MATH PRE BOARD 2 D.

-∞

SHOW YOUR SOLUTION 8. Find the area of the lemniscate r2 =

Write it in a long bond paper a2cos2θ
A. a2
1. What is the differential equation of the B. a
family of parabolas having their C. 2a
vertices at the origin and their foci on the x- D. a3
axis?
A. 2xdy – ydx = 0 9. Find the area bounded by the parabola
B. ydx + ydx = 0 sqrt. of x + sqrt. of y = sqrt. of a and
C. 2ydx –xdy = 0 the line x + y = a.
D. dy/dx – x = 0 A. a2
B. a2 /2
2. Find the rthogonal trajectories of the C. a2 /4
family of parabolas y^2 = 2x + C. D. a2 /3
A. y = Ce^x
B. y = Ce^(-x) 10. Ben is two years away from being twice
C. y = Ce^(2x) Ellen’s age. The sum of twice Ben’s age and
D. y = Ce^(-2x) thrice Ellen’s age is 66. Find Ben’s age now.
A. 19
3. A reflecting telescope has a parabolic B. 20
mirror for which the distance from the C. 16
vertex to the focus is 30 ft. If the distance D. 21
across the top of the mirror is 64 in.,
how deep is the mirror of the center? 11. What percentage of the volume of a
A. 32/45 in. cone is the maximum volume right
B. 30/43 in. circular cylinder that can be inscribed in it?
C. 32/47 in. A. 24%
D. 35/46 in. B. 34%
C. 44%
4. Simplify (1 – tan2x) / (1 + tan2x) D. 54%
A. sin 2x
B. cos 2x 12. A balloon rising vertically, 150 m from an
C. sin x observer. At exactly 1 min, the
D. cos x angle of elevation is 29 deg 28 min. How
fast is the balloon using at that
5. Evaluate L { t^n }. instant?
A. n!/s^n A. 104m/min
B. n!/s^(n+1) B. 102m/min
C. n!/s^(n-1) C. 106m/min
D. n!/s^(n+2) D. 108m/min

6. Simplify 12 cis 45 deg + 3 cis 15 deg. 13. A conic section whose eccentricity is
A. 2 + j less than one (1) is known as:
B. sqrt. of 3 + j2 A. a parabola
C. 2 sqrt. Of 3 + j2 B. an ellipse
D. 1 + j2 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
A. 9/2 C. which passes inside the conic
B. π D. all of the above
C. ∞
15. A die and a coin are tossed. What is the squares and folding the edges. Find the
probability that a three and a head will volume of the largest box.
appear? A.592 cu.in.
A. 1/4 B.529 cu.in.
B. 1/2 C.696 cu.in.
C. 2/3 D.689 cu.in.
D.1/12
23. A retailer bought a number of ball pens
16. Find the integral of 12sin5xcos5xdx if for P90 and sold all but 3 at a profit P2 per
lower limit = 0 and upper limit = pi/2. ball pen. With the total amount received she
A. 0.8 could buy 15 more ball pens than before.
B.0.6 Find the cost per ball pen.
C.0.2 A. P2
D.0.4 B. P3
C.P4
17. 12 oz of chocolate is added to 10 oz of D.P5
flavoring is equivalent to
A.1 lb and 8 oz 24. What is –i^i?
B. 1 lb and 6 oz A.4.81
C.1 lb and 4 oz B.-4.81
D.1 lb and 10 oz C.0.21
D.-0.21
18. The Ford company increased its assets
price from 22 to 29 pesos. What is 25. A balloon travel upwards 6m, North and
the percentage of increase? 8m, East. What is the distance
A.24.14% traveled from the starting point?
B.31.82% A. 7
C.41.24% B. 10
D.28.31% C.14
D. 20
19. Find the area bounded by outside the
first curve and inside the second 26. What do you call the integral divided by
curve, r = 5, r = 10sinθ the difference of the abscissa?
A. 47.83 A. average value
B.34.68 C. abscissa value
C.73.68 B. mean value
D.54.25 D. integral value

