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x x 20. What do you call a sequence of numbers where each of which,
1. The function f(x) = − is
1− 2x 2 after the first, is obtained by adding to the proceeding number a
A. an even but not odd function* constant number called common difference?
B. an odd but not even function A. Arithmetic Sequence* C. Geometric Sequence
C. a both even and odd function B. Harmonic Sequence D. Fibonacci Sequence
D. a neither even nor odd function 21. If you buy one ticket in the Provincial Lottery, then the probability
2. State the domain and the range of the function y=-2√(x-3). that you will win a prize is 0.11. If you buy one ticket each month
A. D=[3,∞); R=(-∞,3] C. D=[3,∞); R=(-∞,0]* for five months, what is the probability that you will win at least
B. D=[0,∞); R=(-∞,0) D. D=[3,∞); R=(-∞,-3] one prize?
A. 0.55 C. 0.44*
3. Determine the domain of the function f(x) = 5 + 9 − x 2 . B. 0.45 D. 0.56
A. [-3,3]* C. (3,3) 22. The probability of a defect of a collection of bolts is 5%. If a man
B. -3<x<3 D. −3 < x ≤ 3 picks 2 bolts, what is the probability that he doe not pick 2
defective bolts?
4. Determine the range of the function f(x) = 5 + 9 − x 2 . A. 0.950 C. 0.0025
A. y | 5 ≤ y ≤ 8 * C. y | y ≥ 5 B. 0.9975* D. 0.9025
B. y | 0 ≤ y ≤ 8 D. y | y ≥ 8 23. If the probability that Niko will be alive in 20 years is 0.7 and the
probability that Jia will be alive in 20 years is 0.9, what is the
5. Find the horizontal asymptote of the function f(x)=3/(x2-1).
probability that neither will be alive in 20 years?
A. y=0 (x-axis)* C. y=-1
A. 0.03* C. 0.06
B. y=1 D. y=3 B. 0.02 D. 0.01

6. Find the horizontal asymptote of the function f(x)=(2x+1)/(x+3). 24. Suppose a die is loaded so that a “1” face lands uppermost 3
A. y=3 C. y=2* times as often as any other face while all the other faces occur
B. y=-4 D. y=1 equally often. What is the probability of a “1” on a single toss?
A. 7/8 C. 3/8*
7. Linda can mow a certain lawn with her riding lawn mower in 4 B. 1/8 D. 1/6
hours. When Linda uses the riding mower and Rebecca operates
the push mower, it takes them 3 hours to mow the lawn. How 25. A committee is composed of six Democrats and five
long would it take Rebecca to mow the lawn by herself using the Republicans. Three of the Democrats are men, and 3 of the
push mower? Republicans are men. If a man is selected for chairman, what is
A. 10 hrs. C. 15 hrs. the probability that he is a Republican?
B. 12 hrs.* D. 8 hrs. A. 2/3 C. 5/11*
B. 1/2 D. 3/5
8. Which of the following notations represents a singleton set?
26. Find the probability that a number less than 5 will result in a
A. {} C. { ∅ }*
single toss of a die given that the toss resulted in an even
B. {1,1,1,1} D. {x,y,z}
number.
A. 1/2 C. 3/4
9. Which of the following notations represents an empty set?
B. 2/3* D. 3/5
A. ∅ * C. { ∅ }
27. Three machines X, Y and Z produce respectively 50%, 30% and
B. {1} D. {0}
20% of the total number of items of a factory. The percentages of
10. If two sets A and B are disjoint, then A ∩B is defective output of these machines are respectively 3%, 4% and
A. {0} C. {}* 5%. If an item is selected at random, what is the probability that
B. {1} D. {0,1} the item is defective?
A. 1.3% C. 2.5%
11. What is the cardinality of the set of positive odd integers less B. 3.7%* D. 4.3%
than 10?
A. 5* C. 7 28. Box 1 contains 30 red and 70 white balls, box 2 contains 50 red
B. 4 D. 10 and 50 white balls, and box 3 contains 75 red and 25 white balls.
The three boxes are all emptied into a large box, and a ball is
12. What is the Cardinality of the Power set of the set {0, 1, 2}. selected at random. If the selected ball is red, what is the
A. 8* C. 6 probability that it came from box 1?
B. 7 D. 9 A. 0.193* C. 0.122
13. What is the Cartesian product of A = {1, 2} and B = {a, b}? B. 0.256 D. 0.109
A. {(1, a), (1, b), (2, a), (b, b)} 29. If the probability that a basketball player sinks the basket at 3-
B. {(1, 1), (2, 2), (a, a), (b, b)} point range is 2/5, determine the probability of shooting 5 out of 8
C. {(1, a), (2, a), (1, b), (2, b)}* attempts.
D. {(1, 1), (a, a), (2, a), (1, b)} A. 12.4%* C. 25.4%
14. What is the Cartesian product of A = {1, 2} and B = {a, b}? B. 10.4% D. 16.8%
A. {(1, a), (1, b), (2, a), (b, b)} C. {(1, 1), (2, 2), (a, a), (b, b)} 30. A man is dealt a poker hand (5 cards) from an ordinary deck of
B. {(1, a), (2, a), (1, b), (2, b)}* D. {(1, 1), (a, a), (2, a), (1, b)} playing cards. In how many ways can he be dealt with a straight
15. What is the Cardinality of the Power set of the set {0, 1, 2}. flush?
A. 8 C. 6 A. 36 C. 27
B. 7 D. 9 B. 40* D. 45

