MATH PRE BOARD 2
SHOW YOUR SOLUTION
Write it in a long bond paper
1. What is the differential equation of the family of parabolas
having their
vertices at the origin and their foci on the x-axis?
A. 2xdy – ydx = 0
B. ydx + ydx = 0
C. 2ydx –xdy = 0
D. dy/dx – x = 0
2. Find the rthogonal trajectories of the family of parabolas y^2 =
2x + C.
A. y = Ce^x
B. y = Ce^(-x)
C. y = Ce^(2x)
D. y = Ce^(-2x)
3. A reflecting telescope has a parabolic mirror for which the
distance from the
vertex to the focus is 30 ft. If the distance across the top of the
mirror is 64 in.,
how deep is the mirror of the center?
A. 32/45 in.
B. 30/43 in.
C. 32/47 in.
D. 35/46 in.
4. Simplify (1 – tan2x) / (1 + tan2x)
A. sin 2x
B. cos 2x
C. sin x
D. cos x
5. Evaluate L { t^n }.
A. n!/s^n
B. n!/s^(n+1)
C. n!/s^(n-1)
D. n!/s^(n+2)
6. Simplify 12 cis 45 deg + 3 cis 15 deg.
A. 2 + j
B. sqrt. of 3 + j2
C. 2 sqrt. Of 3 + j2
D. 1 + j2
A. 9/2
B. π
C. ∞
D. -∞
8. Find the area of the lemniscate r2 = a2cos2θ
A. a2
B. a
C. 2a
D. a3
9. Find the area bounded by the parabola sqrt. of x + sqrt. of y =
sqrt. of a and
the line x + y = a.
A. a2
B. a2 /2
C. a2 /4
D. a2 /3
10. Ben is two years away from being twice Ellen’s age. The sum
of twice Ben’s age and thrice Ellen’s age is 66. Find Ben’s age
now.
A. 19
B. 20
C. 16
D. 21
11. What percentage of the volume of a cone is the maximum
volume right
circular cylinder that can be inscribed in it?
A. 24%
B. 34%
C. 44%
D. 54%
12. A balloon rising vertically, 150 m from an observer. At exactly 1
min, the
angle of elevation is 29 deg 28 min. How fast is the balloon using
at that
instant?
A. 104m/min
B. 102m/min
C. 106m/min
D. 108m/min
13. A conic section whose eccentricity is less than one (1) is
known as:
A. a parabola
B. an ellipse
C. a circle
D. a hyperbola
14. A tangent to a conic is a line
A. which is parallel to the normal
B. which touches the conic at only one point
C. which passes inside the conic
D. all of the above
15. A die and a coin are tossed. What is the probability that a three
and a head will appear?
A. 1/4
B. 1/2
C. 2/3
D.1/12
16. Find the integral of 12sin5xcos5xdx if lower limit = 0 and upper
limit = pi/2.
A. 0.8
B.0.6
C.0.2
D.0.4
17. 12 oz of chocolate is added to 10 oz of flavoring is equivalent
to
A.1 lb and 8 oz
B. 1 lb and 6 oz
C.1 lb and 4 oz
D.1 lb and 10 oz
18. The Ford company increased its assets price from 22 to 29
pesos. What is
the percentage of increase?
A.24.14%
B.31.82%
C.41.24%
D.28.31%
19. Find the area bounded by outside the first curve and inside the
second
curve, r = 5, r = 10sinθ
A. 47.83
B.34.68
C.73.68
D.54.25
20. In two intersecting lines, the angles opposite to each other are
termed as:
A. opposite angles
C. horizontal angles
B. vertical angles
D. inscribed angles
21. 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. 2932 c.u.
B. 2392 c.u.
C. 2229 c.u.
D. 2292 c.u.
22. A cardboard 20 in x 20 in is to be formed into a box by cutting
four equal
squares and folding the edges. Find the volume of the largest box.
A.592 cu.in.
B.529 cu.in.
C.696 cu.in.
D.689 cu.in.
23. A retailer bought a number of ball pens for P90 and sold all but
3 at a profit P2 per ball pen. With the total amount received she
could buy 15 more ball pens than before. Find the cost per ball pen.
A. P2
B. P3
C.P4
D.P5
24. What is –i^i?
A.4.81
B.-4.81
C.0.21
D.-0.21
25. A balloon travel upwards 6m, North and 8m, East. What is the
distance
traveled from the starting point?
A. 7
B. 10
C.14
D. 20
26. What do you call the integral divided by the difference of the
abscissa?