20. In two intersecting lines, the angles 27. Water is running out of a conical funnel
opposite to each other are termed as: at the rate of 1 cubic inch per sec.
A. opposite angles If the radius of the base of the funnel is 4 in.
C. horizontal angles and the altitude is 8 in., find the
B. vertical angles rate at which the water level is dropping
D. inscribed angles when it is 2 in. from the top.
` A. -1/pi in./sec
21. The area in the second quadrant of the B. -2/pi in./sec
circle x^2 + y^2 = 36 is revolved B. -1/9pi in./sec
about the line y + 10 = 0. What is the volume C. D.-2/9pi in./sec
generated?
A. 2932 c.u. 28. How many inches is 4 feet?
B. 2392 c.u. A. 36
C. 2229 c.u. B. 48
D. 2292 c.u. C. 12
D. 56
22. A cardboard 20 in x 20 in is to be formed
into a box by cutting four equal
29. A rectangular trough is 8 ft. long, 2 ft. C.5/4
across the top, and 4 ft. deep. If water flows D. 4/5
in at a rate of 2 cu. ft./min., how fast is the
surface rising when the water 36. Find the length of the arc of 6xy = x^4 +
is 1 ft. deep? 3 from x = 1 to x = 2.
A. 1/5 ft./min A.12/17
B. 1/8 ft./min B.17/12
C. 1/6 ft./min C.10/17
D. 1/16 ft./min D.17/10