16. Which of the following relationships represents the dual of the 31. How many answer patterns are there in a standard multiple
Boolean property x + x'y = x + y? choice exam with 10 items?
A. x’(x+y’)=x’y’ C. x*x’+y=xy A. 2,405,657 C. 1,230,872
B. x(x’+y)=xy* D. x’(xy’)=x’y’ B. 1,048,576* D. 2,251,200

17. Determine the dual of the Boolean (a+b)ab. 32. A student is to answer 8 out of 10 questions on an exam. How
A. a2b+b2a C. ab+a+b* many choices does he have if he must answer at least 4 of the
B. (ab)(a+b) D. a+b(ab) first 5questions?
A. 40 C. 30
18. How many answer patterns are there in a standard multiple B. 35* D. 45
choice exam with 10 items?
A. 2,405,657 C. 1,230,872 33. If the odds against an event occurring are 8:3, what are the odds
B. 1,048,576* D. 2,251,200 in favor of the event occurring?
A. 3:8* C. 11:3
19. A student is to answer 8 out of 10 questions on an exam. How B. 11:8 D. 8:5
many choices does he have if he must answer at least 4 of the
first 5questions? 34. For a complex number z = 3 + j4 the modulus is
A. 40 C. 30 A. 3 C. 4
B. 35* D. 45 B. 5 * D. 6

