number.
3. In the expression an , the number n is referred to as the
A. power
B. exponent
C. degree
D. index
4. The polynomial 2x3 y + 8xyz4 – 3x2y3 has a degree of
A. 6
B. 3
C. 4
D. 8
5. The equations x + y = 2 and 2x + 2y = 8 are examples of equations which are
A. dependent
B. independent
C. conditional
D. inconsistent
6. A non-terminating and non-periodic decimal is
A. rational
B. prime
C. irrational
D. imaginary
7. The probability that an event is certain to occur is
A. greater than one
B. less than one
C. equal to one
D. equal to zero
8. Radicals can be added to form a single radical if they have the same radicand and the same
A. power
B. exponent
C. index
D. coefficient
9. A set of elements that is taken without regard to the order in which the elements are arranged
is called a
A. sequence
B. permutation
C. combination
D. progression
10. If b = 0, then the number a + bi is
A. complex
B. real
C. imaginary
D. irrational
11. How many prime numbers are there between 200 and 210?
A. one
B. three
C. none
D. two
12. In the expression
A. power
B. exponent
C. index
D. radicand
, the number n is called the
13. A harmonic progression is a sequence of numbers such that the reciprocals of the numbers
will form a
A. geometric progression
B. arithmetic progression
C. infinite progression
D. finite progression
14. If a, b and c are rational numbers and if b2 – 4ac is positive but not perfect square, then the
roots of the quadratic equation ax2 + bx + c = 0 are
A. real, irrational and unequal
B. real, rational and unequal
C. real, rational and equal
D. real, irrational and unequal
15. The equation xy = 0 implies that
A. x = 0 and y = 0
B. x = 0 or y = 0
C. x = 0 and y is not equal to zero
D. x = 0 or y is not equal to zero
16. Which of the following events are mutually exclusive?
A. event “Ace” and event “black card”
B. event “Queen” and event “heart”
C. event “Ten” and event “Spade”
D. event “diamond” and event “club”
17.Which of the following best describes (-3)1/2?
A. irrational number
B. pure imaginary number
C. natural number
D. complex number
18. It is a sequence of numbers such that the successive terms differ by a constant.
A. geometric progression
B. arithmetic progression
C. harmonic progression
D. infinite progression
19. If the discriminant of a quadratic equation is greater than zero, the roots of the equation are
A. real and equal
B. real and distinct
C. complex and unequal
D. imaginary and distinct
20. Which of the following terms is not rational in x?
A. 6x2
B. -4x
C. x4
D.
21. In the theory of sets, the relation (A ᴗ B)’ = A’ ᴖ B’ expresses which of the following laws
on set operations?
A. De Morgan’s Law
B. Involution Law
C. Complement Law
D. Identity Law
22. The set of odd integers is closed under the operation of
A. addition
B. subtraction
C. multiplication
D. division
23. Which of the following law states that the factors of a product may be grouped in any manner
without affecting the result?
A. commutative law
B. associative law
C. distributive law
D. inverse law
24. Which of the following terms has a degree of 4?
A. x4 y
B. xy4
C. 4xy
D. xy3
25. The product of two conjugate complex numbers is
A. a real number
B. an imaginary number
C. zero
D. an irrational number
26. The statement “The examinees are not more than 30 years old.” implies that the examinees
are
A. less than 30 years old
B. at least 30 years old
C. 30 years old or less
D. 30 years old or more
27. The closure property of numbers is not satisfied by the set of all integers under the operation
of
A. addition
B. multiplication
C. subtraction
D. division
28. The conditional probability of B given A is denoted symbolically by P(B/A). If P(B/A) =
P(B), then the events A and B are
A. dependent
B. independent
C. mutually inclusive
D. disjoint
29. What is the value of k that will make x2 – 28x + k a perfect square trinomial?
A. 100
B. 121
C. 144
D. 196
30. The roots of 6x2 + 7x + 34 = 0 are
A. real and equal
B. real and unequal
C. complex and unequal
D. pure imaginary
31. What is the conjugate of -6
A. 6
B. -6
C. 6i
D. -6i
32. Which of the following is true?
A.
B. (a + b)2 = a2 + b2
C. a / (b – c) = a/b – a/c
D.
33. Which of the following cannot be a probability value?
A. (0.99)4
B. 88/100
C.
D. (0.5)-1
34. How many subsets has the set {c, u, t, e}?
A. 12
B. 14
C. 16
D. 18
35. Using the remainder theorem, find the remainder when x6 – x + 1 is divided by x – 2.
A. 61
B. 62
C. 63
D. 64
36. What is the sum of the numerical coefficient of (2x – y)20?
A. zero
B. one
C. greater than one
D. less than one
37. How many subsets of one or more elements can be formed from a set containing 12
elements?
A. 4,096
B. 4,095
C. 4,094
D. 4,093
38. What is the product of
and
?
A. 6i
B. -6i
C. 6
D. -6
39. Which of the following is an irrational number?
A. (16)3/4
B. 0.0075
C. 1.36363636...
D. 3(5)1/2
40. Two prime numbers which differ by 2 are called prime twins. Which of the following pairs
of numbers are prime twins?
A. 1 and 3
B. 7 and 9
C. 17 and 19
D. 13 and 15
41. If A ᴖ B ᴖ C is not equal to zero, then which of the following notations refers to the set of
elements found in A and B but not in C?
A. A ᴖ B ᴗ C’
B. A ᴗ B ᴖ C’
C. A ᴖ B ᴖ C’
D. A ᴗ B ᴗ C’
42. Which of the following sequence is a geometric progression?
A. 16, 12, 8, ...
B. 16, 8, 2, ...
C. 16, 12, 9, ...
D. 16, 14, 12, ...
43. The point (x, y) where x = 2 and y = -x is in what quadrant?
A. first
B. second
C. third
D. fourth
44. Experiment:
A die is tossed.
Event:
A prime number results.
Which of the following is not an outcome of the event?
A. 1
B. 2
C. 3
D. 5
45. The logarithmic equation equivalent to 1/a = bc is
A. logc(1/a) = b
B. logb(1/a) = c
C. logc b = 1/a
D. logb c = 1/a
46. If P(A) = 0.78 and P(B) = 0.35, what is P(A’) + P(B’) ?
A. 0.83
B. 0.85
C. 0.87
D. 0.89
47. Which of the following is a polynomial in x ?
A. x -2 + x + 4
B.
+ 3x + 5
C. x3 + 2x + 3
D. 4/x + 3x + 1
48. If a set A has 1,024 subsets, how many elements does A contain?
A. 8
B. 9
C. 10
D. 11
49. Which of the terms in the expansion (y3 + y -1)10 will involve y2 ?
A. 6th term
B. 7th term
C. 8th term
D. 9th term
50. P(A) = 0.60 and P(B’) = 0.30 while P(AᴖB) = 0.15, find P(AᴖB’) by using Venn Diagram.
A. 0.90
B. 0.30
C. 0.45
D. 0.75
51. Evaluate (i – 1)8.
A. 16
B. -16
C. 16i
D. -16i
52. If 16 is 4 more than 3x, then 2x – 5 =
A. 2
B. 3
C. 4
D. 5
53. In the series 2, -4, 8, -16, x, -64, ..., what is x?
A. -24
B. -32
C. 24
D. 32
54. If a, b, 2b, -a, ... is an arithmetic progression, find the next term.
A. 2b – 3a
B. 3b – 2a
C. 2b + 3a
D. 2b + a
55. In how many ways can 6 boys be seated in a row?
A. 520
B. 620
C. 720
D. 820
56.
is true only if
A. x > 2y
B. x = 2y
C. x <= 2y
D. x >= 2y
57. Find the fourth proportional to 3, 5 and 21.
A. 27
B. 56
C. 65
D. 35
58. Give the value of –(-1/27)-2/3
A. 9
B. -9
C. 1/9
D. -1/9
59. Simplify (a -1 + b -1) / (ab) -1
A. ab
B. b + a
C. 1/ab
D. a/b
60. If a die is thrown once, what is the probability of getting a prime number?
A. 1/3
B. ¼
C. ½
D. 1/6
61. Which of the following are similar radicals
A.
and
B.
and
C.
and
D.
and
62. Evaluate x = log 2 8
A. 4
B. 3
C. 2
D. 1
63. What is the greatest common factor (GCF) of 48 and 72 ?
A. 12
B. 24
C. 36
D. 42
64. If x, y and 5x are three consecutive terms of an arithmetic progression whose sum is 81, find
x.
A. 9
B. 10
C. 11
D. 12
65. If f(x) = 2x3 – 3x + 1, then f(1) =
A. 0
B. 1
C. 2
D. 3
66. Find the sum to infinity of 3 -1, 3 -3, 3 -5, ...
A. 1/8
B. 3/8
C. 7/8
D. 5/8
67. Find the value of x if
3 2
1 x
= 10
A. 3
B. 4
C. 5
D. 6
68. Find the value of x in the series 1, 8, 27, x, 125, ...
A. 100
B. 81
C. 30
D. 64
69. Find the least common multiple (LCM) of 72x3y2, 108x2 y3 and 9x2 y
A. 108x3y3
B. 648x3 y3
C. 972x3 y3
D. 216x3y3
70. Evaluate (-1/27)-2/3 + (-1/32)-2/5
A. 6.25
B. 3.25
C. 9.25
D. 7.25
71. For what values of x will (x+3) < 2(2x+1)?
A. x=3
B. x>1/3
C. x<1/3
D. x=0
72. In how many ways can a man choose one or more of 7 ties?
A. 128
B. 127
C. 126
D. 125
73. If i =
, solve for x and y if x+2+4i = 5+(y-3)i
A. -3, 7
B. 3, -7
C. 3, 7
D. -3, -7
74. Determine the number of word of five different letters each that can be formed with the
letters of the word “VOLTAGE”.
A. 5,040
B. 2,520
C. 4,050
D. 2,520
75. If log A 10 = 25, find log10 A.
A. 3
B. 4
C. 5
D. 6
76. The set notation A-B is called the relative complement of B in A. It is equivalent to which of
the following
A. A ᴗ B’
B. A ᴖ B’
C. A’ ᴖ B
D. A’ ᴗ B
77. What is the sum of the first five prime numbers?
A. 17
B. 18
C. 28
D. 29
78. How many straight lines are determined by 8 points?
A. 28
B. 56
C. 36
D. 64
79. From a group of 10 men, in how many ways can we select a group of 6 men?
A. 120
B. 210
C. 200
D. 60
80. Find x if x = log2 (1/64)
A. -6
B. -5
C. -4
D. -3
81. If (x-2)i = y-3i, solve for x.
A. -4
B. -3
C. -2
D. -1
82. If 1/a, 1/b and 1/c are consecutive terms of an arithmetic progression, then b equals
A. 2ac/(a+c)
B. ac/(a+c)
C. (a+c)/2ac
D. (a+c)/ac
83. What is the third proportional to y/x and 1/x?
A. x/y
B. xy
C. y
D. 1/xy
84. (1/2) –x(8) –y is equal to
A. 23xy
B. 43y-x
C. 2x-3y
D. 4xy
85. A father is 27 years older than his son and 10 years from now, he will be twice as old as his
son. How old is his son now?
A. 15
B. 16
C. 17
D. 18
86. Find the 7th term of the geometric progression
,
,
, ...
A.
B.
C.
D.
87. If x = (2)^(log2 x), find the value of x.
A. 3
B. 4
C. 5
D. 6
88. In how many ways can 3 boys be seated in a room where there are 7 seats?
A. 200
B. 205
C. 210
D. 215
89. A card is drawn from a deck of 52 cards. What is the probability of drawing an ace?
A. 0.0763
B. 0.0765
C. 0.0767
D. 0.0769
90. If log10 x = -1/n, then x is equal to
A. 101/n
B. 10-1/n
C. 10-n
D. -10-n
91. The constant remainder when x30 – x + 5 is divided by x + 1 is
A. 7
B. 6
C. 8
D. 5
92. Find the mean proportional between
and
.
A. 6
B. 4
C. 8
D. 5
93. In how many ways can a poster be colored if there are 5 different colors available?
A. 30
B. 29
C. 28
D. 31
94. If 3x = 4y, then 3x2/4y2 is equal to
A. 16/9
B. 4/3
C. ¾
D. 27/64
95. Find the 8th term of 5x+1, 52x+1, 53x+1, ...
A. 56x+1
B. 57x+1
C. 58x+1
D. 59x+1
96. Find the larger of two numbers if their sum is 190 and the smaller number is 3/7 of the larger
number.
A. 132
B. 133
C. 134
D. 135
97. In how many ways can 6 boys be seated at a round table?
A. 120
B. 110
C. 100
D. 90
98. For what value of x is 2 x+4 equal to 1/16
A. 6
B. -6
C. 8
D. -8
99. If x:6 = y:2 and x-y = 12, find y
A. 8
B. 2
C. 4
D. 6
100. Three balls are drawn from a bag containing 5 white balls and 4 red balls. What is the
probability that the balls drawn are all white?
A. 5/42
B. 3/42
C. 7/42
D. 9/42
101. If x3/4 = 8, then x =
A. 16
B. 14
C. 12
D. 10
102. How many 4-digit numbers can be made by using the digits from 1 to 9 if no digit is
repeated in each number?
A. 3,204
B. 3,024
C. 3,240
D. 3,402
103. Find the sum of the infinite geometric series 64 – 16 + 4 - ...
A. 256/5
B. 256/3
C. 256/2
D. 256/4
104. If 2x = 8x-1, solve for x.
A. ½
B. 3/2
C. 1
D. 2
105. A tank can be filled by one pipe in 6 hrs and by another in 8 hrs. If both pipes are open, how
long will it take them to fill the tank?
A. 2 hr
B. 2.5 hr
C. 3 hr
D. 3.5hr
106. The 100th term of the series 1.01, 1.00, 0.99, ... is
A. 0.0002
B. 0.002
C. 0.02
D. 0.2
107. How many consecutive numbers beginning with 5 must be taken for their sum to be equal to
95?
A. 12
B. 11
C. 10
D. 9
108. If 2log2x + log2 4 = 1, find x.
A.
B.
C.
D.
109. Find the next tem in the harmonic progression whose first three terms are 1/3, 2/7 and 1/4.
A. 1/9
B. 2/9
C. 5/9
D. 4/9
110. A card is drawn from a deck of 52 cards. What is the probability of drawing a black king?
A. 1/25
B. 1/26
C. 1/27
D. 1/28
111. In how many ways can the position of President, Vice-President and Secretary be filled in a
club of 12 members if no person is to hold more than one position?
A. 1,230
B. 1,320
C. 1,203
D. 1,302
112. How many arrangement can be made from the letters of the word “RESISTORS” when all
are taken at a time?
A. 30,240
B. 20,340
C. 40,320
D. 40,230
113. The odds are 13 to 8 in favor of winning the first prize of a lottery. What is the probability
of winning that prize?
A. 0.691
B. 0.617
C. 0.619
D. 0.671
114. For a geometric progression for which the first term is x+y and the common ratio is the
reciprocal of the first term, find the 10th term.
A. (x+y) -5
B. (x+y) -6
C. (x+y) -7
D. (x+y)-8
115. Twelve boys go out for tennis. How many matches are required if each boy is to play all the
others exactly once?
A. 64
B. 66
C. 68
D. 62
116. An urn contains 5 white balls, 4 black balls and 3 red balls. If 3 balls are drawn
simultaneously, find the probability that all are white balls.
A. 1/20
B. 1/21
C. 1/22
D. 1/23
117. How many committees can be formed from a group of 9 persons by taking any member at
any time?
A. 411
B. 511
C. 611
D. 711
118. A bag contains 4 black balls and 6 red balls. Two balls are drawn at random. What is the
probability that the balls drawn are both black?
A. 0.131
B. 0.133
C. 0.134
D. 0.135
119. How many liters of pure alcohol must be added to 10 liters of 15 percent alcohol solution in
order to obtain a mixture of 25 percent alcohol?
A. 1/3
B. ½
C. ¼
D. 1/5
120. What is the probability of getting a sum of 5 by throwing two dice once?
A. 0.111
B. 0.112
C. 0.113
D. 0.115
121. The first term of a geometric progression is 3 and the last term is 48. If each term is twice
the previous term, find the sum of the geometric progression.
A. 93
B. 92
C. 91
D. 90
122. How many numbers of two different digits each can be formed by using the digits 1,3,5,7,9?
A. 18
B. 20
C. 22
D. 24
123. What is the sum of the coefficients of (x + y + z) 5 ?
A. 240
B. 241
C. 242
D. 243
124. In a single throw of a pair of dice, what is the probability of obtaining a total of 9?
A. 1/8
B. 1/9
C. 1/7
D. 1/6
125. Evaluate i113 + 4i84 + i3
A. 4
B. -4
C. 4 + 2i
D. 4 – 2i
126. What is the number of permutation of the letters in the word “CHACHA”?
A. 80
B. 85
C. 90
D. 95
127. What is the sum of the first one hundred positive odd integers?
A. 13,000
B. 12,000
C. 11,000
D. 10,000
128. In the arithmetic progression -9, -2, 5, ... which term is 131?
A. 20
B. 21
C. 22
D. 23
129. Fnd the sum of the first fifty positive multiples of 12.
A. 15,300
B. 15,200
C. 15,100
D. 15,000
130. In a geometric progression, if the first term is x2 and the common ratio is x4, which term is
x18?
A. 4th
B. 5th
C. 6th
D. 7th
131. When f(x) = (x+3(x-4)+4 is divided by x-k, the remainder is k. The values of k are
A. 2 and -4
B. -2 and 4
C. 3 and 4
D. -3 and 4
132. From a group of 6 men and 5 women, in how many ways can we select a group of 4 men
and 3 women?
A. 25
B. 39
C. 150
D. 420
133. In a single throw of pair of dice, what is the probability of obtaining a total greater than 9?
A. 1/6
B. 1/3
C. 2/3
D. ½
134. A man can finish a certain job in 10 days. A boy can finish the same job in 15 days. If the
man and the boy plus the girl can finish the job in 5 days, how long will it take the girl to finish
the job alone?
A. 30
B. 45
C. 15
D. 35
135. In how many ways can 4 boys and 4 girls be seated at a round table if each girl is to sit
between two boys?
A. 256
B. 16
C. 144
D. 96
136. If f(x) = x – 1 and f(g(x)) = 4, find g(x).
A. 4
B. 5
C. 6
D. 7
137. If John is 10 percent taller than Peter and Peter is 10 percent taller than May, then John is
taller than May by
A. 18%
B. 20%
C. 21%
D. 22%
138. A bag contains 4 white balls and 5 black balls. Four balls are drawn in succession and not
replaced. Find the probability that the first two balls are white and the last two balls are black.
A. 3/63
B. 4/63
C. 5/63
D. 6/63
139. If xy = 12, xz = 15 and yz = 20, find the value of xyz.
A. 60
B. 55
C. 50
D. 45
140. A committee of 4 is to be selected by lot from a group of 6 men and 4 women. What is the
probability that it will consist exactly of 2 men?
A. 2/7
B. 1/7
C. 3/7
D. 4/7
141. Solve for x in the equation (x+3):10 = (3x-2):8
A. 1
B. 2
C. 3
D. 4
142. If
, find the value if x3y.
A. 8
B. 32
C. 64
D. 128
143. If four electricians can earn P465.80 in 7 days, how much can 14 carpenters paid at the
same rate earn in 12 days?
A. P2,749.80
B. P2,974.80
C. P2,479.80
D. P2,749.80
144. How many four-digit even numbers can be written by using the digits 1 up to 9?
A. 1,144
B. 1,244
C. 1,344
D. 1,444
145. If 1 + x + x2 + ... = ¾, find the value of x.
A. -1/2
B. -1/3
C. -1/4
D. -1/5
146. If it takes A twice as it takes B to do a piece of work and if working together, they can do
the work in 6 days, how long would it take B to do it alone?
A. 8
B. 9
C. 7
D. 6
147. What is the probability that a coin will turn up heads twice in 6 tosses of the coin?
A. 15/64
B. 14/64
C. 13/64
D. 12/64
148. Simplify 3n – 3n-1 – 3n-2.
A. 5(3n-2)
B. 3(3n-2)
C. 33n-2
D. 33-n
149. If the odds are 5:3 that Juan will receive P5,000 in a Math contest, find his mathematical
expectation.
A. P2,135
B. P2,315
C. P3,215
D. P3,125
150. Find the sum of all integers between 90 and 190 if each integer is exactly divisible by 17?
A. 847
B. 857
C. 867
D. 887
151. If P(n,3) = 60(n,5), find n.
A. 7
B. 6
C. 5
D. 8
152. Solve for x in the equation
A. 2
B. 4
C. 8
D. 7
153. Two balls are drawn from a bag containing 9 balls numbered from 1 to 9. Find the
probability that both balls drawn are numbered even.
A. ¼
B. 1/5
C. 1/6
D. 1/7
154. What is the coefficient of the term containing x4 in the expansion of (2x+x -1)8 ?
A. 19,270
B. 19,720
C. 17,920
D. 17,290
155. The bob of a pendulum swings through an arc of 24cm long on its first swing. If each
successive swing is approximately 5/6 the length of the preceding swing, find the approximate
total distance it travels before coming to rest.
A. 121 cm
B. 114 cm
C. 144 cm
D. 169 cm
156. If 6x2 + 36x + k = 6(x+a)2, what is the value of k?
A. 12
B. 18
C. 54
D. 36
157. Two dice are rolled. Find the probability that the sum of the two dice is greater than 10.
A. 1/11
B. 1/12
C. 1/13
D. 1/14
158. If z = 2 + i and w = i – 2, find (z-w)/(z+w).
A. 2i
B. -2i
C. i
D. –i
159. A card is chosen from a deck of 52 cards. In how many ways can a spade or a ten be
chosen?
A. 14
B. 15
C. 16
D. 17
160. Simplify
.
A. a7/8
B. a1/8
C. a2/3
D. a5/6
161. Determine the 7th term of the arithmetic progression 3xy – y, 2xy, xy +y, ...
A. 5y – 3xy
B. 5y + 3xy
C. 5x – xy
D. 5x + xy
162. How many arithmetic means must be inserted between 1 and 36 so that the sum of the
resulting arithmetic progression will be 148?
A. 5
B. 6
C. 7
D. 8
163. Transform the logarithmic equation 4(log x)2 + 9(log y)2 = 12 (log x) (log y) to its
equivalent cartesion form.
A. x3 = y2
B. x2 = y3
C. 3x = 2y
D. 2x = 3y
164. Given 3 dots and 3 dashes. How many code words of exactly 6 symbols can be formed?
A. 18
B. 20
C. 22
D. 24
165. Given
1
x
0
2
-2
1
6
1
-1
= 25, find x.
A. 1
B. 2
C. 3
D. 4
166. A card is drawn from a deck of 52 cards. What is the probability of drawing an ace or a
spade?
A. 17/52
B. 16/52
C. 15/52
D. 14/52
167. A man invested P4,000 at a certain rate of interest and P7,200 at 2% less than the first rate.
The yearly income from both investments is P640. Find the rate of interest for P4,000
A. 5%
B. 6%
C. 8%
D. 7%
168. Rationalize (2+i)/(3-i)
A. i/2
B. (5+i)/2
C. (1+i)/2
D. (1-i)/2
169. If the first term and third term of a harmonic progression are 5/21 and 5/23 respectively,
find the 6th term.
A. 26/5
B. 27/5
C. 24/5
D. 23/5
170. A boat travels 12km downstream and 16km upstream in 12 hours. If the rate of the current
is 3kph, what is the rate of the boat in still water?
A. 7 kph
B. 8 kph
C. 9 kph
D. 10 kph
171. Two people are selected randomly from a group of 4 men and 4 women. The probability
that a man and a woman are selected is
A. 4/7
B. 2/7
C. ¼
D. 3/7
172. Find the 5th term of (x2 – 3y)5 without expanding.
A. 403x2 y4
B. 402x2 y4
C. 404x2 y4
D. 405x2 y4
173. The equation whose roots are the reciprocals of the roots of 3x2 – 7x – 20 = 0 is
A. 20x2 + 7x + 3 = 0
B. 20x2 – 7x + 3 = 0
C. 20x2 + 7x – 3 = 0
D. 20x2 - 7x – 3 = 0
174. Find the 50th term of 1 + i, 2 + 4i, 3 + 7i, ...
A. 47 + 148i
B. 48 + 148i
C. 49 + 148i
D. 50 + 148i
175. Fifty liters of acid solution contains 22% water. How many liters of water must be added to
the solution so that the resulting mixture will be 60% acid?
A. 15
B. 10
C. 20
D. 12
176. From the letters a, e, i, o, r, s, t, how many arrangements of 5 different letters each can be
formed if each arrangement involves 2 consonants and 3 vowels?
A. 100
B. 121
C. 144
D. 169
177. Six coins are tossed. What is the probability that exactly two of them are heads?
A. 0.423
B. 0.342
C. 0.234
D. 0.243
178. There are 0 defective per 1000 items of product in a long run. What is the probability that
there is one and only one defective in a random lot of 100?
A. 0.2770
B. 0.2707
C. 0.2077
D. 0.2207
179. The value of k which will make 8x2 + 8kx + 3k + 2 a perfect square trinomial is
A. 5
B. 6
C. 3
D. 2
180. The 3rd term of an arithmetic progression is 4 and the 9th term is -14. Find the 5th term.
A. -2
B. -3
C. -4
D. -5
181. If x:y:z = 4:-3:2 and 2x + 4y – 3z = 20, find x.
A. -8
B. -6
C. -4
D. -2
182. Two brothers are respectively 5 and 8 years old. In how many years will the ratio of their
ages be 3:4?
A. 3
B. 4
C. 5
D. 6
183. If f(x) = (x+2)/(x-2) and G(y) = y+2, find g(f(3)).
A. 4
B. 5
C. 6
D. 7
184. How many consecutive even integers beginning with 4 must be taken for their sum to equal
648?
A. 20
B. 22
C. 24
D. 26
185. Determine the value of the given determinant
1
1
1
1
0
1
0
0
0
1
1
1
0
0
1
1
A. 1
B. 0
C. -2
D. -1
186. At what time after 3 o’clock will the minute hand of a clock be as far in front of the 5
o’clock mark as the hour hand is behind that mark?
A. 32.21
B. 32.31
C. 32.41
D. 32.51
187. If 2x = 4y and 8y = 16z, find x/z.
A. 1/3
B. 2/3
C. 8/3
D. 5/3
188. If n is a perfect square, what is the next larger perfect square?
A. n2 + 2n + 1
B. n2 + n + 1
C. n2 + 1
D. n +
+1
189. If 2 men can repair 6 machines in 4 hours, how many men are needed to repair 18 machines
in 8 hours?
A. 6
B. 5
C. 4
D. 3
190. In how many ways can 9 books be arranged in a shelf so that 4 books are always together?
A. 2,880
B. 3,024
C. 14,400
D. 17,280
191. If the roots of (2k +2)x2 + (4 – 4k)x + k – 2 = 0 are reciprocals to each other, find the value
of k.
A. -2
B. -3
C. -4
D. -5
192. The 3rd term of a geometric progression is 5 and the 6th term is -40. Find the 8th term.
A. -140
B. -150
C. -160
D. -170
193. The equation x2 – 4kx + 10 – 6k = 0 will have two equal roots if the value of k is
A. 5/3
B. -5/2
C. -5/3
D. 5/2
194. The arithmetic mean of a set of 50 numbers is 38. Two numbers of the set, namely 45 and
55, are discarded. What will be the arithmetic mean of the remaining set of numbers?
A. 35.5
B. 36.5
C. 37.5
D. 38.5
195. The sum of two numbers is 37. If the larger is divided by the smaller, the quotient is 3 and
the remainder is 5. Find the smaller number.
A. 6
B. 7
C. 8
D. 9
196. In how many ways can 6 boys be seated at a round table if two particular boys must always
sit together?
A. 42
B. 44
C. 46
D. 48
197. Maria is twice as old as Ana was when Maria was as old as Ana is now. If Maria is 24 years
old now, how old is Ana now?
A. 18
B. 17
C. 16
D. 15
198. If 1/x = a + b and 1/y = a – b, then x – y is equal to
A. 2b/(b2 – a2)
B. 2a/(b2 – a2)
C. 2b/(a2 – b2)
D. 2a/(a2 – b2)
199. The 4th term of a geometric progression is 81 and the 7th term is 9. What is the 10th term?
A. 1
B. 2
C. 3
D. 4
200. If log 2 = x and log 3 = y, find log(1.2)
A. 2x + y -1
B. 2x + y +1
C. 2x – y + 1
D. 2x – y – 1
201. How many consecutive terms must be taken from the sequence 3, -6, 12, -24, ... for the sum
to equal 8,193?
A. 11
B. 12
C. 13
D. 14
202. At what time between 7 and 8AM will the minute hand and the hour hand of a clock be
opposite one another?
A. 7:05:23
B. 7:05:25
C. 7:05:27
D. 7:05:29
203. A man can do a job with his son in 30 days. If after working together for 12 days, the son
worked alone and finished the job in 24 more days, how long will it take the son to do the job
alone?
A. 40 days
B. 45 days
C. 50 days
D. 55 days
204. How many arrangements can be made from the letters of the word “TRANSIENTS” when
all are taken at a time?
A. 435,600
B. 453,600
C. 436,500
D. 463,500
205. Find the middle term of (x1/3 – y1/3)12 without expanding.
A. -924x2 y2
B. 924x2 y2
C. -492x2 y2
D. 492x2 y2
206. A bag contains 6 red balls, 4 white balls and 8 black balls. If 3 balls are drawn at random,
determine the probability that 2 balls are white and one is red.
A. 0.0414
B. 0.0441
C. 0.0144
D. 0.0141
207. If 102x = 4 find 106x-1.
A. 4.4
B. 5.4
C. 6.4
D. 7.4
208. Find the sum of the first n terms of 2, 8, 14, ...
A. n(3n+1)
B. n(2n+3)
C. n(n+3)
D. n(3n-1)
209. In how many ways can 4 men be selected out of 12 men if 2 of the men are to be excluded
from every selection?
A. 66
B. 45
C. 210
D. 495
210. Three drawn from a pack of 52 cards. Determine the probability that all cards drawn are of
the same suit.
A. 0.0516
B. 0.0518
C. 0.0520
D. 0.0522
211. If log x = 2(1 – log 2), find x.
A. 5
B. 10
C. 15
D. 25
212. Mary is three times as old as Ricky. Three years ago, she was four times as old Ricky was
then. Find the sum of their ages now.
A. 32
B. 24
C. 38
D. 36
213. An air plane went 360 miles in 2 hours with the wind and flying back the same route, took 3
hours and 36 minutes against the wind. What was its speed in still air?
A. 60 mph
B. 120 mph
C. 140 mph
D. 160 mph
214. A ball is dropped from a height of 28 cm. If it always rebounds ½ of the height from which
it falls, how far does it travel after the fifth bounce?
A. 372 cm
B. 374 cm
C. 376 cm
D. 378 cm
215. A tank can be filled by one pipe in 16 minutes, by a second pipe in 24 minutes and can be
drained by a third pipe in 48 minutes. If all pipes are open, in how many minutes can the tank be
filled?
A. 10
B. 12
C. 14
D. 16
216. If 3x 3y = 27 and 2x + y = 5, find x.
A. 2
B. 3
C. 4
D. 5
217. The amount of P300,000 is divided into 3 parts in the ratio 2:5:8 and these parts are invested
at 2%, 4% and 6% respectively. Find the income from the 6% investment.
A. P6,600
B. P7,600
C. P8,600
D. P9,600
218. Find the sum of the first 12 terms of an arithmetic progression whose 7th term is 5/3 and
with a common difference of -2/3
A. 22
B. 24
C. 26
D. 28
219. By use of 3 different red flags and 4 different green flags, how many signals can formed by
flying all the flags from seven positions on a pole if the same color are to be consecutive?
A. 144
B. 288
C. 70
D. 84
220. In a throw of two dice, the probability of obtaining a total of 10 or 12 is
A. 1/16
B. 1/12
C. 1/9
D. 1/18
221. Find x so that x-1, x+2 and x+8 are the first three terms of a geometric progression.
A. 4
B. 3
C. 5
D. 2
222. If
, find x.
A. 5/9
B. 9/5
C. 7/9
D. 9/7
223. Find the sum of the infinity of 1 – ½ + ¼ - 1/8 + ...
A. 1/3
B. 2/3
C. ¼
D. ¾
224. If log 2 = x and log 3 = y, find log4 9
A. x/y
B. xy
C. y/x
D. xy
225. A man drives a certain distance at 50kph and a second man drives the same distance in 20
minutes less time at 60kph. Find the distance traveled.
A. 130 km
B. 120 km
C. 110 km
D. 100 km
226. How many numbers between 200 and 500 can be formed by using the digits 0, 1, 2, 3, 4, 5
if each digit must not be repeated in any number?
A. 120
B. 80
C. 60
D. 100
227. A bag contains 10 white balls and 5 black balls. If 3 balls are drawn in succession without
replacement, find the probability that the balls are drawn in the order black, black and white.