30. Five tables and eight chairs cost $115; 37. A certain radioactive substance has half-
three tables and five chairs cost life of 3 years. If 10 grams are
$70. Determine the total cost of each table. present initially, how much of the substance
A. $15 remain after 9 years?
B. $30 A.2.50g
C. $25 B.5.20g
D. $20 C. 1.25g
D.10.20g
31. Find the 16th term of the arithmetic
sequence; 4, 7, 10,…….. 38. A cubical box is to built so that it holds
A. 47 125 cu. cm. How precisely should
B. 46 the edge be made so that the volume will be
C. 49 correct to within 3 cu. cm.?
D. 48 A.0.02
B.0.03
32. Find the slope of the line through the C.0.01
points (-2, 5) and (7, 1). D.0.04
A. 9/4
B. -9/4 39.
C. 4/9 Find the eccentricity of the ellipse when the
D. -4/9 length of its latus rectum is 2/3
of the length of its major axis.
33. For what value of k will the line kx +5y = A.0.62
2k have a y B. 0.64
-intercept 4? C.0.58
A. 8 D.0.56
B. 7
C. 9 40. Find k so that A = <3, -2> and B =<1, k>
D.10 are perpendicular.
A. 2/3
34. If a bug moves a distance of 3pi cm B.3/2
along a circular arc and if this arc C.5/3
subtends a central angle of 45 degrees, D.3/5
what is the radius of the circle?
A. 8 41. Find the moment of inertia of the area
B. 12 bounded by the curve x^2 = 8y, the
C. 14 line x = 4 and the x-axis on the first quadrant
D. 16 with respect to y-axis.
A.25.6
35. Two vertices of a rectangle are on the B. 21.8
positive x-axis. The other two C.31.6
vertices are on the lines y = 4x and y = -5x + D.36.4
6. What is the maximum possible
area of the rectangle? 42. Find the force on one face of a right
A.2/5 triangle of sides 4m and altitude of 3m. The
B.5/2
altitude is submerged vertically with the 4m 49. The probability of John’s winning
side in the surface. whenever he plays a certain game is 1/3.
A.62.64 kN If he plays 4 times, find the probability that
B.58.86 kN he wins just twice.
C.66.27 kN A.0.2963
D.53.22 kN B.0.2936
C.0.2693
43. In how many ways can 6 people be D.0.2639
seated in a row of 9 seats?
A. 30,240 50. A man row upstream and back in 12
B. 30,420 hours. If the rate of the current is 1.5
C.60,840 kph and that of the man in still water is 4 kph,
D. 60,480 what was the time spent
downstream?
44. The arc of a sector is 9 units and its A.1.75 hr
radius is 3 units. What is the area of B.2.75 hr
the sector? C.3.75 hr
A.12.5 D. 4.75 hr
B.13.5
C.14.5 51. If cot A = -24/7 and A is in the 2nd
D.15.5 quadrant, find sin 2A.
A.336/625
45. The sides of a triangle are 195, 157, and B.-336/625
210, respectively. What is the area of the C.363/625
triangle? D. -363/625
A.73,250
B.10,250 52. The volume of a square pyramid is 384
C.14,586 cu. cm. Its altitude is 8 cm. How
D.11,260 long is an edge of the base?
A.11
46. A box contains 9 red balls and 6 blue B.12
balls. If two balls are drawn in C.13
succession, what is the probability that one D.14
of them is red and the other is
blue? 53. The radius of the circle x^2 + y^2
A.18/35 – 6x + 4y – 3 = 0 is
B.18/37 A.3
C.16/35 B.4
D.16/37 C.5
D.6
47. A car goes 14 kph faster than a truck
and requires 2 hours and 20 minutes 54. If the planes 5x – 6y - 7z = 0 and 3nx +
less time to travel 300 km. Find the rate of 2y – mz +1 = 0
the car. A.-2/3
A.40 kph B. -4/3
B.50 kph C.-5/3
C.60 kph D.-7/3
D.70 kph
55. If the equation of the directrix of the
48. Find the slope of the line defined by y – parabola is x – 5 = 0 and its focus is at (1, 0),
x = 5. find the length of its latus rectum.
A.1 A.6
B.1/4 B.8
C.-1/2 C.10
D.5 D.12
56. If tan A = 1/3 and cot B = 4, find tan (A + A. 5
B). B. 1
A. 11/7 C. 2
B. 7/11 D.3
C. 7/12
D. 12/7 64. If sin3A = cos6B then:
A. A + B = 180 deg
57. A club of 40 executives, 33 like to smoke C. A - 2B = 30 deg
Marlboro, and 20 like to smoke B. A + 2B = 30 deg
Philip Morris. How many like both? D. A + B = 30 deg
A. 13
B. 10 65. What is the area between y = 0, y = 3x^2,
C. 11 and x = 2?
D. 12 A. 8
B. 12
58. The area of the rhombus is 264 sq. cm. C. 24
If one of the diagonals is 24 cm D.6
long, find the length of the other diagonal.
A. 22 66. The volume of the sphere is 36pi cu. m.
B. 20 The surface area of this sphere in
C. 26 sq. m is:
D. 28 A. 36pi
B. 24pi
59. How many sides have a polygon if the C. 18pi
sum of the interior angles is 1080 D. 12pi
degrees?
A. 5 67. The vertex of the parabola y^2 – 2x + 6y
B. 6 + 3 = 0 is at:
C. 7 A. (-3, 3)
D. 8 B. (3, 3)
C. (3, -3)
60. The line segment connecting (x, 6) and D. (-3, -3)
(9, y) is bisected by the point (7, 3).
Find the value of x and y. 68. Add the following and express in meters:
A. 5, 0 3 m + 2 cm + 70 mm
B. 4, 0 A. 2.90 m
C. 5, 2 B. 3.14 m
D.4,1 C. 3.12 m
D.3.09m
61. What is the height of the parabolic arch
which has span of 48 ft. and having a height 69. A store advertised on sale at 20 percent
of 20 ft. at a distance of 16 ft. from the off. The sale price was $76. What
center of the span? was the original price? A. $95