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35. Find the mean, median and mode respectively of the following 52. A rope attached to a boat is being pulled in at a rate of 10 ft/s. If
numbers: 13, 13, 14, 12, 11, 10, 9, 11, 8, 11, 5, and 15. the water is 20 ft below the level at which the rope is being
A. 10, 10, 11 C. 10, 11, 11 drawn in, how fast is the boat approaching the wharf when 36 ft
B. 10, 11, 12 D. 11, 11, 11 of rope are yet to be pulled in?
A. 12.02 ft/s* C. 1.202 ft/s
36. Refer to the following set of numbers: -3,1,2,2,4,4,4,8,23. The
B. 21.20 ft/s D. 22.01 ft/s
mode is _____ and the variance is ______.
A. 4, 48.2* C. 1, 28.2 53. Sand is pouring to form a conical pile such that its altitude is
B. 3, 24.8 D. 8, 48.2 always twice its radius. If the volume of a conical pile is
increasing at the rate of 25 pi cu. Ft./min, how fast is the radius is
37. A wall 8 feet high is 3.375 feet from a house. Find the shortest
increasing when the radius is 5 feet?
ladder that will reach from the ground to the house when leaning
A. 0.5 ft/min* C. 5 ft/min
over the wall.
B. 0.5pi ft/min D. 5pi ft/min
A. 16.525 ft C. 14.625 ft
B. 15.625 ft* D. 17.525 ft 54. A coat of paint of thickness 0.01 inch is applied to the faces of a
cube whose edge is 10 inches, thereby producing a slightly
38. A tree is broken over by a windstorm. The tree was 90 feet high
larger cube. Estimate the number of cubic inches of paint used.
and the top of the tree is 25 feet from the foot of the tree. What is
A. 4 C. 3
the height of the standing part of the tree?
B. 6* D. 5
A. 48.47 ft C. 41.53 ft *
B. 45.69 ft D. 44.31 ft 55. The position of a particle moving along a straight line at any time
t is given by s(t)=2t3-4t2+2t-1. What is the acceleration of the
39. An observer wishes to determine the height of the tower. He
particle when t=2?
takes the sights at the top of the tower from A and B, which are
A. 32 C. 16*
50 ft apart, at the same elevation on a direct line with the tower.
B. 8 D. 4
The vertical angle at point A is 30 degrees and at point B is 40
degrees. What is the height of the tower? cos2 x − 1
56. Evaluate: lim .
A. 85.60 ft C. 110.29 ft x→0 2x sin x
B. 143.97 ft D. 92.54 ft* A. -1 C. -1/2*
40. What conic section is represented by the equation B. 2 D. 0
3x 2 + 2y 2 + 6xy − 4y = 10 ? 57. For what value of x will the tangent lines to y=lnx and y=2x2 be
A. Parabola C. Ellipse parallel?
B. Hyperbola* D. Plane A. 0 C. 1/4
B. 1/2* D. 2
41. If the vertex of y = 2x^2 + 4x + 5 will be shifted 3 units to the left
and 2 units downward, what will be the new location of the 58. Let f(x) = ebx , g(x) = eax . Find the value of b such that
vertex?
A. (-2,1) C. (-3, 1) ⎛ f(x) ⎞ f '(x)
Dx ⎜ = .
B. (-5, -1) D. (-4, 1) ⎝ g(x) ⎟⎠ g'(x)
