EXAM 2018, questions and answers PDF

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MATHEMATICS PRE-BOARD EXAMINATION: SUNDAY, JULY 22, 2018 1. Joseph gave 1/4 of his candies to Joy and Joy gave 1/5 of what she got to Tim. If Tim received 2 candies, how many candies did Joseph have originally? a. 30 b. 20 c. 50 d. 40 2. What conic section is described by the equation 4x 2 – y2 + 8x + 4y = 15? a. Parabola b. Hyperbola c. Circle d. Ellipse 3. Find the maximum area of rectangle which can be inscribed in an ellipse having the equation x2 + 4y2 = 4. a. 4 b. 3 c. 2 d. 5 4. If the general equation of the conic Ax2 +Bxy + Cy2 + Dx + Ey + F=0. If B2 – 4AC >0, the equation describes a ______________. a. Ellipse b. Hyperbola c. Parabola d. Circle 5. Determine the equation that expresses that G is proportional to k and inversely proportional to C and z. Symbols a,b, and c are constant. cK

a. G =GG b. G = c. d.

a

bc cK G = zC bc G =zK

6. The chord passing through the focus of the parabola and is perpendicular to its axis is termed as: a. Axis b. Latus rectum c. Directrix d. Translated axis 7. What is the equation of the hyperbola with a focus at (-3 - 3sqrt. of 13, 1) asymptotes intersecting at (-3,1) a. 4x2 - 9y2 + 54x + 8y - 247=0

b. 4x2 + 9y2 + 54x - 8y + 284=0 c. 9x2 - 4y2 + 54x + 8y - 247=0 d. 9x2 + 4y2 + 54x - 8y - 284=0 8. Find the ratio of the sides of triangle if its sides form n arithmetic progression and one of the angles is 90 degrees. a. 4:5:6 b. 1:2:3 c. 3:4:5 d. 2:3:4 9. The area enclosed by the ellipse 4x2 + 9y2 = 36 is revolved about the line x=3, what is the volume generated? a. 370.3 b. 360.1 c. 355.3 d. 365.1 10. The polynomial x2 + 4x + 4 is the area of a square floor. What is the length of its side? a. x + 2 b. x – 2 c. x + 1 d. x – 1 11. Given a conic section, if B2 – 4AC =0, it is called? a. Circle b. Parabola c. Hyperbola d. Ellipse 12. Find the height of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius 10 cm. a. 11.55 cm b. 14.55 cm c. 12.55 cm d. 18.55 cm 13. The length of the latus rectum of the parabola y = 4px2 is: a. 4p b. 2p c. P d. -4p 14. 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 b. 179 c. 181 d. 184

15. What is the length of the shortest line segment in the first quadrant drawn tangent to the ellipse b2x2 +a2y2 = a2b2 and meeting to the coordinate axes? a. a/b b. a + b c. ab d. b/a 16. Find the radius of the circle inscribed in the triangle determined by the lines y= x + 4, y= x -4 and y = 7x -2. a. 5/sqrt. of 2 b. 5/(2 sqrt. of 2) c. 3/(sqrt. fo 2 d. 3/(2 sqrt. of 2) 17. Find the moment of inertia of the area boundd by the parabola y^2 = 4x and the line x = 1, with respect to the x-axis. a. 2.133 b. 1.333 c. 3.333 d. 4.133 18. What is the unit vector which is orthogonal both 9i + 9j and 9i + 9k? a. b. c. d.

𝐢 √3 𝐢 3 𝐢

𝐣

𝐣 √3

+3 +

√3 𝐢 3

+

− 𝐣

𝐣

√3

−3 −

+ 𝐤 3

− 𝐤

𝐤 √3 𝐤 √3

3

19. Express in polar form: -3-4i −1 a. 5e−i (pi+tan 4/3) −1 b. 5ei (pi+tan 4/3) −1 c. √5 e−i (pi+tan 4/3) −1 d. √5 ei (pi+tan 4/3) 20. The axis of the hyperbola through its foci is known as: a. Conjugate axis b. Transverse axis c. Major axis d. Minor axis 21. Describe the locus represented by | z + 2i| + | z − 2i| = 6 . a. Circle b. Parabola c. Ellipse d. Hyperbola 22. If the radius of the sphere is increased by a factor of 3, by what factor does the volume of the sphere chance?

a. 9 b. 18 c. 27 d. 54 23. Evaluate the ∫(7 x3 − 4x2 )dx. 7 x^4 4x^2 a. + 3 +C b. c. d.

