Refresher Exam
MATHEMATICS
Name:______________________________________
Direction: Solve the following problem and select the letter of the best answer.
1. An isosceles right triangle has its perimeter 10.2426 ft. What is the area of the triangle?
A. 2 ft2
C. 3 ft2
2
B. 4.5 ft
D. 4 ft2
2. The following are parts of a triangle A = 30 deg, a = 39.167 cm, b = 60 cm. The perimeter to the nearest cm is
A. 186
C. 150
B. 176
D. 134
3. Immediately after Mt. Pinatubo showed telltale signs of activity, the PHILVOCS setup stations to monitor the volcano. Two such
stations were located at points A and B 7 kilometers apart on the same horizontal plane as the base of the volcano, B being closer
to the volcano. From A, the angle of elevation of the top of Mt. Pinatubo is 80. At the height of Mt Pinatubo’s wrath, the vertical
smoke emitted from the volcano’s crater subtended an angle of 640 on each station A and B. Assuming the smoke and the two
stations to be on the same vertical plane, find the height of Mount Pinatubo and the height of the smoke at that instant.
A. 1.56 km, 43.81 km
C. 1.18 km, 28.43 km
B. 2.31 km. 26.35 km
D.1.73 km, 36.23 km
4. An equilateral triangle is inscribed a circle of radius 10m, determine the length of the side of the triangle
A. 14.14
C. 5
B. 17.32
D. 8.66
5. What is the value of x in trigonometric substitution with an integrand involving (x2 - a2 )?
A. a sec θ
C. a sin θ
B. a tan θ
D. a cos θ
6. Find the derivative of cosh3 4x
A. -12 cosh2 4x sinh 4x
C. 3 cosh 2 4 x sinh 4x
2
B. 4 cosh 4x sinh 4x
D. 12 cosh2 4x sinh 4x
7. A circle is inscribed in a 3-4-5 right triangle. How long is the line segment joining the points of tangency of the “3–side” and the “5–
side”?
A. 1.28
C. 1.46
B. 1.35
D. 1.79
8. If I just sold my two books at P990 each. On one I gained 10% and on the other one I lost 10%. Then,
A. I lost P20
C. I gained P19.8
B. I gained P198
D. none of these
9. Find the remainder when (x12 + 2 ) is divided by (x – sq. rt. of 3).
A. 652
C. 231
B. 851
D. 731
10. If the polynomial (x cube) + 4(x square) – 3x + 8 is divided by x – 5, the remainder is
A. 175
C. 140
B. 218
D. 200
11. A job can be done in as many days as there are men in the group. If the number of men is reduced by 3, the job will be delayed by
4 days. How many men are there originally in the group?
A. 6
C. 20
B. 12
D. 30
12. Three numbers are in the ratio 2:5:8. If their sum is 60, find the smallest of the numbers.
A. 8
C. 6
B. 4
D. 12
13. Given the three vertices of a parallelogram (1, 3), (0, 0), and (4, 0). Find the possible location/coordinate of the fourth vertex.
A. (– 3, 3)
C. (4, – 3)
B. (5, 4)
D. (4, 5)
14. The following are the vertices of a triangle, (5, 4), (2, – 3), (– 2, 1). Find the area of the triangle.
A. 20
C. 22
B. 26
D. 28
15. A father and his son can dig a well if the father works 6 hours and his son works for 12 hours or they can do the job if the father
works 9 hours and the son works 8 hours. How long will it take for the son to dig the well alone?
A. 60
C. 20
B. 40
D. 30
16. Find the angle of rotation of the curve given the equation xy – 3x – 2y + 5 = 0.
A. 30
C. 60
B. 45
D. 50
17. The equation r = a is the polar equation of a
A. line
C. circle
B. hyperbola
D. ellipse
18. Find the radius of the sphere whose equation is x2+ y2 + z2 = 6 x + 8 z.
A. 5
C. 9
B. 2
D. 4
19. Find the equation of the circle passing thru points (0, 4) and (3, 7) and having its center on the line 2x – y + 4 = 0.
A . ( x – 7)2+ ( y – 3)2 = 8 C. ( x – 4)2 + ( y – 5)2 = 6
B. ( x – 2)2 + ( y – 3)2 = 7 D. ( x – 1)2 + ( y – 6)2 = 5
20. Find the sum of all odd numbers between 100 and 1,000
A. 247,500
C. 248,600
B. 249,700
D. 246,400
21. The general second degree equation has the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. If B2– 4AC = 0, the equation describes
A. a circle
C. an ellipse
B. a parabola
D. a hyperbola
22. Determine the value of B when 3x + 3y – 7 = 0 is perpendicular to the line 2x + By – C = 0
A. – 2
C. – 4
B. – 3
D. – 5
23. Find the coordinates of the centroid of the area enclosed by y = x – x2 and the y = 0 about the x-axis.
A. (1/2, 3/7)
C. (1/2,1/10)
B. (1/3, 2/5)
D. (1/3, 2/3)
24. Find the volume generated by revolving the area bounded by y = x3, y = 8, x = 0 about the y–axis.
A. 64pi/7
C. 768pi/7
B. 512pi/5
D. 96pi/5
25. Find the area enclosed by the curve y2 = x2 – x4
A. 3/8 sq. units
C. 4/5 sq. units
B. 2/3 sq. units
D. 4/3 sq. units
26. Find the ordinate of the centroid of a plane area bounded by the parabola y = 4 –x 2 and the x-axis.
A. 0.247
C. 2.80
B. 3.552
D. 1.60
27. A bicycle travels along a straight road. At t o’clock, it is (t squared) miles from the end of the end of the road. Compute its average
velocity in mi/hr from 1:00 to 4:00.
