PREBOARD EXAM TOLENTINO AND ASSOCIATES MATHEMATICS SURVEYING AND TRANSPORTATION ENGINEERING
1. The radius of the cone is increased by 2.7% and its height is reduced by 0.9%. Determine the percentage change in its volume. a. 4.52% b. 6.89% c. 7.90% d. 8.34% Situation 1: 2. Where is the center of the circle 3x2+ 3y2 + 2x – 4y – 21 = 0 a. ( -1/8, 2/4) b.( -1/3, 2/3) c. ( -1/4, 3/3) d. ( -1/8, 1/3) 3. What is the radius a. 7.60 b. 5.78
c. 4.76
d. 2.98
4. The 1st term of a GP is 160 and the common ratio is 3/2. How many consecutive terms must be taken to give a sum of 2110? a. 5 b. 6 c. 7 d. 4 5. A company sells 80 units and makes P80 profit. It sells 110 units and makes P 140 profit. If the profit is a linear function of the number of units sold, what is the average profit per unit if the company sells 250 units? a. P 1.76 b. P 1.68 c. P 1.66 d. P 1.86 6. Given Station Elevation (m) Distance (km ) Alpha 680 m Alpha to Bravo = 12 km Bravo 645 m Bravo to Charlie = 15 km Charlie 620 m Situation 2 6. Compute the elevation of the line of sight at station at station Bravo with the instrument placed at station Alpha such that station Charlie would be visible from station Alpha considering the effect of curvature and refraction correction. a. 641.27 b. 758.33 c. 822.03 d. 856.66 7. Assuming that station Bravo will obstruct the line of sight from station Alpha while observing station Charlie and a 4 meter tower is constructed on top of station Bravo, compute the height of equal towers at station Alpha and station Charlie in order that both 3 stations will still be intervisible. 1
a. 8.992
b. 7.727
c. 6.042
Situation 3 From the closed traverse shown Lines Bearings 1-2 S 100 00 E 2-3 N 560 00 E 3-4 N 630 00 W 4-5 ------------5-1 S 33000 W
d. 5.673
Distances 485 780 975 ----890
8. The bearing of 4 -5 is closest to a. N 600 59’ E b. N 680 44’ E
c. N 710 74’ E
9. The distance 4 -5 is closest to a. 812.45 m b. 711.90 m c. 691.55 m
d. 534.77 m
10. The area of 3 – 4 4 -5 and 5 -1 is nearest to a. 78160.28 m2 b. 67139.28 m2 c. 99169.28 m2
d. N 730 24’ E
d. 88231.28 m2
11. A man leaves his house at 8:00 AM and traveling at an average speed of 2 kph, arrives at his office 3 min ahead of the expected time. Had he left his house at 8:30 am and traveled at an average speed of 3kph, he will arrive 6 min late of the expected time. Find the distance that he had traveled. a. 2.4 km b. 1.8 km c. 2.1 km d. 2.4 km 12. A ball is dropped from a height of 120ft and continuously rebounds to 2/3 of the distance it falls. What is the total distance traversed by the ball when it comes to rest? a. 1000ft b. 600 ft c. 800ft d. 750 ft 13. A cask containing 20 liters of wine was emptied 1/5 of its contents and then filled with water. If this is done 6 times, how many liters of wine will remain in the cask? a. 5.242liters b. 5.811 liters c. 6.242liters d. 6.134 liters 14. A train , one hour after starting, meets with an accident which detains it an hour , after which it proceeds at 3/5 of its former rate and arrived 3 hrs after the time. Had the accident happened 50 km farther on the line, it would have arrived 1.5hrs sooner. Find the length of the journey. 2
a. 87.91 km c. 88.89 km
b. 92.23 km d. 83.34 km
SITUATION 4 15. The length of a simple curve having a degree of 40 is 210 m. The middle ordinate of the curve is nearest to a. 24.08 m b. 19.03 m c. 17.67 m d. 16.55 m 16 The external distance of the curve is nearest to a. 20.38 m b. 21.44 m c. 22.48 d. 23.67 m 17. The area of the fillet of the curve is nearest to a. 4532 m b. 3422 m c. 2931 m d. 1424 m SITUATION 5 A compound curve has a common tangent of 84.5 m which makes an angle of 160 and 200 with the tangents of the 1st and the 2nd curve respectively. The length of the tangent to the 2nd curve is 42 m. 18. The radius of the 1st curve is closest to a. 205.5 m b. 302.4 m c. 409.2 m d. 506. m 19. The radius of the 2nd curve is nearest to a. 238.19 m b. 334.11 m c. 445.28 m d. 522.12 m 20. The length of the curve from PC to PT is nearest to a. 167.59 m b. 233.67 m c. 332.67 m d. 483.12 m SITUATION 6 A symmetrical parabolic summit curve connects two grades of 6% and -4%. It is to pass thru point P on a curve at station 25 + 140 having an elevation of 98.134 m. If the elevation of the grade of intersection is 100 with stationing of 25 + 160 , 20. The length of the curve is a. 120 m b. 140 m c. 160 m d. 112 m 21. The stationing of the highest point of the curve is closest to a. 25 + 172 b. 25 + 140 c. 25 + 145