4 7 x^4

4 7 x^4 4 7 x^4 4

− + −

4x^2 3 4x^3 3 4x^3 3

+C +C +C

24. Describe the locus represented by | z − 3| − | z + 3| = 4. a. Ellipse b. Circle c. Hyperbola d. Parabola 25. Melissa is 4 times as old as Jim. Pat is 5 years older than Melissa. If Jim is y, how old is Pat? a. 4y + 5 b. y + 5 c. 5y + 4 d. 4 + 5y 26. A conic section whose eccentricity, is less than one is known as: a. A parabola b. An ellipse c. A circle d. A hyperbola 27. Two lines passing through the point (2,3) make an angle of 45 degrees with each other. If the slope of one of the lines is 2, find the slope of the other. a. -2 b. -1 c. -3 d. 0 28. From the top of a building the angle of depression of the foot of a pole is 48 deg 10 min. From the foot of a building the angle of elevation of the top of a pole is 18 deg 50 min. Both building the pole are on a level ground. If the height of a pole is 4 m, how high is the building? a. 13.10 m b. 12.10 m c. 10.90 m d. 11.60 m 29. The locus of a point which moves so that the sum of its distances between two fixed two points is constant is called?

a. Ellipse b. Parabola c. Circle d. Hyperbola 30. Totoy is 5 feet 11 inches tall and Nancy is 6 feet 5 inches tall. How much taller is Nancy than totoy? a. 1 foot 7 inches b. 1 foot c. 7 inches d. 6 inches 31. If log 64 x = 3/2 , find x. a. 512 b. 521 c. 253 d. 258 32. What is the product of -9p3r and 2p-3r? a. 18p4r + 27p3r2 b. -18p4r + 27p3r2 c. 18p3r + 27p2r3 d. -18p3r + 27p2r3 33. Evaluate ∫ a. b. c. d.

1 (x2 3 1 2

+

x3 √x2 +25 25)3/2

dx, using trigonometric substitution x =5 tan θ. − 25(x2 + 25)1/2 + C

(x + 25)3/2 + 25(x2 + 25)1/2 + C

3 25 (x2 3 25 (x2 3

+ 25)3/2 − 25(x2 + 25)1/2 + C + 25)3/2 + 25(x2 + 25)1/2 + C

34. Michael’s favourite cake recipe calls for 0.75 pounds of flour: he has a 5 pound bag. He want to make several cakes of the school bake sale. How many can he make? a. 5 b. 6 c. 7 d. 8 35. Find the minimum amount of tin sheet that can be made into a closed cylinder having a volume of 108 cu. Inches in square inches. a. 125 b. 137 c. 150 d. 120 36. A chord of a circle 10 ft in diameter is increasing at the rate of 1ft/s. Find the rate of change of a smaller arc subtended by the chord when the chord is 8 long. a. 5/2 ft/min b. 2/5 ft/min

c. 5/3 ft/min d. 3/5 ft/min 37. Find the centroid of a semicircular area of a radius a. a. 2a/π b. 4a/π c. 2a/3π d. 4a/3π 38. An equilateral triangle with side “a” is revolved about its altitude. Find the volume of the solid generated. a. 0.32 a3 b. 0.23 a3 c. 0.41 a3 d. 0.14 a3 39. If the area bounded by the parabolas y= x2 – C2 and y = C2 - x2 is 576 square units, find the value of C. a. 5 b. 6 c. 7 d. 8 40. Solve y’ – 5y’ + 4y = sin 3x. 1 a. y = 25 ( 3 cos 3𝑥 − sin 3𝑥 ) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥 b. y = c. y = d. y =