A. 0.0733
B. 0.0723
C. 0.0743
D. 0.0713
228. Find the term free of x in the expansion of (x2 – x -1)9 .
A. 64
B. 84
C. 96
D. 48
229. How many different signals; each consisting of 6 flags hung in a vertical line can be formed
from 4 identical red flags and 2 identical blue flags?
A. 12
B. 13
C. 15
D. 16
230. Juan can do a job in 6 days and Pedro can do the same job in 10 days. If Juan worked for 2
days and Pedro joined him, in how many days more will the two boys finished the job together?
A. 3.5
B. 3
C. 2.5
D. 2
231. What must be the value of x in the arithmetic progression x-7, x-2, x+3, ... so that its 10th
term will be 40?
A. 4
B. 3
C. 2
D. 1
232. The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number
is 4/7 of the original number, Determine the original number.
A. 57
B. 75
C. 93
D. 84
233. Find the probability that a couple with three children have exactly two boys.
A. 0.375
B. 0.365
C. 0.345
D. 0.335
234. The sum of three consecutive odd integers is 75. Find the largest integer.
A. 25
B. 27
C. 24
D. 29
235. In how many ways can a man choose 2 or more of 5 ties?
A. 31
B. 26
C. 20
D. 10
236. A boat can travel 10 kph in still water. It can travel 60 km downstream in the same time that
it can travel 40 km upstream. What is the rate of the current?
A. 2 kph
B. 2.5 kph
C. 3 kph
D. 3.5 kph
237. The first term of a geometric progression is 160 and the common ratio is 2/3. How many
consecutive terms must be taken to give a sum of 2,110?
A. 7
B. 6
C. 5
D. 4
238. At what time after 4PM will the 6 o’clock mark bisect the angle formed by the minute and
the hour hand?
A. 4:36.62
B. 4:36.72
C. 4:36.82
D. 4:36.92
239. If the product of two positive numbers is 14 and their sum is 6, find the sum of their
reciprocals.
A. 3/7
B. 5/7
C. 4/7
D. 2/7
240. In how many ways can 6 plus signs and 4 minus signs be written on a straight line?
A. 24
B. 744
C. 360
D. 210
241. John and Jack can do a job in 4 hours and the working rate of John is twice that of Jack.
How many hours would it take John to work alone?
A. 5
B. 6
C. 7
D. 10
242. How many arrangements can be made out of the letters of the word “CONSTITUTION”?
A. 9,979,200
B. 9,799,200
C. 7,999,200
D. 2,979,900
243. A fair coin is tossed 3 times. Find the probability of getting either 3 heads or 3 tails.
A. 1/8
B. ½
C. 3/8
D. ¼
244. In how many ways can 4 boys and 3 girls be seated in a row of 7 seats with the girls always
in consecutive seats?
A. 620
B. 720
C. 820
D. 920
245. A tank can be filled by one pipe in 16 minutes; by another pipe in 24 minutes and can be
drained by a third pipe in 48 minutes. If all pipes are open at the same time, in how many
minutes can the tank be filled?
A. 12
B. 14
C. 10
D. 16
246. In how many ways can a panel of 5 judges make a majority decisions?
A. 18
B. 25
C. 14
D. 16
247. A man drove from station A to station B, 60 km away, at an average speed of 30kph and
return to A at an average speed of 20 kph. What was the average speed for the whole journey?
A. 22 kph
B. 23 kph
C. 24 kph
D. 25 kph
248. A man can do a job in 8 days. After the man has worked for 3 days, his son joins him and
together they complete the job in 3 more days. How long will it take the son to do the job alone?
A. 12
B. 11
C. 13
D. 10
249. A tunnel is one kilometer long. A train 250 meters long is passing through the tunnel at
25kph. How long will it take the train to completely pass the tunnel?
A. 6 min
B. 4 min
C. 5 min
D. 3 min
250. If two dice are cast, what is the probability that the sum will be less than 6?
A. 1/15
B. 1/16
C. 5/18
D. 2/19
251. If (3)^(log3 x) = 4x3, find the value of x.
A. 1/3
B. ½
C. ¼
D. 1/5
252. The probability that both Vic and Ric can solve a certain puzzle is 0.95. The probability that
Vic alone can solve the same puzzle is 0.98. What is the probability that Ric can solve the puzzle
given the Vic does?
A. 0.9361
B. 0.9694
C. 0.9310
D. 0.9496
253. In how many ways can 8 students be divided into 4 groups of 2?
A. 2,250
B. 2,502
C. 2,205
D. 2,520
254. A lottery has a first prize of P10,000,000. Suppose only 8,000,000 tickets are sold and you
have bought 40 tickets, what is your mathematical expectation?
A. P30
B. P40
C. P50
D. P60
255. Thrice the sum of two numbers is 30 and the sum of their squares is 52. Find the product of
the numbers.
A. 22
B. 24
C. 26
D. 28
256. What is the harmonic mean between 3/8 and 4?
A. 24/35
B. 23/35
C. 22/35
D. 21/35
257. Find the third proportional to 4n2 and 2mn2.
A. mn
B. (mn)2
C. 2/m
D. m/2
258. One basket contains 5 apples and 2 oranges and a second basket contains 4 apples and 3
oranges. If a fruit is taken from one of the two baskets at random, what is the probability that it is
an orange?
A. 0.4133
B. 0.4143
C. 0.4153
D. 0.4163
259. If
A. 3/14
B. 3/15
C. 3/16
D. 3/17
, find the value of x.
260. The sum of the first three terms of an arithmetic progression is -3 while the sum of the first
five terms of the same arithmetic progression is 10. Find the first term.
A. -5
B. -4
C. -3
D. -2
261. The product of n P n-r and r P 1 is equal to
A. n Pn-r+1
B. nPn+r-1
C. nPn-1
D. nPr-1
262. In how many ways can 12 books be divided among four students so that each student
receive 3 books?
A. 390,660
B. 396,600
C. 366,900
D. 369,600
263. Ann is eleven times as old as Beth. In a certain number of years, Ann will be five times as
old as Beth and five years after that, Ann will be three times as old as Beth. How old is Beth
now?
A. 20
B. 21
C. 22
D. 23
264. Juan after working on a job for 2 hours was helped by Jose and it took 3 hours more for
them working together to finish the job. Had they worked together from the start, it would only
require 4 hours to finish the job. How long would it take Juan to finish the job alone?
A. 6 hr
B. 7 hr
C. 8 hr
D. 9 hr
265. Find the common ratio of a geometric progression whose first term is 1 and for which the
sum of the first 6 terms is 28 times the sum of the first 3 terms.
A. 2
B. 3
C. 4
D. 5
266. If x:y:z = 2:5:7 and 4x – y + 2z = 51, find z.
A. 21
B. 6
C. 15
D. 8
267. A bag contains an assortment of red and blue balls. If two balls are drawn from it at random,
the probability that 2 red balls are drawn is 5 times the probability that 2 blue balls are draw.
Furthermore, the probability that one ball of each color is drawn is 6 times the probability that 2
blue balls are drawn. How many red balls are there in the bag?
A. 3
B. 4
C. 5
D. 6
268. Solve for x in the equation x + 3x + 5x + 7x + ... + 49x = 625.
A. ¼
B. ½
C. 1
D. 2
269. The sum of the digits of a three-digit number is 14. The units digit is half the tens digit. If
the digits are reversed, the resulting number is 198 more than the original number. Find the
original number.
A. 563
B. 842
C. 284
D. 921
270. Te arithmetic mean of two positive numbers exceeds their geometric mean by 2. Find the
smaller number if it is 40 less than the larger number.
A. 90
B. 101
C. 121
D. 81
271. Juan’s age on his birthday in 1989 is equal to the sum of the digits of the year 19Juan’s age
on his birthday in 1989 is equal to the sum of the digits of the year 19XY in which he was born.
If X and Y satisfy the equation X – Y – 6 = 0, find the age of Juan in 1990.
A. 18 yr
B. 19 yr
C. 20 yr
D. 21 yr
272. A group of neighbors plan to pay equal amount in order to buy a small power mower which
costs P4,800. If the adding 2 more neigbor to the original group, cost to each is reduced by P120.
Find the number of neighbor in the original group.
A. 4
B. 6
C. 8
D. 10
273. A chemist mixed 40 ml of 8% hydrochloric acid with 60 ml of 12% hydrochloric acid
solution. She used a portion of this solution and replaced it with distilled water. If the new
solution tested 5.2% hydrochloric acid, how much of the original solution did she used?
A. 50 ml
B. 40 ml
C. 70 ml
D. 60 ml
274. There are two copies each of 3 different books. In how many ways can they be arranged on
a shelf?
A. 80
B. 85
C. 90
D. 95
275. A coin is tossed 6 times. What are the odds in favor of getting at least 3 head?
A. 18:11
B. 19:11
C. 20:11
D. 21:11
276. A bag contains 10 red balls. 30 white balls, 20 black balls and 15 yellow balls. If 2 balls are
drawn, replacement being made after each drawing, find the probability that only one is red.
A. 0.2211
B. 0.2311
C. 0.2411
D. 0.2511
277. In how many ways can 9 people cross a river riding 3 bancas whose maximum capacity is
2,4 and 5 respectively?
A. 5,346
B. 3,654
C. 6,453
D. 4,536
278. A realtor bought a group of lots for S90,000. He then sells them at a gain of S3,750 per lot
and has a total profit equal to the amount he received for the last 4 lots sold. How many lots were
originally in the group?
A. 10
B. 11
C. 12
D. 13
279. A man left Manila for Baguio City at past 9AM. Between 4 to 5 hours, he arrived at Baguio
and noticed the minute and hour hands of his wrist watch interchanged in position. At what time
did the man arrived at Baguio?
A. 1:45.53 PM
B. 1:45.63 PM
C. 1:45.73 PM
D. 1:45.83 PM
280. A and B can do a piece of work in 42 days, B and C in 31 and A and C in 20 days. In how
many days can all of them do the work together?
A. 18.86 days
B. 18.76 days
C. 18.66 days
D. 18.56 days
281. Using the relation nCr-1 + nCr = n+1Cr , find the value of y given that 89C63 –88 C63 =x Cy
A. 60
B. 61
C. 62
D. 63
282. A class of 40 students took examination in Mathematics and English. If 30 passed in
English, 36 passed in Mathematics and 2 fails in both subjects, the number of students who
passed both subjects is
A. 26
B. 28
C. 29
D. 30
283. There is a 30% chance of rain today. If it does not rain today, there is a 20% chance of rain
tomorrow. If it rains today, there is a 50% chance of rain tomorrow. What is the probability that
it rains tomorrow?
A. 0.27
B. 0.28
C. 0.29
D. 0.26
284. A group consists of n boys and n girls. If two of the boys are replaced by two other girls,
then 49% of the group members will be boys. Find the value of n.
A. 49
B. 51
C. 98
D. 100
285. If P(n,r) = 840 and C(n,r) = 35, find the value of r.
A. 2
B. 3
C. 4
D. 5
286. If three sticks are drawn from 5 sticks whose lengths are 1, 3, 5, 7 and 9, what is the
probability that they will form a triangle?
A. 0.24
B. 0.21
C. 0.30
D. 0.36
287. A passenger train x times as fast as a freight train takes x times as long to pass when
overtaking the freight train as it takes to pass when two trains are going in opposite directions.
What is the value of x?
A. 2.31
B. 2.41
C. 2.51
D. 2.61
288. Three card are drawn from a deck of 52 cards without replacement. Find the probability that
all are of the same color.
A. 0.2353
B. 0.3523
C. 0.3323
D. 0.2335
289. The simplest form of [(n+1)!]2 / n!(n-1)! is
A. n2
B. n(n+1)
C. n+1
D. n(n+1)2
290. How many products can be formed from the numbers 2,3,4,5,6,7 by taking two or more
numbers at a time?
A. 57
B. 64
C. 59
D. 69
291. Juan is thrice as old as Jose was when Juan was as old as Jose is now. When Jose becomes
twice as old as Juan is now, together they will be 78 years. How old is Juan now?
A. 12
B. 14
C. 16
D. 18
292. If z2 = 24 + 10i, find z.
A. 5+i or 5-i
B. 5+i or -5-i
C. 5-i or -5+i
D. 5-i or -5-i
293. There are three candidates A, B and C for mayor in a town. If the odds that candidate A will
win are 7:5 that of B are 1:3, what is the probability that candidate C will win?
A. 0.25
B. 0.13
C. 0.17
D. 0.21
294. The first term of an arithmetic progression is 6 and the tenth term is 3 times the second
term. What is the common difference?
A. 1
B. 2
C. 3
D. 4
295. A basket contains 3 red balls and 2 white balls while a second basket contains 2 red balls
and 5 white balls. A man selected a basket at random and picked a ball and placed it on the other
basket. Then another ball is drawn from the second basket. Find the probability that both balls he
picked are of the same color.
A. 601/1680
B. 701/1680
C. 801/1680
D. 901/1680
296. If 7 coins are tossed once, find the probability of tossing at most 6 heads.
A. 0.9911
B. 0.9922
C. 0.9933
D. 0.9944
297. A businessman travelled 1,110km to attend a company conference. He drove his car 60 km
to an airport and flew the rest of the way. His plane speed is 10 times that of his car. If he flew
45 minutes longer than he drove, how long did he fly?
A. 1.25 hr
B. 1.45 hr
C. 1.75 hr
D. 1.50 hr
298. Maria, Ana, Cora, Cely, Jose, Juan and Pedro are participating in elections for four student
officers: President, Vice-President, Secretary and Treasurer. What is the probability that a girl
becomes a President and a boy Vice-President?
A. 0.8257
B. 0.2857
C. 0.5827
D. 0.7285
299. A tank can be filled separately in 10 and 15 minutes respectively by tow pipes. When a third
pipe was used simultaneously with the first two pipes, the tank can be filled in 4 minutes. How
long would it take the third pipe alone to fill the tank?
A. 12 min
B. 11 min
C. 10 min
D. 9 min
300. In a single toss of a pair of dice, find the probability of tossing a total of at most 5.
A. 0.278
B. 0.268
C. 0.258
D. 0.248
301. In how many ways can 9 different books be divided among three boys A, B and C so that
they receive 4, 3 an 2 books respectively?
A. 1,600
B. 3,600
C. 1,260
D. 2,460
302. A family budget provides an expenditures of P5,100 per month for food. If the amount
alloted for meat is P300 more than that of milk and if the allotment for other food is twice as
much as that for meat and milk, find the amount alloted for milk.
A. P600
B. P650
C. P700
D. P750
303. Find the probability of throwing 11 each time in 3 tosses of 2 dice.
A. 0.00017
B. 0.00170
C. 0.01700
D. 0.17000
304. A 2.5-liter container has a mixture of 25% alcohol. How many liters of the mixture must be
drained out and replaced with pure alcohol in order to obtain a mixture containing 40% alcohol?
A. 0.35
B. 0.40
C. 0.45
D. 0.50
305. If the sum of two numbers is 1 and their product is also 1, find the sum of their cubes.
A. -3
B. -1
C. -2
D. -4
306. On a trip, a man noticed that his car averaged 21 km per liter of gasoline except for the days
he used the air conditioning and then it averaged only 17 km per liter. If he used 91 liters of
gasoline to drive 1,751 km, on how many of these kilometers did he used the air conditioning?
A. 480 km
B. 580 km
C. 680 km
D. 780 km
307. The 2nd, 4th and 8th terms of an arithmetic progression are themselves in geometric
progression. Find the common ratio of the geometric progression.
A. 1
B. 2
C. 3
D. 4
308. The 4th term of a geometric progression is 343 and the 6th term is 16,807. Find the 8th term.
A. 853,243
B. 835,432
C. 824,533
D. 823,543
309. Mary is twice as old as Ann was when Mary was as old as Ann is now. If Mary is 20 years
old now, how old is Ann now?
A. 15
B. 16
C. 17
D. 18
310. The tens digit of a two-digit number is one third of the units digit. When the digits are
reversed, the new number exceeds twice the original number by 2 more than the sums of the
digits. Find the units digit.
A. 5
B. 6
C. 2
D. 3
311. Determine the common difference of an arithmetic progression whose sum to n terms is 2n 2
+ 3n.
A. 7
B. 5
C. 4
D. 6
312. A man sold a book at 105% of the marked price instead of discounting the marked price by
5%. If he sold the book at P4.20, what was the discounted price for which he should have sold
the book?
A. P2.80
B. P3.80
C. P3.00
D. P2.50
313. If logb y = 2x + logb x, find y.
A. y = 2xbx
B. y = xbx
C. y = xb2x
D. y = x2 bx
314. The probability of Juan’s winning whenever he plays a certain game is 1/3. If he plays 4
times, find the probability that he wins at most twice.
A. 0.86
B. 0.94
C. 0.89
D. 0.79
315. A tank can be filled by pipe A in half the time that pipe B can empty the same tank. When
both pipes are operating, the tank can be filled in 1 hour and 12 minutes. Find the time for pipe A
to fill the tank alone.
A. 0.60 hr
B. 0.50 hr
C. 0.40 hr
D. 0.30 hr
316. How many terms of the arithmetic progression 9,11,13,... must be added in order that the
sum should equal the sum of the first nine terms of the geometric progression 3,-6,12,-24,...?
A. 18
B. 19
C. 20
D. 21
317. How many arithmetic means must be inserted between 1 and 36 so that the sum of all
numbers in the resulting progression will be 148?
A. 4
B. 3
C. 5
D. 6
318. The head of a fish measures 22cm long. The tail is as long as the head and half the body and
the body is as long as the head and tail. How long is the fish?
A. 172 cm
B. 174 cm
C. 176 cm
D. 178 cm
319. From a 2-liter vessel containing water a certain amount was drained and replaced with pure
alcohol. Later from the mixture, the same amount was drained and again replaced with pure
alcohol. What amount was removed each time if the resulting mixture has 36% alcohol?
A. 0.50 L
B. 0.40 L
C.0.30 L
D. 0.20 L
320. An urn contains 4 red marbles and 8 black marbles. A marble is drawn from the urn and a
marble of the other color is then put into the urn. A second marble is drawn from the urn. Find
the probability that the 2nd marble is red.
A. 5/18
B. 7/24
C. 13/36
D. 1/12
321. If a man was left with 10 hectares fewer than 40% of his land after selling 6 hectares more
than 70% of his land, how many hectares of land did he initially own?
A. 30
B. 40
C. 50
D. 60
322. The 3rd term of an arithmetic progression is 4 and the 9th term is -14. Find the sum of the
first six terms.
A. 21
B. 19
C. 17
D. 15
323. A number of 5 different digits is written at random by use of the digits 1,2,3,4,5,6,8. Find
the probability that the number will have even digits at each end.
A. 0.2578
B. 0.2785
C. 0.2857
D. 0.2587
324. Without expanding (4x -2 – (x/2) )7, find the term involving x -2
A. 130x-2
B. 140x-2
C. 150x-2
D. 160x-2
325. Juan and Pedro can do a job together in 4 days. If the working rate of Juan is twice that of
Pedro, how long would it take for Pedro to do the job alone?
A. 12 days
B. 10 days
C. 8 days
D. 6 days
326. A man decided to build a wire fence along one straight side of his property. He planned to
place the posts 6 feet apart but after he bought the posts and the wire, he found that he had
miscalculated. He had 5 posts too few. However, he discovered that he could do with the posts
he had by placing them 8 feet apart. How ling was the side of the plot?
A. 100 ft
B. 110 ft
C. 120 ft
D. 125 ft
327. Solve the given system for z:
x – y + 6z = -15
3y – 2z = 18
5x + 2z = -8
A. 5
B. -1
C. ½
D. -3/2
328. The positive value of x so that x, x2 – 5 and 2x will be in harmonic progression is
A. 6
B. 5
C. 4
D. 3
329. If a number of 6 different digits is written at random by using the digits 1,2,3,4,5,6,7, find
the probability that the number will be even.
A. 1/7
B. 2/7
C. 3/7
D. 4/7
330. Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 4 blue balls
if each selection consists of 3 balss of each color.
A. 600
B. 700
C. 800
D. 900
331. What is the probability of drawing a face card on the first selection from a deck of 52 cards
without replacement drawing an ace on the second selection?
A. 0.0181
B. 0.0118
C. 0.0811
D. 0.0188
332. A boy is two years more than twice as old as his brother. The two boys together are 17 years
older than their sister who is 7 years younger than the older boy. How old is the sister?
A. 10
B. 15
C. 18
D. 22
333. The 3rd term of a harmonic progression is 15 and the 9th term is 6. Find the 11th term.
A. 4
B. 8
C. 5
D. 7
334. It takes 10 second for two trains to pass each other when moving in opposite directions. If
they move in the same direction, the faster train could completely pass the slower train in 2
minutes. If the faster train is 120 meters long and the slower train is 130 meters long, find the
rate of the faster train.
A. 11 m/s
B. 12 m/s
C. 13 m/s
D. 14 m/s
335. A bag contains 9 balls numbered 1 to 9. Two balls are drawn at random. Find the
probability that one is odd and the other is even.
A. 7/9
B. 5/9
C. 6/9
D. 4/9
336. Three boys A, B and C working together can do a job in a certain number of hours. If A can
do the job in 1 day more and B alone can do the same job in 6 days more while C alone can do
the job twice as much time, in how many days would the three boys finish the job?
A. 2
B. 3
C. 4
D. 5
337. A man drove 156 km at a constant rate of speed. If he had driven 9 km more per hour, he
would have made the trip in 45 minutes less time. What was his initial speed?
A. 28 kph
B. 39 kph
C. 42 kph
D. 50 kph
338. Find the positive value of x so that 4x, 5x+4 and 3x2-1 will be in arithmetic progression.
A. 5
B. 4
C. 3
D. 2
339. In how many ways can 3 men be selected out of 15 men if 2 of the men are to be excluded
from every selection?
A. 284
B. 286
C. 288
D. 282
340. A bag contains 5 red balls and 6 white balls. If we draw 4 balls, find the probability that at
least 3 balls are white.
A. 0.2578
B. 0.4385
C. 0.3485
D. 0.3845
341. The sum of the squares of two consecutive even integers is 340. Find the larger even
integer.
A. 12
B. 14
C. 16
D. 18
342. Find the 10th term in an arithmetic progression where the first term is 3 and whose 1 st, 4th
and 13th terms form a geometric progression.
A. 21
B. 22
C. 23
D. 24
343. Town A is 11 kilometers from town B.A boy walks from A to B at the rate of 3kph and a
man starting at the same time walks from B to A at 4kph. When will they be 2kilometer apart
after meeting each other along the way?
A. 1.66 hr
B. 1.76 hr
C. 1.86 hr
D. 1.96 hr
344. A man can do a job in 8 hours, a boy can do the same job in 12 hours and a girl can do it in
16 hours. How long will it take them to do the job if the man and the boy work together for one
hour and then the boy and the girl finish the job?
A. 6.54 hr
B. 6.45 hr
C. 6.34 hr
D. 6.43 hr
345. Three groups of men A, B and C assemble 96 machines. Group A assembles 3 machines, B
4 machines and C 5 machines per day. If B works twice as many days as A and if C works 1/3 as
many days as both A and B together, how many days does group A work?
A. 6
B. 8
C. 10
D. 12
346. The probability that Juan will win a game of chess whenever he plays is ¼. If he plays
twice, what are the odds that he wins the 1st game and loses the 2nd game?
A. 0.2108
B. 0.2208
C. 0.2308
D. 0.2408
347. A bag contains 3 black balls, 7 white balls, 6 black cubes and 14 white cubes. Find the
probability of drawing a ball and a black object.
A. 0.10
B. 0.15
C. 0.20
D. 0.25
348. A man is 4 years older than his wife and 5 times as old as his son. When the son was born,
the age of the wife was 6/7 the age of her husband. Find the age of the son now/
A. 6 yr
B. 7 yr
C. 8 yr
D. 9 yr
349. Equal volumes of different liquids evaporate at different but constant rates. If the first is
totally evaporated in 6 weeks and the second is 5 weeks, when will the second be ½ the volume
of the first?
A. 27/7
B. 29/7
C. 33/7
D. 30/7
350. A piece of work can be done by women in 11 days and 30 men in 7 days. In how many days
can the work be done by 22 women and 21 men?
A. 4
B. 5
C. 3
D. 2
351. It is a measure of dispersion which depends only on two scores in the entire distribution.
A. mean deviation
B. quartile deviation
C. variance
D. range
352. The most numerous or most common value in a series of values is called the
A. range
B. mode
C. mean
D. median
353. Which of the following numbers has two significant figure?
A. 0.039
B. 3,009
C. 39.00
D. 30.09
354. What do you call each possible outcome of an experiment?
A. trial
B. sample
C. event
D. variate
355. Which of the following is a measure of central tendency whose magnitude depends directly
on the size of the scores in the group?
A. arithmetic mean
B. harmonic mean
C. median
D. mode
356. An arrangement of raw numerical data in ascending or descending order of magnitude is
called
A. data
B. array
C. frequency
D. category
357. It is halfway between the upper limit and lower limit of a class interval.
A. average
B. class mark
C. mean
D. boundary
358. A tabulation of data showing the number of times a score or group of scores appears is
called
A. normal distribution
B. Poisson distribution
C. frequency distribution
D. probability distribution
359. The probability of an impossible event is
A. zero
B. unity
C. infinity
D. undefined
360. It is a variable that can only assume designated values.
A. continuous variable
B. parameter
C. discrete variable
D. variate
361. If two events have no outcome in common, then they are said to be
A. dependent
B. independent
C. mutually exclusive
D. mutually inclusive
362. The class interval 50 – 52 theoretically includes all measurement from 49.5 and 52.5 and
these end points are called the
A. class limits
B. class boundaries
C. class marks
D. class widths
363. The total area under a probability curve is always
A. equal to unity
B. greater than one
C. between zero and one
D. less than one
364. The range is a measure of
A. variability
B. deviation
C. central tendency
D. distribution
365. When 130 numbers are arranged in an array, the median corresponds to the
A. 65th number in the array
B. the mid number of the array
C. the mean of the 65th and the 66th numbers in the array
D. the average of the sum of all numbers in the array
366. It is a measure of dispersion which depends upon the deviation of all scores from the mean.
A. average deviation
B. quartile deviation
C. mean deviation
D. standard deviation
367. Which of the following is not a continuous variable?
A. weight of a body
B. height of a body
C. temperature of an object
D. number of girls
368. How many significant figures has the number 10.01?
A. one
B. two
C. three
D. four
369. It represents that point in the data where one half of the scores fall below that point and one
half falls above it.
A. average
B. median
C. midscore
D. mean
370. Which of the following events are mutually exclusive?
A. events “ace or black card”
B. events “king or red card”
C. events “queen or face card”
D. events “diamond or black Jack”
371. If H is the event of getting a head by tossing a coin and N is the event of getting a prime
number by tossing a die, then which of the following means “The probability of getting a tail
given that a prime number has come up on the die.”?
A. p(H’ and N)
B. p(H’ or N)
C. p(H’/N)
D. p(N/H’)
372. In probability theory, what do you call the set of all possible outcomes of an experiment?
A. event
B. trial
C. sample space
D. variate
373. If A and B are two events and p(A and B) = p(A)p(B), then the two events are said to be
A. complementary
B. independent
C. disjoint
D. dependent
374. The highest score in a distribution minus the lowest score is called the
A. class mark
B. median
C. range
D. standard score
375. Which of the following is commonly used to illustrate the government income and
expenditure?
A. pie chart
B. frequency polygon
C. histogram
D. scattergram
376. It refers to the number of times a score occurs in a sample.
A. outcome
B. mode
C. average
D. frequency
377. It refers to the facts and figures collected on some characteristics of a sample.
A. array
B. data
C. population
D. histogram
378. In set theory, the sum A + B is denoted by
A. A ᴗ B
B. A ᴖ B
C. A = B
D. A ↔ B
379. The most widely used measure of dispersion is the
A. mean deviation
B. standard deviation
C. quartile deviation
D. average deviation
380. The number of favorable outcomes divided by the total amount of outcomes is called
A. permutation
B. certainty
C. probability
D. frequency
381. The sum of the squared deviation about the mean is called
A. variate
B. variable
C. variance
D. value
382. In statistics, which of the following is a qualitative variable?
A. number grade in a card
B. letter grade in a card
C. number of people
D. salary of a teacher
383. The positive square root of the variance is equal to
A. quartile deviation
B. mean deviation
C. standard deviation
D. average deviation
384. Which of the following is true?
A.
B.
C.
D.
385. It is equal to the absolute difference between the observations in a sample and the mean
divided by the total number of observations in the sample.
A. arithmetic mean
B. root mean square
C. quartile deviation
D. mean deviation
386. When the sample is large and the variable is quantitative which of the following measures
of central tendency has a distinct advantage in terms of accuracy?
A. geometric mean
B. arithmetic mean
C. median
D. mode
387. When a coin is tossed 8 times in succession, head appeared 3 times and tail 5 times in the
following order HTTTHHTT. In how many other orders could they have appeared?
A. 53
B. 54
C. 55
D. 56
388. In a single toss of a pair of dice, the probability of obtaining a sum of 6 is
A. 5/36
B. 7/36
C. 4/36
D. 6/36
389. A point in the distribution of scores at which 50% of the score fall below and 50% fall
above.
A. mode
B. mean
C. median
D. range
390. If a coin is tossed 100 times, find the theoretical standard deviation.
A. 4
B. 2
C. 3
D. 5
391. If a die is thrown 3 times, what is the probability that all throws show 6?
A. 1/8
B. ¼
C. 3/8
D. ¾
392. If A and B are two independent events and P(A) = 0.9 and P(not B) = 0.2, find P(not A or
B).
A. 0.8
B. 0.9
C. 0.7
D. 0.6
393. A boy has an average of 85 in four subjects. What grade must he make in the fifth subject so
that his average will be 87?
A. 93
B. 94
C. 95
D. 96
394. A Poisson distribution is given by p(X) = [(0.7) X e-0.7] / X!. Find p(2).
A. 0.1172
B. 0.1217
C. 0.1127
D. 0.1721
395. If the probability of a defective bolt is 0.20, how many bolts are expected to be defective if
there are a total of 600 bolts?
A. 100
B. 105
C. 115
D. 120
396. When a test was given, the probability of getting a score of 85 was 0.70. If 40 students took
the test, what is the expected number of students who will get a score of 85?
A. 28
B. 27
C. 26
D. 25
397. Consider two independent events A and B. If P(A) = 0.85 and P(B’) = 0.35, find P(A’ and
B).
A. 0.0795
B. 0.0975
C. 0.0597
D. 0.0759
398. The grades of an examinee in a board examination in three subjects A, B and C were 70, 76
and 82 respectively. If the weights accorded to these grades are 25, 35 and 45 respectively, what
is the mean grade of the examinee?
A. 80
B. 79
C. 81
D. 78
399. The ages of 8 people are 17, 50, 19, 43, 20, 36, 21 and 29. Find the median
A. 24
B. 26
C. 27
D. 25
400. Find the mode for the following numbers: 16,29,19,27,18,20,27,24,19,27.
A. 19
B. 27
C. 18
D. 24
401. Find the mean of 67,53,50,76,66,81,69,77,91.
A. 73
B. 72
C. 71
D. 70
402. The odds that a new product will succeed are estimated as being 5:3. Find the probability
that the product will succeed.
A. 0.625
B. 0.652
C. 0.562
D. 0.626
403. Determine the root mean square (RMS) of the numbers 2.7, 3.2, 3.8, 4.3.
A. 1.55
B. 2.55
C. 3.55
D. 4.55
404. If the variable x assumes that values 1, 3 and 5 while those of the variable y are 2, 4 and 6,
calculate the value of
.
A. 184
B. 188
C. 187
D. 186
405. If P(A) = 0.25 and P(B) = 0.35 and if A and B are not mutually exclusive events, find P(A
or B).
A. 0.0875
B. 0.0714
C. 0.6000
D. 0.5125
406. The number of minutes, a girl spent in making 6 phone calls was 3, 8, 9, 11, 15 and 20
minutes. Find the mean number of calls.
A. 10 min
B. 11 min
C. 12 min
D. 13 min
407. In a basketball game, Jawo is given two free throws. Based on his previous record, the
probability that his first free throw will be successful is 0.75 and the probability that he will be
successful on both throws is 0.55. If Jawo is successful on the first throw, what is the probability
that he makes the second throw?
A. 0.71
B. 0.73
C. 0.74
D. 0.75
408. For a sample which consists of the values 45, 50, 55, 60 and 65, the average deviation is
A. 5
B. 4
C. 6
D. 7
409. If
and
, find
.
A. 18
B. 16
C. 15
D. 17
410. Find the geometric mean of 2, 3, 3, 5, 7 and 8.
A. 4.121
B. 4.131
C. 4.141
D. 4.151
411. Of 300 students, 100 are currently enrolled in mathematics and 80 are currently enrolled in
Physics. These enrolment figures include 30 students who are enrolled in both subjects. What is
the probability that a randomly chosen student will be enrolled in either Mathematic or Physics?