d. 25 + 156
22. The elevation at station 25 + 120 is closest to a. 68.442 b. 79.872 c. 97.433
d. 102.546
SITUATION 7 3
The center height of the road at station 7 + 110 is 2 m fill while at station 7 + 160 is 1.2 m cut. From station 7 + 110 to the other station , the ground makes a uniform slope of 4.8%. 23. The slope of the new road is a. 0.016 b. 1.442 c. 2.641 d. 3.124 24. Distance in which the fill is extended is a. 31.25 m b. 33.42 c. 35.26 d. 38.46 25. Stationing of the point where the fill is extended id a. 71 + 112.25 b. 71 + 123.25 c. 71 + 141.25
d. 71 + 152.25
26. The resistance of a wire varies directly with its length and inversely with its area. If a certain piece of wire 10 m long and 0.1 cm in diameter has a resistance of 100 ohms, what will be the resistance if it is uniformly stretched so that its length becomes 12 m. a. 120 b. 130 c. 140 d. 150 27. A stack of bricks has 61 in the bottom layer, 58 bricks in the second layer, 55 bricks in the 3 rd layer and so on until there are 10 bricks in the last layer. How many bricks are there altogether a. 510 b. 529 c. 639 d. 71 28. The base of a truncated prism is a rectangle with length twice its width. The corner edges have heights of 12 m, 12 m , 16 m and 16 m respectively. If the volume of the prism is 8,200 cu m, find the length of the base. a.33.24 m b. 34.23 m c. 35.25 m d. 36.34 m 29. The perimeter of a triangle ABC is 400. If angle A is 300 and angle B is 580, find the measure of side AC. a. 170.285 b. 180.214 c. 190.546 d. 193.472 30. A regular hexagon is inscribed in a circle having an area of 150 cm2. Find the area of the circle not covered by the hexagon. a. 27.326 cm2 b. 29.344 cm2 c. 31.253 cm2 d. 33.642 cm2 31. The sum of the digits of a 3 digit number is 14. The hundreds digit being 4 times the unit digit. If 594 is subtracted from the number, the order of the digits will be reversed. Find the ten’s digit of the number. a. 1 b. 2 c. 3 d. 4 32. An arch 18 m high has the form of a parabola with vertical axis. The length of a horizontal beam placed across the arch 8 m from the top is 64 m. Find the width of the arch at the bottom. a. 69.674
b. 72.442
c. 74.267
d. 76.987
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33. How much of a 7% solution should be mixed with an appropriate amount of a 12% solution to get 5 liters of a 10% solution? a. 2 b. 3 c. 4 d. 5
34. Find the area bounded by 𝒚𝟐 = 𝟑𝒙 + 𝟗 and 𝒚𝟐 = 𝟗 − 𝟑𝒙 a. 22 b. 24 c. 26 35. If sin A = ¾ sin( A + B ) = ? a. 0.374 36.
d. 28
cos A < 0 , cos B = - 1/3 tan B > 0 b. 1.892
c. 2.768
d. 3.452
What is the first period of y = 4 tan 4x ? a. π b. π/2 c. π/4 d. π/8
37. What is the amplitude of 3 cos2 x + 5 sin 2x a. 5.83 b. 8 c. -2 d. 15 38. Find the volume of the solid generated by rotating the curve 𝟒𝒙𝟐 + 𝟗𝒚𝟐 = 𝟑𝟔along the line 4x + 3y = 20. a. 1184.34 u3 b. 3482.32 u3 c. 6713.37 u3
d. 8913.84 u3
39. Two cars begin a trip from the same point P. If car A travel north at 30 kph and car B travel west at 40 mph, how fast is the distance changing 2 hrs later. a. 45 mph b. 50 mph b 55 mph d. 60 mph 40. Determine the equation of the line passing ( -1, 3) and perpendicular to the line 4x – y + 5 = 0.
a. x + 4y = 11
b. x + 4y = 14
c. x + 4y = 16
d. x + 4y = 18
41. If steel ball is immersed in an 8 cm diameter cylinder, it displaces water to a depth of 2.25 cm. What is the radius of the ball?