1

25 1 50 1 50

( 3 sin 3𝑥 − cos 3𝑥 ) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥 ( 3 cos 3𝑥 − sin 3𝑥 ) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥 ( 3 sin 3𝑥 − cos 3𝑥 ) + 𝐶1 𝑒 𝑥 + 𝐶2 𝑒 4𝑥

41. A car is travelling at a rate of 36 m/s towards a statue of height 6 m. What is the rate of change of a distance of the car towards the top of the statue when it is m from the statue? a. 32.4 m/s b. 39.6 m/s c. 26.6 m/s d. 28.8 m/s 42. A fencing is limited to 20 ft length. 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 43. Evaluate Laplace transform of t cos kt. a. s2/(s2 + k2)2 b. k2/(s2 + k2)2 c. (-s2 + k2)/(s2 + k2)2 d. (s2 – k2)/(s2 + k2)2

44. Carmela and Marian got summer jobs at the ice cream shop and were supposed to work 15 hours per week each for 8 weeks. During that time, Marian was ill for one week and Carmela took her shifts. How many hours did Carmela work during the 8 weeks? a. 120 b. 135 c. 150 d. 185 45. Manuelita had 75 stuffed animals. Her grandmother gave 15 of them to her. What percentage of the stuffed animals did her grandmother give her? a. 20% b. 15% c. 25% d. 10% 46. Find the coordinates of an object that has been displaced from the poin (-4, 9) by the vector 4i – 5j. a. (0, 4) b. (0, -4) c. (4, 0) d. (-4, 0) 47. A triangle has two congruent sides and the measure of one angle is 40 degrees. Which of the following types of triangle is it? a. Isosceles b. Equilateral c. Right d. Scalene 48. The parabola defined by the equation 3y2 + 4x = 0 opens ________________ -a. Upward b. Downward c. To the left d. To the right 49. If a place on the earth is 12 degrees south of the equator, find its distance in nautical miles from the north pole. a. 6,021 b. 6,102 c. 6,210 d. 6,120 50. If the standard deviation of a population is 9, the population variance is: a. 9 b. 3 c. 21 d. 81 51. Simplify the equation sin2 𝜃 (1 + cos2 𝜃 ) a. 1

b. sin2 𝜃 c. tan2 𝜃 d. cos2 𝜃 52. What is the complement of a 60-degree angle? a. 120 degrees b. 30 degrees c. 40 degrees d. 20 degrees 53. If 2xy - y2 = 3, find y” a. 2/(x – y)4 b. -2/(x – y)4 c. 3/(x – y)3 d. -3/(x – y)3 54. The Rotary Club and the Jaycees Club had a joint part, 120 members of the Rotary Club and 100 members of the Jaycees Club also attended but 30 of those attended are members of both clubs. How many persons attended the part? a. 220 b. 190 c. 150 d. 250 55. Two numbers have a harmonic mean of 9 and a geometric mean of 6. Determine the arithmetic mean. a. 1/4 b. 4 c. 1/9 d. 9 56. Find the force on one face of a right triangle of sides 4 m and altitude of 3 m. The altitude is submerged vertically with the 4 m side in the surface. a. 58.86 kN b. 62.64 kN c. 53.22 kN d. 66.67 kN 57. An airplane flying with wind, took 2 hours to travel 1000 km and 2.5 hours in flying back. What was the wind velocity in kph? a. 40 b. 50 c. 60 d. 70 58. In how many ways can 6 people be lined up to get on a bus, if a certain 3 persons insist on following each other? a. 72 b. 144 c. 480