A. 0.45
B. 0.50
C. 0.55
D. 0.60
412. On a single roll of a die, what are the odds of rolling either an even number or a 5?
A. 2:1
B. 3:1
C. 4:1
D. 5:1
413. Find the standard deviation of 4, 7, 8, 9 and 12.
A. 2.61
B. 2.81
C. 2.41
D. 2.31
414. If the median (Md) is 57.22 and the mean (M) is 55.78, find the mode (Mo) by using the
empirical formula M-Mo = 3(M-Md).
A. 30.10
B. 40.10
C. 50.10
D. 60.10
415. Evaluate
by using the summation formulas.
A. 10
B. 11
C. 12
D. 13
416. Calculate the harmonic mean of the numbers 2, 4, 5 and 7.
A. 3.55
B. 3.66
C. 3.77
D. 3.88
417. In an electric company, the probability of passing an IQ test is 0.75. If ten applicants took
the test, what is the theoretical standard deviation of the group?
A. 1.57
B. 1.47
C. 1.37
D. 1.27
418. A student has test scores of 75, 83 and 78. The final test counts half the total grade. What
must be the minimum(integer) score of the final test so that the average is 80?
A. 83
B. 82
C. 84
D. 81
419. Out of 10,000 men, the probability that a man picked at random weighs over 86 kg is 0.25
and the probability that the man weighs less than 61 kg is 0.15. What is the probability that a
man picked at random weighs between 61 kg and 86 kg?
A. 0.55
B. 0.60
C. 0.65
D. 0.70
420. The amount X of money a certain author earns is shown in the following probability
function:
X:
P1,000
P1,200
P1,600
P2,000
P2,400
P(X):
0.20
0.22
0.24
0.21
0.13
What is the probability that the author will earn more than P1,500?
A. 0.66
B. 0.34
C. 0.58
D. 0.80
421. If the probability of a defective bolt is 0.10, find the standard deviation of defective bolts in
a total of 500 bolts.
A. 6.71
B. 7.61
C. 5.71
D. 6.17
422. An urn contains 3 white balls and 2 black balls. If two balls are drawn at random, what is
the probability that the two balls drawn are of different colors?
A. 4/5
B. 2/5
C. 1/5
D. 3/5
423. On the final examination on Algebra, Juan was informed that he received a standard score
of 1.4. If the standard deviation of the examination grades is 10 and the mean is 72, find the
examination grade if Juan.
A. 85
B. 86
C. 84
D. 83
424. A die is tossed 6 times. Using the binomial probability formula, determine the probability of
rolling the number 5 four times.
A. 0.00804
B. 0.00480
C. 0.08004
D. 0.00840
425. For the probability distribution given below, find the mean.
X:
-10
-20
P(X):
1/5
3/10
A. 6
B. 7
C. 5
D. 8
30
1/2
426. The probability that a man will be alive in 20 years is 0.68 and the probability that his wife
will be alive in 20 years is 0.45. What is the probability than both will be alive in 20 years?
A. 0.360
B. 0.306
C. 0.630
D. 0.603
427. In problem 426, find the probability that at least one of them will be alive in 20 years.
A. 0.428
B. 0.482
C. 0.824
D. 0.842
428. If a pack of 52 cards is cut, what is the probability that it shows a king, a jack, a spade or an
ace?
A. 0.3421
B. 0.2431
C. 0.1432
D. 0.4231
429. A box contains 4 red marbles, 8 white marbles and 12 blue marbles. If 3 marbles are drawn,
what is the probability that one of each color is drawn?
A. 0.1897
B. 0.1987
C. 0.1798
D. 0.1879
430. A lottery has one prize of P100,000, two prizes of P50,000, five prizes of P25,000 and ten
prices of P10,000. If there are 100,000 ticket sold, what is the expected value of a ticket?
A. P4.00
B. P4.25
C. P4.50
D. P4.75
431. Out of 800 families with 4 children each, how many of these families would have at least
one boy?
A. 600
B. 650
C. 700
D. 750
432. What is the probability of obtaining a sum of 11 when 3 dice are tossed?
A. 0.145
B. 0.135
C. 0.125
D. 0.115
433. From 5 men and 6 women, a committee consisting of 3 men and 2 women is to be formed.
How many different committees can be formed if 2 men must be on the committee?
A. 35
B. 40
C. 45
D. 50
434. Two students A and B were informed that they received standard scores of 2.6 and -0.8
respectively on the final examinations in Physics. If their examination grades were 83 and 62
respectively, find the standard deviation of the examination grades.
A. 15
B. 13
C. 11
D. 9
435. In how many ways can 30 boys be selected out of 100 boys? Hint: Use Stirling’s
approximation to n!
A. 24
B. 25
C. 26
D. 27
436. Find the probability of winning the first prize of a state lottery in which one is required to
choose six of the numbers 1, 2, 3, ..., 45 in any order.
A. 1.52 x 10-7
B. 1.42 x 10-7
C. 1.32 x 10-7
D. 1.22 x 10-7
437. If 3 percent of the electric bulbs manufactured by a company are defective, find the
probability that in a sample of 100 bulbs, 5 will be defective by using Poisson distribution.
A. 0.105
B. 0.103
C. 0.101
D. 0.107
438. A bag contains 3 white balls and 4 red balls. Each of three boys A, B and C, named in that
order, draws a ball without replacement. The first to draw a red ball receives P70. Determine the
mathematical expectation of C.
A. P6.00
B. P8.00
C. P10.00
D. P7.00
439. Find the probability of getting between 2 and 5 heads inclusive in 8 tosses of a fair coin.
A. 0.8203
B. 0.8302
C. 0.8230
D. 0.8032
440. Five sealed envelopes are placed in a box, three of them containing P50 bill each and two of
them containing P100 bill each. Another box has ten sealed envelopes, six of them containing
P50 bill each and four of them containing P100 bill each. If a box is selected at random and an
envelope is drawn from it, what is the probability that it contains a P100 bill?
A. 3/5
B. ¾
C. 2/3
D. 2/5
441. A die is tossed 8 times. What is the probability of tossing 5 and 6 twice?
A. 0.044
B. 0.064
C. 0.054
D. 0.034
442. Given the probability distribution
X:
8
P(X):
1/4
Find the expected value of x2 or E(x2)
A. 283
B. 273
C. 263
D. 253
15
1/3
16
3/8
24
1/6
443. If a man buys a lottery ticket, he can win first prize of P30,000,000 or a second prize of
P20,000 with probabilities of 1.9 x 10-7 and 4.1 x 10-5 respectively. What should be a fair price to
pay the ticket?
A. P5.52
B. P6.52
C. P7.52
D. P8.52
444. A box contains 3 red balls and 7 black balls. A person selects a ball at random and the color
is noted. Then the ball is replaced. After shaking the box, a second ball is drawn and followed by
the same procedure until five drawings were made. What is the probability that of the 5 balls
drawn, 2 were red?
A. 0.3078
B. 0.3708
C. 0.3087
D. 0.3807
445. Three towns A, B and C are equidistant from each other. A car travels from A to B at
40kph, from B to C at 50 kph and from C to A at 60 kph. Determine the average speed for the
entire trip. (Hint: The average speed is equal to the harmonic mean of the given speeds.)
A. 44.65 kph
B. 46.65 kph
C. 48.65 kph
D. 59.65 kph
446. An airplane travels distances of 1,500 mi, 2,000 mi and 3,200 mi at speeds of 120 mph, 150
mph and 200 mph respectively. Find the average speed of the plane.
A. 160 mph
B. 150 mph
C. 140 mph
D. 130 mph
447. Given the following frequency distribution:
Class Interval
5–7
8 – 10
11 – 13
14 – 16
17 – 19
Find the arithmetic mean.
A. 10.95
B. 11.95
C. 12.95
D. 13.95
Frequency
8
14
18
11
9
448. In problem 447, find the median.
A. 11.73
B. 11.63
C. 11.83
D. 11.53
449. In problem 447, find the standard deviation.
A. 3.73
B. 3.63
C. 3.53
D. 3.43
450. In problem 447, find the coefficient of variance.
A. 31.21 %
B. 41.21 %
C. 51.21 %
D. 61.21 %
451. Out of 50 numbers, 8 were 10’s, 12 were 7’s, 15 were 16’s, 10 were 9’s and the remainder
were 15’s. Find the mean.
A. 13.38
B. 10.38
C. 11.38
D. 12.38
452. The probability that a man will be alive in 25 years is 3/5 and the probability that his wife
will be alive in 25 years is 2/3. Find the probability that one of them will be alive in 25 years.
A. 4/15
B. 1/5
C. 2/5
D. 7/15
453. Three marbles are drawn without replacement from an urn containing 4 red marbles and 6
white marbles. If X is a random variable that denotes the total number of red marbles, construct a
table showing the probability distribution and find the variance of the distribution.
A. 0.54
B. 0.56
C. 0.58
D. 0.52
454. Three teachers in mathematics reported mean examination grades of 2.45, 2.25 and 1.85 in
their classes which consisted of 35, 28 and 20 students respectively. Determine the mean grade
of the classes.
A. 2.22
B. 2.24
C. 2.26
D. 2.28
455. A fair die is tossed 6 times. Find the probability that one 2, two 3’s and three 4’s turn up.
A. 0.0013
B. 0.0015
C. 0.0017
D. 0.0019
456. If it rains, an umbrella salesman can earn P780 per day. If it is fair, he can lose P156 per
day. What is his mathematical expectation if the probability of rain is 0.30?
A. P120.80
B. P122.80
C. P124.80
D. P126.80
457. A continuous random variable X that can be assume values only between X = 2 and X = 8
inclusive has a density function p(X) = a(X + 3) where a is a constant. Find the value of a.
A. 1/45
B. 1/46
C. 1/47
D. 1/48
458. In problem 457, find P(X-4).
A. ¾
B. 3/5
C. 3/7
D. 3/8
459. A factory supervisor finds that 20 percent of the bolts produced by a machine will be
defective. If 5 bolts are chosen at random, find the probability that at most 2 bolts will be
defective.
A. 0.9214
B. 0.9421
C. 0.9124
D. 0.9412
460. A box contains 5 white balls, 3 red balls and 2 black balls. A ball selected at random from
the box, its color noted and then the ball is replaced. Find the probability that out of 5 balls
selected in this manner, 2 are white balls, 2 are red balls and 1 is a white ball.
A. 0.115
B. 0.125
C. 0.135
D. 0.145
461. Compute the standard deviation for a binomial distribution in which out of 60 bolts, 42 bolts
are found to be defective.
A. 3.5496
B. 3.6549
C. 3.4596
D. 3.9546
462. Joey took examinations in algebra, physics, chemistry and english and scored 84, 79, 88 and
93 respectively. If the mean grade in algebra is 80, in physics 75, in chemistry 85 and in english
90 and if the standard deviation are 8, 6, 4 and 5 in algebra, physics, chemistry and english
respectively, in which subject was his relative standing higher? Hint: Calculate the standard
grade corresponding to each subject and compare.
A. algebra
B. physics
C. chemistry
D. english
463. A bag contains 8 one-centavo coins, 6 ten-centavo coins, 4 twenty five-centavo coins and 2
one-peso coins. The coins are placed one each in uniform boxes. What is the mathematical
expectation of a person drawing a box at random?
A. 11.65
B. 12.65
C. 13.65
D. 14.65
464. If the variance of a sample is 29 and its arithmetic mean is 11, find the root mean square.
A. 11.26
B. 12.25
C. 13.24
D. 10.36
465. In a company, the mean earnings per hour is P180. If the mean earning paid to male nd
female employees were P200 and P150 respectively, determine the percentage of male employed
by the company.
A. 50%
B. 55%
C. 60%
D. 65%
466. A box contains 10 red balls, 15 orange balls, 20 blue balls and 30 green balls. Two balls are
drawn in succession replacement being made after each drawing. Find the probability that at
least one ball is blue
A. 103/225
B. 104/225
C. 105/225
D. 106/225
467. A bag contains 1 red marble and 7 white marbles. A marble is drawn from the bag. After its
color has been noted, it is put back into the bag and another marble is drawn from the bag. Using
Poisson approximation, find the probability that in 8 such drawings, a red ball is selected 3 times.
A. 0.0631
B. 0.0541
C. 0.0451
D. 0.0316
468. A bag contains 9 tickets numbered from 1 to 9 inclusive. If 3 tickets are drawn from the box
one at a time, find the probability that they are drawn in the order odd, odd, even or even, eve,
odd.
A. 7/18
B. 5/18
C. 4/18
D. 3/18
469. Between 1 and 3 pm, the average number of phone calls per minute coming into the switch
board of a company is 2. Using Poisson approximation, find the probability that during one
particular minute there will be 4 phone calls.
A. 0.0702
B. 0.0802
C. 0.0902
D. 0.0602
470. A box contains a very large number of red, white, blue and yellow balls in the ratio 1:2:3:4.
Find the probability that in 10 drawing, 9 yellow balls and 1 red ball will be drawn
A. 0.00026
B. 0.00036
C. 0.00046
D. 0.00056
471. In how many ways can 8 persons be seated at a round table if a certain 2 persons are not to
sit next to each other?
A. 3,600
B. 4,600
C. 5,600
D. 6,600
472. How many sums of money each consisting 3 or more coins can be formed from 6 different
kinds of coins?
A. 40
B. 41
C. 42
D. 43
473. There are 5 different chemistry books, 4 different physics books and 2 different history
books to be placed on a shelf with the books of each subject kept together. Find the number of
ways in which the books can be placed.
A. 54,360
B. 64,350
C. 34,560
D. 45,630
474. Evaluate
A. ¾
B. ¼
C. 2/3
D. 1/3
475. Find the area bounded by the curve y = 2x – x2 and the x-axis.
A. 1/3
B. 2/3
C. 4/3
D. ¾
476. The integral of secn y tan y dy is
A. (secn+1 y)/(n+1)
B. (secn y)/n + C
C. tan y + C
D. (sec2n y)/(2n) + C
477. Use the Wallis’ formula to evalute
A. 8/693
B. 9/693
C. 10/693
D. 11/693
476. If f(x) = x + 3 and g(x) = (x+1) 2, find
A. 2.4139
B. 2.4319
C. 2.3491
D. 2.1943
477. If the integral of
A. 1
B. 2
C. 3
D. 4
dx from x = 0 to x = y is equal to 14/3, find y.
478. Find the integral of 2dx / x3 from x = 0 tp x = infinity.
A. ½
B. 1/3
C. ¼
D. 1/5
479. The arc of the curve
the surface generated.
A. 3.33
B. 4.33
C. 5.33
D. 6.33
480. If
from x=0 to x=1 is revolved about the x-axis. Find the area of
, find k.
A. 0
B. 1
C. 2
D. 3
481. A 30-m long cable weighing 15N/m is to be wound about a windlass. Find the work done.
A. 6750 joules
B. 7650 joules
C. 6507 joules
D. 5760 joules
482. The area bounded by 4x2 + 9y2 = 36 is revolved about the line y = 6 – x. Use Pappus’
theorem to find the volume of the solid generated.
A. 501.4
B. 502.5
C. 503.6
D. 504.7
483. Evaluate
A. 0.271
B. 0.371
C. 0.471
D. 0.571
484. A particle moves along a straight line with velocity v given at time t by v = 12 t2 m/s. Find
the distance traveled by the particle in the first 5 seconds.
A. 300 m
B. 400 m
C. 500 m
D. 600 m
485. The value of
is equal to
A. 0
B. 1
C. -1
D. 2
486. If the area bounded by y = x2, x=k (k>0) and the x-axis is equal to 8/3, find k.
A. -1
B. 1
C. 2
D. -3
487. Evaluate
A. [(4x2+1)3/2]/20 + C
B. [(4x2+1)3/2]/8 + C
C. [(4x2+1)5/2]/20 + C
D. [(4x2+1)5/2]/8 + C
.
488. The length of the arc of the curve y = ln sec x from x = 0 to x = pi/3 is
A. 1.4170
B. 1.3170
C. 1.2170
D. 1.1170
489. If
and
evaluate
A. 3
B. 7
C. 6
D. 5
490. Find the area bounded by y=x2-1 and y=3.
A. 31/3
B. 32/3
C. 35/3
D. 37/3
,
491. Integrate
A.
B.
C.
D.
492. Find the moment of inertia with respect to the x-axis of the area bounded by y2 = 4x, y = 4
and x = 0.
A. 21.2
B. 31.2
C. 41.2
D. 51.2
493. Find the y-coordinate (ŷ) of the centroid of the first-quadrant area under the curve y = ex
between x = 0 and x = 1.
A. 0.91
B. 0.93
C. 0.95
D. 0.97
494. Evaluate
A. 1.7726
B. 1.7627
C. 1.6772
D. 1.6727
495. Find the integral of
from x = 0 to x = 1.
A. pi/6
B. pi/7
C. pi/8
D. pi.9
496. Find the area bounded by y2 = 1 – x, y = x -2, y=1 and y=-1.
A. 7/3
B. 8/3
C. 10/3
D. 11/3
497. If the second-degree equation Ax2 + Bxy + Cy2 +Dx + Ey + F = 0 represents a real conic
and B2 – 4AC is positive, then it is
a. ellipse
b. circle
c. parabola
d. hyperbola
498. If the slopes of two lines are equal and their y-intercepts are different, then the lines are
a. intersecting
b. parallel
c. coincident
d. perpendicular
499. A line with inclination between 0° and 90° has
a. zero slope
b. no slope
c. positive slope
d. negative slope
500. The parabola x2 – 4x + 2y – 6 = 0 opens
a. downward
b. upward
c. to the right
d. to the left
501. The locus of a point on a circle which rolls without slipping on a straight line is called
a. strophoid
b. trochoid
c. astroid
d. cycloid
502. If b2 – 4ac < 0, then the graph of y = ax2 + bx + c
a. crosses the x-axis once
b. crosses the x-axis twice
c. does not cross the x-axis
d. touches the x-axis once
503. The point (4,y) where y < 0 lies in quadrant
a. I
b. II
c. III
d. IV
504. The slope of a vertical line is
a. zero
b. one
c. 90°
d. undefined
505. The graph of y2 – 1 = 0 is
a. a pair of parallel lines
b. a pair of intersecting lines
c. a parabola
d. a point
506. The curve y = x3 is symmetric with respect to
a. the z-axis
b. the y-axis
c. the origin
d. both axes
507. The polar equation of the line parallel to the polar axis and 4 units above it is
a. r = 4cscθ
b. r = 4secθ
c. r = 4sinθ
d. r = 4cosθ
508. The equation y2 + 12y + 36 = 0 represents
a. two parallel lines
b. two intersecting lines
c. a point
d. a straight line
509. If C = 0, then the graph of the line Ax + By + C = 0
a. is parallel to the x-axis
b. is parallel to the y-axis
c. crosses the positive x-axis
d. passes through the origin
510. If the inclination θ of a line is an obtuse angle, then the tangent of θ is
a. positive
b. negative
c. zero
d. infinity
511. Which of the following as no graph?
a. x2 + y2 – 9 = 0
b. x2 + y2 + 9 = 0
c. x2 – y2 – 9 = 0
d. x2 – y2 + 9 = 0
512. The ellipse is symmetric with respect to
a. the x-axis only
b. the y-axis only
c. the origin only
d. both axes and the origin
513. The circle x2 + y2 = 100 has a radius of
a. 25
b. 30
c. 10
d. 50
514. If the eccentricity of a conic is 3/5, then it is
a. an ellipse
b. a circle
c. a parabola
d. a hyperbola
515. The graph of the polar equation r(2 + 4sinθ) = 3 is
a. a circle
b. a hyperbola
c. a parabola
d. an ellipse
516. if a line slants downward to the right, then it has
a. negative slope
b. positive slope
c. no slope
d. zero slope
517. the equation of the directrix of the parabola x2 =16y is
a. x + 4 = 0
b. x – 4 = 0
c. y – 4 = 0
d. y + 4 = 0
518. the locus of a point such that its radius vector is proportional to its vectorial angle is
called the
a. Conchoid of Nicomedes
b. Spiral of Archimedes
c. Cissoid of Diocles
d. Folium of Descartes
519. If A = 0 and B∙C ≠ 0, then the line Ax + By + C = 0 is
a. parallel to the x-axis
b. parallel to the y-axis
c. perpendicular to the x-axis
d. coincident with the y-axis
520. The graph of the equation 4y2 = 8 – x2 is
a. a circle
b. an ellipse
c. a parabola
d. a hyperbola
521. If the directed distance from a point to the line is negative, then which of the following is
true?
a. The point and the origin are not on the side of the line.
b. The point and the origin are on the opposite sides of the line.
c. The point is below the line.
d. The point is above the line.
522. It is the locus of a point which moves in a plane so that the sum of its distance from two
fixed points is constant.
a. a circle
b. a parabola
c. an ellipse
d. a hyperbola
523. If M is a point that is 1/3 of the distance from point A to point B, then M divides the line
segment AB in the ratio
a. 1:3
b. 1:2
c. 2:3
d. 1:4
524. Which of the following is the polar equation of a limacon?
a. r = 1 + sinθ
b. r = 2(1 – sinθ)
c. r = 2 – sinθ
d. r = 2sinθ
525. A line will have a positive slope under which of the following conditions?
a. positive x-intercept and positive y-intercept
b. negative x-intercept and positive y-intercept
c. negative x-intercept and negative y-intercept
d. both b and c
526. If two lines with slopes m1 and m2 are perpendicular to each other, then which of the
following relations is true?
a. m1 = m2
b. m1m2 = -1
c. m1/m2 = -1
d. m1 – m2 = 1
527. The graph of y2 + 4x = 0 has symmetry with respect to the
a. x-axis only
b. y-axis only
c. origin only
d. all of a, b and c
528. If the eccentricity of a conic is greater than one, then it is a
a. an ellipse
b. a circle
c. a parabola
d. a hyperbola
529. The graph of Ax2 + Cy2 + Dx +Ey +F = 0 where A and C are not both zero is a parabola
if
a. AC = 0
b. AC > 0
c. AC < 0
d. AC ≠ 0
530. Which of the following curves is symmetric with respect to the x-axis?
a. y2 = 2x3
b. y = 2x3
c. xy = 2
d. y = 3x2
531. the graph of a limacon r = a + bcosθ has an inner loop if
a. a = b
b. 0 < a/b < 0
c. ab = 1
d. 0 < b/a < 1
532. Which of the following is the equation of a pair of parallel lines?
a. y2 – x2 = 0
b. x2 + y2 +7 = 0
c. y2 + 4y = 0
d. x2 – 6x + 9 = 0
533. Which of the following is an equation of a pair of semicubical parabola?
a. y = x3/2
b. y = x1/2
c. y = x4
d. y = 1/x
534. The graph of 3x2 – y = y2 + 6x is
a. a parabola
b. an ellipse
c. a circle
d. a hyperbola
535. The equation Ax2 + Cy2 + Dx + Ey +F = 0 is an ellipse if
a. both A and C are not zero, A = C and they have the same sign
b. neither A nor C is zero, A ≠ C and they have the same sign
c. both A and C are not zero, A = C and they have opposite signs
d. neither A nor C is zero, A ≠ C and they have opposite signs
536. The distance between the foci of an ellipse 6x2 + 2y2 = 12
a. 4
b. 5
c. 6
d. 7
537. The distance between the directrices of an ellipse in problem 40 is
a. 5
b. 6
c. 7
d. 8
538. What is the polar equation of the line passing through (3, 0°) and perpendicular to the
polar axis?
a. r = 3cscθ
b. r = 3secθ
c. r = 3cosθ
d. r = 3sinθ
539. Find the equation of the radical axis of the following circles:
C1:
x2 + y2 – 5x +3y -2 = 0
C2:
x2 + y2 + 4x – y – 7 = 0
a. 9x + 4y – 5 = 0
b. 9x – 4y + 5 = 0
c. 9x – 4y – 5 = 0
d. 9x + 4y + 5 = 0
540. Find the distance between the points A(-3,0) and B(-4,7).
a.
b.
c.
d.
541. If the slope of the line determined by the points (x,5) and (1,8) is -3, find x.
a. 2
b. 1
c. 0
d. 3
542. The focus of the parabola y2 = 4x is at
a. (4,0)
b. (0,4)
c. (1,0)
d. (0,1)
543. The inclination of the line determined by the points (2,5) and (1,8) is
a. 106.41°
b. 107.42°
c. 108.43°
d. 109.44°
544. The length of the latus rectum of 27x2 + 36y2 = 972 is
a. 8
b. 9
c. 10
d. 11
545. The slope of the line through the points (-4,-5) and (2,7) is
a. 2
b. -2
c. 3
d. -3
546. The equivalent of x2 + y2 – y = 0 in polar form is
a. r = 2cosθ
b. r = 2sinθ
c. r2 = 2sinθ
d. r2 = 2cosθ
547. The area of the ellipse x2/64 + y2/16 = 1 is
a. 30π
b. 31π
c. 32π
d. 33π
548. Find the equation of the ellipse which has the line 2x – 3y = 0 as one of its asymptotes.
a. 2x2 – 3y2 = 6
b. 3y2 – 2y2 = 6
c. 4x2 – 9y2 = 36
d. 9y2 – 4x2 = 36
549. The transverse axis of the hyperbola 36x2 – 25y2 = 900 is
a. 13
b. 12
c. 11
d. 10
550. The parabola y = 3x2 – 6x + 5 has its vertex at
a. (0,5)
b. (1,2)
c. (-1,14)
d. (2,5)
551. The line 4x – 6y + 14 = 0 is coincident with the line
a. 2x = 3y – 7
b. 2x = 3y + 7
c. 4x = 6y + 14
d. 4x = 14 – 6y
552. Determine the axis of symmetry of the parabola (y + 5) 2 = 24x
a. y = 5
b. y = -5
c. x = 5
d. x = -5
553. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate
axes.
a. 2
b. 3
c. 4
d. 5
554. The directrix of the parabola is y = 5 and its focus is at (4,-3). What is the latus rectum?
a. 14
b. 15
c. 16
d. 17
555. Find the equation of the circle containing the point (1,-4) and center at the origin.
a. x2 + y2 = 16
b. x2 + y2 = 17
c. x2 + y2 = 18
d. x2 + y2 = 19
556. Find the equation of the line containing the point (2,-3) and is parallel to the line 3x + y –
5 = 0.
a. 3x + y – 1 = 0
b. 3x + y – 4 = 0
c. 3x + y – 2 = 0
d. 3x + y – 3 = 0
557. The distance from the point (2,1) to the line 4x – 3y + 5 = 0 is
a. -2
b. 2
c. -3
d. 4
558. If the slope of the line (k + 1)x + ky – 3 = 0 is -2, find k.
a. 2
b. 1
c. -3
d. -2
559. Write the equation of the line with x-intercept -6 and y-intercept 3.
a. x + 2y – 6 = 0
b. x – 2y – 6 = 0
c. x – 2y + 6 = 0
d. x + 2y + 6 = 0
560. Write the equation of the tangent line to the circle x2 + y2 = 80 at the point in the first
quadrant where x = 4.
a. x – 2y – 20 = 0
b. x – 2y + 20 = 0
c. x + 2y – 20 = 0
d. x + 2y + 20 = 0
561. If the distance between (8,7) and (3,y) is 13, what is the value of y?
a. -5 or 19
b. 5 or 19
c. 5 or -19
d. -5 or -19
562. If the major axis of an ellipse is twice its minor axis, find its eccentricity.
a. 0.965
b. 0.866
c. 0.767
d. 0.668
563. The center of the circle x2 + y2 – 18x +10y +25 = 0 is
a. (9,5)
b. (-9,5)
c. (-5,9)
d. (9,-5)
564. Compute the area of the polygon with vertices at (6,1), (3,-10), (-3,-5) and (-2,0).
a. 60
b. 50
c. 40
d. 30
565. A line with the equation y = mx + k passes through the points (-1/3,-6) and (2,1). Find m.
a. 2
b. 3
c. 4
d. 5
566. Find the tangential distance from the point (8,5) to the circle (x – 2)2 + (y – 1)2 = 16.
a. 7
b. 8
c. 9
d. 6
567. Find the equation of the line through (-1,3) and is perpendicular to the line 5x – 2y + 3 =
0.
a. 2x + 5y – 13 = 0
b. 2x + 5y – 12 = 0
c. 2x + 5y – 11 = 0
d. 2x + 5y – 10 = 0
568. Find the distance between the two lines represented by the two linear equations 4x – 3y –
12 = 0 and 4x – 3y + 8 = 0.
a. 8
b. 6
c. 5
d. 4
569. The distance between the points (sinθ,cosθ) and (cosθ,-sinθ) is
a. 1
b. 2
c.
d.
570. Find the equation of the line parallel to 3x + 4y + 2 = 0 and -3 units from it.
a. 3x + 4y + 13 = 0
b. 3x + 4y – 13 = 0
c. 3x + 4y + 17 = 0
d. 3x + 4y – 17 = 0
571. If the circle has its center (-3,1) and passes through (5,7), then its radius is
a. 7
b. 8
c. 9
d. 10
572. Find the area of the triangle whose vertices lie at A, B and C whose coordinates are (4,1),
(6,2) and (2,-5), respectively.
a. 4
b. 5
c. 6
d. 7
573. Express y3 = 4x2 in polar form
a. r = 4cot2θcscθ
b. r = 4cotθcsc2θ
c. r = 4cot2θcsc2θ
d. r = 4cotθcscθ
574. If the slopes of the lines L1 and L2 are 3 and -1 respectively, find the angle between them
measured counterclockwise from L1 to L2.
a. 64.33°
b. 36.43°
c. 63.43°
d. 43.36°
575. What is the length of the latus rectum of a hyperbola with foci at (-3,15) and (-3,-5) and a
transverse axis equal to 12?
a. 44/3
b. 54/3
c. 64/3
d. 74/3
576. If the line through (-1,3) and (-3,-2) is perpendicular to the line through (-7,4) and (x,2),
find x if x is positive.
a. 3
b. 2
c. 4
d. 1
577. Determine k so that the line y = kx – 3 will be parallel to the line 4x + 12y = 12.
a. 1/2
b. 1/3
c. -1/3
d. -3
578. Find the equation of the parabola with focus at (0,8) and directrix y + 8 = 0.
a. x2 = -32y
b. x2 = 32y
c. y2 = -32x
d. y2 = 32x
579. find the tangent of the angle from the line through (-2,-3) and (4,3) to the line through (1,6) and (3,-2)
a. 3
b. 4
c. 2
d. 1
580. The second-degree equation
19x2 + 6xy 11y2 + 20x – 60y +80 = 0
represents a conic. To remove the xy-term, we rotate the coordinate axes through an
angle of
a. 16.40°
b. 17.41°
c. 18.43°
d. 19.45°
581. Find the value of k given that the slope of the line joining (3,1) and (5,k) is 2.
a. 2
b. 3
c. 4
d. 5
582. If the focus of a parabola is at (-6,0) and its vertex is at (0,0), the equation of its directrix
is
a. x + 6 = 0
b. x – 6 = 0
c. x + 3 = 0
d. x – 3 = 0
583. For what value of k is the line 6y + (2k – 1 )x = 12 perpendicular to the line 3y – 2x = 6?
a. 5
b. 4
c. 3
d. 2
584. The circumference of the circle x2 + y2 – 8x +2y + 8 = 0 is
a. 18.85
b. 17.85
c. 16.85
d. 15.85
585. If the perpendicular distance from the line kx – 3y + 15 = 0 to the point (2,1) is -4, find k.
a. -4
b. -3
c. -2
d. -1
586. The eccentricity of the hyperbola 16(y – 6)2 – 9(x – 7)2 = 144 is equal to
a. 4/3
b. 5/3
c. 7/3
d. 9/4
587. If the tangent of the angle from the line through (6,y) and (-4,2) to the line through (6,6)
and (3,0) is 8/9, find the value of y if y is positive.
a. 4
b. 5
c. 6
d. 7
588. Find the equation of the line which passes through the point (8,3) and forms with the
coordinate axes a triangle of area 54.
a. 4x + 3y – 41 = 0
b. 2x + 4y – 28 = 0
c. 5x + 2y – 46 = 0
d. 3x + 4y – 36 = 0
589. If P0(x0,y0) is such that P1P0/P0P2 = 7/6 where P1(2,5) and P2(5 ,-1), find x0.
a. 45/13
b. 46/13
c. 47/13
d. 18/13
590. Find the polar equation of the line perpendicular to θ = 20° and passing through the point
(6,20°).
a. r = 6sec(θ + 20°)
b. r = 6sec(θ – 20°)
c. r = -6sec(θ + 20°)
d. r = -6sec(θ – 20°)
591. Determine b so that x2 + y2 + 2x – 3y – 5 = 0 and x2 + y2 + 4x + by + 2 = 0 are
orthogonal.
a. 10/3
b. 11/3
c. 13/3
d. 14/3
592. If the value of the invariant B2 – 4AC is negative, then the second-degree equation Ax2 +
Bxy +Cy2 + Dx + Ey + F = 0 represents either an ellipse or
a. a pair of parallel lines
b. two intersecting lines
c. a point
d. a line
593. Find the distance between the points (4,40°) and (4,220°).
a. 7
b. 8
c. 10
d. 9
594. Identify the locus of the curve whose parametric equations are x = 3sinθ, y = 2cosθ.
a. a circle
b. a parabola
c. an ellipse
d. a hyperbola
595. Find the equation of the line through the midpoint of AB where A(-3,1), B(2,-1) and is
perpendicular to AB.
a. 10x + 4y + 5 = 0
b. 10x + 4y – 5 = 0
c. 10x – 4y + 5 = 0
d. 10x – 4y – 5 = 0
596. Find the length of the tangent line from the point P(4,-7) to the circle x2 + y2 – 10x – 4y +
25 = 0.
a.
b.
c.
d.