42. What is the equation of the line, in the xy-plane, passing through a2 a2
the point (6,4) and parallel to the line with parametric equations x A. C.
a2 − 1 a2 + 1
= 5t + 4 and y = t – 7?
A. 5y – x = 14 * C. 5x – y = 26 a2 a −1
B. * C.
B. 5y – 4x = -4 D. 5x – 4y = 14 a −1 a2
43. The equations fro two lines are 3y – 2x = 6 and 3x + ky = -7. For 59. A 20-lb weight is being raised by a 100-ft rope weighing 1/2 lb/ft.
what value of k will the two lines be parallel? Determine the work needed to raise the weight 50 ft.
A. -9/2 * C. 9/2 A. 2,875 ft-lb* C. 8,275 ft-lb
B. -7/3 D. 7/3 B. 5,875 ft-lb D. 7,875 ft-lb
44. Find the distance from the plane 2x + y – 2z + 8 = 0 to the point 60. If a spring is stretched 1/2 inch when a 2 pound force is applied
(-1,2,3) to it, how much work is done in stretching the spring and
A. 1/3 C. 2/3 * additional 1/2 in?
B. 4/3 D. 5/3 A. 2/3 inch-pound C. 5/3 inch-pound
45. What value of c in the open interval (0,4) satisfies the Mean B. 2.5 inch-pound D. 3/2 inch-pound*
Value Theorem for f(x) = 3x + 4 ?. 61. A vertical floodgate in the form of a rectangle 6 feet long in the
A. 0 C. 3/5 horizontal dimension and 4 feet deep has its upper edge 2 feet
B. 5/3* D. 2 below the surface of the water. Find the force, which it must
withstand?
46. If the average value of f(x) =x3 +bx- 2 on [0,2] is 4,find b. A. 4900 lbs C. 6000 lbs*
A. 3 C. 2 B. 3800 lbs D. 8900 lbs
B. 5 D. 4*
62. The area enclosed by the ellipse 4x^2 + 9y^2 = 36 is revolved
47. Find the equation of the tangent line to the curve y = x 2 − x + 3 about the line x = 3, what is the volume generated?
at the point (2,5). A. 370.3 C. 355.3*
A. 3x-y-1=0* C. x+3y-17=0 B. 360.1 D. 365.10
B. 3x+y+1=0 D. 2x-3y+5=0 63. Find the centroid of the upper half of the circle x^2 + y^2 = 9.
A. (0, 3/pi) C. (0,5/pi)
48. Find the equation of the normal line to the curve y = x − x + 3
2
B. (0,4/pi)* D. (0, 6/pi)
at the point (2,5).
A. 3x-y-1=0 C. x+3y-17=0* 64. Water is flowing into a conical cistern at the rate of 8 m3/min. If
B. 3x+y+1=0 D. 2x-3y+5=0 the height of the inverted cone is 12 m and the radius of its
circular opening is 6m. How fast is the water level rising when
49. Find the equation of the tangent line to the ellipse: the water is 4m depth?
4x 2 + 9y 2 = 40 at point (1,2). A. 0.64 m/min * C. 0.46 m/min
A. 2x+9y-20=0* C. 9x+2y-15=0 B. 4.6 m/min D. 6.4 m/min
B. 2x-3y+10=0 D. 2x-9y+20=0
50. If a function is differentiable on an open interval containing c, 65. Find the area inside the curve: r = cosθ , and outside the
then the graph at this point is concave _______if f”(c)>0. curve: r = 1− cosθ .
A. upward* C. downward
B. right D. left π π
A. 3− * C.
3 3
51. Find the point of inflection of the curve y = x^3 + 3x^2 – 1.
A. (-1,1) * C. (-2,3) π π
B. 3+ D.
B. (0,-1) D. (-3,-1) 3 3