A. 4
C. 5
B. 3
D. 2
28. A billboard 60 ft high is mounted on a 20 ft building. How far should a man stand from the building so he can have the
best view of the billboard?.
A. 40 ft
C. 35 ft
B. 50 ft
D. 25 ft
29. A card is drawn from an ordinary deck of 52 cards. Find the probability that the card is a face card and heart..
A. 11/26
C. 3/18
B. 3/52
D. ¼
30. What is the value of x in trigonometric substitution with an integrand involving ( a 2 – x2 )?
A. asec θ
C. a sin θ
B. atan θ
D. a cos θ
31. A tetrahedron is a regular solid with equilateral triangles for each of the 3 surfaces. If each side is 10 cm, What is the volume of
the tetrahedron?
A. 107.21 cu cm
C. 117.85 cu cm
B. 119.67 cu cm
D. 183.33 cu cm
32. A closed rectangular r box is 6cm by 8cm by 10cm is painted 0.1 cm thick. Find the cc of paint used up.
A. 4.1
C. 18.8
B. 37.6
D. 9.4
33. Find the principal value of the complex expression (1 + j2) raise to the power of (3+j4)
A. 0.134 + j0.034
C. 0.129 + j0.34
B. -2.013 + j0.129
D. 2.236 – j0.034
34. Evaluate the value of log (-5)
A. 5 + j 1.364
C. 0.7 + j1.364
B. 15.71 + j0.434
D. 1.196 + j.0434
35. Evaluate cos (0.492 + j0.942)
A. -1.032 + j0.541
C. 3.12 + j1.54
B. 1.302 – j0.514
D. 1.48 + j0.01
36. Write in the form a + bi the expression i 321- i 742 + i720
A. 2i + 1
C. 2 + i
B. - I + 1
D. 1 + I
37. Three sides of a trapezoid are each 8cm long. How long is the fourth side when the area of the trapezoid has the greatest value
A. 10cm
C. 15 cm
B. 12 cm
D. 16 cm
38. Four different coloured flags can be hung in a row to make a coded signal. How many signals can be made if a signal consists of
the display of one or more flags?
A. 16
C. 24
B. 15
D. 1
39. The position of the particle in inches along the x-axis is given by the expression x (t) = 24t2 – t3 + 10. Determine the average
velocity of the particle in inches per sec at t = 3 sec.
A. 63
C. 72
B. 36
D. 54
40. Determine the divergence of the vector
V = (x2y) i + (-xy) j + (xyz) k coordinates (3,2,1)
A. 9
C. 7
B. 11
D. 15
41. Convert the spherical -coordinate equation θ = 60o to is rectangular-coordinate equation
A. x2 + y2 = z5/4
C. x2 + y2 = 3z3
2
2
4
B. x + y = z /3
D. x2 + y2 = 3z2
42. The Laplace transform of [ 1 – exponent (-at)] / a is …............
A. 1/s(s+ a)
C. ½ (s-a)
B. 1/[(s square ) + (a square) D. 1/(s+a) square
43. The Laplace transform of of cos ώt is:
A. s/ [(s square ) + (w square )]
C. w/ / (s+w)
B. w/ [ (square) + ((w square )]
D. s/ (s + w)
44. Simply (3 – i) 2 – 7 (3-i)+ 10
A. - (3 +1)
C. (3-1)
B. (3 +1)
D. -(3-1)
45. What is the simplified form complex expression of (4.33 + j2.5) square?
A. 12.5 + j21.65
C. 15 + j 20
B. 20 + j 20
D. 21.65 + j 12.5
46. Find the volume generated by revolving a rectangle of sides ‘a’ and ‘b’ about the side a.
A. πab2
C. πba2
2
B. πab /2
D. πba2/2
47. Given vector A = 4j + 10k and vector B = 2i + 3j, Find the projection of A on B
A. 3.328
C. 3.528
B. 3.428
D. 3.628
48. The distance of a body travels is a function of time, t and is defined by x(t) = 18t + 9t 2. What is the velocity at t = 3 sec?
A. 18
C. 54
B. 36
D. 72
49. A weight is attached to one end of a 29 ft rope which passes over a pulley 17 ft above the ground. A man keeping his hand at 5 ft
above the ground grasp the other end and walks away at the rate of 3 ft/s. Find the speed of the weight rising when the man is 9 ft
from the point directly below the pulley?
A. 1.4 fps
C. 1.7 fps
B. 1.5 fps
D. 1.8 fps
50. The probabilities that Gary will win the preliminary, semi-final, and final contest in singing are 3/8, 1/6, and 1/12, respectively.
Failure in any one contest prohibits participation in the following one. Find the probability that he will reach the final contest.
A. 1/12
C. 1/16
B. 13/24
D. 1/192