A. average value
C. abscissa value
B. mean value
D. integral value
27. Water is running out of a conical funnel at the rate of 1 cubic
inch per sec.
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/pi in./sec
B. -2/pi in./sec
B. -1/9pi in./sec
C. D.-2/9pi in./sec
28. How many inches is 4 feet?
A. 36
B. 48
C. 12
D. 56
29. A rectangular trough is 8 ft. long, 2 ft. across the top, and 4 ft.
deep. If water flows in at a rate of 2 cu. ft./min., how fast is the
surface rising when the water
is 1 ft. deep?
A. 1/5 ft./min
B. 1/8 ft./min
C. 1/6 ft./min
D. 1/16 ft./min
30. Five tables and eight chairs cost $115; three tables and five
chairs cost
$70. Determine the total cost of each table.
A. $15
B. $30
C. $25
D. $20
31. Find the 16th term of the arithmetic sequence; 4, 7, 10,……..
A. 47
B. 46
C. 49
D. 48
32. Find the slope of the line through the points (-2, 5) and (7, 1).
A. 9/4
B. -9/4
C. 4/9
D. -4/9
33. For what value of k will the line kx +5y = 2k have a y
-intercept 4?
A. 8
B. 7
C. 9
D.10
34. If a bug moves a distance of 3pi cm along a circular arc and if
this arc
subtends a central angle of 45 degrees, what is the radius of the
circle?
A. 8
B. 12
C. 14
D. 16
35. Two vertices of a rectangle are on the positive x-axis. The
other two
vertices are on the lines y = 4x and y = -5x + 6. What is the
maximum possible
area of the rectangle?
A.2/5
B.5/2
C.5/4
D. 4/5
36. Find the length of the arc of 6xy = x^4 + 3 from x = 1 to x = 2.
A.12/17
B.17/12
C.10/17
D.17/10
37. A certain radioactive substance has half-life of 3 years. If 10
grams are
present initially, how much of the substance remain after 9 years?
A.2.50g
B.5.20g
C. 1.25g
D.10.20g
38. A cubical box is to built so that it holds 125 cu. cm. How
precisely should
the edge be made so that the volume will be correct to within 3 cu.
cm.?
A.0.02
B.0.03
C.0.01
D.0.04
39.
Find the eccentricity of the ellipse when the length of its latus
rectum is 2/3
of the length of its major axis.
A.0.62
B. 0.64
C.0.58
D.0.56
40. Find k so that A = <3, -2> and B =<1, k> are perpendicular.
A. 2/3
B.3/2
C.5/3
D.3/5
41. 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
42. Find the force on one face of a right triangle of sides 4m and
altitude of 3m. The altitude is submerged vertically with the 4m
side in the surface.
A.62.64 kN
B.58.86 kN
C.66.27 kN
D.53.22 kN
43. In how many ways can 6 people be seated in a row of 9 seats?
A. 30,240
B. 30,420
C.60,840
D. 60,480
44. The arc of a sector is 9 units and its radius is 3 units. What is
the area of
the sector?
A.12.5
B.13.5
C.14.5
D.15.5
45. The sides of a triangle are 195, 157, and 210, respectively.
What is the area of the triangle?
A.73,250
B.10,250
C.14,586
D.11,260
46. A box contains 9 red balls and 6 blue balls. If two balls are
drawn in
succession, what is the probability that one of them is red and the
other is
blue?
A.18/35
B.18/37
C.16/35
D.16/37
47. A car goes 14 kph faster than a truck and requires 2 hours and
20 minutes
less time to travel 300 km. Find the rate of the car.
A.40 kph
B.50 kph
C.60 kph
D.70 kph
48. Find the slope of the line defined by y – x = 5.
A.1
B.1/4
C.-1/2
D.5
49. The probability of John’s winning whenever he plays a certain
game is 1/3.
If he plays 4 times, find the probability that he wins just twice.
A.0.2963
B.0.2936
C.0.2693
D.0.2639
50. A man row upstream and back in 12 hours. If the rate of the
current is 1.5
kph and that of the man in still water is 4 kph, what was the time
spent
downstream?
A.1.75 hr
B.2.75 hr
C.3.75 hr
D. 4.75 hr
51. If cot A = -24/7 and A is in the 2nd quadrant, find sin 2A.
A.336/625
B.-336/625
C.363/625
D. -363/625
52. The volume of a square pyramid is 384 cu. cm. Its altitude is 8
cm. How
long is an edge of the base?