a. 3.098 cm
b. 4.762
c. 5.196 cm
d. 6.462 cm
SITUATION 8 Studies show that 90% of married women claim that their husband’s mother is the biggest bone of contention in their marriages ( sex and money are lower rated areas of contention). Suppose that 6 women are having coffee together one morning. 42. What is the probability that no more than three of them dislike their mother in law? 5
Solution: a. 0.01585
b. 2.01932
c. 3.12983
d. 4.98212
43. What is the probability that at least 3 of them dislike their mother in law? a. 0.9987 b. 2.3427 c. 3.7844 d. 4.3477 SITUATION 𝝅 iF. If 𝐬𝐢𝐧−𝟏 𝟑𝒙 − 𝟒𝒚 = 𝟐
𝝅
and 𝐜𝐨𝐬 −𝟏 𝒙 − 𝒚 = 𝟑 ,
44.x is a. 1
b. 3
c. 5
d. 7
45. y is a. ¼
b. ½
c. 1/3
d. 1/8
46. 𝐬𝐢𝐧−𝟏(𝒙 − 𝒚) a. π/7
b. π/5
c. π/6
d. π/8
47. A curve f(x) is concave upward at (-1,3) , concaves up and concave downward at ( 3, 7). Which of the following is true? a. f(5) > 0 , max point at x = 5 b. f’(1) < 0, min points at x = 1 c. f’(3) > 0, point of inflection is at x = 3. d. f’’(3) < 0, point of inflection is at x = 3. 48. A hemispherical tank having a top diameter of 40 ft is filled with oil having a density of 52.4 lb/ft3. Find the work done in pumping all the water to the top of the tank. a.. 7,234,668.2 ftlbs b. 6,584,778.2 ftlbs c. . 5,523,228.2 ftlbs d. .4,524,733.2 ftlbs Solution:
SITUATION 9 Given triangle ABC, how many possible triangles can be formed from the following conditions? Solution: 49. AB = 25 AC = 36 angle A = 280 a. 1 b. 2 c. 3 d. 4 50. AB = 18 AC = 25 and angle C = 420
a. 1
b. 2
c. 3
d. 4
51. AB = 40 AC = 35 and angle B = 650 6
a. 1
b. 2
c. 3
d. 4
SITUATION 10 The following points are at the terminal sides of the angles in standard position. Determine the required trigonometric function. Solution: 52. Terminal point at (8,1). sin θ = ? Solution: a. 0.124 b. 2.342 c. 4.987 d. 6.239 53. Terminal point at ( 3, -8) cos θ = ? Solution: a. 0.3511 b. 3.209 c. 4.287 d. 2.439 54. Terminal Point at ( 3, -8) tan θ =? Solution: a. -8/3 b. -8/7 c. -7/2 d. 6/2 SITUATION 11 A tank has the form of a segment of a sphere of diameter 40 m. Its upper base is 8 m from the center and its lower base is 14 meter from the center of the sphere. Determine the following. 55. The area of the upper base. a. 1055.57 b. 1409.57 c. 1723.57 d. 2343.42 56. The curved surface of the tank. a. 2,764.60 m2 b. 5,254.60 m2 c. 6,544.64 d. 7,982.21 57. The volume of the tank in cubic meters. a. 24,236.34 m3 b. 25,903.34 m2 c. 26,283.35 m2 d. 27,225.77 m2 SITUATION 12 58-60 𝟏𝟒
Give the binomial 𝒙 + 𝟐𝒙−𝟐 . Determine: 58. 3rd term of the expansion. a.𝟏𝟐𝟒𝒙𝟖 b.𝟐𝟑𝟒𝒙𝟖 59. numerical coefficient of the 5th term. a. 12,223 b.16,106 60. The sum of exponents. a. 105 b.-105
c. 𝟑𝟔𝟒𝒙𝟖
d. 𝟒𝟐𝟓𝒙𝟖
c. 18, 103
d. 19,674
c. 109
d. -109
SITATIUON 13 One of the diameters of a frustum of a sphere is 9.8 cm while the other is 10.4 cm. If the thickness of the frustum of a sphere is 4.2 cm, find the following:
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61. volume of the frustum a. 234.77
b. 301.18
c.375.59
62. radius of the sphere a. 5.483
b. 6.789
c. 7.549
d. 8.552
63. area of the zone formed a. 78.23
b. 87.55
c. 98.275
d. 119.96
d. 456.09
64. A particle is moving along the x axis with velocity of 𝒗 𝒕 = 𝐬𝐢𝐧 𝒕 − 𝐜𝐨𝐬 𝒕 t> 0. What is the maximum acceleration over the interval [ 0, 2π ]. a. 𝟐 b. 𝟑 c. 𝟒 d. 𝟔 65.Two measurements for angle of elevations were made on the top of an inaccessible cliff, one from point A and another from point B. Point B is 425 nearer to the cliff than A but 25.6 m lower in elevation. If the angles of elevation from A and B are 18.650 and 36.20 respectively, find the elevation of the top of the cliff if A has an elevation of 625.4m. a. 674.355 m b. 785.322 m c. 842.566 m d. 913.506 m 66. A certain bacteria grows and multiplies exponentially and follows the formula 𝑨 = 𝑨𝟎 𝒆𝟎.𝟓𝟖𝟒𝒕where A is the final number of bacteria after time t. 𝑨𝟎 is the initial number of bacteria and t is the time elapse. If there were originally 4 bacteria , in how many hours will the bacteria increased to 2500? a. 13.033 hrs b. 12.229 hrs c. 11.024 hrs d. 10.088 67. The height of a rectangular box is 10 cm. The length increase at the rate of 2 cm/s, its width decreases at the rate of 4 cm/s. When the length is 8 cm and the width is 6 cm, the rate in cm3/s at which the volume is changing is ?