597. Find the equation of the circle with center at the midpoint of A(4,2), B(-1,-2) and having
a radius 3.
a. 4x2 + 4y2 + 12x + 27 = 0
b. 4x2 + 4y2 – 12x – 27 = 0
c. 4x2 + 4y2 + 12x – 27 = 0
d. 4x2 + 4y2 – 12x + 27 = 0
598. Write the polar equation of the circle with center (-5,π) and radius 5.
a. r = 5cosθ
b. r = -5cosθ
c. r = 10cosθ
d. r = -10cosθ
599. Give the Cartesian equation of the line whose parametric equations are x = 2t – 1, y = 3t
+ 5 where t is the parameter.
a. 3x – 2y + 13 = 0
b. 3x + 2y – 13 = 0
c. 3x – 2y – 13 = 0
d. 3x + 2y + 13 = 0
600. Find the equation of the line through (6,-3) and parallel to the line through (2,8) and (5,1).
a. 3x + y + 15 = 0
b. 3x – y – 15 = 0
c. 3x – y – 15 = 0
d. 3x – y + 15 = 0
601. The vertices of a triangle are A(4,6), B(2,-4) and C(-4,2). Find the length of the median
of the triangle from the vertex C to the side AB.
a.
b.
c.
d.
602. Find the equation of the circle containing (1,-4) and center at the origin.
a. x2 + y2 = 14
b. x2 + y2 = 15
c. x2 + y2 = 16
d. x2 + y2 = 17
603. If AB is perpendicular to CD and A(-1,0), B(2,5), C(3,-1), D(-3,a), find the value of a.
a. 13/4
b. 13/5
c. 13/6
d. 13/7
604. Find the equation of the line through (4,0) and is parallel to the altitude from A to BC of
the triangle A(1,3), B(2,-6) and C(-3,0).
a. 5x + 6y + 20 = 0
b. 5x – 6y – 20 = 0
c. 5x + 6y – 20 = 0
d. 5x – 6y + 20 = 0
605. Find the equation of the circle which has the line joining (4,7) and (2,-3) as diameter.
a. (x – 2)2 + (y – 3)2 = 26
b. (x – 2)2 + (y – 3)2 = 27
c. (x – 2)2 + (y – 3)2 = 28
d. (x – 2)2 + (y – 3)2 = 29
606. Write the equation of the line with x-intercept -6 and y-intercept 3.
a. x – 2y – 6 = 0
b. x + 2y + 6 = 0
c. x – 2y + 6 = 0
d. x + 2y – 6 = 0
607. Find the abscissa of the point P0 which divides P 1P2 in the ratio P1P0/P0P2 = r1/r2 were
P1(2,5), P2(6,-3), r1 = 3, r2 = 4.
a. 25/7
b. 26/7
c. 27/7
d. 28/7
608. Find the equation of the conic with eccentricity 7/4 and foci at (7,0) and (-7,0).
a. x2/33 + y2/16 = 1
b. x2/16 + y2/33 = 1
c. x2/33 – y2/16 = 1
d. x2/16 – y2/33 = 1
609. Find the equation of the line passing trough (2,-3) and is parallel to the line 3x – y = 5.
a. 3x + y – 2 = 0
b. 3x + y – 3 = 0
c. 3x + y – 4 = 0
d. 3x + y – 5 = 0
610. If the slope of the line (k + 1)x + ky – 3 = 0 is arctan(-2), find the value of k.
a. 1
b. 2
c. 3
d. 4
611. Find the equation of the line parallel to 5y – 5x + 12 = 0 and contains the point (0,-3).
a. x – y + 3 = 0
b. x + y – 3 = 0
c. x – y – 3 = 0
d. x + y + 3 = 0
612. Find k so that the circle x2 + y2 + 2kx + 4y – 5 = 0 will pass through the point (5,1).
a. -3/2
b. -5/2
c. -7/2
d. -9/2
613. Find the equation of the line through the points (-7,-3) and (-1,9).
a. 2x – y + 11 = 0
b. 2x + y – 11 = 0
c. 2x + y + 11 = 0
d. 2x – y – 11 = 0
614. The equation of the parabola with vertex (-1,2) and directrix at x = -3 is
a. (y – 2)2 = 8(x + 1)
b. (y + 2)2 = 8(x + 1)
c. (x + 1)2 = 8(y + 2)
d. (x – 1)2 = -8(y + 2)
615. Find the length of the latus rectum of a parabola with focus at (-2,-6) and directrix x – 2 =
0.
a. 6
b. 4
c. 8
d. 10
616. Write the equation of the line tangent to the circle
x2 + y2 + 14x + 18 y – 39 = 0
at the point in the second quadrant where x = -2.
a. 5x + 12y + 26 = 0
b. 5x – 12y – 26 = 0
c. 5x + 12y – 26 = 0
d. 5x – 12y + 26 = 0
617. The two points on the line 2x + 3y + 4 = 0 which are at a distance 2 from the line 3x + 4y
– 6 = 0 are
a. (7,-6) and (-11,6)
b. (-88,-8) and (-16,-16)
c. (64,-44) and (4,-4)
d. (-44,64) and (10,-10)
618. Find the equation of the line which forms with the axes in the first quadrant a triangle of
area 2 and whose intercepts differ by 3.
a. x + 4y – 4 = 0
b. x – 4y + 4 = 0
c. x + 4y + 4 = 0
d. x – 4y – 4 = 0
619. What is the locus of a point which moves so that its distance from the line x = 8 is twice
its distance from the point (2,8)?
a. a circle
b. an ellipse
c. a parabola
d. a hyperbola
620. Write the polar equation of a line which passes through the points (2,π/2) and (-1,0).
a. r(2cosθ + sinθ) – 2 = 0
b. r(2cosθ – sinθ) – 2 = 0
c. r(2cosθ + sinθ) + 2 = 0
d. r(2cosθ – sinθ) + 2 = 0
621. The line segment with end points A(-1,-6) and B(3,0) is extended beyond point A to a
point C so that C is 4 times as far from B as from A. find the abscissa of point C.
a. -5/3
b. -7/3
c. -8/3
d. -4/3
622. A semi-elliptic arch is 20-ft high at the center and as a span of 50-ft. find the height of the
arch at a point 10-ft from one end of the base.
a. 14 ft
b. 15 ft
c. 16 ft
d. 17 ft
623. If the slope of a line 3x + y – 5 + k(x + 2y – 3) = 0 is 11/3, find k.
a. -4/5
b. -3/5
c. -2/5
d. -1/5
624. The equation of the ellipse with vertices at (-3,-2) and (1,-2) and which passes through (2,-1) is
a. x2 + 3y2 + 2x + 12y + 9 = 0
b. 3x2 + y2 + 2x + 12y – 9 = 0
c. x2 + 3y2 – 2x + 12y + 9 = 0
d. 3x2 + y2 – 2x + 12y – 9 = 0
625. Find the diameter of the ellipse 9x2 + 16y2 = 144 defined by the system of parallel chords
of slope 2.
a. 9x – 32y = 0
b. 9x + 32y = 0
c. 32x – 9y = 0
d. 32x + 9y = 0
626. The locus of 4x2 + 4xy + y2 + 2x + y – 2 = 0 is a pair of parallel lines. What is the slope
of each line?
a. -1
b. -2
c. 1
d. 2
627. Find the area of a triangle with one vertex at the pole and the two others are (5,60°) and
(4,-30°).
a. 13
b. 12
c. 11
d. 10
628. Given A(3,7), B(-6,4), C(-2,8) and D(-7,0). Find the tangent of the angle measured
counterclockwise from AB to CD.
a. 17/23
b. 18/23
c. 19/23
d. 20/23
629. Find the equation of the hyperbola with vertices at (4,0) and (-4,0) and asymptotes y = 2x
and y = -2x.
a.
x2/64 – y2/16 = 1
b. x2/16 – y2/64 = 1
c.
y2/64 – x2/16 = 1
d.
y2/64 – x2/64 = 1
630. The equation of the perpendicular bisector of the line segment joining the points (2,6) and
(-4,3) is
a. x + 2y – 8 = 0
b. 4x + 2y – 5 = 0
c. x – 2y + 10 = 0
d. 4x + 2y – 13 = 0
631. Assume that power cables hang in a parabolic arc between two pole 100-ft apart. If the
poles are 40-ft high and if the lowest point on the suspended cable is 35-ft above the
ground, find the height of the cable at a point 20-ft from the pole.
a. 34.8 ft
b. 35.8 ft
c. 36.8 ft
d. 37.8 ft
632. Transform the rectangular equation (x2 + y2)3 = 4x2 y2 into polar coordinates.
a. r = 2sinθ
b. r = sin2θ
c. r = 2cosθ
d. r = cos2θ
633. What is the eccentricity of an equilateral hyperbola?
a.
b.
c. 1.5
d. 2
634. Find the equation of the locus of a point which moves so that its distance from (4,0) is
equal to two thirds of its distance from the line x = 9.
a. 9x2 – 5y2 = 180
b. 5x2 – 9y2 = 180
c. 9x2 + 5y2 = 180
d. 5x2 + 9y2 = 180
635. Find the equation of the line through the point which divides A(-1,-1/2), B(6,3) in the
ratio AP/PB = 3/4 and through the point Q which is equidistant from C(1,-1), D(-3,1)
and E(-1,3).
a. x – 8y – 6 = 0
b. x – 8y + 6 = 0
c. x + 8y + 6 = 0
d. x + 8y – 6 = 0
636. Find the equation of the line tangent to the hyperbola 9x2 – 2y2 = 18 at the point (-2,3).
a. 3x + y + 3 = 0
b. 3x – y + 3 = 0
c. 3x + y – 3 = 0
d. 3x – y – 3 = 0
637. For the conic 2x2 – xy + x + y – 5 = 0, find the equation of the diameter defined by the
cords of slope ½.
a. 7x + 2y – 3 = 0
b. 7x + 2y + 3 = 0
c. 7x – 2y + 3 = 0
d. 7x – 2y – 3 = 0
638. The equation of the hyperbola with foci at (0,9) and (0,-9) and conjugate axis 10 units is
a. x2/56 – y2/25 = 1
b. x2/25 – y2/56 = 1
c. y2/56 – x2/25 = 1
d. y2/25 – x2/56 = 1
639.
An arch is in the form of a semi-ellipse with major axis as the span. If the span is 24.4 m
and the maximum eight is 9.2 m, find the height of the arch at a point 4.6 m from the
semi-minor axis.
a. 6.9 m
b. 5.9 m
c. 8.9 m
d. 7.9 m
640. If the area of the quadrilateral with vertices at (-5,-1), (x,2), (10,-4) and (-2,7) is 78.5, find
x if x is positive.
a. 5
b. 6
c. 7
d. 8
641. Find the value of k so that the radius of the circle x2 + y2 – kx + 6y – 3 = 0 is equal to 4.
a. 3
b. 4
c. 5
d. 6
642. A parabolic segment is 32 dm high and its base is 16 dm. What is the focal distance?
a. 0.5 dm
b. 0.4 dm
c. 0.6 dm
d. 0.3 dm
643. Write the equation of the hyperbola conjugate to the hyperbola 4x2 – 3y2 + 32x + 18y +
25 = 0.
a. 4x2 – 3y2 + 32x + 18y – 49 = 0
b. 4x2 – 3y2 + 32x + 18y – 36 = 0
c. 4x2 – 3y2 + 32x + 18y – 16 = 0
d. 4x2 – 3y2 + 32x + 18y – 64 = 0
644. Find the abscissa of the point P on the line segment AP for A(-8,4) and B(-13,6) if AP:PB
= 3:2.
a. -10
b. -11
c. -9
d. -12
645. Find the point on the parabola x2 = 16y at which there is a tangent with a slope ½.
a. (8,4)
b. (-8,4)
c. (4,1)
d. (-4,1)
646. What is the equation of the line tangent to the hyperbola
if the slope
of the line is 2?
a. 2x + y + 23 = 0
b. 2x + y – 23 = 0
c. 2x – y + 23 = 0
d. 2x – y – 23 = 0
647. Find the eccentricity of an ellipse whose latus rectum is 2/3 of the major axis.
a. 0.58
b. 0.68
c. 0.78
d. 0.88
648. The vertices of a triangle are (2,4), (x,-6) and (-3,5). If x is negative and the area of the
triangle is 28.5, find x.
a. -5
b. -6
c. -4
d. -7
649. A parabolic arch has a span of 20 m and a maximum height of 15 m. how high is the arch
4 m from the center of the span?
a. 10.6 m
b. 11.6 m
c. 12.6 m
d. 13.6 m
650. Determine the value of k so the following circles are orthogonal:
C1:
x2 + y2 + 2x – 3y – 5 = 0
C2:
x2 + y2 + 4x + ky + 2 = 0
a.
11/2
b.
13/3
c.
14/3
d.
10/3
651. An ellipse has its foci at (0,c) and (0,-c) and its eccentricity is ½. Find the length of the
latus rectum.
a. 2c
b. 3c
c. 4c
d. 5c
652. The earth’s orbit is an ellipse with eccentricity 1/60. If the semi-major axis of the orbit is
93M miles and the sun is at one of the foci, what is the shortest distance between the
earth and the sun?
a. 89.43M mi
b. 90.44M mi
c. 91.45M mi
d. 92.46M mi
653. If the length of the latus rectum of an ellipse is ¾ of the length of its minor axis, then its
eccentricity is
a.
0.46
b.
0.56
c.
0.66
d.
0.76
654. If the point P(9,2) divides the line segment from A(6,8) to B(x,y) such that AP:AB =
3:10, find y.
a. -11
b. -10
c. -9
d. -12
655. Find the rectangular equation for the curve whose parametric equations are x = 2cosθ, y =
cos2θ.
a. x2 = 2(y + 1)
b. x2 = 2(y – 1)
c. y2 = 2(x + 1)
d. y2 = 2(x – 1)
656. A parabolic arch spans 200-ft wide. How high must the arch be above the stream to give
a minimum clearance of 40-ft over a tunnel in the center which is 120-ft wide?
a. 60.5 ft
b. 61.5 ft
c. 62.5 ft
d. 63.5 ft
657. In the parabola x2 = 4y, an equilateral triangle is inscribed with one vertex at the origin.
Find the length of each side of the triangle.
a. 13.86
b. 12.85
c. 11.84
d. 10.83
658. The foci of a hyperbola are (4,3) and (4,-9) and the length of the conjugate axis is
.
Find its eccentricity.
a. 1.3
b. 1.5
c. 1.7
d. 1.9
659. Find the length of the common chord of the curves whose equations are x2 + y2 = 48 and
x2 + 8y = 0.
a.
b.
c.
d.
660. The point (8,5) bisects a chord of the circle whose equation is x2 + y2 – 4x + 8y = 110.
Find the equation of the cord.
a. 3x + 2y = 0
b. 3x – 2y = 14
c. 2x + 3y = 31
d. 2x – 3y = 1
661. Find the length of the latus rectum of the parabola with focus at (-2,-6) and directrix x – 2
= 0.
a. 8
b. 7
c. 6
d. 4
662. Find the distance between (1,2,-5) and (-1,-1,4).
a.
b.
c.
d.
663. What is the distance from the origin to the point (4,-3,2)?
a.
b.
c.
d.
664. Find the direction numbers of the line through (4,-1,-3) and (0,1,4).
a. 4,-2,-7
b. -4,2,-7
c. -4,-2,7
d. -4,2,-7
665. The direction numbers of two lines are 2,-1,4 and -3,y,2 respectively. Find y if the lines
are perpendicular to each other.
a. -1
b. 3
c. -2
d. 2
666. Transform p = 6θ to spherical coordinates.
a. r2 – z2 = 36θ2
b. r2 – z2 = 6θ
c. r2 + z2 = 36θ2
d. r2 + z2 = 6θ
667. the surface described by the equation 4x2 + y2 + 26z = 100 is an
a. elliptic hyperboloid
b. elliptic paraboloid
c. ellipsoid
d. elliptic cone
668. Find the Cartesian coordinates of the point having the cylindrical coordinates (3,π/2,5).
a. (5,0,3)
b. (3,0,5)
c. (0,5,3)
d. (0,3,5)
669. Find the cylindrical coordinates of the point having the rectangular coordinates (4,4,-2).
a. (
b. (
c. (
d. (
670. The distance of the point (-4,5,2) from the x-axis is
a.
b.
c.
d.
671. The equivalent of (3,4,5) in the cylindrical coordinate system is
a. (5,31.53°,5)
b. (5,51.33°,5)
c. (5,53.13°,5)
d. (5,35.31°,5)
672. If one end of a line is (-2,4,8) and its midpoint is (1,-2,5), find the x-coordinate of the
other end.
a. 4
b. 3
c. 5
d. 6
673. Find the value of k such that the plane x + ky – 2z – 9 = 0 shall pass through the point
(5,-4,-6).
a. 2
b. 1
c. 3
d. 4
674. The locus of 9x2 – 4z2 – 36y = 0 is a/an
a. elliptic cone
b. hyperbolic paraboloid
c. parabolic cylinder
d. ellipsoid
675. The trace of x2 + 4z2 – 8y = 0 on the xy-plane is
a. a hyperbola
b. an ellipse
c. a parabola
d. a point
676. The locus of y2 + z2 – 4x = 0 has symmetry with respect to
a. xz-plane only
b. yz- and xy-planes
c. z-axis
d. xz- and xy-planes
677. If the plane curve b2x2 + a2y2 = a2b2 is revolved about the x-axis, the surface generated is
a/an
a. ellipsoid of revolution
b. hyperbolic paraboloid
c. paraboloid of revolution
d. parabolic cylinder
678. The rectangular coordinates for the point whose cylindrical coordinates are (6,120°,-2)
are
a. (3,3 ,-2)
b. (2,3
,-3)
c. (-3,3
,-2)
d. (-2,3
,-3)
679. Which of the following has a locus that is a hyperbolic paraboloid?
a. x2 + y2 – 2z = 0
b. x2 + 5z2 – 6y = 0
c. z2 – 2y2 + 4x = 0
d. 4x2 + y2 – 4z = 0
680. Find the z-coordinate of the midpoint of the segment whose end points are (4,5,6) and (3,1,2).
a. 3
b. 4
c. 5
d. 6
681. The traces of the surface
a.
b.
c.
d.
on the coordinate planes are
circles
ellipses
parabolas
hyperbolas
682. Transform the equation θ = tanφ to cylindrical coordinates.
a. r = zθ
b. z = rθ
c. θ = rz
d. r = zφ
683. Which of the following is a quadric cone?
a. x2 – y2 – 4z2 = 0
b. x2 – y2 – 4z = 0
c. x2 + y2 – 4z2 = 0
d. x2 + y2 – 4z = 0
684. Transform z 2r = 1 to spherical coordinates.
a. pcosφ – 2sinφ = 1
b. p(sinφ – 2cosφ) = 1
c. cosφ – 2psinφ = 1
d. p(cosφ – 2sinφ) = 1
685. If z = 0 in the equation 2y2 + 3z2 – x2 = 0, then the trace of the surface on the xy-plane is
a
a. pair of parallel lines
b. pair of intersecting lines
c. line
d. point
686. Find the cylindrical coordinates for the point (6,3,2).
a. (
b. (
c. (
d. (
687. A line makes an angle of 45 degrees with the x-axis and an angle of 60 degrees with the
y-axis. What angle does it make with the z-axis?
a. 30°
b. 45°
c. 60°
d. 55°
688. Two directions cosines of a line are 1/3 and -2/3. What is the third?
a. 2/3
b. 4/3
c. 5/3
d. 7/3
689. A line makes equal angles with the coordinate axes. Find the angle.
a. 44.64°
b. 54.74°
c. 64.84°
d. 74.94°
690. Find the distance of the point (6,2,3) from the x-axis.
a.
b.
c.
d.
691. What
a.
b.
c.
d.
is the locus of any equation of the form x2 + y2 = f(z)?
hyperboloid of revolution
ellipsoid of revolution
paraboloid of revolution
cylinder of revolution
692. The radius of the sphere x2 + y2 + z2 – 6x + 4z – 3 = 0 is
a. 2
b. 3
c. 5
d. 4
693. The direction numbers of two lines are 2,-1,k and -3,2,2 respectively. Find k if the lines
are perpendicular.
a. 4
b. 2
c. 5
d. 3
694. Find the equation of the locus of a point which moves so that it is 4 units in front of the
xz-plane.
a. y +4 = 0
b. z – 4 = 0
c. x + 4 = 0
d. y – 4 = 0
695. The equation x2 + z2 = 5y is a paraboloid of revolution that is symmetric with respect to
a. x-axis
b. y-axis
c. z-axis
d. origin
696. The equation of the plane through the point (-1,2,4) and parallel to the plane 2x – 3y – 5z
+ 6 = 0.
a. 2x – 3y – 5z + 27 = 0
b. 2x – 3y – 5z + 26 = 0
c. 2x – 3y – 5z + 28 = 0
d. 2x – 3y – 5z + 29 = 0
697. Find the distance of the point (6,2,3) from the z-axis.
a.
b.
c.
d. 7
698. A line drawn from the origin to the point (-6,2,3). Find the angle which the line makes
with the z-axis.
a. 147°
b. 149°
c. 151°
d. 150°
699. Find the length of the line segment whose end points are (3,5,-4) and (-1,1,2).
a.
b.
c.
d.
700. Find the locus of a point whose distance from the point (-3,2,1) is 4.
a. x2 + y2 + z2 + 6x – 4y – 2z + 3 = 0
b. x2 + y2 + z2 + 6x – 4y – 2z – 4 = 0
c. x2 + y2 + z2 + 6x – 4y – 2z + 1 = 0
d. x2 + y2 + z2 + 6x – 4y – 2z – 2 = 0
701. Find the center of the sphere x2 + y2 + z2 – 6x + 4y – 8z = 7.
a. C(3,2)
b. C(-3,2)
c. C(3,-2)
d. C(-3,-2)
702. Find the rectangular coordinates for the point (4,210°,30°).
a. (
b. (
c. (
d. (
703. The vertices of a triangle are A(2,-3,1), B(-6,5,3) and C(8,7,-7). Find the length of the
median drawn from A to BC.
a.
b.
c.
d.
704. Find the angle between the line L1 with direction numbers 3,4,1 and the line L2 with
direction numbers 5,3,-6.
a. 55.41°
b. 60.51°
c. 65.61°
d. 70.71°
705. Find spherical coordinates for the point (-2,2,-1).
a. (3,315°,109.5°)
b. (3,240°,107.5°)
c. (3,300°,110°)
d. (3,215°,100°)
706. Find the distance from the plane 2x + 7y + 4z – 3 = 0 to the point (2,3,3).
a.
b.
c.
d.
707. Transform psinφsinθtanθ = 5 to rectangular coordinates.
a. x2 = 5y
b. y2 = 5x2
c. y2 = 5x
d. y = 5x2
708. Two direction angles of a line are 45 degrees and 60 degrees. Find the third direction
angle.
a. 30°
b. 35°
c. 40°
d. 45°
709. Find m so that the plane 5x – 6y – 7z = 0 and the plane 3x + 2y – mz + 1 = 0 are parallel.
a. -5/3
b. -7/3
c. -4/3
d. -2/3
710. Transform y2 = 4ax to cylindrical coordinates.
a. rcosθtanθ = 4a
b. rcosθcotθ = 4a
c. rsinθtanθ = 4a
d. rsinθcotθ = 4a
711. The triangle with vertices (3,5,-4),(-1,1,2) and (-5,-5,-2) is
a. equilateral
b. isosceles
c. right
d. equiangular
712.
The sphere x2 + y2 + z2 – 2x + 6y +2z – 14 = 0 has a radius
a. 2
b. 4
c. 5
d. 3
713. Find the x-coordinate of a point which is 10 units from the origin and has direction
cosines cosβ = 1/3 and cosγ = -2/3.
a. 19/3
b. 20/3
c. 17/3
d. 22/3
714. Give the equivalent spherical coordinates of (3,4,6).
a. (
b.
c.
d. (
715. If the line L1 has direction numbers x,-2x3 and line L2 has direction numbers -2,x,4 and if
L1 is perpendicular to L2, find x.
a. 5
b. 4
c. 3
d. 2
716. Find the cosine of the angle between the line directed from (3,2,5) to (8,6,2) and the line
directed from (-4,5,3) to (-3,4,3).
a. 1/12
b. 1/10
c. 1/11
d. 1/13
717. Find the angle between the planes 3x – y + z – 5 = 0 and x + 2y + 2z + 2 = 0.
a. 69.42°
b. 70.43°
c. 71.44°
d. 72.45°
718. Find the coordinates of the point P(x,y,z) which divides the line segment P 1P2 where
P1(2,5,-3) and P2(-4,0,1) in the ratio 2:3.
a. (2/5,-3,-7/5)
b. (-2/5,3,7/5)
c. (-2/5,3,-7/5)
d. (-2/5,-3,-7/5)
719. Find the Cartesian coordinates of the point having the spherical coordinates
(4,
.
a. (
b. (
c. (
d. (
720. Find the equations of the line through (2,-1,3) and parallel to the x-axis.
a. y + 1 = 0, z – 3 = 0
b. y – 1 = 0, z + 3 = 0
c. y – 1 = 0, z – 3 = 0
d. y + 1 = 0, z + 3 = 0
721.
Give the polar coordinates for the point (1,-2,2).
a. (3,48.2°,131.8°,70.5°)
b. (3,70.5°,131.8°,48.2°)
c. (3,48.2°,70.5°,131.8°)
d. (3,131.8°,70.5°,48.2°)
722. Transform the equation cosγ = p(cos2α – cos2β) to rectangular coordinates.
a. y = x2 – z2
b. x = y2 – z2
c. z = x2 – y2
d. z = x2 + y2
723. A point P(x,y,z) moves so that its distance from the z-axis is 4 times its distance from the
x-axis. Find the equation of the locus.
a. 15y2 + 16z2 – x2 = 0
b. 15y2 – 16z2 + x2 = 0
c. 15y2 – 16z2 – x2 = 0
d. 15y2 + 16z2 + x2 = 0
724. Write the equation in rectangular coordinates of p = 5acosφ.
a. x2 – y2 + z2 = 5az
b. x2 + y2 – z2 = 5az
c. x2 – y2 – z2 = 5az
d. x2 + y2 + z2 = 5az
725. The rectangular coordinates for the point (2,90°,30°,60°) is
a. (0,
b. (0,
c. (1,
d. (1,
726. Find the equations of the line through (1,-1,6) with direction numbers 2,-1,1.
a. x = 2z + 11, y = z – 5
b. x = 2z – 11, y = z + 5
c. x = 2z – 11, y = 5 – z
d. x = 2z + 11, y = 5 – z
727. If the angle between two lines with direction numbers 1,4,-8 and x,3x-6 respectively is
arccos(62/63),find x.
a. 4
b. 5
c. 2
d. 3
728. Find the polar coordinates of the point (0,-2,-2)
a. (2
b. (2
c. (
d. (
729. Find the point where the line through the points (3,-1,0) and (1,3,4) pierces the xz-plane.
a. (1,0,1)
b. (1.5,0,1)
c. (2,0,1)
d. (2.5,0,1)
730. Find the equation of the plane such that the foot of the perpendicular from the origin to
the plane is (-6,3,6).
a. 2x + y + 2z – 27 = 0
b. 2x – y – 2z + 27 = 0
c. 2x – y + 2z + 27 = 0
d. 2x + y – 2z – 27 = 0
731. Find angle A of the triangle whose vertices are A(4,6,1), B(6,4,0) and C(-2,3,3).
a. 112.39°
b. 111.38°
c. 110.37°
d. 109.36°
732. Find the equation of the plane that passes through (3,-2,1), (2,4,-2) and (-1,3,2).
a. 21x + 13y + 19z – 56 = 0
b. 21x + 13y – 19z – 56 = 0
c. 21x + 13y + 19z + 56 = 0
d. 21x – 13y – 19z – 56 = 0
733. Find the acute angle between the lines x + y + z + 1 = 0, x – y + z + 1 = 0 and x – y – z –
1 = 0, x + y = 0.
a. 71.20°
b. 72.21°
c. 73.22°
d. 74.23°
734. Find the equation of the plane through the point (-1,2,3) and perpendicular to the line for
which cosα = 2/3, cosβ = -1/3, cosγ = 2/3.
a. 2x – y + 2z – 2 = 0
b. 2x – y – 2z + 2 = 0
c. 2x + y – 2z – 2 = 0
d. 2x + y + 2z – 2 = 0
735. Find the area of the triangle with vertices (1,3,3), (0,1,0) and (4,-1,0).
a.
b.
c.
d.
736. If the acute angle between the planes 2x – y + z – 7 = 0 and x + y + kz – 11 = 0 is 60°,
find k.
a. 4
b. 3
c. 1
d. 2
737. Transform the cylindrical coordinates (8,120°,6) to spherical coordinates.
a. (10,120°,53.13°)
b. (11,120°,53.13°)
c. (12,120°,53.31°)
d. (10,120°,51.33°)
738. Find the locus of the point equidistant from the plane y = 7 and the point (0,5,0).
a. x2 – z2 + 4y – 24 = 0
b. x2 – z2 – 4y + 24 = 0
c. x2 + z2 + 4y – 24 = 0
d. x2 + z2 – 4y + 24 = 0
739. Find the direction numbers of the line 2x – y + 3z + 4 = 0, 3x + 2y – z + 7 = 0.
a. 5,-11,7
b. -5,11,7
c. -5,7,11
d. 5,-7,11
740. Find the equation of the plane perpendicular to the line joining (2,5,-3) and (4,-1,0) and
which passes through the point (1,4,-7).
a. 2x – 6y – 3z + 43 = 0
b. 2x + 6y – 3z + 43 = 0
c. 2x – 6y + 3z + 43 = 0
d. 2x + 6y + 3z + 43 = 0
741. Find the equation of the line which passes through (-1,-3,6) and which is parallel to the
plane 4x – 9y + 7z + 2 = 0.
a. 4x – 9y + 7z – 65 = 0
b. 4x + 9y + 7z – 65 = 0
c. 4x – 9y – 7z + 65 = 0
d. 4x + 9y – 7z + 65 = 0
742. Find the value of m so that the line passing through (-m,-1,2) and (0,2,4) be perpendicular
to the line through (1,m,1) and (m+1,0,2).
a. 1 or 5
b. 1 or 4
c. 1 or 3
d. 1 or 2
743. Find the acute angle between the line
and the line
.
a.
b.
c.
d.
744. If the angle between the planes 2x – 3y + 6z = 18 and 2x – y + kz = 12 is arccos(19/21),
find k.
a. 4
b. 3
c. 2
d. 1
745.
A plane contains the point P1(4,-4,2) and is perpendicular to the line segment from P1 to
P2(0,6,6). Find the equation of the plane.
a. 2x + 5y + 2z – 24 = 0
b. 2x + 5y – 2z + 24 = 0
c. 2x – 5y + 2z + 24 = 0
d. 2x – 5y – 2z – 24 = 0
746. A
line
whose
parametric
equations
are
is
perpendicular to the plane 2x + ky + 12z = 3. Find the value of k.
a. -3
b. -4
c. -5
d. -6
747. Write the equations of the line through (-2, 2, -3) and (2, -2, 3).
a. x – y = 0, 3y + 2z = 0
b. x + y = 0, 3y + 2z = 0
c. x – y = 0, 3y – 2z = 0
d. x + y = 0, 3y – 2z = 0
748. Find the equation of the paraboloid with vertex at (0, 0, 0), axis along the y-axis and
passing through (1, 1, 1) and (3/2, 7/12, 1/2).
a. x2 + 5z2 = 6y
b. x2 + 6z2 = 5y
c. 5x2 + z2 = 6y
d. 6z2 + z2 = 5y
749. Find the equation of the plane determined by the points (6,-4,1), (0,1,-3) and (2,2,-7).
a. x + 2y – z + 1 = 0
b. x – 2y + z – 1 = 0
c. x + 2y + z + 1 = 0
d. x – 2y – z – 1 = 0
750. What is the locus of the moving point, the difference of whose distance from (0,0,3) and
(0,0,-3) is 4?
a.
b.
c.
d.