d. 120 59. If 3x3y = 27 and 2x + y = 5, find x. a. 3 b. 4 c. 2 d. 1 60. Find the work done in moving an object along a vector a = 3i + 4j if the force applied is b = 2i + j . a. 8 b. 9 c. 10 d. 12 61. If the line 3x – ky – 8 = 0 passes through the point (-2, 4), then k is equal to a. -7/2 b. -5/2 c. -3/2 d. -1/2 62. What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu.m., if the error of the computed volume is not to exceed 0.03 cu.m? a. 0.002 b. 0.003 c. 0.0025 d. 0.001 63. A man can do a job in 8 days. After the man has worked for 3 days, his sons joins him and together they are complete the job in 3 more days. How long will it take the son to do the job alone? a. 12 days b. 10 days c. 13 days d. 11 days 64. The probability that a randomly chosen sales prospects will make a purchase is 0.18. If a salesman calls on 5 prospects, what is the probability that the salesman will make exactly 3 sales? a. 0.0392 b. 0.0239 c. 0.0329 d. 0.0293 65. If sec2 A = 5/2, then 1 – sin2 A = a. 0.20 b. 0.30 c. 0.40 d. 0.50 66. What is the angle between the diagonal of a cube and one of its edges?

a. 44.74 ° b. 54.74 ° c. 64.74 ° d. 74.74 ° 67. The line 3x – 4y = 5 is perpendicular to the line a. 3x – 4y = 1 b. 4x – 3y = 2 c. 4x + 3y = 3 d. 3x + 4y = 4 68. If the plane 3x + 2y – 3z = 0 is perpendicular to the plane 9x – 3ky + 5z -1 = 0, find the value of k. a. 2 b. -2 c. 3 d. -3 69. A solid has circular base of radius r. Find the volume of the solid if every planar section perpendicular to a fixed diameter is a semicircle. a. 1.26r3 b. 2.09r3 c. 2.51r3 d. 4.19r3 70. Find the y – intercept of the line given by the equation 7 x + 4y = 8. a. 2 b. -2 c. 3 d. -3 71. Find the area inside the cardioid r = 1 + cos 𝜃 and outside the circle r = 1. a. 2.97 b. 2.79 c. 2.85 d. 2.58 72. A person had a rectangular-shaped garden with sides of lengths 16 feet and 9 feet. The garden was changed into a square design with the same area as the original rectangularshaped garden. How many feet in length are each of the sides of the new square-shaped garden? a. 7 b. 9 c. 12 d. 16 73. Which of the following rope lengths is longest? a. 1 meter b. 1 yard c. 32 inches

d. 85 centimeters 74. Martin, a motel housekeeper, has finished cleaning about 40% of the 32 room he’s been assigned. About how many more rooms does he have left to clean? a. 29 b. 25 c. 21 d. 19 75. A horse is tied to a post with a twenty-foot rope. What is the longest part that the horse can walk? a. 20 feet b. 40 feet c. 62.83 feet d. 125.66 feet 76. Doming wants to know the height of the telephone pole. He measures his shadow, which is 3 feet long, and the pole’s shadow, which is 10 feet ling. Doming’s height is 6 feet. How tall is the pole? a. 40 feet b. 30 feet c. 20 feet d. 10 feet 77. A weight of 60 pounds rest on the end of an 8-foot lever and is 3 feet from the fulcrum. What weight must be placed on the other end of the lever to balance the 60-pound weight? a. 36 pounds b. 32 pounds c. 40 pounds d. 46 pounds 78. A number is 1 more than the twice another. Their squares differ by 176. What is the larger number? a. 9 b. 7 c. 15 d. 16 79. The sides of a right triangle is in arithmetic progression whose common difference is 6 cm. Its area is: a. 216 sq. cm b. 270 sq. cm c. 360 sq. cm d. 144 sq. cm 80. A tank has 100 liters of brine with 40 N of dissolved salt. Pure ware enters the thank at the rate of 2 liters per minute and the resulting mixture leaves the tank at the same rate. When will the concentration in the tank be 0.20 N/L ? a. 24.6 min