a. -230 cm3/s.
b. -220 cm3/s.
c. -210 cm3/s
.d. -200 cm3/s.
SITUATION 14 Electrical Resistance of metal are dependent on temperature. For a certain given wire at t deg C, the resistance R in ohms may be computed by 𝑹 = 𝑹𝟎 (𝟏+∝ 𝒕)where𝑹𝟎 is the resistance at 00 C and a is the temperature coefficient of resistance in /degC. 68. Solve the value of 𝑹𝟎 in ohms if R = 30 ohms at 500C and R = 35 ohms at 1000C. a.19 b. 25 c. 33 d. 41 69. Find ∝ a. 0.004 b. 2.009 c. 3.002 d. 4.212 8
𝟑
70. At what point on the graph 𝒚 = 𝟐𝒙𝟐 is its tangent line perpendicular to the line 2x + 3y = 6 ? Solution: a. ( 1/2, 1/4) b. ( 1/4,1/4) c. ( 3/5, 1/4). d. ( 4/4, 1/4). 71. Find the volume generated when the region bounded by 𝒚 = 𝒙 , the lines x = 1 and x = 4 is revolved around the x axis. a.
𝟏𝟓𝝅 𝟐
b.
𝟏𝟔𝝅 𝟐
c.
𝟏𝟕𝝅 𝟐
d.
𝟏𝟖𝝅 𝟐
72. A 4.2 cm by 4.2 cm square pyramid has sloping edges of 15 cm each. Find the total surface area. a.
452.2 cm2
b. 312.3 cm2
c. 255.3 cm2
d. 142.4 cm2
73. A certain physical characteristic of solid yields the following equation. c = a + bt. When c = 52 , t = 100 and when c = 172, t= 400. Find a. 8 b. 9 c. 11 d. 12 74. If the domain of y = 2x + 1 is [ -2, 3] Which is not in the range. a. -4 b. 0 c. -2 d. 7 75. Find the perimeter of r = 4 sin θ. Perimeter a. 3π. b. 4π. c. 5π. d. 6π. Given the polar curve 𝒓 = 𝟒(𝟏 − 𝐬𝐢𝐧 𝜽). 76. Find the Cartesian form of the equation. a. 𝒙𝟐 + 𝒚𝟐 = 𝟒 𝒙𝟐 + 𝒚𝟐 − 𝟒𝒚 b. 𝒙 + 𝒚 = 𝟒 𝒙𝟐 + 𝒚𝟐 − 𝟒𝒚 c. 𝒙𝟐 + 𝒚𝟐 = 𝟒 𝒙 + 𝒚 − 𝟒𝒚 d. 𝒙𝟐 + 𝒚 = 𝟒 𝒙𝟐 + 𝒚 − 𝟒𝒚 77. Find the slope of the curve when θ = 300. a. 0 b. 1 c. 2
d. 3
SITUATION 15 LED lamps are packages in boxes of 200. If production is known to produce 1.5% defective lamps on the average, determine the probability that a box chosen at random will contain 78.no defective lamps. a. 3.224 b. 2.235 c. 1.789 d. 0.0487 79. 2 defective led lamps. a. 0.2246 b. 1.8923 c. 2.6753 d. 3.8672 80. more than 3 defective lamps. a. 5.2219 b. 4.0921 c. 2.0912 d. 0.5785
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SITUATION 16 Light bulbs having a mean life of 2400hrs and standard deviation of 62 hrs are used for a consignment of 4000 bulbs. 81. Determine the number of bulbs likely to have a life in excess of 2500 lbs. a. 5.2313 b. 4.2286 c. 2. 1233 d.0.05338 82. Determine the percentage of bulbs with a life between 2300 hrs to 2500 hrs. a. 5.8841 b. 4.2217 c. 2.7632 d. 0.8923
83. In a certain triangle ABC, A = 950, B = 500 and C = 350. Which of the following expression correctly defines the lengths of the sides of a triangle? a. b. c. d.