A.11
B.12
C.13
D.14
53. The radius of the circle x^2 + y^2
– 6x + 4y – 3 = 0 is
A.3
B.4
C.5
D.6
54. If the planes 5x – 6y - 7z = 0 and 3nx + 2y – mz +1 = 0
A.-2/3
B. -4/3
C.-5/3
D.-7/3
55. If the equation of the directrix of the parabola is x – 5 = 0 and
its focus is at (1, 0), find the length of its latus rectum.
A.6
B.8
C.10
D.12
56. If tan A = 1/3 and cot B = 4, find tan (A + B).
A. 11/7
B. 7/11
C. 7/12
D. 12/7
57. A club of 40 executives, 33 like to smoke Marlboro, and 20 like
to smoke
Philip Morris. How many like both?
A. 13
B. 10
C. 11
D. 12
58. The area of the rhombus is 264 sq. cm. If one of the diagonals
is 24 cm
long, find the length of the other diagonal.
A. 22
B. 20
C. 26
D. 28
59. How many sides have a polygon if the sum of the interior
angles is 1080
degrees?
A. 5
B. 6
C. 7
D. 8
60. The line segment connecting (x, 6) and (9, y) is bisected by the
point (7, 3).
Find the value of x and y.
A. 5, 0
B. 4, 0
C. 5, 2
D.4,1
61. What is the height of the parabolic arch which has span of 48 ft.
and having a height of 20 ft. at a distance of 16 ft. from the center
of the span?
A. 30 ft.
B. 40 ft.
C. 36 ft.
D.34ft.
62. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x –
By + 2 =0.
A. 2
B. 3
C. 4
D.5
63. The value of x + y in the expression 3 + xi = y + 2i is;
A. 5
B. 1
C. 2
D.3
64. If sin3A = cos6B then:
A. A + B = 180 deg
C. A - 2B = 30 deg
B. A + 2B = 30 deg
D. A + B = 30 deg
65. What is the area between y = 0, y = 3x^2, and x = 2?
A. 8
B. 12
C. 24
D.6
66. The volume of the sphere is 36pi cu. m. The surface area of
this sphere in
sq. m is:
A. 36pi
B. 24pi
C. 18pi
D. 12pi
67. The vertex of the parabola y^2 – 2x + 6y + 3 = 0 is at:
A. (-3, 3)
B. (3, 3)
C. (3, -3)
D. (-3, -3)
68. Add the following and express in meters: 3 m + 2 cm + 70 mm
A. 2.90 m
B. 3.14 m
C. 3.12 m
D.3.09m
69. A store advertised on sale at 20 percent off. The sale price was
$76. What
was the original price? A. $95
B. $96
C. $97
D.$98
70. Find the equation of the straight line which passes through the
point (6, -3)
and with an angle of inclination of 45 degrees.
A. x + y = 8
B. x – y = 8
C. x + y = 9
D. x – y = 9
71. A freight train starts from Los Angeles and heads for Chicago
at 40 mph.
Two hours later a passenger train leaves the same station for
Chicago
traveling at 60 mph. How long will it be before the passenger train
overtakes
the freight train?
A. 3 hrs.
B. 5 hrs.
C. 4 hrs.
D. 6 hrs.
72. The number of board feet in a plank 3 inches thick, 1 ft. wide,
and 20 ft.
long is:
A. 30
B. 60
C. 120
D. 90
73. Boyles’s law states that when a gas is compressed at constant
temperature, the product of its pressure and volume remains
constant. If the pressure gas is 80 lb/sq.in. when the volume is 40
cu.in., find the rate of change of pressure with respect to volume
when the volume is 20 cu.in.
A. -8
B. -10
C. -6
D.-9
74. Find the average rate of change of the area of a square with
respect to its
side x as x changes from 4 to 7.
A. 8
B. 11
C. 6
D. 21
75. How many cubic feet is equivalent to 100 gallons of water?
A. 74.80
B. 1.337
C. 13.37
D. 133.7
76. A merchant purchased two lots of shoes. One lot he purchased
for $32 per pair and the second lot he purchased for $40 per pair.
There were 50 pairs in the first lot. How many pairs in the second
lot if he sold them all at $60 per pair and made a gain of $2800 on
the entire transaction?