A. 30 ft. B. $96
B. 40 ft. C. $97
C. 36 ft. D.$98
D.34ft.
70. Find the equation of the straight line
62. Determine B such that 3x + 2y – 7 = 0 is which passes through the point (6, -3)
perpendicular to 2x –By + 2 =0. and with an angle of inclination of 45
A. 2 degrees.
B. 3 A. x + y = 8
C. 4 B. x – y = 8
D.5 C. x + y = 9
63. The value of x + y in the expression 3 + D. x – y = 9
xi = y + 2i is;
71. A freight train starts from Los Angeles B. 40
and heads for Chicago at 40 mph. C. 70
Two hours later a passenger train leaves the D. 60
same station for Chicago
traveling at 60 mph. How long will it be 77. The diagonal of a face of a cube is 10 ft.
before the passenger train overtakes The total area of the cube is
the freight train? A. 300 sq. ft.
A. 3 hrs. B. 150 sq. ft.
B. 5 hrs. C. 100 sq. ft.
C. 4 hrs. D. 200 sq. ft.
D. 6 hrs.
78. A ship is sailing due east when a light is
72. The number of board feet in a plank 3 observed bearing N 62 deg 10 min E. After
inches thick, 1 ft. wide, and 20 ft. the ship has traveled 2250 m, the light bears
long is: N 48 deg 25 min E. If the course is
A. 30 continued, how close will the ship approach
B. 60 the light?
C. 120 A. 2394 m
D. 90 B. 2934 m
C. 2863 m
73. Boyles’s law states that when a gas is D. 1683 m
compressed at constant
temperature, the product of its pressure and 79. If f(x) = 1/(x – 2), (f g)’(1) = 6 and g’(1) =
volume remains constant. If the pressure -1, then g(1) =
gas is 80 lb/sq.in. when the volume is 40 A.-7
cu.in., find the rate of change of pressure B. -5
with respect to volume when the volume is C. 5
20 cu.in. D. 7
A. -8
B. -10 80. Find the work done by the force F = 3i +
C. -6 10j newtons in moving an object
D.-9 10 meters north.
A.104 40 J
74. Find the average rate of change of the B. 100 J
area of a square with respect to its C.106 J
side x as x changes from 4 to 7. D. 108.60 J
A. 8
B. 11 81. The volume of a frustum of a cone is
C. 6 1176pi cu.m. If the radius of the lower base
D. 21 is 10m and the altitude is 18m, compute the
lateral area of the frustum of a cone
75. How many cubic feet is equivalent to A.295pi sq. m.
100 gallons of water? B. 691pi sq. m.
A. 74.80 C.194pi sq. m.
B. 1.337 D. 209pi sq. m.
C. 13.37
D. 133.7 82. In an ellipse, a chord which contains a
focus and is in a line perpendicular
76. A merchant purchased two lots of shoes. to the major axis is a:
One lot he purchased for $32 per pair and A.latus rectum
the second lot he purchased for $40 per pair. B. minor axis
There were 50 pairs in the first lot. How C. focal width
many pairs in the second lot if he sold them D. major axis
all at $60 per pair and made a gain of $2800
on the entire transaction? 83. With 17 consonant and 5 vowels, how
A. 50 many words of four letters can be
four letters can be formed having 2 different she is idle. At the end of 25 days she nets
vowels in the middle and 1 $450. How many days did she work?
consonant (repeated or different) at each A.21 days
end? B. 22 days
A.5780 C. 23 days
B. 5785 D.24 days
C. 5790
D. 5795 91. Francis runs 600 yards in one minute.
What is his rate in feet per second?
84. Evaluate tan2 (j0.78). A.25
A.0.653 B. 30
B.-0.653 C.35
C.0.426 D.40
D. -0.426
92. For a complex number z = 3 + j4 the
85. A particle moves along a line with modulus is:
velocity v = 3t^2 – 6t. The total distance A.3
traveled from t = 0 to t = 3 equals B. 4
A.8 C. 5
B. 4 D. 6
C. 2
D. 16 93. Which of the following is an exact DE?
A. (x^2 + 1)dx – xydy = 0
86. An observer at sea is 30 ft. above the C. 2xydx + (2 + x^2)dy = 0
surface of the water. How much of B. xdy + (3x – 2y)dy = 0
the ocean can he sea? D. x^2 ydy – ydx = 0
A.124.60 sq. mi.
C. 154.90 sq. mi. 94. There are 8 different colors, 3 of which
B.142.80 sq. mi. are red, blue and green. In how
E. 132.70 sq. mi many ways can 5 colors be selected out of
the 8 colors if red and blue are
87. There are three consecutive integers. always included but green is excluded?
The sum of the smallest and the A.12
largest is 36. Find the largest number. B.11
A.17 C. 10
B. 18 D.9
C.19
D. 20 95. Five cards are drawn from a pack of 52
well – shuffled cards. Find the
88. If y = sqrt. of (3 – 2x), find y. probability that 3 are 10’s and 2 are queens.
A.1/sqrt. of (3 – 2x) A. 1/32
C. 2/sqrt. of (3 – 2x) B. 1/108,290
B. -1/sqrt. of (3 – 2x) C. 1/54,350
D. -2/sqrt. of (3 – 2x) D.1/649,740

89. The logarithm of MN is 6 and the 96. From past experience, it is known 90%
logarithm of N/M is 2, find the value of of one year old children can distinguish their
logarithm of N. mother’s voice from the voice of a similar
A.3 sounding female. A random sample of 20
B. 4 one year’s old are given this voice recognize
C. 5 test. Find the probability that all 20 children
D.6 recognize their mother’s voice.
A. 0.122
90. A woman is paid $20 for each day she B. 1.500
works and forfeits $5 for each day 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