751. Find the piercing point in the xy-plane of the line x + y – z – 3 = 0, x + 2y + z – 4 = 0.
a. (1,2,0)
b. (1,0,2)
c. (2,0,1)
d. (2,1,0)
752. Find the acute angle between the line
0.
a.
b.
c.
d.
and the plane 2x – 2y + z – 3 =
25.3°
26.4°
27.5°
28.6°
753. Find the equation of the plane through (1,-2,3) and perpendicular to the line of
intersections of the plane 3x + 2y – 2z = 12 and x + 2y + 2z = 0.
a. 2x – 2y – z – 9 = 0
b. 2x – 2y + z – 9 = 0
c. 2x + 2y – z + 9 = 0
d. 2x + 2y + z + 9 = 0
754. A plane contains the points (3,1,7) and (-3,-2,3) and as an x-intercept equal to three times
its z-intercepts. Find the equation of the plane.
a. x + 6y – 3z + 18 = 0
b. x – 6y – 3z + 18 = 0
c. x – 6y + 3z – 18 = 0
d. x + 6y – 3z – 18 = 0
755. Find the acute angle between the lines through the points (-2,3,1) and (4,6,7) and the
plane x + 4y + z – 10 = 0.
a. 35.64°
b. 36.74°
c. 37.84°
d. 38.94°
756. Find the equation of the plane which contains the line x – 2y + z = 1, 2x = y – z and is
perpendicular to the plane 3x + 2y – 3z = 0.
a. 9x – 6y + 5z – 1 = 0
b. 9x + 6y – 5z + 1 = 0
c. 9x + 6y – 5z – 1 = 0
d. 9x + 6y + 5z + 1 = 0
757. Find the equation of the plane which is perpendicular to the xy-plane and which passes
through (2,-1,0) and (3,0,5).
a. x + y + 3 = 0
b. x – y – 3 = 0
c. x + y – 3 = 0
d. x – y + 3 = 0
758. Find the acute angle between the lines
and 2x + 2y + z – 4 = 0, x – 3y +
2z = 0.
a. 46°24’
b. 47°25’
c. 48°26’
d. 49°27’
759. Find the equations of the line through (2,-3,4) and perpendicular to the plane 3x – y + 2z
= 4.
a. x = 3y – 7, z = 2y – 2
b. x = 3y + 7, z = 2y + 2
c. x = -3y – 7, z = -2y – 2
d. x = -3y + 7, z = -2y + 2
760. Find the point of intersection of the plane 3x + 2y + z = 1 and the line
a.
b.
c.
d.
.
(1,0,1)
(1,1,0)
(-1,1,0)
(1,-1,0)
761. Transform 3x2 – 3y2 = 8z to spherical coordinates.
a. 2psin2φcos2θ = 8pcosφ
b. 2psin2φcos2θ = 8pcosφ
c. 2p2sin2φcos2θ = 8pcosφ
d. 2p2sin2φcos2θ = 8pcosφ
762. Find the equation of the sphere whose center is (2,1,-1) and which is tangent to the plane
x – 2y + z + 7 = 0.
a. x2 + y2 – 4z – 2y + 2z = 0
b. x2 + y2 – 4z + 2y + 2z = 0
c. x2 + y2 + 4z – 2y – 2z = 0
d. x2 + y2 + 4z + 2y – 2z = 0
763. If the line
k.
a.
b.
c.
d.
is parallel to the plane 6x + ky – 5z – 8 = 0, find the value of
2
3
-2
-3
764. Find the equation of the plane that is perpendicular to the yz-plane and having 5 and -2 as
its y- and z-intercepts respectively.
a. 2y + 5z – 10 = 0
b. 2y – 5z – 10 = 0
c. 2y + 5z + 10 = 0
d. 2y – 5z + 10 = 0
765. Find the angle between the line with direction numbers 1,-1,-1 and the plane 3x – 4y + 2z
– 5 = 0.
a. 32.42°
b. 34.22°
c. 42.32°
d. 43.22°
766. Find the equation of the locus of a point whose distance from the xy-plane is equal to its
distance from (-1,2,-3).
a. x2 + y2 – 2x + 4y – 6z – 14 = 0
b. x2 + y2 – 2x – 4y + 6z + 14 = 0
c. x2 + y2 + 2x + 4y – 6z – 14 = 0
d. x2 + y2 + 2x – 4y + 6z + 14 = 0
767. Given the points A(k,1,-1), B(2k,0,2) and C(2+2k,k,1). Find k so that the line segment
AB shall be perpendicular to the line segment BC.
a. 3
b. 1
c. 2
d. 4
768. The angle between two lines with direction numbers 4,3,5 and x,-1,2 respectively is 45
degrees. Find x.
a. 4
b. 5
c. 2
d. 3
769. At the minimum point, the slope of the tangent line to a curve is
a. positive
b. negative
c. zero
d. infinity
770. A curve y = f(x) is concave downward if the value of y’’ is
a. negative
b. positive
c. unity
d. zero
771. The point where the concavity of a curve changes is called the
a. maximum point
b. minimum point
c. inflection point
d. tangent point
772. If the 1st derivative of a function is a constant, then its graph is
a. a point
b. a line
c. a parabola
d. a circle
773. At the minimum point of y = f(x), the value of d2 y/dx2 is
a. zero
b. undefined
c. positive
d. negative
774. If at x = a, f’’(a) is positive, then f’(x) increases as x
a. increases
b. decreases
c. becomes infinite
d. becomes zero
775. If the first derivative of a function is a constant, then the function is
a. sinusoidal
b. exponential
c. linear
d. quadratic
776. A function f(x) is said to be an even function if its graph is symmetric with respect to
a. the x-axis
b. the y-axis
c. the origin
d. both axes
777. Which of the following is an odd function?
a. f(x) = xcosx
b. f(x) = xsinx
c. f(x) = ecosx
d. f(x) = sin2x
778. The notation f’(x) was invented by
a. Leibniz
b. Newton
c. Wallis
d. Lagrange
779. At the inflection point of y = f(x) where x = a,
a. f”(a) < 0
b. f”(a) = 0
c. f”(a) > 0
d. f”(a) = ∞
780. If a function f(x) is concave downward on the interval (1,10), then f(8) and f(3)
a. may be true
b. cannot be true
c. must be true
d. is never true
781. If a tangent to a curve y = f(x) is horizontal at x = a, then f’(a) is
a. positive
b. negative
c. zero
d. infinity
782. For a function y = f(x), if f”(x) = -f(x), then the function is
a. logarithmic
b. exponential
c. transcendental
d. sinusoidal
783. Which of the following notations is an open interval?
a. (-3,4)
b. [-3,4]
c. [-3,∞)
d. (-∞,4)
784. The graph of y = x5 – x will cross the x-axis
a. twice
b. 3 times
c. 4 times
d. 5 times
785. The derivative of an increasing function f(x) must be
a. strictly positive
b. always positive
c. nonnegative
d. negative
786. If the function f(x) increases at x = a, then which of the following is definitely true?
a. f'(a) = 0 or f’(a) > 0
b. f’(a) = 0 or f’(a) < 0
c. f’(a) ≠ 0 or f’(a) > 0
d. f’(a) ≠ 0 or f’(a) < 0
787. At the maximum point, the value of the 2nd derivative of a function is
a. positive
b. negative
c. zero
d. infinite
788. At the inflection point, the value of y” is
a. zero
b. positive
c. negative
d. unity
789. Which of the following functions will have an inflection point?
a. y = x4
b. y = x3
c. y = x2
d. y = x
790. The function y = f(x) has a maximum value of x = 2 if f’(2) = 0 and f”(2) is
a. equal to zero
b. less than zero
c. greater than zero
d. unity
791. At the maximum point, the tangent line is
a. slanting upward
b. oblique
c. horizontal
d. vertical
792. Which of the following is true?
a. ∞ – ∞ = 0
b. ∞ + ∞ = ∞
c. ∞/∞ = ∞
d. both a and b
793. Which of the following functions is neither even nor odd?
a. h(x) = x2
b. g(x) = x3
c. f(x) = x2 + x
d. t(x) = x3 + x
794. Find the rate of change of the volume of a cube with respect to its side when the side is 6
cm.
a. 108 cm3/cm
b. 107 cm3/cm
c. 106 cm3/cm
d. 105 cm3/cm
795. If f(x) = e –x+1, then f’(1) is equal to
a. 0
b. 1
c. -1
d. ∞
796. If f(x) = Aekx, f(0) = 5 and f(3) = 10, find k.
a. 0.1184
b. 0.1285
c. 0.1386
d. 0.1487
797. The function
a.
b.
c.
d.
is discontinuous at x =
1 or -3
1 or -2
-1 or 2
-1 or 3
798. Find the slope of the line tangent to y = 4/x at x = 2.
a. 1
b. -1
c. 2
d. -2
799. If y = cos24x, find dy/dx.
a. 2cos4x
b. 2sin4x
c. -4sin8x
d. -8sin4x
800. Evaluate the limit of ln(1 – x)/x as x approaches zero.
a. 0
b. -1
c. 1
d. ∞
801. Evaluate
a.
b.
c.
d.
∞
0
½
2
.
802. The rate of change of the area of a circle with respect to its radius when the diameter is
6cm is
a. 4π cm2/cm
b. 5π cm2/cm
c. 6π cm2/cm
d. 7π cm2/cm
803. At what point of the curve y = x3 + 3x are the values of y’ and y” equal?
a. (0,0)
b. (-1,-4)
c. (2,14)
d. (1,4)
804. If f(x) = ln x and g(x) = log x and if g(x) = kf(x), find k.
a. 0.4433
b. 0.3434
c. 0.3344
d. 0.4343
805. If N(x) = sin x – sin θ and D(x) = x – θ, find the limit of N(x)/D(x) as x approaches θ.
a. sinθ
b. cosθ
c. zero
d. no limit
806. Given z2 + x2 + y2 = 0, find
a.
b.
c.
d.
807. What
a.
b.
c.
d.
x/z
–x/z
z/x
–z/x
is the 50th derivative of y = cosx
sinx
–sinx
cosx
–cosx
808. Which of the following has no horizontal asymptote?
a.
b.
c.
d.
809. If f(x) =
a.
b.
c.
d.
if f(x) = x – 2 and g(x) = x2 – 1.
∞
0
½
¼
811. Evaluate
a.
b.
c.
d.
.
infinity
unity
zero
undefined
810. Evaluate
a.
b.
c.
d.
, find
.
0
∞
1
e
812. If z = xy2 + yx3, find zxyx.
a. 6yx
b. 6x
c. 3xy
d. 3x2
813. If y = x2, find ∆y – dy when x = 2 and dx = 0.01.
a. 0.0001
b. 0.001
c. 0.0002
d. 0.002
814. If f(x) = x3 + 2x, find f”(2).
a. 10
b. 11
c. 12
d. 13
815. The motion of a particle along the x-axis is given by the equation x = 2t 3 – 3t2. Find the
velocity of the particle when t = 2.
a. 10
b. 9
c. 11
d. 12
816. Find x for which the line tangent to the parabola y = 4x – x2 is horizontal.
a. 4
b. -4
c. 2
d. -2
817. The slope of the tangent to y = 2 – x2 at the point (1,1) is
a. -2
b. -1
c. 0
d. -4
818. If y = sin2x, the derivative dy/dx is equal to
a. cos2x
b. sin2x
c. 2cosx
d. 2sinx
819. If y = x3 – 2x2 + 3x – 1, then d2 y/dx2 is equal to
a. 6x
b. 6x + 4
c. 6x – 4
d. 3x – 4
820. If y = x2 – 2x and x changes from 2 to 2.01, find ∆y.
a. 0.0102
b. 0.0210
c. 0.0120
d. 0.0201
821. The radius R of a circle is increasing at the rate of 1cm per sec. how fast is the area
changing when R = 4cm?
a. 8π cm2/s
b. 10π cm2/s
c. 6π cm2/s
d. 12π cm2/s
822. Find the slope of y = 1 – x3 at the point where y = 9.
a. -11
b. -12
c. -10
d. -13
823. If an error of 1 percent is made in measuring the edge of a cube, what is the percentage
error in the computed volume?
a. 3%
b. 2%
c. 4%
d. 5%
824. Find the derivative of y with respect to x of y = xlnx – x.
a. 1
b. x
c. lnx
d. lnx – 1
825. For what value of x will the curve y = x3 – 3x2 + 4 be concave upward?
a. 1
b. 2
c. 3
d. 4
826. How fast does the diagonal of a cube increase if each edge of the cube increases at a
constant rate of 5cm/s?
a. 6.7 cm/s
b. 7.7 cm/s
c. 8.7 cm/s
d. 9.7 cm/s
827. If f(x) = tanx – x and g(x) = x3, evaluate the limit of f(x)/g(x) as x approaches zero.
a. 0
b. ∞
c. 3
d. 1/3
828. Find the 3rd derivative of y = xlnx.
a. -1/x
b. -1/x2
c. -1/x3
d. -1
829. Evaluate
a.
b.
c.
d.
.
∞
1
e
1/e
830. If xy3 + x3y = 2, find dy/dx at the point (1,1).
a. 1
b. -1
c. 2
d. -2
831. The tangent line to the curve y = x3 at the point (1,1) will intersect the x-axis at x =
a. 2/3
b. 4/3
c. 1/3
d. 5/3
832. If y = ex + xe + xx, find y’ at x = 1.
a. e +1
b. e – 1
c. 2e + 1
d. 2e – 1
833. Evaluate
a.
b.
c.
d.
.
0
∞
½
1
834. Find the value of x for which y = x3 – 3x2 has a minimum value.
a. 1
b. 2
c. 0
d. -2
835. Find the angle of intersection between the curve y = x2 and x = y2.
a.
b.
c.
d.
836. If z = xy2, and x changes from 1 to 1.0, and y changes from 2 to 1.98, find the
approximate change in z.
a. -0.0202
b. -0.0303
c. -0.0404
d. -0.0505
837. A ball is thrown vertically upward from a roof 112-ft above the ground. The height s of
the ball above the roof is given by the equation
s = 96t -16t2
where s is measured in ft and the time t in sec. calculate its velocity wen it strikes the
ground.
a. -130 fps
b. -128 fps
c. -126 fps
d. -124 fps
838.
If y = ln(tanhx), find dy/dx.
a. 2sech2x
b. 2sech2x
c. 2csch2x
d. 2coth2x
839.
Find the approximate surface area of a sphere of radius 5.02 cm.
a. 317 sq. cm
b. 315 sq. cm
c. 313 sq. cm
d. 311 sq. cm
840.
Find the value of x for which y = x5 – 5x3 – 20x – 2 will have a maximum point.
a. -1
b. -2
c. 1
d. 2
841.
A man is walking at a rate of 1.5 m/s toward a street light which is 5 m above the level
ground. At what rate is the tip of his shadow moving if the man is 2 m tall?
a. -1.5 m/s
b. -2.5 m/s
c. -3.5 m/s
d. -5 m/s
842.
If y = ln(x2ex), find y”.
a. -1/x2
b. -2/x2
c. -1/x
d. -2/x
843.
Find the radius of curvature of y = x3 at the point (1,1).
a. 3.25
b. 4.26
c. 5.27
d. 6.25
844.
A particle moves along the circumference of a circle of radius 10-ft in such a manner
that its distance measured along the circumference from a fixed point at the end of t sec
is given by the equation s = t2. Find the angular velocity at the end of 3 seconds.
a. 0.40 rad/s
b. 0.50 rad/s
c. 0.60 rad/s
d. 0.70 rad/s
845.
Find the point on the curve y = x3 – 3xfor which the tangent line is parallel to the x-axis.
a. (-1,2)
b. (2,2)
c. (1,2)
d. (0,0)
846.
If y = 1/2tan2x + ln(cosx), find y’.
a. tan3x
b. tanx – sinx
c. tanxsec2 x
d. 0
847.
If S = 4πR2, find ∆S – dS when R = 2 and ∆R = 0.01.
a. 0.0021
b. 0.0102
c. 0.0210
d. 0.0012
848.
Find two numbers whose sum is 8 if the product of one number and the cube of the other
is a maximum.
a. 3 and 5
b. 4 and 4
c. 2 and 6
d. 1 and 7
849.
Find the approximate height of the curve y = x3 – 2x2 + 7 at the point where x = 2.98.
a. 14.8
b. 15.7
c. 16.6
d. 17.5
850.
If y =
, find x for which dy/dx = 0.
a.
b.
c.
d.
851.
Te volume of a cube is increasing at the rate of 6 cm3/min. How fast is the surface
increasing when the length of each edge is 12 cm?
a. 3 cm2/min
b. 4 cm2/min
c. 2 cm2/min
d. 5 cm2/min
852.
If u =
, find the approximate change in u as x changes from 10 to 10.02 and
y changes from 4 to 4.01.
a. -0.00170
b. -0.00701
c. -0.00107
d. -0.00017
853.
Find the equation of the line tangent to y = x2 – 3x – 5 and parallel to the line y = 3x – 2.
a. y = 3x – 14
b. y = 3x – 13
c. y = 3x – 12
d. y = 3x – 11
854.
A garden is in the form of an ellipse with semi-major axis 4 and semi-minor axis 3. If
the axes are increased by 0.18 unit each, find the approximate increase in the area.
a. 3.92
b. 3.94
c. 3.96
d. 3.98
855.
Find the relative error in the computed area of an equilateral triangle due to an error of 3
percent in measuring the edge of the triangle.
a. 0.05
b. 0.06
c. 0.07
d. 0.08
856.
A body is thrown vertically upward from the ground. After 2 seconds, its velocity is 10
ft/sec. Find its initial velocity.
a. 54 fps
b. 64 fps
c. 74 fps
d. 84 fps
857.
In problem 345, find the rate at which the length of the shadow of the man is shortening.
a. -1 cm/s
b. -1.5 cm/s
c. -2 cm/s
d. -2.5 cm/s
858.
A rectangular field is fenced off, an existing wall being used as one side. If the area of
the field is 7,200 sq. ft, find the least amount of fencing needed.
a. 250 ft
b. 240 ft
c. 230 ft
d. 220 ft
859.
The side of an equilateral triangle is increasing at the rate of 0.50 cm/s. Find the rate at
which its altitude is increasing.
a. 0.334 cm/s
b. 0.443 cm/s
c. 0.433 cm/s
d. 0.343 cm/s
860.
Find C co that the line y = 4x + 3 is tangent to the curve y = x2 + C.
a. 3
b. 4
c. 5
d. 6
861.
At what acute angle does the curve y = 1 – 1/2x2 cut the x-axis?
a. 34.54°
b. 44.64°
c. 54.74°
d. 64.84°
862.
The angle θ, made by a swinging pendulum with the vertical direction, is given at time t
by the equation θ = asin(bt + c), where a, b and c are constants. Find the angular
acceleration at time t.
a. –a2θ
b. –b2θ
c. –aθ
d. –bθ
863.
If y =
a.
b.
c.
d.
find y’ at x = 5.
1/13
1/14
1/15
1/16
864.
Find the equation of the line with slope -1/2 and tangent to the ellipse x2 + y2 = 8.
a. x + 2y – 4 = 0
b. x – 2y + 4 = 0
c. x + 2y + 4 = 0
d. x – 2y – 4 = 0
865.
Find the second derivative (y”) of 4x2 + 9y2 = 36 by implicit differentiation.
a. -16y3/9
b. -16/9y3
c. -9y3/16
d. -9/16y3
866.
Approximate the root of 3x + x – 2 = 0 by Newton’s Method of Approximation.
a. 0.420
b. 0.419
c. 0.421
d. 0.418
867.
The volume of a sphere is increasing at the rate of 6 cm3/hr. at what rate is its surface
area increasing when the radius is 40 cm?
a. 0.30 cm2/hr
b. 0.40 cm2/hr
c. 0.50 cm2/hr
d. 0.60 cm2/hr
868.
If f(x) = ex – e-x – 2x and g(x) = x – sinx, evaluate the limit of f(x)/g(x) as x approaches
zero.
a. ∞
b. 0
c. 1
d. 2
869.
Find the point of inflection of y = 4 + 3x – x3.
a. (1,6)
b. (0,4)
c. (-2,4)
d. (2,2)
870.
Find the volume of the largest right circular cone that can be cut from a sphere of radius
R.
a. 1.421 R3
b. 1.124 R3
c. 1.241 R3
d. 1.412 R3
871.
If s = x2 + 2y2 + 3z2 and x +y +z = 5, find the minimum value of s.
a. 148/11
b. 149/11
c. 150/11
d. 151/11
872.
The cost of fuel per hour in operating a luxury liner is proportional to the square of its
speed and is Php. 12,000.00 per hour for a speed of 10-kph. Other costs amount to Php.
48,000.00 per hour independent of the speed. Calculate the speed at which the cost per
kilometer is a minimum.
a. 35 kph
b. 30 kph
c. 25 kph
d. 20 kph
873.
Find the slope of the tangent to the curve
a.
b.
c.
d.
at the point (1,1).
-1/5
-2/5
-3/5
-4/5
874.
If y = 1/2x(sin(lnx) – cos(lnx)), find dy/dx.
a. sin(lnx)
b. cos(lnx)
c. –sin(lnx)
d. –cos(lnx)
875.
If x = et and y = 2e-t, find d2 y/dx2.
a. 4e-t
b. 4e-2t
c. 4e-3t
d. 4e-4t
876.
Two corridors 6 m and 4 m wide respectively, intersect at right angles. Find the length
of the longest ladder that will go horizontally around the corner.
a. 13 m
b. 14 m
c. 15 m
d. 16 m
877.
An angle φ of a right triangle is given by the equation φ = arcsin(y/x). If x is increasing
at the rate of 1 in/sec and y is decreasing at 0.10 in/sec, how fast is φ changing?
a. -0.06892 rad/sec
b. -0.08926 rad/sec
c. -0.09268 rad/sec
d. -0.06928 rad/sec
878.
Find the maximum capacity of a conical vessel whose slant height is 9 cm.
a. 293.84 cm3
b. 283.94 cm3
c. 284.93 cm3
d. 294.83 cm3
879.
If the semi-axes of the ellipse 4x2 + 9y2 = 36 are each increased by 0.15 cm, find the
approximate increase in its area.
a. 2.36 cm2
b. 2.46 cm2
c. 2.56 cm2
d. 2.66 cm2
880.
If y = 4/(2x – 1)3, find y” at x = 1.
a. 190
b. 191
c. 192
d. 193
881.
The side of an equilateral triangle increases at the rate of 2 cm/hr. At what rate is the
area of the triangle changing at the instant when the side is 4 cm?
a.
b. 4
c. 5
d. 6
882.
Find the value of x and y which satisfy 2x + 3y = 8 and whose product is a minimum.
a. 1 and 2
b. 3 and 2/3
c. 3/2 and 5/3
d. 2 and 4/3
883.
If ln(ln y) + ln y = ln x, find dy/dx.
a.
b.
c.
d.
884.
If x = 2sinθ, y = 1 – 4cosθ, then dy/dx is equal to
a. 2cotθ
b. 2tanθ
c. 2cscθ
d. 2secθ
885.
The upper and lower edges of a picture frame hanging on a wall are 8 feet and 2 feet
above an observer’s eye level respectively. How far from the wall must the observer
stand in order that the angle subtended by the picture is a maximum?
a. 3.5 ft
b. 4 ft
c. 4.5 ft
d. 5 ft
886.
If x increases at the rate of 30 cm/s, at what rate is the expression (x + 1) 2 increasing
when x becomes 6 cm?
a. 400 cm2/s
b. 410 cm2/s
c. 420 cm2/s
d. 430 cm2/s
887.
Find the radius of a right circular cylinder of maximum volume that can be inscribed in a
right circular cone of radius R.
a. R/3
b. R/2
c. 3R/4
d. 2R/3
888.
Find the area of the triangle bounded by the coordinate axes and the tangent to the
parabola y = x2 at the point (2,4).
a. 2
b. 3
c. 4
d. 5
889.
What is the maximum value of y = 3sinx + 4cosx ?
a. 8
b. 7
c. 6
d. 5
890.
Find the maximum point of the curve y = 4 + 3x – x3.
a. (-2,6)
b. (0,4)
c. (1,6)
d. (-3,22)
891.
Water flows into a cylindrical tank at the rate of 20 m3/s. How fast is the water surface
rising in the tank if the radius of the tank if the radius of the tank is 2 m?
a. 5/π
b. 6/π
c. 3/π
d. 4/π
892.
If (0,4) and (1,6) are critical points of y = a + bx + cx3, find the value of c.
a. 1
b. 2
c. -1
d. -2
893.
Intensity of light is proportional to the cosine of the angle of incidence and inversely
proportional to the square of the distance from the source of light. A lamp is directly
over the center of a circular table of radius 3 feet. How high above should the lamp be
placed so that there will be maximum illumination around the edge of the table?
a. 2.18 ft
b. 2.16 ft
c. 2.14 ft
d. 2.12 ft
894.
Find the value of x so that the determinant given below will have a minimum value.
a.
b.
c.
d.
5
6
7
8
895.
Find the area of the largest triangle that can be formed by the tangent to the curve y = e -x
and the coordinate axes.
a. 1/e
b. 2/e
c. 3/e
d. 4/e
896.
A bus company planning a tour knows from experience that at Php. 20.00 per person, all
30 seats in the bus will be taken but for each increase of Php. 1.00, two seats will
become vacant. The expenses of the tour are Php. 100.00 plus Php. 11.00 per person.
What price should the company charge to maximize the profit?
a. Php. 23.00
b. Php. 24.00
c. Php. 25.00
d. Php. 26.00
897.
An isosceles triangle has legs 26 cm long. The base decreases at the rate of 12 cm/s.
Find the rate of change of the angle at the apex when the base is 48 cm.
a. -1.4 cm/s
b. -1.3 cm/s
c. -1.2 cm/s
d. -1.1 cm/s
898.
Find the weight of the heaviest cylinder that can be cut out from a sphere which weighs
12 kg.
a. 4.93 kg
b. 5.93 kg
c. 6.93 kg
d. 7.93 kg
899.
If
find dy/dx.
a.
b.
c.
d.
eaxcosbx
eaxsinbx
–eaxcosbx
-eaxsinbx
900.
A weight is attached to one end of a 29-m rope passing over a small pulley 17 m above
the ground. A man keeping his hand 5 m above the ground holds the other end of the
rope and walks away at a rate of 3 m/s. How fast is the weight rising at the instant when
the man is 9 m from the point directly below the pulley?
a. 1.2 m
b. 1.4 m
c. 1.6 m
d. 1.8 m
901.
A right triangle as a hypotenuse of length 13 and one leg of length 5. Find the area of the
largest rectangle that can be inscribed in the triangle if it has one side along the
hypotenuse of the triangle.
a. 15
b. 16
c. 17
d. 18
902.
Evaluate
a.
b.
c.
d.
903.
.
∞
1
e-2
e2
The sum of two numbers is K. Find the minimum value of the sum of their cubes.
a. K3
b. K3/2
c. K3/3
d. K3/4
904.
A chord of a circle 4 m in diameter is increasing at the rate of 0.60 m/min. Find the rate
of change of the smaller arc subtended by the chord when the chord is 3 m long.
a. 0.81 m/min
b. 0.71 m/min
c. 0.91 m/min
d. 0.61 m/min
905.
A manufacturer estimates that he can sell 1,000 units of a certain product per week if he
sets the price per unit at Php. 3.00 and that his sale will rise by 100 units with each Php.
0.10 decrease in price. Find his maximum revenue.
a. Php. 3,000
b. Php. 4,000
c. Php. 5,000
d. Php. 6,000
906.
The volume of a pyramid is increasing at the rate of 30 cm3/s and the area of the base is
increasing at the rate of 5 cm2/s. How fast is the altitude increasing at the instant when
the area of the base is 100 cm2 and the altitude is 8 cm?
a. 0.50 cm/s
b. 0.40 cm/s
c. 0.60 cm/s
d. 0.70 cm/s
907.
A closed right circular cylinder has a surface area of 100 cm2. What sould be its radius
in order to provide the largest possible volume?
a. 3.320 cm
b. 2.330 cm
c. 3.203 cm
d. 2.303 cm
908.
A ship 5 km from a straight shore and travelling at the rate of 36 kph is moving parallel
to the shore. How fast is the ship coming closer to a fort on the shore when it is 13 km
from the fort?
a. 34.24 km
b. 33.23 km
c. 32.21 km
d. 31.20 km
909.
The sum of the base and the altitude of a trapezoid is 36 cm. Find the altitude if its area
is to be maximum.
a. 18 cm
b. 20 cm
c. 19 cm
d. 17 cm
910.
Find the equation of the line parallel to the line x + 2y = 6 and tangent to the ellipse x 2 +
4y2 = 8 in the first quadrant?
a. x + 2y + 4 = 0
b. x – 2y + 4 = 0
c. x + 2y – 4 = 0
d. x – 2y – 4 = 0
911.
A sector with perimeter of 24 cm is to be cut from a circle. What should be the radius of
the circle if the area of the sector is to be a maximum?
a. 6 cm
b. 7 cm
c. 5 cm
d. 4 cm
912.
Find the equation of the line tangent to the curve y = x3 – 6x2 at its point of inflection.
a. 3x + y + 2 = 0
b. 3x – y + 2 = 0
c. 3x + y – 2 = 0
d. 3x – y – 2 = 0
913.
Find the radius of a right circular cylinder of greatest lateral surface area that can be
inscribed in a sphere of radius 4.
a. 2.53
b. 2.63
c. 2.73
d. 2.83
914.
Evaluate
a.
b.
c.
d.
zero
one
infinity
none
.
915.
Two posts 30 m apart are 10 m and 15 m high respectively. A transmission wire passing
through the tops of the post is used to brace the posts at a point on level ground between
them. How far from the 10-m post must that point be located in order to use the least
amount of wire?
a. 10 m
b. 11 m
c. 12 m
d. 13 m
916.
Three sides of a trapezoid are each 8 cm long. How long is the fourth side when the area
of the trapezoid has the largest value?
a. 14 cm
b. 15 cm
c. 16 cm
d. 17 cm
917.
A spherical iron ball 8 inches in diameter is coated with a layer of ice of uniform
thickness. If the ice melts at a rate of 10 cu in per min, how fast is the outer surface of
the ice decreasing when the ice is 2 inches thick?
a. -3.39 in2/min
b. -3.33 in2/min
c. -3.36 in2/min
d. -3.31 in2/min
918.
A circular filter paper of radius 15 cm is folded into a conical filter, the radius of whose
base is x. Find the value of x for which the conical filter will have the greatest volume.
a. 11.25 cm
b. 12.25 cm
c. 13.25 cm
d. 14.25 cm
919.
Water flows out of a hemispherical tank at the constant rate of 18 cu cm per min. If the
radius of the tank is 8 cm, how fast is the water level falling when the water is 4 cm
deep?
a. -0.1491 cm/min
b. -0.1941 cm/min
c. -0.1194 cm/min
d. -0.1149 cm/min
920.
Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 2.
a. 5.2
b. 6.3
c. 4.1
d. 3.8
921.
Sand is poured at the rate of 10 ft 3/min so as to form a conical pile whose altitude is
always equal to the radius of its base. At what rate is the area of the base increasing
when its radius is 5 ft?
a. 3 ft3/min
b. 4 ft3/min
c. 5 ft3/min
d. 6 ft3/min
922.
Find the altitude of the largest right circular cone that can be cut from a sphere of radius
R.
a. 7R/3
b. 5R/3
c. 4R/3
d. 8R/3
923.
A light is placed 3 ft above the ground and 32 ft from a building. A man 6 ft tall walks
from the light toward the building at the rate of 6 ft/sec. Find the rate at which the length
of his shadow is decreasing when he is 8 ft.
a. -1 fps
b. -1.5 fps
c. -2 fps
d. -2.5 fps
924.
An open box is made by cutting squares of side x inches from four corners of a sheet of
cardboard that is 24 inches by 32 inches and then folding up the sides. What should x be
to maximize the volume of the box?
a. 16.3 in
b. 15.2 in
c. 13.8 in
d. 14.1 in
925.
Let f be a function defined by f(x) = Ax2 + Bx + C with the following properties: f(0) =
2, f’(2) = 10 and f”(10) = 4. Find the value of B.
a. 1
b. 2
c. 3
d. 4
926.
A rectangle has its base on the x-axis and its two upper corners on the curve y = 2(1 –
x2). What is the maximum perimeter of the rectangle?
a. 4
b. 5
c. 6
d. 7
927.
Find the maximum vertical distance between y = cosx and y =
sinx over the interval
[0,2].
a. 1.5
b. 2
c. 2.5
d. 3
928.
A baseball diamond is a square 90 ft on the side. A runner travels from home plate to
first base at the rate of 20 ft/sec. how fast is the runner’s distance from the second base
changing when the runner is halfway to the first base?
a.
b.
c.
d.
929.
If the line to the curve y = x – lnx at x = a, passes through the origin, find a.
a. 2.72
b. 2.83
c. 2.91
d. 2.69
930.
Find the radius of curvature of the ellipse 4x2 + 5y2 = 20 at (0,2).
a. -1.5
b. -2.5
c. -3.5
d. -4.5
931.
If sin(x/y) = y/x, find dy/dx.
a. x/y
b. –x/y
c. y/x
d. –y/x
932.