b. 34.7 min c. 44.8 min d. 54.9 min 81. The base of an isosceles triangle is 20.4 and the base angles are 48°20’. Find the altitude of the triangle. a. 9.8 b. 10.8 c. 11.6 d. 12.7 82. A lady gives a party dinner party for six guests. In how many ways they be selected from among 10 friends if the two of the friends will not attend the party together? a. 112 b. 128 c. 140 d. 160 83. A rubber ball is dropped from a height of 81 m. Each time it strikes the ground, it rebounds two-thirds of the distance through which it last feel. Find the total distance it travels in coming to rest? a. 243 m b. 162 m c. 405 m d. 324 m 84. Find the diameter of a pulley which is driven at 360 rpm by a belt moving at 40 ft/s. a. 2.12 ft b. 1.11 ft c. 2.43 ft d. 1.24 ft 85. Find the volume generated by the circle x2 + y2 =25 if it is revolved about the line 4x + 3y = 40. a. 3,498 c.u. b. 3,948 c.u. c. 4,624 c.u. d. 4,426 c.u. 86. A ranch has cattle and horses in a ratio of 9:5. If there are 80 more head of cattle than horses, how many animals are on the ranch? a. 140 b. 168 c. 238 d. 280 87. The first term of a geometric sequence is 375 and the fourth term is 192. Find the common ratio. a. 5/4 b. 4/5

c. 3/2 d. 2/3 88. In how many ways can a person choose 1 or more of 4 electrical appliances? a. 16, b. 15 c. 12 d. 20 89. The probability that a certain man will be alive 25 years hence is 3/7, and the probability that his wife will be alive 25 years hence is 4/5. Determine the probability that 25 years hence, only the man will be alive. a. 12/35 b. 4/35 c. 31/35 d. 3/35 90. 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. (1,-2) c. (2,1) d. (2,-1) 91. (1 – 2i)-1 can be written as: 1 1 a. 5 + 𝑖 5 b.

1

5

− 1

2

5

c. − − 3 1

d. − + 3

𝑖 2 𝑖 3 2 3

𝑖

92. Find the y – intercept of the line tangent to the parabola x = 2y2 at the point (2,1). a. -7 b. 7 c. 3/2 d. ½ 93. A growth curve is given by A = 10e2t. At what value of t is A = 100? a. 5.261 b. 3.070 c. 1.151 d. 0.726 94. If the short leg of a right triangle is 5 units ling and the long leg is 7 units long, find the angle opposite the short leg, in degrees. a. 26.3 b. 28.9 c. 31.2 d. 35.5

95. Express 2 sin2 𝜃 as a function of cos 2𝜃 . a. cos 2𝜃 – 1 b. cos 2𝜃 + 1 c. cos 2𝜃 + 1 d. 1 - cos 2𝜃 96. The x – and y – axes are the asymptotes of a hyperbola that passes through the point (2,2). Its equation is a. x2 – y2 = 0 b. xy = 4 c. y2 – x2 = 0 d. x2 + y2 = 4 97. If the area of the equilateral triangle is 4(sqrt. of 3), find the perimeter. a. 16 b. 12 c. 18 d. 14 98. Find the area bounded by the curve r = 8 cos 𝜃 a. 50.27 b. 12.27 c. 8 d. 67.02 99. What is the length of the transverse axis of the hyperbola whose equation is 9y2 – 16x2 = 144? a. 6 b. 9 c. 8 d. 7 100. Find the area bounded by x = 2y – y2 and the axis. a. 4/3 b. 5/3 c. 2/3 d. 1/3

ANSWERS 1. D 2. B 3. A 4. B 5. C 6. B 7. C 8. C 9. C 10. A 11. B 12. A 13. A 14. C 15. B 16. B 17. A 18. C 19. B 20. B 21. C 22. C 23. D 24. C 25. A 26. B 27. C 28. A 29. A 30. D 31. A 32. B

33. A 34. B 35. A 36. C 37. D 38. B 39. B 40. C 41. D 42. B 43. D 44. B 45. A 46. A 47. A 48. C 49. D 50. D 51. A 52. B 53. D 54. B 55. B 56. A 57. B 58. B 59. C 60. C 61. A 62. C 63. A 64. A 65. C 66. B

67. C 68. A 69. B 70. A 71. B 72. C 73. A 74. D 75. D 76. C 77. A 78. C 79. A 80. B 81. C 82. C 83. C 84. A 85. B 86. D 87. B 88. B 89. D 90. A 91. A 92. D 93. C 94. D 95. D 96. B 97. B 98. A 99. C 100.

A...