A. 50
B. 40
C. 70
D. 60
77. The diagonal of a face of a cube is 10 ft. The total area of the
cube is
A. 300 sq. ft.
B. 150 sq. ft.
C. 100 sq. ft.
D. 200 sq. ft.
78. A ship is sailing due east when a light is observed bearing N
62 deg 10 min E. After the ship has traveled 2250 m, the light
bears N 48 deg 25 min E. If the course is continued, how close will
the ship approach the light?
A. 2394 m
B. 2934 m
C. 2863 m
D. 1683 m
79. If f(x) = 1/(x – 2), (f g)’(1) = 6 and g’(1) = -1, then g(1) =
A.-7
B. -5
C. 5
D. 7
80. Find the work done by the force F = 3i + 10j newtons in moving
an object
10 meters north.
A.104 40 J
B. 100 J
C.106 J
D. 108.60 J
81. The volume of a frustum of a cone is 1176pi cu.m. If the radius
of the lower base is 10m and the altitude is 18m, compute the
lateral area of the frustum of a cone
A.295pi sq. m.
B. 691pi sq. m.
C.194pi sq. m.
D. 209pi sq. m.
82. In an ellipse, a chord which contains a focus and is in a line
perpendicular
to the major axis is a:
A.latus rectum
B. minor axis
C. focal width
D. major axis
83. With 17 consonant and 5 vowels, how many words of four
letters can be
four letters can be formed having 2 different vowels in the middle
and 1
consonant (repeated or different) at each end?
A.5780
B. 5785
C. 5790
D. 5795
84. Evaluate tan2 (j0.78).
A.0.653
B.-0.653
C.0.426
D. -0.426
85. A particle moves along a line with velocity v = 3t^2 – 6t. The
total distance
traveled from t = 0 to t = 3 equals
A.8
B. 4
C. 2
D. 16
86. An observer at sea is 30 ft. above the surface of the water.
How much of
the ocean can he sea?
A.124.60 sq. mi.
C. 154.90 sq. mi.
B.142.80 sq. mi.
E. 132.70 sq. mi
87. There are three consecutive integers. The sum of the smallest
and the
largest is 36. Find the largest number.
A.17
B. 18
C.19
D. 20
88. If y = sqrt. of (3 – 2x), find y.
A.1/sqrt. of (3 – 2x)
C. 2/sqrt. of (3 – 2x)
B. -1/sqrt. of (3 – 2x)
D. -2/sqrt. of (3 – 2x)
89. The logarithm of MN is 6 and the logarithm of N/M is 2, find the
value of
logarithm of N.
A.3
B. 4
C. 5
D.6
90. A woman is paid $20 for each day she works and forfeits $5 for
each day
she is idle. At the end of 25 days she nets $450. How many days
did she work?
A.21 days
B. 22 days
C. 23 days
D.24 days
91. Francis runs 600 yards in one minute. What is his rate in feet
per second?
A.25
B. 30
C.35
D.40
92. For a complex number z = 3 + j4 the modulus is:
A.3
B. 4
C. 5
D. 6
93. Which of the following is an exact DE?
A. (x^2 + 1)dx – xydy = 0
C. 2xydx + (2 + x^2)dy = 0
B. xdy + (3x – 2y)dy = 0
D. x^2 ydy – ydx = 0
94. There are 8 different colors, 3 of which are red, blue and green.
In how
many ways can 5 colors be selected out of the 8 colors if red and
blue are
always included but green is excluded?
A.12
B.11
C. 10
D.9
95. Five cards are drawn from a pack of 52 well – shuffled cards.
Find the
probability that 3 are 10’s and 2 are queens.
A. 1/32
B. 1/108,290
C. 1/54,350
D.1/649,740
96. From past experience, it is known 90% of one year old children
can distinguish their mother’s voice from the voice of a similar
sounding female. A random sample of 20 one year’s old are given
this voice recognize test. Find the probability that all 20 children
recognize their mother’s voice.
A. 0.122
B. 1.500
C. 1.200
D. 0.222
97. When the ellipse is rotated about its longer axis, the ellipsoid is
A. spheroid
B. oblate
C. prolate
D. paraboloid
98. If the distance between points A(2, 10, 4) and B(8, 3, z) is
9.434, what is the value of z?
A. 4
B. 3
C. 6
D. 5
99. A line with equation y = mx + b passes through (-1/3, -6) and (2,
1). Find
the value of m.
A. 1
B. 3
C. 4
D. 2
100. For the formula R = E/C, find the maximum error if C = 20
with possible error 0.1 and E = 120 with a possible error of 0.05.
A. 0.0325
B. 0.0275
C. 0.0235
D. 0.0572