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66. Find the total area enclosed by the curve: r 2 = cosθ . B. 4.275 grams D. 3.125 grams *
A. 4 C. 2* 83. Find the general solution of y” + 6y’ + 9y = x + 1.
B. π/4 D. 2π A. y = (C1x + C2x2) e-3x + 1/27 + x/9
B. y = (C1 + C2x) e-3x + 1/27 + x/9 *
67. Find k so that A = (3, -2) and B = (1, k) are perpendicular. C. y = (C1x + C2x2) e3x + 1/27 + x/9
A. 2 C. ½ D. y = (C1 + C2x)e3x + 1/27 + x/9
B. 3 D. 3/2* 84. Find A for which y = Ae^x will satisfy y” – 2y’ – y = 4e^x
A. -1 C. -2 *
68. The equation r = 2sinθ − cosθ in rectangular coordinates is B. -3 D. -4
given by:
7 7 7

85. If ∫1 ∫ ∫
A. x 2 + y 2 + x − 2y = 0 * C. x 2 + y2 + 2x − y = 0 f(x)dx = 4 g(x)dx = 2 [3f(x) + 2g(x) + 1]dx.
and 1 , find 1
B. x 2 − y2 + x − 2y = 0 D. y2 − x 2 − x + 2y = 0 A. 22 * C. 23
B. 24 D. 25
69. Identify the graph of an ellipsoid.
86. Find the equation of the circle with center at (-3+2i) and
x 2 y2 z2 x 2 y2 z 2 radius 5.
A + + =1 * C. + − =1
a2 b2 c 2 a2 b2 c 2 A. z − 2 + 3i = 5 C. z + −2i = 5
x 2 y2 z 2 x 2 y2 z 2
B. 2 − 2 − 2 = 1 D. 2 + 2 − 2 = 0 B. z − 2 − 3i = 5 D. z − (−3 + i) = 5
a b c a b c
∞
(−1)n x 2n 87. Find the volume of a parallelepiped with sides A=3i-j, B=j+2k, C=
70. Given that cos x = ∑ what is cos(3x 2 ) ?
n=0 (2n)! i+5j+4k.
∞ ∞
A. 15 C. 32
(−1)n x 4n (−9x)2n
A. ∑ C. ∑ B. 29 D. 20*
n=0 (2n)! n=0 (2n)! 88. Evaluate jj.
∞
(−1)n x 2n ∞
(−9)n x 4n A. e-π/2 C. e-π/3
B. ∑ D. ∑ * B. 1 D. -1
n=0 (2n)! n=0 (2n)!
89. If φ=x2yz3 and A= xzi-y2j+2x2yk, find div(φA).
71. What is the coefficient of x3 in Taylor series for f(x)=lnx about
A. 3x2yz4-3x2y2z3+6x4y2z2* C. 3x2yz4i-3x2y2z3j+6x4y2z2k
x=1?
B. 3x2y-3x2y2z3+6x4z2 D. 3x2yi-3x2y2z3j+6x4z2k
A. 1/6 C. 2/3
B. 2 D. 1/3* 90. The equation xy = c represents the family of all equilateral
∞ hyperbolas with center at the origin. Find the equation of the
1 2n
72. If the Maclaurin series for f(x) is ∑
n=0 n!
x , find f(x). family of curves that intersect the curves of the given family at
right angles.
A. e x C. e − x A. y2-x2=C1* C.y2=C1x
2
B. y2-2x2=C1 D. 2y2-x2=C1
B. e x D. ln x
∞
91. Which of the following equations is not exact?
xn
73. Find the radius of convergence of ∑ n! .
n=0
A. y 2dt + (2yt + 1)dy = 0 C. ydx-xdy=0*
B. (x+siny)dx+(xcosy-2y)dy=0 D. none of these
A. 0 C. 1
B. 3 D. ∞* 92. Determine the integrating factor of the differential equation ydx-
xdy=0.
74. The coefficient of the x^4 in the Taylor series for e^2x about x = A. –x2 C. -1/x2*
0 is B. 1/x2 D. x2
A. 3/4 C. 2/3 *
B. 1/24 D. 1/2 93. Find the height of the right circular cylinder of maximum volume
∞ V that can be inscribed in a sphere of radius R.
(−1) (x − 2)
n n
75. Determine ROC and the IOC of ∑
n=0 3n
. A. 3R/2
B. 2R/√3*
C. √3 (R/2)
D. 2/3R
A. ROC=3; IOC=(-1,5)* C. ROC=1; IOC=(1,3) 94. Find the r.m.s. value of y=x2+3 on the interval x=1 and x=3.
B. ROC=2; IOC=(0,4) D. ROC=4; IOC=(-2,6) A. 52.9 C. 59.2
B. 6.79 D. 7.69*
76. Determine the Wronskian of ex , cosx, sinx.
A. 3ex C. -3ex 95. Find the area of the triangle with vertices at P(2,3,5), Q(4,2,-1)
x
B. 2e * D. ex and R(3,6,4).
A. 10.32* C. 21.25
77. Find the acute angle between the vectors z1 = 3- 4i and z2 = -4 B. 12.11 D. 25.43
+ 3i.
A. 17 deg and 17 min C. 15 deg and 15 min
96. Which of the following is not a transcendental number?
B. 16 deg and 16 min* D. 18 deg and 18 min
A. π C. √2*
78. If z1 = 1 – i and z2 = -2 + 4i evaluate z1^2 + 2z1 – 3. B. e D. Both A & B
A. -1 + 4i C. -1 – 4i*
97. Determine the curl of the vector function F(x,y,z)=3x2i+7exyj.
B. 1 – 4i D. 1 + 4i
A. 7exy C. 7exyi
79. Find the area of the polygon with vertices at 2 + 3i, 3 + i, -2 – 4i – x
B. 7e yj D. 7exyk*
4 – i – 1 + 2i.
A. 47/5 C. 45/2 98. Determine the Laplace transform of a unit step function.
B. 47/2* D. 45/4 A. 1/s C. s
80. Evaluate lim
z→2i
(i z^4 + 3z^2 - 10i) B. 1/s
2
D. s
2

A. -12 + 6i* C. 12 + 6i 99. Find the principal root of 𝒍𝒏 (𝟑 + 𝒋𝟒)

B. 12 – 6i D. -12 – 6i A. 1.857∠92.25° C. 1.857∠-29.95°
81. Find the work done in moving an object along a vector a = 3i + B. 1.857∠29.95° * D. -1.857∠29.95°
4j if the force applied is b = 2i + j 100. Evaluate 𝒍𝒐𝒈(−𝟐 + 𝟑𝒊)
A. 8 C. 9
A. 0.3 + 𝑗1.364 C. −0.87– 𝑗2.331
B. 10 * D. 12
B. −0.3 − 𝑗1.364 D. 0.557 + 𝑗0.938*
82. A certain chemical decomposes exponentially. Assume that 200
grams becomes 50 grams in 1 hour. How much will remain after GOD BLESS!
3 hours?
A. 1.50 grams C. 6.25 grams

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