Water is running into a right circular cone with vertical angle equal to 60 degrees (at the
bottom) at the rate of 2 cubic feet per second, and at the same time water is leaking out
at a rate which is 4.8 times the square root of its depth. How high will the water rise?
a. 0.1637 ft
b. 0.1367 ft
c. 0.1673 ft
d. 0.1736 ft
933.
If
,
a.
b.
c.
d.
and
evaluate
25
26
27
28
934.
Find the area bounded by x = y + 2, x = 1 – y2, y = 1 and y = -1 with or without
integration.
a. 11/3
b. 8/3
c. 7/3
d. 5/3
935.
Find the upper area bounded by the curves r = cscθ and r = 4sinθ.
a. 9.10
b. 10.11
c. 11.12
d. 12.13
936.
If f(x) = x1/2 and g(x) = (2x + 1)5/2, evaluate
a.
b.
c.
d.
from x = ½ to x = 4.
37/324
36/324
35/324
43/324
937.
Find the perimeter of the cardioid r = 1 – cosθ.
a. 7
b. 9
c. 6
d. 8
938.
Find the centroid of the volume of a cone formed by revolving about the y-axis the part
of the line
intercepted between the coordinate axes.
a.
b.
c.
d.
(0,1)
(0,2)
(0,3)
(0,4)
939.
A barrel has the shape of an ellipsoid of revolution with equal pieces but off ends. If the
barrel is 10 units long with circular ends of radius 2 units and the midsection of radius 4
units, find the volume of the barrel with or without integration.
a. 100π
b. 110π
c. 120π
d. 130π
940.
Each of the vertical ends of a trough is a parabolic segment with base 4 m and altitude 1
m. Find the force against one of the trough if it is full of water.
a. 11.43 kN
b. 12.44 kN
c. 11.45 kN
d. 10.46 kN
941.
If the trough in problem 444 is 5 m long, how long will it take a 0.50-hp pump to empty
the trough by pumping all of the water to the top of the trough?
a. 2.1 min
b. 1.2 min
c. 1.4 min
d. 2.4 min
942.
Find
axis.
a.
b.
c.
d.
943.
944.
(r/π,0)
(3r/π,0)
(2r/π,0)
(4r/π,0)
Find the moment of inertia of the semicircular arc in problem 446 with respect to its
diameter.
a. r5
b.
r4
c.
r3
d.
r2
If
, find the value of m.
a.
b.
c.
d.
945.
the centroid of a semicircular arc of radius r by placing its diameter along the y-
2
3
4
5
A dog is tied to a 4-m circular tank with a cord 3 m long. The point where the cord is
attached to the tank is at the same level as the dog’s collar. Compute the total area in
which the dog can move.
a. 18.64 m2
b. 16.84 m2
c. 14.85 m2
d. 16.48 m2
946.
If
find f(x).
a.
b.
c.
d.
x3/3
x4/4
x3
x4
947.
An equilateral triangle of side 8 ft is immersed in water with its plane vertical. If one
side is horizontal, and the vertex opposite that side is in the surface of the water, find the
force of pressure on the face of the triangle.
a. 8,500 lb
b. 8,000 lb
c. 7,500 lb
d. 7,000 lb
948.
The area bounded by y = x2 and y = 2 – x2 is revolved about the x-axis. Find the volume
of the solid generated with or without integration.
a. 14π/3
b. 16π/3
c. 17π/3
d. 19π/3
949.
Find the perimeter of the curve x2/3 + y2/3 = 4.
a. 46
b. 47
c. 48
d. 49
950.
Evaluate the integral of cos4xdx from x = -π/2 to x = π/2.
a. 3π/8
b. 4π/5
c. 5π/6
d. 9π/4
951.
Find the surface area generated by revolving the length of the arc of r = 1 + cosθ from 0
to π about the polar axis.
a. 23.13
b. 22.15
c. 21.12
d. 20.11
952.
Evaluate
a.
b.
c.
d.
.
5/3
7/3
2/3
4/3
953.
Find the area of the region that is inside the curve r = 8cosθ but is outside the curve r =
4cosθ with or without integration.
a. 10π
b. 11π
c. 12π
d. 13π
954.
A conoid is a solid having a circular base such that every plane section perpendicular to
the diameter of the base is an isosceles triangle. Find the volume of the conoid having a
radius of 2 m and the altitude of the triangle is 4 m.
a. 6 m3
b. 7 m3
c. 8 m3
d. 9 m3
955.
A rectangular plate 5 ft long and 4 ft wide is submerged in a liquid at an angle of 60
degrees with the vertical. If the liquid weighs w lb per cu ft, find the force of pressure on
the plate if the longer edge is parallel to the surface of the liquid and is 2 ft below the
surface.
a. 45w lb
b. 50w lb
c. 55w lb
d. 60w lb
956.
Find the moment of inertia of the volume of a right circular cylinder with base radius r
and altitude h relative to its base.
a.
b.
c.
d.
957.
Evaluate
a.
b.
c.
d.
.
1/23
1/24
1/25
1/26
958.
Find the area bounded by yx2 = 1, x = 1 and the x-axis.
a. ½
b. 1
c. 3/2
d. 2
959.
If a 10-lb weight could be lifted from the surface of the earth to a height of 4000 miles
above the surface of the earth, how much work would have to be done? Assume the
force of gravitation to vary inversely as the square of the distance from the center of the
earth and take the radius of the earth to be 4000 miles.
a. 20,000 mi-lb
b. 21,000 mi-lb
c. 22,000 mi-lb
d. 23,000 mi-lb
960.
Find the value of
a.
b.
c.
d.
961.
.
0.4049
0.4409
0.4094
0.4904
The cross section of a certain solid made by any plane perpendicular to the x-axis is an
equilateral triangle with the ends of one of its sides on the parabolas y = x2 + 5 and y =
2x2 + 1. Find the volume of this solid between the points of intersection of the parabolas.
a. 12.76
b. 13.77
c. 14.78
d. 15.79
962.
A hole of radius 3 units is bored through the center of a sphere of radius 5 units. Find the
volume of the part of the sphere with or without integration.
a. 278.2
b. 268.1
c. 258.4
d. 248.3
963.
Find the x-coordinate (or ) of the centroid of the area in the first quadrant bounded by
the curves y = 2 – x2 and y = x2.
a. 3/8
b. 1/4
c. 2/3
d. 4/9
964.
Find the length of the arc of the curve r = 2(1 + cosθ) from θ = 0 to θ = π.
a. 6
b. 7
c. 8
d. 9
965.
A solid has a circular base of radius 3 units. Find the volume of the solid if every plane
section perpendicular to a fixed diameter of the base is an isosceles triangle with its
altitude equal to its base. Solve with or without integration.
a. 42
b. 52
c. 62
d. 72
966.
A pit is to be dug in the form of an inverted right circular cone, 4 m deep, and 6 m in
diameter at the surface of the ground. Find the number of kilojoules of work to be done
if the material weighs w kN/cm3.
a. 15πw kJ
b. 14πw kJ
c. 12πw kJ
d. 13πw kJ
967.
A trough 6 m long as its vertical cross section in the form of an isosceles trapezoid. The
upper and lower bases are 6 m and 4 m respectively and its altitude is 2 m. if the trough
is full of liquid with specific weight 9.81 kN per cu m, find the forces against the slant
side of the trough.
a. 111.41 kN
b. 121.51 kN
c. 131.61 kN
d. 141.71 kN
968.
Evaluate
a.
b.
c.
d.
.
4.9348
4.3894
4.4938
4.8439
969.
The stretch of a spring is proportional to the force applied. If a force of 5 pounds
produces a stretch of one-tenth the original length, how much work will be done in
stretching the spring to double its original length? (Let L = original length)
a. 20L
b. 22L
c. 24L
d. 25L
970.
Find the volume of the ring-shaped solid generated by revolving about the x-axis the
portion of the plane bounded by the line y = 5 and the parabola y = 9 – x2.
a. 342.24
b. 442.34
c. 542.44
d. 642.54
971.
A uniform chain that weighs 4 N/m has a leaky 15-L bucket attached to it. The bucket
contains a liquid that weighs 9 N/L. If the bucket is full when 8 m of the chain is out and
half full when no chain is out, how much work was done in winding the chain on a
windlass. Assume that the liquid leaks out at a uniform rate.
a. 893 J
b. 938 J
c. 398 J
d. 839 J
972.
Find the volume of the torus generated by revolving a circle of radius r about a line on
the same plane of the circle and whose distance is 2r from the center of the circle. Solve
with or without integration.
a. 4πr2
b. 4πr3
c. 4π2r2
d. 4π2r3
973.
A plate in the shape of a right triangle is submerged vertically in the water and the base
3 m long is in the surface of the water. Find the altitude of the triangle if the force due to
the water pressure against one face of the plate is 50w kN where w is the specific weight
of the water.
a. 8 m
b. 9 m
c. 10 m
d. 11 m
974.
Find the volume of the torus generated by revolving about the x-axis the area bounded
by x2 + (y – 4)2 = 4.
a. 315.83
b. 314.73
c. 313.63
d. 312.53
975.
The cross section of a deep well containing mineral water is a circle of radius 1.2 m. the
cost of pumping the water to an outlet at the top of the well is 2 pesos per joule of work.
The mineral water weighs 9810 newton per cubic meter. If the surface of the water is
one meter below the top of the well and the water is sold 50,000 pesos per cubic meter,
find the depth to which the water is to be pumped out to realize maximum profit.
a. 2.35 m
b. 2.45 m
c. 2.55 m
d. 2.65 m
976.
A wedge is cut from a circular tree whose diameter is 2 m by a horizontal cutting plane
up to the vertical axis and another cutting plane which is inclined by 45 degrees from the
previous plane. Find the volume of the wedge with or without integration.
a. 3/5 m3
b. 2/3 m3
c. 3/4 m3
d. 2/5 m3
977.
Find the moment of inertia of a circle 5 cm in diameter about an axis through its
centroid.
a. 30.68 cm4
b. 31.58 cm4
c. 32.48 cm4
d. 33.38 cm4
978.
Find the moment of inertia of the circle in problem 481 relative to the line tangent to the
circle.
a. 76.47 cm4
b. 77.57 cm4
c. 78.67 cm4
d. 79.77 cm4
979.
Find the perimeter of the astroid whose parametric equations are x = acos 3t, y = asin3t.
a. 5a
b. 6a
c. 7a
d. 8a
980.
The axes of two right circular cylinders of equal radii 9 cm each intersect at right angles.
Find the volume of the common part of the cylinders.
a. 3666 cm3
b. 3777 cm3
c. 3888 cm3
d. 3999 cm3
981.
A hemispherical tank is full of oil weighing 7.85 kN/m3. The oil is to be pumped to the
top of the tank. Find the work done if the radius of the tank if 0.60 m.
a. 0.799 kJ
b. 0.688 kJ
c. 0.577 kJ
d. 0.466 kJ
982.
Find the area of one loop of the curve r2 = 8cos2θ.
a. 3
b. 4
c. 5
d. 6
983.
Find
of the centroid of the solid generated by revolving about the y-axis, the first
quadrant area bounded by y2 = 12x, x = 3 and y = 0.
a. 2.3
b. 2.5
c. 2.7
d. 2.9
984. The angle between 90 degrees and 180 degrees has
A. negative cotangent and cosecant
B. negative sine and tangent
C. negative secant and tangent
D. negative sine and cosine
985. It is defined as the angle subtended by a circular arc whose length is equal to the radius of
the circle.
A. mil
B. radian
C. degree
D. grade
986. In what quadrant does an angle terminate if its cosine and tangent are both negative?
A. first
B. second
C. third
D. fourth
987. Which of the following angles in standard position is a quadrantal angle ?
A. 540 degrees
B. 480 degrees
C. -135 degrees
D. -390 degrees
988. It is an angular unit that is equal to 1/6400 of four right angles.
A. mil
B. grade
C. radian
D. rpm
989. Relative to a right triangle ABC where C = 90 degrees, which of the following is not true ?
A. sin A = cos B
B. tan A = cot B
C. cos A = sec B
D. csc A = sec B
990. If the value of sin A is a negative fraction, then angle A terminates in
A. quadrants II and III
B. quadrants I and III
C. quadrants III and IV
D. quadrants II and IV
991. The secant is the cofunction of
A. sine
B. cosine
C. cotangent
D. cosecant
992. Which of the following is an undirected distance ?
A. The distance of a point from the x-axis.
B. The distance of a point from the y-axis.
C. The distance of a point from the origin.
D. The distance of a point from a line.
993. Which of the following systems of angle measurements uses the degree as the unit of
measure?
A. circular system
B. mil system
C. sexagesimal system
D. grade system
994. In what quadrant will angle A terminate if sec A is positive and csc A is negative.
A. I
B. II
C. III
D. IV
995. Which of the following relations is not true ?
A. sinx = (tanx/secx)
B. (cotx/cscx) = (sinx/tanx)
C. cotx = cscx cosx
D. (secx/tanx) = (cosx/cotx)
996. Within what limits between between 0 degrees and 360 degrees must the angle θ lie if cos θ
= -2/5 ?
A. between 0 degrees and 180 degrees
B. between 90 degrees and 180 degrees
C. between 90 degrees and 270 degrees
D. between 90 degrees and 360 degrees
997. The coreference angle of any angle A is the positive acute angle determined by the terminal
side of A and the y-axis. What is the coreference angle of 290 degrees ?
A. 70 degrees
B. 50 degrees
C. 30 degrees
D. 20 degrees
998. A measure of 3200 mils is equal to
A. 90 deg
B. 45 deg
C. 180 deg
D. 120 deg
999. The value of vers θ is equal to
A. 1 - cosθ
B. 1 - sinθ
C. 1 + cosθ
D. 1 +sinθ
1000. To find the interior angles of a triangle whose sides are given, use the law of
A. sine
B. cosine
C. tangent
D. secant
1001. The point P(x,y) where x  0 and y > 0 is located in quadrant
A. I or IV
B. II or III
C. I or II
D. III or IV
1002. Which of the following relations is true for any angle θ ?
A. sin(-θ) = sin θ
B. sec(-θ) = sec θ
C. tan(-θ) = tan θ
D. csc(-θ) = csc θ
1003. Coversine A is equal to
A. 1 - cosA
B. 1 - sin A
C. 1 + cosA
D. 1 + sin A
1004. The terminal side of -1,500 degrees will lie in quadrant
A. one
B. two
C. three
D. four
1005. Which of the following is false as the angle A increases from 0 degrees to 90 degrees ?
A. sin A increases from zero to one
B. tan A increases from zero to infinity
C. cos A decreases from one to zero
D. sec A decreases from one to infinity
1006. Which of the following functions is positive if angle A terminates in the second quadrant ?
A. csc A
B. tan A
C. sec A
D. cos A
1007. An angle in standard position and whose terminal side falls along one of the coordinate
axes is called a
A. reference angle
B. vertical angle
C. quadrantal angle
D. central angle
1008. Which of the following pairs of angles in standard positions are coterminal angles ?
A. 710 degrees and -10 degrees
B. 120 degrees and 60 degrees
C. -240 degrees and 30 degrees
D. 325 degrees and -40 degrees
1009. The gradient of the line in the figure is
A. tan θ
B. -1/tan θ
C. -tan θ
D. cot θ
1010. Which of the following is true in quadrants III and IV ?
A. negative cosecant
B. positive sine
C. negative cotangent
D. positive tangent
1011. Which of the following is not a first quadrant angle ?
A. 450 degrees
B. 60 degrees
C. -330 degrees
D. -120 degrees
1012. If tan θ > 0 and cosθ < 0, then θ is a
A. first quadrant angle
B. second quadrant angle
C. third quadrant angle
D. fourth quadrant angle
1013. If an angle is in the standard position and its measure is 215 degrees, the its reference
angle is
A. 25 degrees
B. 30 degrees
C. 35 degrees
D. 40 degrees
1014. In the second quadrant, which of the following is true ?
A. The tangent and secant are positive
B. The sine and cosecant are positive
C. The cotangent and cosecant are positive
D. The sine and tangent are positive
1015. In what quadrant can we locate the point (x, -4) if x is positive ?
A. I
B. II
C. III
D. IV
1016. In what quadrants do the secant and cosecant of an angle have the same algebraic sign?
A. II and IV
B. I and II
C. I and III
D. III and IV
C. 60 degrees
D. 90 degrees
C. 45 degrees
D. 60 degrees
C. 10/13
D. 12/13
1017. If cos 3A + sin A = 0, find the value of A.
A. 30 degrees
B. 45 degrees
1018. If tan A = 2 and tan B = 1/2, find A + B.
A. 90 degrees
B. 30 degrees
1019. If sin x = 5/13 , find sin 2x.
A. 120/169
B. 25/169
1020. If cot θ = square root of 3 and cos θ < 0, find csc θ.
A. 2
B. -2
C. 1/2
D. -1/2
1021. If sin A = -5/13 and A in quadrant III, find cot A.
A. 12/5
B. -12/5
C. 5/12
D. -5/12
C. 17/19
D. 8/17
C. 4 pi
D. 6 pi
1022. Find the value of sin(Arecos 15/17).
A. 8/9
B. 8/2
1023. The cosecant of 960 degrees is equal to
A. -2( square root of 3 / 3)
B. 2( square root of 3 / 3)
C. 1/2
D. -1/2
1024. If sin 3A = cos 6B, then
A. A - 2B = 90 degrees
B. A + 2B = 90 degrees
C. A + B = 180 degrees
D. A + 2B = 30 degrees
1025. What is the period of y = 3 sin(x/2) ?
A. 2 pi
B. 3 pi
1026. If the product of cot 2θand cot 68 degrees is equal to unity, find θ.
A. 13 degrees
B. 12 degrees
C. 11 degrees
D. 10 degrees
1027. Sec A - cos A is identically equal to
A. sin A cot A
B. cos A tan A
C. sin A tan A
D. cos A cot A
1028. Simplify ( sin θ/ 1 - cos θ) - ( 1 + cos θ/ sin θ)
A. sin²θ
B. cos²θ
C. 1
D. 0
C. 2
D. 1/2
C. 24/25
D. -24/25
C. sinθ
D. 2tanθ
1029. If tan x = 1/2 and tan y = 1/3, find tan (x + y).
A. 1
B. 2/3
1030. If cos θ= 3/5 and θ in quadrant IV, find cos2θ
A. 7/25
B. -7/25
1031. simplify (sinθ + cosθtanθ)/(cosθ)
A. tanθ
B. 2cotθ
1032. If Arcsin(2x) = 30 degrees, find x.
A. 0.20
B. 0.25
C. 0.3
D. 0.35
1033. If sin 40 degrees + sin 20 degrees = sin θ, find the value of θ.
A. 20 degrees
B. 60 degrees
C. 80 degrees
D.120 degrees
1034. The angle that is equal to one half of its supplement is
A. 60 degrees
B. 90 degrees
C. 80 degrees
D. 45 degrees
1035. Find the equivalent value of y in the equation y = (1 + cos 2θ) / (cot θ)
A. sin2θ
B. cos2θ
C. sinθ
D. cosθ
1036. If tan A = -3 and tan B = 2/3, find tan(A - B).
A. -11/9
B. -10/9
C. -13/9
D. -12/9
1037. If cos 65 degrees + cos 55 degrees = cos θ, find the θ in radians.
A. 1.832
B. 1.658
C. 0.7853
D. 0.0873
C. 42 degrees
D. 62 degrees
C. sin 2x
D. cos 2x
1038. If tan (A / 4) = cot A, find A.
A. 52 degrees
B. 72 degrees
1039. Simplify cos^4 x- sin^4 x
A. cos 4x
B. sin 4x
1040. If tan 4x = cot 6y, then
A. 2x - 3y = 45 degrees
B. 2x + 3y = 45 degrees
1041. Simplify
C. 4x - 6y = 90 degrees
D. 6y - 4x = 90 degrees
Arctan(1/3) + Arctan(1/5)
A. Arctan (7/4)
B. Arctan (4/7)
C. Arctan (8/15)
D. Arctan (1/15)
1042. If sin A =3.5x and cos A = 5.5x, find angle A.
A. 32.47 degrees
B. 33.47 degrees
C. 34.47 degrees
D. 35.47 degrees
1043. If the tangent of an angle x is 3/4, find the value of the cosine of 2x.
A. 0.60
B. 0.28
C. 0.8
D. 0.38
1044. Find the angle which a 9-m ladder will make with the ground if it is leaned against a
window still 6m high.
A. 21.8 degrees
B. 31.8 degrees
C. 41.8 degrees
D. 51.8 degrees
1045. The expression (1 -sinx) / (cosx) is equal to
A.tanx
B.1
C.(1 - cosx)/sinx
D.(cosx) / (1 + sinx)
1046. A tree 30 m long casts a shadow 36 m long. Find the angle of elevation of the sun.
A. 39.41 degrees
B. 39.51 degrees
C. 39.81 degrees
D. 39.61 degrees
1047. Which of the following is true ?
A. tan(180 degrees + θ) = - tanθ
B. tan(180 degrees - θ) = -tan θ
C. tan(90 degrees + θ) = -tanθ
D. tan(270 degrees - θ) = - tanθ
1048. Express 3i + 5 + (square root of -16) in the standard form.
A. 5 - 7i
B. 5 + 7i
C. -5 + 7i
D. -5 - 7i
1049. Write (square root of 2) cis 135 degrees in rectangular form.
A. 1 -i
B. -1 + i
C. -1 - i
D. 1 + i
1050. Give the conjugate of 2 + (square root of -25) in the standard form.
A. 2 - 5i
B. 2 + 5i
C. -2 + 5i
D. -2 -5i
1051. For the trigometric function y = a sin(bx +c), the absolute value of the ratio c/b is called
A. amplitude
B. period
C. argument
D. phase shift
1052. If sin2x sin4x = cos2x cos4x, find the value of x.
A. 13°
B. 14°
C. 15°
D. 16°
C. 0.1536
D. 0.1538
C. 19°
D. 17°
1053. If sin θ = 3.5x and cos θ = 5.5x, find x.
A. 0.1532
B. 0.1534
1054. Find θ if 2tan θ = ( 1 - tan² θ) cot 56° .
A. 18°
B. 16°
1055. Solve for x if Arctan ( 1 – x ) + Arctan ( 1 + x ) = Arctan ( 1/8 ).
A. 2
B. 4
C. 6
D. 3
1056. If A + B = 180°, then which of the following is true ?
sin A = sin B
cos A = cos B
tan A = tan B
A. (1) only
1057. Simplify
B. (2) only
C. (3) only
D. all of them
(sin ½x – cos ½x) ²
A. 1 + sin x
B. 1 – sin x
C. 1 + cos x
D. 1 – cos x
C. 60°
D. 90°
1058. Find the value of Arctan 2cos(Arcsin √3/ 2) .
A. 30°
B. 45°
1059. If sin A = -7/25 where 180° < A < 270°, find tan(A/2).
A. -1/5
B. -5
C. -1/7
D. -7
1060. If sin²x + y = m and cos²x + y = n, find y.
A. (m + n + 1)/2
B. (m + n – 1)/2
C. (m+n)/2 – 1
D. (m+n)/2 +1
1061. Given cos θ = √3/2, find the value of 1 - tan² θ.
A. -2
B. -1/3
C. ½
D. 2/3
1062. What is the value of A between 270° and 360° if 2sin² A – sin A = 1 ?
A. 290°
B. 275°
C. 300°
D. 330°
1063. Evaluate ( sin 0° + sin 1° + sin 2° + … + sin 90°) / ( cos 0° + cos 1° + cos 2° + … + cos
90°)
A. 0
B. 1
C. 2
D. 3
1064. If the supplement of an angle θ is 5/2 of its complement. Find the value of θ.
A. 30°
B. 25°
C. 20°
D. 15°
1065. Express -4 - 4√3 i in trigonometric form.
A. 8 cis 120°
B. 8 cis 240°
C. 8 cis 150°
D. 8 cis 300°
1066. If cos A = -15/17 and A is in quadrant III, find cos ½ A.
A. 0.29054
B. 0.24125
C. 0.24254
D.0.24354
1067. If sin A = 3/5 and cos B = 5/13, find sin (A + B).
A. 0.388
1068. Simplify
A. cot x
B. 0.865
C. 0.650
D. 0.969
C. tan 2x
D. 1
(sin 2x) / ( 1 + cos 2x)
B. tan x
1069. A pole which leans to the sun by 10° 15’ from the vertical casts a shadow of 9.43 m on the
level ground when the angle of elevation of the sun is 54°50’. The length of the pole is
A. 15.3 m
B. 16.3 m
C. 17.3 m
D. 18.3 m
1070. Triangle ABC has sides a, b and c. If a = 75 m, b = 100 m and the angle opposite side a is
32°, find the angle opposite side c.
A. 93°
B. 80°
C. 103°
D. 100°
1071. If the cosine of angle x is 3/5, then the value of the sine of x/2 is
A. 0.500
B. 0.361
C. 0.215
D. 0.447
C. 12°
D. 13°
C.-25/7
D.-24/7
C. 36/85
D.37/85
1072. If 82° + 0.35x = Arctan( cot 0.45x ), find x.
A. 11°
B. 10°
1073. If sec A = -5/4, A in quadrant II, find tan 2A.
A.24/7
B.25/7
1074. Evaluate cos( Arcsin 3/5 + Arctan 8/15 )
A. 34/85
B. 35/85
1075. If sin x = ¼ , find the value of 4sin(x/2)cos(x/2).
A. 1/8
B. 1/3
C. ½
D. 1/6
1076. If Arcsin( x – 2 ) = π/6, find x.
A. 5/4
B. 5/3
C. 5/2
D. 5/6
1077. The trigonometric expression ( 1 - tan²x ) / ( 1 + tan²x ) is equal to
A. sin1/2x
B. sin2x
C. cos1/2x
D. cos2x
1078. If x + y = 90°, then ( sinx tan y ) / ( sin y tan x ) is equal to
A. tanx
B. 1/tanx
C. –tanx
D. -1/tanx
1079. Twelve round holes are bored through a square piece of steel plate. Their centers are
equally spaced on the circumference of a circle 18 cm in diameter. Find the distance between the
centers of two consecutive holes.
A. 4.33 cm
B. 4.44 cm
C. 4.55 cm
D. 4.66 cm
1080. Two sides and the included angle of a triangle are measured to be 11 cm, 20 cm and 112°
respectively. Find the length of the third side.
A. 26.19 cm
B. 24.14 cm
C. 23.16 cm
D. 22.15 cm
1081. The rationalized value of ( 4 - 4√3 i ) / ( -2√3 + 2i ) is
A. √3 + i
B. -√3 + i
C. -√3 – i
D. √3 – i
C. 0.281
D. 0.291
1082. If Arctan(2x) + Arctan(x) = π/4, find x.
A. 0.261
B. 0.271
1083. A ladder leans against the wall of a building with its lower end 4 m from the building.
How long is the ladder if it makes an angle of 70° with the ground?
A. 12.3 m
B. 13.5 m
C. 11.7 m
D. 10.8 m
1084. Find the product of (4cis120°)(2cis30°) in rectangular form.
A. -4(√3 + i)
B. -4(√3 – i)
C. 4(√3 + i)
D. 4(√3 – i)
C. 2
D. 1
1085. Solve for x if x = (tanθ + cotθ) ² sinθ - tan²θ
A. 4
B. 3
1086. If ysinx = a and ycosx = b, find y in terms of a and b.
A. a + b
B. a² + b²
C. √a² + b²
D. √a + b
C. 1/5
D. ½
1087. If tan(Arctanx + Arctan ¼) = 7/11, find x.
A. 1/3
B. ¼
1088. if tanθ = √3, θ in quadrant III, find the value of (1 + cosθ) / (1 – cosθ).
A. ½
B. ¼
C. 1/3
D. 1/5
1089. From the top of a lighthouse 37 m above sea level, the angle of depression of a boat is 15°.
How far is the boat from the lighthouse?
A. 138.1 m
B. 137.2 m
C. 136.3 m
D. 135.4 m
1090. The angles B and C of a triangle ABC are 50°30’ and 122°09’ respectively and BC = 9,
find the length of AB.
A. 57.36
B. 58.46
C. 59.56
D. 60.66
1091. If the product of csc(x/2) and cos(x/3 + 60°) is equal to 1, find the value of x.
A. 46°
B. 36°
C. 26°
D. 16°
C. 1/5
D. 1/6
1092. If Arctanx + Arctan(1/3) = 45°, find x.
A. ½
B. ¼
1093. If cscθ = 2 and cosθ < 0, then ( secθ + tanθ ) / ( secθ – tanθ ) =
A. 2
B. 3
C. 4
D. 5
1094. Evaluate [6( cos80° + isin80° ) / 3( cos35° + isin35° )]
A. √2 ( 1 + i )
B. √2 ( 1 – i )
C. 2 ( 1 + i )
D. 2 ( 1- i )
C. 21°
D. 20°
1095. If sin(x + 10°) = cos3x, then x =
A. 23°
B. 22°
1096. If cos(x + y) = 0.17 and cosx = 0.50, find sin y.
A. 0.2355
B. 0.3455
C. 0.4344
D. 0.4233
1097. If sin A + sin B = 1 and sin A – sin B = 1, find A.
A. 60°
B. 70°
C. 80°
D. 90°
1098. At a certain instant, a lighthouse is 4 miles north of a ship which is traveling directly east.
If after 10 minutes, the bearing of the lighthouse is found to be North 21 degree 15 minutes
West, find the speed of the ship in miles per hour.
A. 11.3 mph
B. 10.3 mph
C. 9.3 mph
D. 8.3 mph
C. cot A
D. sin A
B. -8
C. 8i
D. -8i
B. 16i
C. -16
D. 16
1099. Simplify ( sec A + csc A ) / ( 1 + tan A )
A. csc A
B. sec A
1100. Evaluate [2(cos60° + isin60°)]³
A. 8
1101. Evaluate (1 + i)^8
A. -16i
1102. Two buildings with flat roofs are 15 m apart. From the edge of the roof of the lower
building, the angle of elevation of the edge of the roof of the taller building is 32°. How high is
the taller building if the lower building is 18 m high?
A. 26.4 m
B. 27.4 m
C. 28.4 m
D. 29.4 m
1103. If two sides of a triangle are each equal to 8 units and the included angle is 70°, find the
third side.
A. 6.15
B. 7.16
C. 8.17
D. 9.18
1104. Express sin(2Arccosx) in terms of x.
A. 2x√1 + x²
B. 3x√1 + x²
C. 2x√1 - x²
D. 3x√1 - x²
1105. Transform Arctanx + Arctany = pi/4 into an algebraic equation
A. x + xy + y = 1
B. x + xy –y = 1
C. x – xy + = 1
D. x – xy-y =1
1106. A tower 28.65 m high is situated on the bank of a river. The angle of depression of an
object on the opposite bank of the river is 25°20’. Find the width of the river.
A. 62.50 m
B. 60.52 m
C. 65.20 m
D. 63.25 m
1107. Two cars start at the same time from the same station and move along straight roads that
form an angle of 30°, one car at the rate of 30 kph and the other at the rate of 40 kph. How far
apart are the cars at the end of half an hour ?
A. 10.17 km
B. 10.27 km
C. 10.37 km
D. 10.47 km
C. -0.80
D. -0.90
1108. Given: sec2θ = √10 and 2θ in quadrant IV
Find : cos4θ
A. -0.60
B. -0.70
1109. The bearing of B from A is N20°E, the bearing of C from B is S30°E and the bearing of A
from C is S40°W. If AB = 10, find the area of triangle ABC.
A. 14.95
B. 13.94
C. 12.93
D. 11.92
1110. Two ships start from the same point, one going south and the other North 28° East. If the
speed of the first ship is 12 kph and the second ship is 16 kph, find the distance between them
after 45 minutes.
A. 17.3 km
B. 18.5 km
C. 19.2 km
D. 20.4 km
1111. If tanθ = ´ and θ is in the 1st quadrant, find tan 4θ.
A. -24/7
B. -20/7
C. -23/7
D.-22/7
1112. Find the height of a tree if the angle of elevation of its top changes from 20° to 40° as the
observer advances 23m toward its base.
A. 138.5 m
B. 148.5 m
C. 158.5 m
D. 159.5 m
C. 19°
D. 18°
C. -33/54
D. -33/53
1113 If 77° + (2x/5) = Arccos(sin x/4) , find x.
A. 21°
B. 20°
1114. Evaluate tan (Arccos(12/13) – Arcsin(4/5))
A. -33/56
B. -33/55
1115. Three times the sine of an angle is equal to twice the square of the cosine of the same
angle. Find the angle.
A. 20°
B. 25°
C. 30°
D. 35°
1116. Stations A and B are 1000 m apart on a straight road running from eat to west. From A, the
bearing of a tower at C is 32° west of north and from B, the bearing of C is 26° north of east.
Find the shortest distance of the tower at C from the road.
A. 243.92 m
B. 253.92 m
C. 263.92 m
D. 273.92 m
1117.If tan35° = y, then (tan145° - tan125°) / (1 + tan145°tan125°) =
A.(1 + y²) / 2y
B.(1 - y²) / 2y
C.(y²-1) / 2y
D. (2y-1)/2y
1118. A tree stands vertically on a hillside which makes an angle of 22° with the horizontal.
From a point 60 ft down the hill directly from the base of the tree, the angle of elevation of the
top of the tree is 55°. How high is the tree ?