AB< BC < CA AC< BC < AB AB< AC < BC BC< AC < AB
84. Find the angle between the curves 𝒙𝟐 + 𝒚𝟐 = 𝟒 and 𝟓𝒙𝟐 + 𝒚𝟐 = 𝟓 at the point of intersection of which x and y are positive. a. 23.76 deg. b. 30.76 deg. c. 32.76 deg. d. 37.76 deg. 85. If 𝒇 𝒙 = a. y = 1
𝟑𝒙𝟐 −𝟔𝒙−𝟗 , 𝒙𝟐 −𝒙−𝟐
give horizontalasymptotes
b. y = 2
c.y = 3
d. y = 4
86. Ernie earned P 2,000 commission on a big deal raising his average commission by P100. Ernie’s new average commission is 900. How many sales has he made so far? a. 10 b. 12 c. 14 d. 16 87. The rate of population growth of a country is proportional to the number of inhabitants. If the population of a certain country now is 40 million and 50 million in 10 years time,What will the population 20 years from now? a. 53.2 m b. 62.5 m c. 65.2 m d. 66.5 m 88. A certain type of bacteria, given a favorable growth medium, doubles in population every6.5 hours. Given that there were approximately 100 bacteria to start with, how many bacteria will there be in a day and a half? a. 4647.74 b. 3155.32 c. 2756.12 d. 1266.23 89. A boat makes 25 mph in still water. It is headed N 450 E. Find the direction of the course of the boat. a. N 54.930 E b. N 44.920 E c. N 41.900 E d. N 40.830 E 10
90. A circle having a radius of 9 cm circumscribes a right triangle with area of 43.23 sq m. If one of the side is 18 cm long, what is the length of the other side? a. 16.03 b. 17.29 c. 23.11 d. 25.64 91. The distance S meters from a fixed point of a vehicle travels in a straight line with a constant 𝟏
acceleration a and is given by: 𝑺 = 𝒖𝒕 + 𝟐 𝒂𝒕𝟐 S is in m, u in m/s and t is in seconds. Given a. b. c. d.
S = 42 when t = 2 and S = 144 when t = 4s, Determine the acceleration. 15 m/s2 16 m/s2 18 m/s2 20 m/s2
92. Find the equation of the tangent line,normal line at P(-1,-4) and ordinate of the vertex of the curve 𝒚 = 𝒙 + 𝟐 (𝒙 − 𝟑)at P(-1,-4). a. 3y + y + 8 = 0 b. 2 + y + 7 = 0 c. 3y + x + 6 = 2 d. 3x + y + 7 = 0 93. The ordinate of the vertex of the curve a. -6.25 b. -5.31
c. 6.25
d. 5.31
SITUATION 18 Two towers A and B are placed at 100 m apart horizontally. The height of tower A is 40 m and that of B is 30 m. 90. At what distance above the ground will the intersection of the lines forming the angles of elevation of the two 94. towers are observed from the bases of the towers A and B respectively. a. 16.22 b. 17.14 c. 18.23 d. 19.15 95. At what distance horizontally is this point located from the tower A? a. 57.14 b. 60.23 c. 62.73 d. 68.12 Consider the curve 𝒚𝟐 = 𝟒 + 𝒙 and the chord AB joining the points A( -4, 0) and B( 0,2). 96. Find the coordinates of the point on the curve where the tangent line is parallel to the chord AB. a. (4,-2) b. (3, -1) c. (2,-3 ) d. (1,-6) SITUATION 19 Consider the curve defined by 𝒙𝟐 + 𝒙𝒚 + 𝒚𝟐 = 𝟐𝟕. 97. . Find the points on the curve where the lines tangent to the curve are vertical. a. (6, -3) b. (5, -2) c. (4,-1) d. (3, 0) 98.The curve is a. parabola b.circle c.hyperbola d.elipse 11
99. The eccentricity is a. 4.22
b. 3.05
c. 2.81
d. 1.41
Let R be the region enclosed by the graphs of y = ln(x2 +1) and y= cos x. 100. Find the area of R. a.0.23 b. 1.17 c. 2..55
d. 3.98
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