A. 56.97 ft
B. 57.96 ft
C. 59.76 ft
D. 57.69 ft
C. 8m² -8m + 1
D. 8m² - 8m -1
1119. If cos 2A = √m , find cos 8A.
A. 8m² + 8m + 1
B. 8m² + 8m – 1
1120. The angle of triangle ABC are in the ratio 5:10:21 and the side opposite the smallest angle
is 5. Find the side opposite the largest angle.
A. 13.41
B. 14.31
C. 13.14
D. 11.43
1121. On the top of a cliff, the farthest distance that can be seen on the surface of the earth is 60
miles. How high is the cliff if the radius of the earth is taken to be 4000 miles ?
A. 0.41 mi
B. 0.43 mi
C. 0.45 mi
D. 0.47
1122. Two towers are of equal height. At a point P on level ground between them, the angle of
elevation of the top of the nearer tower is 60° and at a point M 24 meters directly away from
point P, the angle of elevation of the top of the nearer tower is 45°. How high is each tower ?
A. 20.8 m
B. 19.8 m
C. 18.8 m
D. 17.8 m
1123. A quadrilateral ABCD has its side AB perpendicular to side BC at B and its side AD
perpendicular to side CD at D. If angle BAD equals 60°, AB = 10 m and AD = 12 m, find the
distance (diagonal) from A to C.
A. 11.96 m
B. 12.86 m
C. 13.76 m
D. 14.66 m
1124. The sides of triangle ABC are AB = 5, BC = 12 and AC = 10. Find the length of the line
segment drawn from vertex A and bisecting BC.
A. 5.15
B. 5.25
C. 5.35
D.5.45
1125. Express 1/2 (1 - √3 i ) in trigonometric form.
A. cis 120°
B. cis 240°
C. cis 300°
D. cis 315°
1126. If versinθ = x and 1 – sinθ = ´ , find x if θ < 90°.
A. 0.124
B. 0.134
C. 0.154
D. 0.164
1127. Two points A and B, 150 m apart lie on the same side of a tower on a hill and in a
horizontal line passing directly under the tower. The angles of elevation of the top and bottom of
the tower viewed from B are 42° and 34° respectively and at A, the angle of elevation of the
bottom is 10°. Find the height of the tower.
A. 7.3 m
B. 8.3 m
C. 9.3 m
D. 10.3 m
1128. A point P is at a distance of 4, 5 and 6 from the vertices of an equilateral triangle of side of
x. Find x.
A. 8.5
B. 9.5
C. 7.5
D. 10.5
1129. A quadrilateral ABCD has its sides AB and BC perpendicular to each other at B. Side AD
makes an angle of 45° with the vertical while side CD makes an angle of 70° with the horizontal.
If AB = 15 and BC = 10, find the length of side CD.
A. 31.5
B. 51.5
C. 61.5
D. 41.5
1130. A clock has a minute hand 16 cm long and an hour hand 11 cm long. Find the distance
between the outer tips of the hands at 2:30 o’clock.
A. 19.6 cm
B. 20.6 cm
C. 21.6 cm
D. 22.6 cm
1131. If rcosxsiny = a, rcosxcosy = b and rsinx = c, find r.
A. √a² - b² - c²
B. √a² + b² -c²
C. √a² - b² + c²
D. √a²+b²+c²
1132. From the top of a tower 18 m high, the angles of depression of two objects situated in the
horizontal line with the base of the tower and on the same side, are 30 and 45 degrees. Find the
distance between the two objects.
A. 13.18 m
B. 13.28 m
C. 13.38 m
D. 13.48 m
1133. The sum of the sines of two angles A and B is 3/2 while the sum of the cosines of the
angles is √3 /2 . Find A.
A. 60°
B. 30°
C. 90°
D. 45°
C. ¾
D. 3/6
1134. Evaluate tan( Arcsec √5 – Arccot 2 )
A. 3/7
B. 3/5
1135. What is the greatest distance on the surface of the earth that can be seen from the top of
Mayon volcano which is 2.4 kilometers high if the radius of the earth is 6370 km ?
A. 159.7 km
B. 174.8 km
C. 179.7 km
D. 189.7 km
1136. A pole stands on a plane which makes an angle of 15° with the horizontal. A wire from the
top of the pole is anchored on a point 8 m from the foot of the pole. If the angle between the wire
and the plane is 30 degrees, find the length of the wire.
A. 10.93 m
B. 11.93 m
C. 12.93 m
D. 13.93 m
1137. If sin x + sin y = ½ and cos x – cos y = 1, find x.
A. 15°
B. 20°
C. 25°
D. 30°
1138. A tower standing on level ground is due north of point A and due east of point B. At A and
B, the angles of elevation of the top of the tower are 60° and 45° respectively. If AB = 20 , find
the height of the tower.
A. 18.32 m
B. 17.32 m
C. 16.32 m
D. 15.32 m
C. 90°
D. 45°
1139. If cot(80° - x/2) cot(2x/3) = 1, find x.
A. 30°
B. 60°
1140. If Arctan z = x/2, find cos x in terms of z.
A. (1 + z²) / (1 - z²)
B. (1 - z²) / (1 + z²)
C. (z² + 1) / (z² - 1)
D. (z² - 1) / (z² + 1)
1141. A flagstaff stands on the top of a house 15 m high. From a point on the plane on which
thee house stands., the angles of elevation of the top and bottom of the flagstaff are found to be
60° and 45° respectively. Find the height of the flagstaff.
A. 10.98 m
B. 11.87 m
C. 12.76 m
D. 13.25 m
1142. Two observers 100m apart and facing each other on a horizontal plane, observer at the
same time the angles of elevation of a balloon in their vertical to be 58° and 44°. Find the height
of the balloon .
A. 60.23 m
B. 59.34 m
C. 61.31 m
D. 58.75 m
1143. From a point outside an equilateral triangle, the distances of the vertices are 10 m, 18 m
and 10 m respectively. Find the side of the triangle.
A. 19.94 m
B. 20.94 m
C. 21.94 m
D. 22.94 m
1144. A spherical triangle which contains at least one side equal to a right angle is called
A. a right triangle
B. a polar triangle
C. an isosceles triangle
D. a quadrantal triangle
1145. If A, B and C are the angles of a spherical triangle, then which of the following is true ?
A. 180° < A + B + C < 360°
B. 180° < A + B + C < 540°
C. 0° < A + B + C < 360°
D. 0° < A + B + C < 180°
1146. The angular distance of the horizon from the zenith is equal to how many degrees ?
A. 45°
B. 60°
C. 90°
D. 180°
1147. The point on the celestial sphere directly above the observer is called the
A. zenith
B. nadir
C. pole
D. azimuth
1148. The small circle parallel to the equator is called the
A. equinox
B.parallel of latitude C.meridian
D.horizon
1149. If a, b and c are the sides of a spherical triangle, then
A. a + b + c < 180°
B. a + b + c < 360°
C. a + b + c < 540°
D. a+b+c< 90°
1150. The point on the celestial sphere diametrically opposite the zenith is called the
A. south pole
B. nadir
C. azimuth
D. north pole
1151. It is the angular distance of a heavenly body from the celestial equator.
A. declination
B. altitude
C. latitude
D. colatitude
1152. At sunset or at sunrise, the astronomical triangle is
A. an isosceles triangle
B. a quadrantal triangle
C. a right triangle
D. an oblique triangle
1153. The azimuth angle is always less than
A. 90°
B. 180°
C. 360°
D. 540°
1154. A great circle which passes through the celestial poles and a heavenly body B is called the
________ of B.
A. vertical circle
B. hour circle
C. longitude
D. horizon
1155. The angular distance of a point on the celestial sphere from the horizon is called its
A. longitude
B. altitude
C. latitude
D. polar distance
1156. It is the angle at the zenith from the upper branch of the observer's meridian toward the
east to the vertical circle of the heavenly body.
A. quadrantal angle
B. polar angle
C. hour angle
D. azimuth
1157. The zenith distance of a star is the complement of its
A. declination
B. polar distance
C. altitude
D. latitude
1158. Which of the following given sets of parts of a spherical triangle is possible in order to
define the triangle ?
A. A = 55°, B = 65°, C = 60°
B. a = 110°, b = 135°, c = 130°
C. A = 160°, B = 65°, C = 90°
D. a = 120°, b = 150°, c = 60°
1159. The complement of the declination of a star is called the
A. polar distance
B. zenith distance
C. longitude
D. altitude
1160. A 90-degree arc on the terrestrial sphere is equal to how many nautical miles ?
A. 3400
B. 4400
C. 5400
D. 6400
1161. How far in statute miles is a place at latitude 40° N from the equator ?
A. 2764.8
B. 2846.7
C. 2684.7
D. 2486.7
1162. Find the distance in nautical miles between A ( 40°30'N, 60°E ) and B (80°20'S, 60°E)
A. 6250
B. 7250
C. 8250
D. 9250
C. 4964
D. 4496
1163. Express 82°26' in nautical miles.
A. 4946
B. 4694
1164. If a place is 12°S of the equator, find its distance in nautical miles from the north pole.
A. 5130
B. 6120
C. 7110
D. 8100
1165. Find the difference in longitude between the following
places: M(34°54'33" N, 56°12'51" W)
P(30°20'46" N, 87°18'20" W)
A. 31°05'29"
B. 31°06'28"
C. 31°07'27"
D. 31°08'26"
1166. Find the difference in latitude between the places given in problem 22.
A. 4°32'46"
B. 4°33'47"
C. 4°31'48"
D. 4°30'49"
1167. If an observer is 840 nautical miles south of the equator, find his latitude.
A. 12° S
B. 13° S
C. 14° S
D. 15° S
1168. How far apart are two points on the equator one in longitude 40° East and the other in
longitude 150° West ?
A. 190°
B. 180°
C. 170°
D. 160°
1169. Express 3^h 11^m 55^s in angle units.
A. 45°47'58"
B. 58°47'45"
C. 47°45'58"
D. 47°58'45"
1170. Express 260°34' in time units.
A. 17^h 22^m 16^s
B. 17^h 16^m 22^s
C. 17^h 26^m 21^s
D. 17^h 21^m 26^s
1171. The plane of a small circle on a sphere of radius 25 cm is 7 cm from the center of the
sphere. Find the radius of the small circle.
A. 22 cm
B. 23 cm
C. 24 cm
D. 25 cm
1172. Find the area of a spherical triangle ABC on the surface of a sphere of raidus 10 where A =
119°37', B = 38°43' and C = 34°23'.
A. 23.18
B. 22.19
C. 21.16
D. 24.13
C. 15°
D. 16°
1173. An hour-angle of one hour is equal to
A. 14°
B. 13°
1174.The plane of a small circle on a sphere of radius 10 cm is 5 cm from the center of the
sphere. Find the area of the small circle.
A. 55π
B. 65π
C. 75π
D. 85π
1175. If the radius of the earth is 3960 miles, find the radius of a parallel of latitude 50° north.
A. 2445.44 mi
B. 2554.44 mi
C. 2455.44 mi
D. 2545.44 mi
1176. Use Napier's rule to find a formula for finding angle B of a right spherical triangle when
angle A and side c are given.
A. tan B = cos c tan A
B. cot B = sin c tan A
C. cot B = cos c tan A
D. tan B = sin c cot A
1177. Given a right triangle with angles A = 63°15' and B = 135°34'. Find side b.
A. 134.1°
B. 143.1°
C. 131.4°
D. 141.3°
1178. The two sides of a right spherical triangle are 86°40' and 32°41'. Find the angle opposite
the first given side.
A. 88°12'
B. 87°11'
C. 86°10'
D. 85°09'
1179. If the angles of a spherical triangle are A = 74°21' , B = 83°41' and C = 58°39', find side c.
A. 55°54'
B. 54°55'
C. 45°55'
D. 55°45'
1180. The sides of an oblique spherical triangle ABC are given as follows: a = 51°31' , b =
36°47' and c = 80°12'. Find A.
A. 32.35°
B. 33.45°
C. 34.55°
D. 35.56°
1181. Find the distance of Manila(14°36' N, 121°05' E) from Hongkong(22°18' N, 114°10' E) in
kilometers.
A. 1123.42 km
B.1124.32 km
C.1231.24 km
D.1321.42km
1182. If a boat sails N 30° W until the departure is 20 miles, what distance does it sail?
A. 55 mi
B. 50 mi
C. 45 mi
D. 40 mi
1183. A ship in latitude 50° N sails 80 nautical miles due East. Find the resulting change in
longitude.
A. 2.05° E
B. 2.07° E
C. 2.09° E
D. 2.03° E
1184. Find the longitude of an observer if his local apparent time is 10:36:41 AM and the local
Greenwich time is 4:23:12 AM.
A. 93°22'15" E
B. 92°22'15" E
C. 91°22'15" E
D. 90°22'15" E
1185. A ship in latitude 32° N sails due East intil it has made good a difference in longitude of
2°35' . Find the departure.
A. 128.42 nm
B. 129.43 nm
C. 130.44 nm
D. 131.45 nm
1186. Given a spherical triangle ABC with a = 68°27' , b = 87°32' and C = 97°53'. Find c.
A. 96.41°
B. 95.14°
C. 94.61°
D. 93.65°
1187. Find the area of a spherical triangle on the surface of a sphere of radius 10 where a =
140°30', b = 70°15' and C = 116°45'
A. 301.7
B. 370.2
C. 300.7
D. 307.1
1188. If the difference in longitude between two places A and B on the earth is 50° and their
latitudes are each 30° N. Find the distance AB in nautical miles.
A. 2589
B. 2598
C. 2985
D. 2895
1189. A ship leaves A(45°15' N, 140°38' W) and arrives at a place B(48°45' N, 137°12' W). Find
the distance AB in nautical miles using middle latitude sailing.
A. 140.49
B. 140.47
C. 140.45
D. 140.43
1190. An arc of one degree on the surface of the earth is approximately equal to how many
statute miles ?
A. 67.1
B. 68.1
C. 69.1
D. 70.1
1191. How many miles away is Manila(14°36' N, 121°05' E) from San Francisco(37°48' N,
122°24' W) ?
A. 7051
B. 8051
C. 9051
D. 10051
1192. A ship sails on a course between south and east making a difference in latitude of 13
nautical miles and a departure of 20 nautical miles. Find the course of the ship.
A. 54°56'43" E
B. 55°53'84" E
C. 56°58'34" E
D. 58°54'36" E
1193. Leaving point A(49°57' N, 15°16' W) , a ship sails between south and west till the
departure is 38 nautical miles and the latitude is 49°38' N. Find the distance traveled.
A. 42.49 n miles
B. 43.48 n miles
C. 44.47 n miles
D. 45.46 n miles
1194. Find the initial course of a flight from Manila(14°36' N, 121°05' E) to Tokyo(35°40' N,
139°46' E).
A. 35°06'
B. 36°05'
C. 30°56'
D. 30°65'
1195. Given a quadrantal triangle with B = 117°54', a = 95°42' and c = 90°. Find angle A.
A. 95.64°
B. 96.46°
C. 97.54°
D. 94.56°
1196. The initial course of a ship sailing from a place at latitude 40°40' N and longitude 73°58'
W is due east. After it has sailed 600 nautical miles on a great-circle track, find its latitude.
A. 37°54' N
B. 38°54' N
C. 39°54' N
D. 36°54' N
1197. If an airplane is to fly from Manila ( 14°36' N, 121°05' E) to Hongkong(22°18' N, 114°10'
E) at an average speed of 200 nautical miles per hour, how long should the trip take ?
A. exactly 3 hours
B. less than 3 hours
C. almost 3 hours
D. about 3 hours
1198. Find the local apparent time of sunrise at Paris ( lat 48°50' N) when the declination of the
sun is 14°38'.
A. 4:40:31 AM
B. 4:45:30 AM
C. 4:50:41 AM
D. 4:55:40 AM
1199. Find the local apparent time when an observer at latitude 37°52' N finds that the sun's
altitude in the eastern sky is 50°10' and the sun's declination is 12°30' N.
A. 9:46:51 AM
B. 9:45:56 AM
C. 9:56:45 AM
D. 9:41:56 AM
1200. An airplane leaves Guam ( 13°24' N, 144°38' E) with an initial course of 36°40' for a
great-circle track. Locate the point on the great-circle track which is nearest to the north pole.
A. (54°09' N, 80°12' W)
B. (59°04' N, 82°10' W)
C. (45°10' N, 81°02' W)
D. (49°05' N, 80°21' W)
1201. The declination of a star is 22°; its hour angle is 15°10' and the place of observation is
Berlin ( 52°32' N, 13°25' E). Find the altitude of the star.
A. 56.32°
B. 57.32°
C. 58.32°
D. 59.32°
1202. At 8:56 AM, the altitude and declination of the sun are found to be 36°18' and 14°35'
respectively. If the observation is done in the northern hemisphere, find the latitude of the place
of observation.
A. 52°56' N
B. 53°57' N
C. 54°58' N
D. 55°59' N
1203. An airplane flew from Manila (14°36' N, 121°05' E) at an average speed of 300 mph on a
course S 32° E. At what point will it cross the equator ?
A. 130°02' E
B. 140°03' E
C. 150°04' E
D. 160°05' E
1204. A ship sails from A( 38° N, 120° W) on a course 300° for a distance of 140 nautical miles
to point B. Find the position of B by middle latitude sailing method.
A. (107° N, 125°44' W)
B. (106° N, 126°54' W)
C. (108° N, 126°55' W)
D. (109° N, 127°45' W)
1205. Find the azimuth of a star at 5:30 PM at a place whose latitude is 41° if the star's
declination is 24°.
A. 284.18°
B. 274.18°
C. 264.18°
D. 254.18°
1026.Which of the following statements is false ?
A. The diagonals of a rhombus are perpendicular to each other.
B. The diagonals of a rectangle are equal.
C. The diagonals of a rhombus are equal.
D. The diagonals of a parallelogram bisect each other.
1207. The angle inscribed in a semicircle is
A. an obtuse angle
B. an acute angle
C. a straight angle
D. a right angle
1208. Which of the following points is equidistant from the vertices of a triangle ?
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1209. The point of intersection of the altitudes of a triangle is called the
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1210. The point of concurrency of the angle bisectors of a triangle is called the
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1211. The point inside a triangle that is equidistant from its sides is called the
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1212. The point of intersection of the medians of a triangle is called
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1213.The line segment which joins the midpoints of two sides of a triangle is parallel to the third
side and is what part of the third side ?
A. one half
B. one third
C. one fourth
D. two thirds
1214. The sum of the interior angles of a convex polygon of n sides is equal to how many right
angles ?
A. 2(n-1)
B. 2(n-2)
C. 2(n-3)
D.2(n-4)
1215. A convex polygon is a polygon each interior angle of which is less than
A. 45°
B. 60°
C. 180°
D. 90°
1216. Which of the following points is two thirds of the distance from each vertex of a triangle to
the midpoint of the corresponding opposite side ?
A. incenter
B. centroid
C. orthocenter
D.circumcenter
1217. It is a quadrilateral two and only two of whose sides are parallel
A. rectangle
B. rhombus
C. trapezoid
D. parallelogram
1218. It is a quadrilateral whose four sides are equal and with no angle equal to a right angle.
A. rectangle
B. rhombus
C. trapezoid
D. parallelogram
1219. The area of a circle is 6 times its circumference. What is its radius ?
A. 12
B. 11
C. 10
D. 13
1220. In a circle of radius 6, a sector has an area of 15 pi. What is the length of the arc of the
sector ?
A. 3 pi
B. 4 pi
C. 5 pi
D. 6 pi
1221. If the length of a side of a square is increased by 100%, its perimeter is increased by
A. 100 %
B. 150 %
C. 200 %
D. 250 %
1222. The side of a regular hexagon measures 10 cm. The radius of the circumscribing circle is
A. 8 cm
B. 10 cm
C. 12 cm
D. 14 cm
1223. The median of a trapezoid is 8 and one base is 5. How long is the other base ?
A. 13
B. 12
C. 11
D. 10
1224. What is the value of θ in the figure ?
A. 20°
B. 10°
C. 30°
D. 15°
1225. The area of the triangle inscribed in a circle is 40 sq. cm. and the radius of the
circumscribed circle is 7 cm. If two sides of the triangle are 8 cm and 10 cm, find the third side.
A. 10 cm
B. 12 cm
C. 13 cm
D. 14 cm
1226. The altitude of an equilateral triangle is 4. Find the length of each side.
A. 3.62
B. 4.62
C. 5.62
D. 6.62
1227. Find the side of a square whose area is equal to that of a rectangle with sides 32 and 18
cm.
A. 21 cm
B. 22 cm
C. 23 cm
D. 24 cm
1228. A railroad curve is to be laid out on a circle. What radius should be used if the tract is to
change direction by 25° in a distance of 36 m ?
A. 82.51 m
B. 81.52 m
C. 85.21 m
D. 81.25 m
1229. Find the area of a rhombus whose diagonals are 32 cm and 40 cm.
A. 540 cm²
B. 340 cm²
C. 640 cm²
D. 440 cm²
1230. The altitude of a triangle is half the base. Find the base if the area is 64.
A. 15
B. 16
C. 17
D. 18
1231. Find the area of a triangle whose sides are 9, 12 and 15.
A. 54
B. 53
C. 52
D. 51
1232. An isosceles trapezoid has two base angles of 45° and its bases are 6 and 10. Find its area.
A. 12
B. 14
C. 16
D. 18
1233. Find the altitude of a trapezoid of area 180 cm² if the bases are 16 and 14 cm.
A. 11 cm
B. 12 cm
C. 13 cm
D. 14 cm
1234. Find the area of a sector of a circle of radius 10 cm and whose central angle is 15°.
A. 193.32 cm²
B. 194.33 cm²
C. 195.34 cm²
D. 196.35 cm²
1235. Find the length of an arc of a circle of radius 20 which subtends a central angle of 30°.
A. 10 pi/3
B. 11 pi/3
C. 13 pi/3
D. 14 pi/3
1236. Find the length of a chord which is 2 units from the center of a circle of radius 6 units.
A. 6√2
B. 7√2
C. 8√2
D. 9√2
1237. How many sides has a convex polygon if the sum of the measure of its angles is 1080°?
A. 8
B. 7
C. 6
D. 5
1238. What is the measure of each interior angle of a regular pentagon ?
A. 106°
B. 109°
C. 107°
D. 108°
1239. What is the radius of a circle if its circumference is equal to its area ?
A. 4
B. 3
C. 2
D. 1
1240. What is the radius of a circle if the length of a 72° arc is 4π ?
A. 11π
B. 10π
C. 9π
D. 8π
1241. Find the area of a parallelogram of sides 15 and 16 if one of its angles is 60°.
A. 206.82
B. 207.85
C. 208.81
D. 205.83
1242. Each side of a rhombus is 7 and one angle is 42° . What is its area ?
A. 30.69
B. 31.59
C. 32.79
D. 33.89
1243. In triangle ABC, if a = 10 and b = 12 and angle C = 150° , find the area of the triangle.
A. 30
B. 31
C. 32
D. 33
1244. The diagonals of a rhombus are 6 cm and 8 cm long. Find the perimeter of the rhombus.
A. 20 cm
B. 24 cm
C. 22 cm
D. 26 cm
1245. The angles between the diagonals of a rectangle is 30° and each diagonal is 12 cm long.
Find the area of the rectangle.
A. 26 cm²
B. 36 cm²
C. 46 cm²
D. 56 cm²
1246. The sides of triangle ABC are a = 14, b = 12 and c = 10. Find the length of the median
from vertex A to side a.
A. 8.34
B. 8.44
C. 8.54
D.8.64
1247. The minute hand of a large clock is 2 m long. Find the distance traveled by the tip of the
minute hand in 5 minutes.
A. pi/4
B. pi/2
C. pi/6
D. pi/3
1248. Find the area of a parallelogram whose sides are 128 and 217 if an included angle is 136°.
A, 16942.38
B. 17492.83
C. 19294.83
D. 18249.38
1249. The area of a sector of a circle, having a central angle of 60° is 24 pi. Find the radius of the
circle.
A. 11
B. 12
C. 13
D. 14
1250. Two circles, each of radius 6 units, have their centers 8 units apart. Find the length of their
common chord.
A. 2√5
B. 3√5
C. 4√5
D. 5√5
1251. What is the apothem of a regular polygon having an area 225 sq. cm. and a perimeter 60
cm?
A. 7.5 cm
B. 6.5 cm
C. 8.5 cm
D. 4.5 cm
1252. Find the area of a regular hexagon of side 3 cm.
A. 22.28 cm²
B. 23.38 cm²
C. 24.48 cm²
D. 25.58 cm²
1253. A triangle has sides 3, 6 and 9. Find the shortest side of a similar triangle whose longest
side is 15.
A. 6
B. 10
C. 8
D. 5
1254. The perimeter of an octagon is 32 and its longest side is 6. What is the longest side of a
similar octagon whose perimeter is 24 ?
A. 3.5
B. 4
C. 4.5
D. 5
1255. In the figure AB = AC. The value of θ is
A. 31
C. 33
B. 32
D. 34
1256. A hexagon is circumscribed about a circle of radius 5. If the perimeter of the hexagon is
38, what is the area of the hexagon ?
A. 75
B. 65
C. 85
D. 95
1257. The circumference of two circles are 6π and 10π. What is the ratio of their areas ?
A. 9/25
B. 8/25
C. 7/25
D.6/25
1258. If a regular polygon has 54 diagonals, then it has how many sides ?
A. 10
B. 11
C. 14
D. 12
1259. If AB is parallel to CD where CD is the diameter of the circle as shown in the figure, find
angle θ.
A. 20°
B. 10°
C. 25°
D. 15°
1260. What is the diameter of a circle that is circumscribed about an equilateral triangle of side
7.4 cm.
A. 8.64 cm
B. 8.54 cm
C. 9.54 cm
D. 9.64 cm
1261. If the perimeter of a rhombus is 40 and one of its diagonals is 12, find the other diagonal.
A. 16
B. 15
C. 18
D. 17
1262. Given a circle as shown. The length of arc AB is
A. 1.527
B. 1.725
C. 1.257
D. 1.275
1263. Find the area of the annulus bounded by the inscribed and circumscribed circles of an
equilateral triangle with a side of length 6.
A. 11π
B. 8π
C. 10π
D. 9π
1264. Find the area of a regular octagon inscribed in a circle whose radius is 10 cm.
A. 822.8 cm²
B. 282.8 cm²
C. 828.2 cm²
D. 228.8 cm²
1265. In the figure shown, find the shaded area.
A. 4π
B. 5π
C. 6π
D. 7π
1266. If the perimeter of a regular hexagon is 24, what is the apothem ?
A. 3√3
B. 4√3
C. 2√3
D. 5√3
1267. The ratio of the sum of the exterior angles to the sum of the interior angles of a polygon is
1:3. Identify the polygon.
A. hexagon
B. heptagon
C. octagon
D. nonagon
1268. A circular sector has a radius of 6 cm and whose central angle is 60°. If it is bent to form a
right circular cone, the radius of the cone is
A. 1 cm
B. 2 cm
C. 3 cm
D. 4 cm
1269. A square is inscribed in a 90° sector of a circle as shown. Find the area of the shaded
region.
A. 1.214
B. 1.412
C. 1.124
D. 1.142
1270. A regular octagon is inscribed in a circle of radius 6. Find the perimeter of the octagon.
A. 34.54
B. 35.64
C. 36.74
D. 37.84
1271. If four angles of a pentagon have measures 100°, 96°, 87° and 97°, find the measure of the
fifth angle.
A. 150°
B. 160°
C. 140°
D. 130°
1272. If the sum two exterior angles of a triangle is 230°, find the measure of the third exterior
angle.
A. 130°
B. 120°
C. 110°
D. 100°
1273. Given are two concentric circles with line segment AB = 10 cm which is always tangent to
the small circle. Find the area of the shaded region (see figure).
A. 50 pi cm²
B. 45 pi cm²
C. 25 pi cm²
D. 30 pi cm²
1274. A circle whose radius is 10 cm is inscribed in a regular hexagon. The area of the hexagon
is
A. 346.4 cm²
B. 634.4 cm²
C. 364.4 cm²
D. 436.6 cm²
1275. The area of a parabolic segment having a base width of 10 cm and a height of 27 cm is
A. 270 cm²
B. 150 cm²
C. 210 cm²
D. 180 cm²
1276. A side of a regular hexagon is 6. What is the circumference of its circumscribed circle?
A. 12 pi
B. 11 pi
C. 13 pi
D. 10 pi
1277. Two chords PQ and RS of a circle meet when extended through Q and S at a point T. If
QP = 7, TQ = 9 , TS = 6, find SR.
A. 16
B. 17
C. 18
D. 19
1278. What is the angle at the center of a circle if the subtending chord is equal to two thirds of
the radius.
A. 39.95°
B. 38.94°
C. 37.93°
D. 36.92°
1279. The area of a rhombus is 250 and one of the angles is 37°25'. What is the length of each
side?
A. 20.18
B. 20.28
C. 20.38
D. 20.48
1280. If in triangle ABC, A = 76°30', B = 81°40' and c = 368, find the diameter of the
circumscribed circle.
A. 989.5
B. 959.8
C. 395.8
D. 958.9
1281. Given a parallelogram ABCD such that AB = 7, AC = 10 and angle BAC = 36°07'. Find
the length of BC.
A. 4.992
B. 5.992
C. 6.992
D. 7.992
1282. What is the diameter of the circle that is circumscribed about an isosceles triangle whose
vertical angle is 18° and the sum of the two equal sides is 18 units ?
A. 7.11
B. 8.11
C.9.11
D.10.11
1283. The diagonals of a quadrilateral are 34 and 56 intersecting at an angle of 67°. Find its area.
A. 837.62
B. 863.72
C. 826.37
D. 876.32
1284. Find the radius of a circle in which is inscribed a regular nonagon whose perimeter is
417.6 cm.
A. 68.37 cm
B. 67.83 cm
C. 63.87 cm
D. 68.73 cm
1285. If each interior angle of a regular polygon has a measure of 160°, how many sides has the
polygon ?
A. 16
B. 17
C. 19
D. 18
1286. The sides of a right triangle are a, b and c where c is the hypotenuse. Find the radius of the
circle that is inscribed in the triangle.
A. 1/2 (a+b+c)
C. 1/2(a-b+c)
B. 1/2(a+b-c)
D. 1/2(a-b-c)
1287. Two sides of a parallelogram are 20 and 30 and the included angle is 36°. Find the length
of the longer diagonal.
A. 74.65
B. 64.75
C. 57.46
D. 47.65
1288. The sides of a triangle are 17, 21 and 28. Find the length of the line segment bisecting the
longest side and drawn from the opposite angle.
A. 11
B. 12
C. 13
D. 14
1289. Two tangent circles of radii 6 and 2 have a common external tangent as shown in the
figure. Find the length of this external tangent.
A. 4√3
B. 5√3
C. 6√3
D. 7√3
1290. PQ and RS are secants of a circle which when extended beyond Q and S at a point T
outside the circle. Given that arc PR = 105° and arc QS = 67°, find the angle QTS.
A. 18°
B. 19°
C. 20°
D. 21°
1291. A bridge across a river is in the form of an arc of a circle. If the span is 40 ft and the
midpoint of the arc is to be 8 ft higher than the ends, what is the radius of the circle?
A. 27 ft
C. 29 ft
B. 28 ft
D. 30 ft
1292. Find the angle formed by the secant and tangent to a circle if one intercepted arc is 30°
more than the other and the secant passes through the center of the circle.
A. 15°
B. 16°
C. 17°
D. 18°
1293. Find the radius of a circle whose area is equal to the area of the annulus formed by two
consecutive circles with radii 5 and 13.
A. 10
B. 11
C. 12
D. 13
1294. A circle in inscribed in an equilateral triangle. If the circumference of the circle is 3, find
the perimeter of the equilateral triangle.
A. 9.246
B. 6.294
C. 2.946
D. 4.962
1295. Given a square ABCD as shown where E is the midpoint of side AD and G is the midpoint
of side BC. If arc DF has its center at E and arc FB has its center at G, find the shaded area.
A. 6
B. 8
C. 10
D. 12
1296. Two concentric circles each contains an inscribed square. The larger square is also
circumscribed about the smaller circle. If the circumference of the larger circle is 12 pi, what is
the circumference of the smaller circle ?
A. 6√2 pi
B. 5√2 pi
C. 4√2 pi
D. 3√2 pi
1297. A regular cross is inscribed in a circle as shown. Find the area ( shaded) between the
regular cross and the circle.
A. 43.44
B. 44.55
C. 45.66
D. 46.77
1298. Point P is a point on the minor arc AB of a circle with center at 0 as shown. If the angle
APB is x degree and angle A0E is y degrees, find an equation connecting x and y.
A. 2x - y = 360°
B. 2x + y = 360°
C. x - 2y = 360°
D. x+2y=360°
1299. The quadrilateral ABCD is inscribed in a circle and its diagonal AC is drawn so that angle
DAC = 34°, angle CAB = 38° and angle DBA = 65°. Find arc AB.
A. 96°
B. 86°
C. 76°
D. 66°
1300. PORS is a quadrilateral that is inscribed in a circle. If angle SQR = 23° and the angle
between the side SP and the tangent line through the point P is 64°, find angle PSR.
A. 86°
B. 87°
C. 88°
D. 89°
1301. The lines TA and TB are tangent to a circle at points A and B respectively. IF angle T =
42° and P is a point on the major arc AB, find angle APB.
A. 69°
B. 68°
C. 67°
D. 66°
1302. A secant and a tangent to a circle intersect an angle of 38° degrees. If the measures of the
arc intercepted between the secant and tangent are in the ratio 2:1, find the measure of the third
arc.
A. 129°
B. 130°
C. 131°
D. 132°
1303. Compute the difference between the perimeters of a regular pentagon and a regular
hexagon if the area of each is 12.
A. 0.31
B. 0.21
C.0.41
D.0.51
1304. The area of the sector of a circle having a central angle of 60° is 24π. Find the perimeter of
the sector.
A. 34.4
B. 35.5
C. 36.6
D.37.7
1305. Find the common area of two intersecting circles of radii 12 and 18 respectively if their
common chord is 14 long.
A. 34.19
B. 35.29
C. 36.39
D. 37.49
1306. In a parallelogram ABCD, the diagonal AC makes with the angle 27°10' and 32°43'
respectively. If side AB is 2.8 m long, what is the area of the parallelogram ?
A. 4.7 m²
B. 5.7 m²
C. 6.7 m²
D. 8.7m²
1307. The sum of the sides of a triangle is 100. The angle at A double that of B and the angle at
B is double that of C. Find side c.
A. 41.5
B. 42.5
C. 43.5
D. 44.5
1308. A diagonal of a parallelogram is 56.38 ft long and makes an angle of 26°13' and 16°24'
respectively with the sides. Find the area of the parallelogram.
A. 595 ft²
B. 585 ft²
C. 575 ft²
D. 565 ft²
1309. Find the area of a regular five-pointed star that is inscribed in a circle of radius 10.
A. 121.62
B. 112.26
C. 122.16
D. 126.21
1310. What is the difference in the areas between an inscribed and a circumscribed regular
octagon if the radius of the circle is 6?
A. 15.27
B. 16.37
C. 17.47
D. 18.57
1311. If BC = 2(AB), what fraction of the circle is shaded?
A. 1/4
B. 1/3
C. 1/2
D. 1/5
1312. In the figure, the small circle is tangent to 4 circular arcs. Find the area of the shaded
region if the radius of the larger circle is 10.
A. 34.94
B. 35.49
C. 31.94
D. 32.49
1313. A regular five-pointed star is inscribed in a circle of radius b cm. Find the area between the
circle and the star.
A. 4.04 b²
B. 3.03 b²
C. 1.01 b²
D. 2.02 b²
1314. From a point outside of an equilateral triangle, the distances of the vertices are 12, 20 and
12 respectively. Find the length of each side of the triangle.
A. 23.95
B. 22.85
C. 21.78
D. 20.68
1315. Using the vertices of a square, four arcs are drawn as shown in the figure. If each edge is
10 units long, find the shaded portion (common area).
A. 21.5
B. 31. 5
C. 41.5
D. 51.5
1316. Assuming that the earth is a perfect sphere of radius 6370 kilometers, a person at a point T
on top of a tower 60 meters high looks at a point P on the surface of the earth. What is the
approximate distance of P from T ?
A. 24.3 km
B. 25.4 km
C. 26.5 km
D. 27.6 km
1317. Each of four circles ( see figure ) is tangent to the other three. If the radius of each of the
smaller circles is 3, what is the radius of the largest circle ?
A. 6.46
B. 6.64
C. 4.64
D.4.46
1318. In the figure, if arc AB = 50°, arc BC = 80° and arc AD = 90°, find θ.
A. 85°
B. 65°
C. 95°
D. 75°
1319. In the figure, if PB = 6, PC = 10, PA = 5 and θ = 30°, find the area of the quadrilateral
ABCD.
A. 21.5
B. 22.5
C. 23.5
D. 24.5
1320. The solid formed by revolving a circle about an external axis in its plane is called.
A. annulus
B. conoid
C. torus
D. prismatoid
1321. The intersection of two faces of a pyramid is called the
A. lateral edge
B. slant height
C. altitude
D. hypotenuse
1322. It is a polyhedron of which one face is a polygon and the other faces are triangles which
have a common vertex.
A. prism
B. pyramid
C. cone
D. prismatoid
1323. The altitude of any of the lateral faces of a regular pyramid is called the
A lateral edge
B.altitude
C.median
D.slant height
1324. It is a polyhedron whose faces are all squares.
A. tetrahedron
B. hexahedron
C. octahedron
D.dodecahedron
1325. The dihedral angle is the angle between two intersecting
A.lines
B. arcs
C.planes
D.chords
1326. Which of the regular polyhedrons has faces that are regular pentagons ?
A. tetrahedron
B. dodecahedron
C. octahedron
D. icosahedron
1327. If the base of a solid is a circle and every section perpendicular to the base is an isosceles
triangle, the solid is called
A. conicoid
B. prismoid
C. conoid
D. astroid
1328. The radius of a sphere that is inscribed in a regular hexahedron of edge e is equal to
A. e/2
B. e/3
C. e/4
D. e/5
1329. It is a polyhedron of which two faces are equal polygons in parallel plane and the other
faces are parallelogram.
A.tetrahedron
B. prism
C.pyramid
D.prismoid
1330. A spherical wooden ball 15 cm in diameter sinks to a depth of 12 cm in a certain liquid.
The area exposed above the liquid is
A. 54 pi cm²
B. 15 pi cm²
C. 45 pi cm²
D. 35 pi cm²
1331. What is the total area of a cube whose edge is 5 cm?
A. 150 cm²
B. 145 cm²
C. 140 cm²
D. 135 cm²
1332. Find the volume of the frustum of a right circular cone whose altitude is 6 and whose base
radii are 2 and 3.
A. 35π
B. 36π
C. 37π
D. 38π
1333. The angle of a lune is 60° and the radius of the sphere is 15 cm. Find the volume of the
spherical wedge whose base is the given lune.
A. 750π cm³
B. 700π cm³
C. 650π cm³
D. 600π cm³
1334. A sphere of radius R is inscribed in a cube of edge e. What is the ratio of the volume of the
sphere to the volume of the cube?
A. 0.6523
B. 0.5236
C. 0.3652
D. 0.2635
1335. The slant height of a right circular cone is 13 and the altitude is 12. Find the radius of the
base.
A. 8
B. 7
C. 6
D. 5
1336. A hemispherical bowl of radius 10 cm is filled with water to a depth of 5 cm. Find the
volume of the water.
A. 615π/3 cm³
B. 620π/3 cm³
C. 625π/3 cm³
D. 630π/3 cm³
1337. The area of a lune is 4π m² and the radius of the sphere is 3 m. Find the angle of the lune.
A. 40°
B. 45°
C. 50°
D. 55°
1338. The volume of a sphere whose diameter is 20 cm is
A. 4198.87 cm³
B. 4179.88 cm³
C. 4188.79 cm³
D.4187.89 cm³
1339 Find the length of the diagonal of a rectangular box whose edges are 6, 8 and 10.
A. 7√2
B. 8√2
C. 9√2
D. 10√2
1340. Find the area of the base of a prism whose volume is 516.6 cu. ft and whose height is 16.4
in.
A. 372 ft²
B. 374 ft²
C. 376 ft²
D. 378 ft²
1341. Find the slant height of a regular pyramid each of whose faces is enclosed by an equilateral
triangle with side 8.
A. 6.73
B. 6.93
C. 6.83
D. 6.63
1342. What is the volume of a pyramid whose altitude is 27 and whose base is a square 8 on a
side ?
A. 756
B. 657
C. 576
D. 675
1343. A concrete pedestal is in the form of a frustum of a regular square pyramid whose altitude
is 1.2 cm and base edges 0.40 m and 0.70 m respectively. Find the volume of the pedestal.
A. 0.372 m³
B. 0.327 m³
C. 0.273 m³
D. 0.723 m³
1344. The base radii of the frustum of a cone are 6 cm and 10 cm respectively. Find the altitude
of the frustum if its volume is 1176π cu. cm ?
A. 16 cm
B. 17 cm
C. 18 cm
D. 19 cm
1345. What is the diameter of a sphere for which its volume is equal to its surface area?
A. 7
B. 6
C. 5
D. 4
1346. Find the volume of a spherical wedge whose angle is 36° on a sphere of radius 6 cm.
A. 28.8π cm³
B. 27.7π cm³
C. 26.6π cm³
D. 25.5π cm³
1347. Find the volume of a right circular cone whose base radius is 8 cm and whose altitude is 15
cm.
A. 320 pi cm³
B. 330 pi cm³
C. 340 pi cm³
D. 350 pi cm³
C. 3.872 m³
D. 7.238 m³
1348. The volume of a sphere of radius 1.2 m is
A. 8.372 m³
B. 2.783 m³
1349. The volume of a square pyramid is 384 cm³ and its altitude is 8 cm. How long is an edge
of the base?
A. 11 cm
B. 12 cm
C. 13 cm
D. 14 cm
1350. Find the altitude of a right prism flow which the area of the lateral surface is 338 and the
perimeter of the base is 13.
A. 25
B. 26
C. 27
D. 28
1351. A conical tank is 10.5 m deep and its circular top has a radius of 5 cm. How many liters of
water will it hold?
A. 260500π
B. 261500π
C. 262500π
D. 263500π
1352. Find the diameter of a sphere whose surface area is 324π.
A. 16
B. 17
C. 18
D. 19
1353. Find the area of a zone of a sphere whose radius is 6 if the altitude of the zone is 2.
A. 21 pi
B. 22 pi
C. 23 pi
D. 24 pi
1354. The volume of a 10-cm high conical paper weight is 180 cm³. The radius of the base is
A. 4.15 cm
B. 4.17 cm
C. 4.19 cm
D. 4.21 cm
1355. The volume of the frustum of a cone which is 25 cm high and whose base radii are 7.5 cm
and 5 cm long respectively is
A. 3108.87 cm³
B. 3107.88 cm³
C. 3170.88 cm³
D.3180.78 cm³
1356. Find the volume of a cube whose total area is 384 cm².
A. 212 cm³
B. 312 cm³
C. 412 cm³
D. 512 cm³
1357. Find the total area of a tetrahedron 3 units on an edge.
A. 8√3
B. 9√3
C. 10√3
D. 11√3
1358. The volume of a pyramid is 256 cm³ and its altitude is 24 cm. Find its base area.
A. 52 cm²
B. 42 cm²
C. 32 cm²
D. 22 cm²
1359. The radii of the bases of the frustum of a right circular cone are 6 and 9 respectively and
the altitude is 4. Find the lateral area.
A. 75π
B. 65π
C. 95π
D. 85π
1360. The altitude of a parallelepiped is 20 and the base is a rhombus with diagonals 10 and 16.
Find the volume of the parallelepiped.
A. 1500
B. 1600
C. 1700
D. 1800
1361. A sphere of radius r just fits into a cylindrical box. Find the empty space inside the box.
A. 2π r³/3
B. 8π r³/9
C. 4π r³/9
D. 20π r³/27
1362. Find the volume of a pyramid having a pentagonal base with sides each equal to 12 cm and
an apothem of 3 cm. The altitude of the pyramid is 36 cm.
A. 2660 cm³
B. 2770 cm³
C. 2880 cm³
D. 2990 cm³
1363. Find the volume of the frustum of a regular triangular pyramid whose altitude is 3 and
whose base edges are 4 and 8 respectively.
A. 25√3
B. 26√3
C. 27√3
D. 28√3
1364. The lateral area of a right circular cone with a radius of 20 cm and a height of 30 cm is
A. 2265.43 cm²
B. 2236.45 cm²
C. 2245.63 cm²
D.2253.46 cm²
1365. The base of a prism is a rectangle with sides 3 and 5. If its lateral area is 64, find its
altitude.
A. 3
B. 4
C. 5
D. 6
1366. Find the number of degrees on a dihedral angle of a regular tetrahedron.
A. 68.33°
B. 69.43°
C. 70.53°
D. 71.63°
1367. Find the volume of a spherical cone in a sphere of radius 17 cm if the radius of the zone is
8 cm.
A.1126π/3 cm³
B.1136π/3 cm³C.1146π/3cm³
D.1156π/3 cm³
1368. Find the volume of a regular square pyramid whose slant height is 10 and whose base edge
is 12.
A. 384
B. 374
C. 364
D. 354
1369. The base of a prism is a rhombus whose sides are each 10 cm and whose shorter diagonal
is 12 cm. If the altitude is 12 cm, find its volume.
A. 1132 cm³
B. 1142 cm³
C. 1152 cm³
D. 1162 cm³
1370. Find the volume of a triangular prism whose altitude is 20 cm and whose sides are 6 cm, 8
cm and 12 cm.
A. 426.61 cm³
B. 421.66 cm³
C. 461.26 cm³
D. 416.62 cm³
C. 2660
D. 2770
1371. Find the volume of the solid as shown.
A. 2330
B. 2440
1372. Find the volume of a spherical segment if the radii of the bases are 3 and 4 respectively
and its altitude is 2.
A. 83.27
B. 87.32
C. 83.72
D. 82.73
1373. A stone is dropped into a circular tub 40 inches in diameter, causing the water therein to
rise 20 inches. What is the volume of the stone ?
A. 6000π in³
B. 7000π in³
C. 8000π in³
D. 9000π in³
1374. The base of a right parallelepiped is a rhombus whose sides are each 10 cm long and one
of whose angles is 60 degrees. If the altitude of the parallelepiped is 4 cm, find its volume .
A. 100√3 cm³
B. 200√3 cm³
C. 300√3 cm³
D. 400√3 cm³
1375. Find the volume of the largest cube that can be out from a circular log whose radius is 30.
A. 76367.53
B. 75567.33
C. 73675.36
D. 77653.36
1376. Find the lateral area of a pyramid whose altitude is 27 cm and whose base is a square 8 cm
on a side.
A. 437.62 cm²
B. 436.72 cm²
C. 432.76 cm²
D. 427.63 cm²
1377. The diagonal of a cube is 2√3. Find its volume.
A. 9
B. 7
C. 8
D. 6
1378. Find the lateral area of the frustum of a regular square pyramid whose base edges are 6 and
12 and whose altitude is 4.
A. 150
B. 160
C. 170
D. 180
1379. If the radius of a sphere is 8 and if a plane passes through the sphere at a distance of 5
from its center. what is the area of the circle of intersection ?
A. 38 pi
B. 39 pi
C. 40 pi
D. 41 pi
1380. Find the lateral area of a right circular cone that can be inscribed in a cube whose volume
is 64.
A. 28.1
B. 26.1
C. 24.1
D.22.1
1381. Find the lateral edge of a regular square pyramid whose slant height is 8 and whose base
edge is 6.
A. 6.54
B. 7.54
C. 8.54
D.9.54
1382. The base edges of a triangular pyramid are 12, 14 and 16. If its altitude is 22, what is the
volume of the pyramid ?
A. 594.64
B. 564.94
C. 544.69
D. 596.44
1383. The volume of the frustum of a right circular cone is 78 pi. The upper base radius is 2 and
the lower base radius is 5. What is the altitude of the frustum ?
A. 5
B. 6
C. 7
D. 8
1384. The volume of a right circular cone having a slant height of 13 and altitude 12 is
A. 100π
B. 150π
C. 200π
D. 250π
1385. Find the lateral area of a regular triangular pyramid whose base edge is 4 and its lateral
edge is 6.
A. 21√2
B. 22√2
C. 23√2
D. 24√2
1386. Find the height of a pyramid whose volume is 35 and whose base is a triangle with sides 4,
7 and 5.
A. 11.72
B. 10.72
C. 8.72
D. 9.72
1387. The radii of the bases of the frustum of a right circular cone are 6 and 9 respectively and
its altitude is 4. Find its lateral area.
A. 75π
B. 85π
C. 95π
D. 65π
1388. Find the volume of a sphere whose surface area is 64π.
A. 256π/3
B. 254π/3
C. 252π/3
D. 250π/3
1389. What is the area of a sphere if a zone on it having an area of 18 and has an altitude of 2?
A. 79 pi
B. 80 pi
C. 81 pi
D. 82 pi
1390. A spherical bowl of radius 8 inches contains water to a depth of 3 inches. Find the volume
of the water in the bowl.
A. 199.72 in³
B. 197.92 in³
C. 179.29 in³
D. 192.27 in³
1391. The volume of a pyramid is 256 cm³ and its altitude is 24 cm. Find the area of its base.
A. 32 cm²
B. 34 cm²
C. 31 cm²
D. 33 cm²
1392. Find the lateral area of a right circular cone if its slant height is 22 and the circumference
of its base is 8.
A. 55
B. 66
C. 77
D. 88
1393. What is the diameter of a sphere for which its volume is equal to its surface area ?
A. 5
B. 6
C. 7
D. 8
1394. The lateral area of a regular pyramid is 2048 and the perimeter of the base is 128. Find the
slant height.
A. 42
B. 22
C. 32
D. 52
1395. The area of the base of a right circular cone is 144π . If its altitude is 14, find its slant
height.
A. 18.44
B. 17.33
C. 16.22
D. 15.11
1396. Find the approximate change in the volume of a cube if each edge x of the cube is
increased by one percent.
A. 0.02 x³
B. 0.03 x³
C. 0.04 x³
D. 0.05 x³
1397. The area of a diagonal section of a cube is 4√2 cm². Find the edge of the cube.
A. 3 cm
B. 2 cm
C. 4 cm
D. 1 cm
1398. Find the volume of the largest circular cylinder that can be inscribed in a cube whose
volume is 64 cu. cm.
A. 13π cm³
B. 14π cm³
C. 15π cm³
D. 16π cm³
1399. The altitude of a square pyramid is 10 and a side of the base is 15. Find the area of a cross
section at a distance of 6 from the vertex.
A. 81
B. 82
C. 83
D. 84
1400. The diameter of one solid ball is 3 times the diameter of another ball of the same material.
If the weight of the smaller ball is 250 pounds, what is the weight of the larger ball ?
A. 6957 lb
B. 6750 lb
C. 6507 lb
D. 6570 lb
1401. Find the volume of a regular tetrahedron whose edges are each equal to 6.
A. 16√2
B. 17√2
C. 18√2
D. 19√2
1402. The lateral area of a regular pyramid is 514.5 and the slant height is 42. Find the perimeter
of the base.
A. 24.5
B. 26.5
C. 22.5
D.28.5
1403. A wedge is cut from a circular tree whose diameter is 2 m by a horizontal plane up to the
vertical axis and another cutting plane which is inclined at 45 degrees from the previous plane.
The volume of the wedge is
A. 1/4
B. 1/2
C. 2/3
D. 3/4
1404. The zone of a spherical cone has a altitude of 2 cm and a radius of 4 cm. Find the volume
of the spherical cone.
A. 115π/3 cm³
B. 110π/3 cm³
C. 105π/3 cm³
D. 100π/3 cm³
1405. The base of a prism is the triangle ABC with A = 35 degrees, B = 68 degrees and c = 25. If
the altitude of the prism is 10, find the volume of the prism.
A. 1607.5
B. 1705.6
C. 1507.6
D. 1076.5
1406. The capacities of two hemispherical tanks are in the ratio 64:125. If 4.8 kg of paint is
required to paint the outer surface of the smaller tank, then how many kg of paint would be
needed to paint the outer surface of the larger tank ?
A. 6.5 kg
B. 7.5 kg
C. 8.5 kg
D. 9.5 kg
1407. Find the volume of a sphere that is circumscribed about a cube of edge 4.
A. 30√3 π
B. 32√3 π
C. 34√3 π
D. 36√3 π
1408. A sphere is inscribed in a right circular cone. The slant height of the cone is equal to the
diameter of its base. If the altitude of the cone is 15, find the surface area of the sphere.
A. 125π
B. 120π
C. 110π
D. 100π
1409. The base of a tetrahedron is a triangle whose sides are 10, 24 and 26. If the altitude of the
tetrahedron is 20, find the area of a cross-section whose distance from the base is 15.
A. 9.5
B. 8.5
C. 7.5
D. 6.5
1410. If the length of each edge of a cube is increased by 3 cm, its volume is increased by 387 cu
cm. Find the length of each edge of the original cube.
A. 5 cm
B. 6 cm
C. 7 cm
D. 8 cm
1411.The lateral area of a regular pyramid is 2048 and the perimeter of the base is 128. If its base
is a regular octagon, find the altitude of the pyramid.
A. 24.5
B. 25.5
C. 26.5
D. 27.5
1412. Find the area illuminated by a candle h meters from the surface of a ball r meters in radius.
A. (2πrh²) / (r+h)
B. (2πrh²) / (r-h)
C. (2πr²h) / (r+h)
D. (2πr²h) / (r-h)
1413. Find the volume of the frustum of a pyramid whose bases are regular hexagons with base
edges 5 cm and 10 cm respectively and the altitude is 15 cm.
A. 2273.31 cm³
B. 2171.33 cm³
C. 2327.13 cm³
D. 2713.32 cm³
1414. What is the volume of the cube if the number of cubic units in its volume is twice the
number of square units in its total surface area ?
A. 1827
B. 1287
C. 1872
D. 1728
1415. Find the lateral area of a regular hexagonal pyramid whose lateral edges are each 13 cm
and whose base has sides 10 cm each.
A. 350
B. 360
C. 370
D. 380
1416. The ratio of the volume of two spheres is 8:27. What is the ratio of their surface areas?
A. 2/9
B. 4/9
C. 5/9
D. 7/9
1417. Each edge of the upper base of the frustum of a regular quadrangular pyramid is 2 less
than an edge of the lower base. Find the edge of the lower base if the slant height is 10 and the
total area is 160.
A. 3
B. 4
C. 5
D. 6
1418. Find the volume of a solid formed by revolving an equilateral triangle with side e about an
altitude.
A. (√3π e³) / 24
B. (√3π e³) / 12
C. (√2π e³) / 24
D. (√2π e³) / 12
1419. If the diameter of a sphere is increased by 40 percent, by what percent is the volume
increased ?
A. 144.7%
B. 147.4%
C. 177.4%
D. 174.4%
1420. The radii of two spheres are in the ratio 3:4 and the sum of their surfaces is 2500. Find the
radius of the smaller sphere.
A. 14
B. 15
C. 16
D. 17
1421. If the ratio of the lateral area of the frustum of a cone to its volume is 15:28, find the
altitude of the frustum if its base radii are 3 and 6 respectively.
A. 6
B. 5
C. 4
D. 3
1422. The lateral area of a right circular cone is 3 times the area of its base. Find the angle at
which the slant height of the cone is inclined with the base.
A. 71.35°
B. 72.15°
C. 70.53°
D. 73.25°
1423. The volume of a rectangular parallelepiped is 162. The three dimensions are in the ratio
1:2:3. Find the total area.
A. 198
B. 197
C. 196
D. 195
1424. The base edge of a square pyramid is 3 m and its altitude is 10 m. Find the area of a
section parallel to the base and 6 m from it.
A. 1.22 m²
B. 1.33 m²
C. 1.44 m²
D. 1.55 m²
1425. The area of the base of a pyramid is 25 and its altitude is 10. What is the distance from the
base of a section parallel to the base whose area is 9 ?
A. 4
B. 3
C. 5
D. 2
1426. The edge of a regular tetrahedron is 5. Find the edge of a cube which has the same volume
as the tetrahedron.
A. 2.35
B. 2.45
C. 2.55
D. 2.65
1427. The segment of a paraboloid of revolution( see figure ) is a solid in which every section
parallel to the base is a circle the radius R of which is the mean proportional between the
distance H from the vertex and the radius r of the base. Find the volume of the segment of
altitude h.
A. 1/2 πr²h
B. 1/3 πr²h
C. 1/2 πrh²
D. 1/3 πrh²
1428. A right circular cone whose slant height is 18 cm and the circumference of whose base is 6
cm is cut by a plane parallel to the base such that the cone is cut off, has a slant height of 4 cm.
Find the lateral area of the frustum formed.
A. 48.3
B. 49.1
C. 50.2
D. 51.3
1429. A tank has the form of a cylinder of revolution whose diameter is 60 cm and whose height
is 244 cm. The tank is in horizontal position and is filled with water to a depth of 46 cm. Find the
approximate number of liters of water in the tank.
A. 566
B. 567
C. 568
D. 569
1430. A solid gas a circular base of radius 20. Find the volume of the solid if every section
perpendicular to a certain diameter is an equilateral triangle.
A. 18475.21
B. 14871.52
C. 17845.12
D. 15781.25
1431. In a cone of altitude h and elliptic base A, every section parallel to the base has an area Ay
= Ay² / h² where y is the distance from the vertex to the section ( see figure ). Find the volume of
the elliptic cone.
A. πabh / 3
B. πabh / 2
C. πabh / 4
D. πabh / 5
1432. Find the total area of a regular hexagonal pyramid whose slant height is 5 ft and whose
base is 4 ft.
A. 105.71 ft²
B. 107.15 ft²
C. 101.57 ft²
D. 110.75 ft²
1433. For the solid shown, every section perpendicular to the edge AB is a circle. If arc ACB is a
semicircle of diameter 18, find the volume of the solid ( see figure ).
A. 342 pi
B. 423 pi
C. 432 pi
D. 243pi
1434. A solid consists of a hemisphere surmounted by a right circular cone. Find the vertical
angle of the cone if the volume of the conical and spherical portions are equal.
A. 51.13°
B. 52.13°
C. 53.13°
D. 54.13°
1435. The slant height of the frustum of a right circular cone makes an angle of 60° with the
larger base. If the slant height is 30 cm and the radius of the smaller base is 5 cm, find the
volume of the frustum.
A. 15283.7 cm³
B. 14283.7 cm³
C. 13283.7 cm³
D.12283.7 cm³
1436. The lateral area of the frustum of a regular pyramid is 336 sq cm. If the lower base is a
square having a side of 8 cm; the upper base is a square of side x cm and its slant height is 12
cm, find the value of x.
A. 6
B. 4
C. 7
D. 5
1437. If the area of the base of a regular hexagonal prism is 3√3 / 2 sq cm and the total area is a
45√3 sq cm, find the volume of the prism.
A. 20.5 cm³
B. 31.5 cm³
C. 21.5 cm³
D. 30.5 cm³
1438. If a cylinder has a lateral area of 88 pi and a volume of 176 pi, what is its total area ?
A. 120 pi
B. 125 pi
C. 130 pi
D. 135 pi
1439. A rectangular prism has a width of 2 cm, a height of 4 cm and a length of 3√3 cm. If its
volume is equal to the volume of a cube with diagonal d, find the value of d.
A. 8 cm
B. 7 cm
C. 6 cm
D. 5 cm
1440. The axes of two right circular cylinders of equal radii 3 m long, intersect at right angles.
Find the volume of their common part.
A. 122 m³
B. 133 m³
C. 144 m³
D. 155 m³
1441. Which of the following statements is false ?
A. Any two integrals of a given function differ by a constant.
B. The integral of secnxdx where n is an odd integer requires integration by parts.
C. If f(x) is an even function, then the integral of f(x)dx from x = -a to x = a is equal
to zero.
D. The key connection between the derivative and integral is known as the fundamental
theorem of calculus.
1442. Which of the following differentials must be integrated by parts ?
A. (lnx/x)dx
B. sin²(3x)dx
C. x²cos(x³)dx
D. (lnx)²dx
1443. The process of finding the function f(x) whose differential f'(x)dx is given, is called
integration or
A. involution
B. evolution
C.antidifferentiation D. exponentiation
B. xex – 1 + c
C. ex – x + c
1444. Evaluate ∫xexdx
A. ex(x-1) + c
D. xex – x + c
1445. For some constant k, the antiderivative of xk is equal to
A. (xk+1)/(k+1)
B. [(xk+1)/(k+1)]+c C. [(x2k) / 2k] +c
D. A or B
1446. The mathematician who first give a modern definition of the definite integral is
A. Riemann
B. Leibniz
C. Newton
D. Gauss
1447. To integrate ∫(xdx) / (1+x4) by the u-substitution method, let u =
A. 1 + x²
B. x²
C. 1 + x4
D. x4
1448. Which of the following is correct ?
A. ∫cos2xdx = -sin2x + c
B. ∫sin2xdx = sin2x + c
C. ∫sin3xdx = [(sin4x) / 4] + c
D. ∫excosxdx = exsinx + c
1449. Who proved that the area under a parabolic arch is 2bh/3 where b is the width of the base
of the arch and h is the height ?
A. Wallis
B. Newton
C. Riemann
D. Archimedes
C. -20/3
D. -28/3
C. lncoshu + c
D. lncothu + c
1450. Evaluate ∫1-1(x2 – 4) dx
A. -25/3
B. -22/3
1451. The antiderivative of tanhudu is
A. lnsinhu + c
B. lnsechu + c
1452. using the theorem of Pappus, find the volume of the torus generated by revolving the area
of the circle x2 + y2 = a2 about the line x = b where b > a.
A. 2π a2b
B. 2π ab2
C. 2π2ab2
D. 2π2a2b
1453. If f(x) = x3 – 1and g(x) = x – 1, evaluate ∫10 [f(x) / g(x) ] dx.
A. 11/6
B. 13/6
C. 10/6
D. 14/6
1454. Find the area bounded by the curve y = e x , the lines x = -1, x = 1 and the x-axis.
A. 2.15
B. 2.25
C. 2.35
D. 2.45
1455. If the area bounded by y = x2 and y = 2 – x2 is revolved about the x-axis and a vertical
rectangular element is taken, the element of volume generated is a
A. disk
B. washer
C. shell
D. torus
1456. If ∫5-2 f(x)dx = 18, ∫5-2 g(x)dx = 5, and ∫5-2 h(x)dx = -11, evaluate∫ 5-2 [f(x)+g(x)h(x)]dx.
A. 32
B. 33
C. 34
D. 35
1457. Find the length of the curve y = coshx from x = -1 to x = 1.
A. 2.15
B. 2.25
C. 2.35
D. 2.45
1458. Evaluate ∫2-1 (2x-(2/x)+(x/2) dx
A. 2.3637
B. 2.3763
C. 2.3367
D. 2.6733
1459. If the area bounded by the parabola y = x2 and the line y = x is revolved about the x-axis,
the volume of the solid formed may be found by using which of the following methods ?
A. washer method only
B. washer or disk method
C. shell or washer method
D. shell or disk method
1460. The differential xnex^2dx is integrable if n is
A. an even integer
B. an odd integer
C. any positive integer
D. any whole number
1461. If a vertical element of area is used in finding the area bounded by the parabolas y = x2 – 7
and y = 1 – x2, then the elemental area dA =
A. (2x2 – 8)dx
B. (8 – 2x2)dx
C. (2x2 – 6)dx
D. (6 – 2x2)dx
C. pi/3
D. pi/4
1462. If ∫x0 sin2ycos2ydy = ¼, then x is equal to
A. pi/2
B. pi/6
1463. If the area bounded by the ellipse 9x2 + 4y2 = 36 is revolved about the line 2x + y = 8 and
a horizontal rectangular element is taken, the element of volume generated is a
A. washer or circular ring
B. cylindrical shell
C. circular disk
D. none of A, B or C
1464. Which of the following cannot be evaluated by the power rule formula ?
A. ∫ (dx) / x2 (1 + (2/x))3
B. ∫ (√1 + sinx)dx / (secx)
C. ∫ (ln(x+1)dx) / (x+1)
D. ∫ (x2 √x2 + 4 ) dx
1465. Evaluate ∫2π0∫ 10 rdrdθ
A. 3π/4
B. π/4
C. π/6
D. 2π/3
1466. Find the area bounded by y = x2, the x-axis and the lines x = 1, x = 3.
A. 26/3
B. 25/3
C. 23/3
D. 20/3
1467.Evaluate the integral of xsin(x2)dx from x = 0 to x = √π
A. -1
B. 0
C. -1/2
D. 1/3
1468. Find the volume of the solid generated by revolving about the x-axis, the area bounded by
y = x3, the x-axis and the line x = 1.
A.π/3
B. π/5
C. π/7
D. π/9
C. ¼
D. 1/5
1469. The integral of e4lnx dx from x = 0 to x = 1 is
A. ½
B. 1/3
1470. Evaluate ∫ sec2xtanx dx
A. ½ tan2 x + c
B. 1/3 sec3 x + c
C. ½ sec2 x + c
D. A or C
1471. To integrate ∫ x2 ex dx by parts, it is wise to choose u =
A. x
B. x2
C. ex
D. xex
1472. Evaluate ∫sin2 xdx
C. 1/3sin3 x + c
D. A or B
A. 1/2(x-sinxcosx) + c
B. ½ x - ¼ sin2x + c
1473. Evaluate ∫10