TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES MANILA PRE BOARD EXAM DAY 3 STUDENT NUMBER_______________________________________________________DATE: September 30, 2011 DIRECTIONS: PROVIDE THE SOLUTION TO SELECT THE CORRECT ANSWER. 1. The cost C, of a product is a function of the quantity x, of the product: C(x) = x2 – 4000x + 50. Find the quantity for which the cost is minimum. a. 1000 b. 1500 c. 2000 d. 3000 2. The integral of a function between certain limits divided by the difference in abscissas between those limits gives the _________________ of the function. a. constant b. derivative c. average d. domain 3. In the figure, the radius of the circle is 1, what is the perimeter of the shaded part of the figure? a. 4Π/3 b. Π c. 2Π/3 d. Π/3 4. A 200 mm pulley , loaded as shown, is keyed to a shaft of 60 mm diameter. Determine the width “b”, of the 70 mm long key if the allowable shearing stress is 60Mpa. a. 14.56 mm b. 15.84 mm c. 16.25 mm d. 17.46 mm 5 KN Figure 6
2 KN/m
Figure 4
Figure 3
Figure 5
1m
5m
5. A 7/8 inch diameter bolt having a diameter at the root of the threads of 0.731 inch is used to fasten two timbers together as shown. The nut is tightened to cause a tensile strength of 18,000 psi in the bolt. Determine the outside diameter of the washers if their inside diameter is 9/8 inches and the bearing stress is limited to 800 psi. a. 3.18 in. b. 3.65 in c. 4.1 in d. 4.3 in. 6. Determine the position of maximum moment in the beam ABC. a. at A b. at B c. at C d. left of C 7. In how many different orders can 7 books be arranged on a shelf if this 3-volume work is not merely not to be separated but must be kept in the proper order. a) 720 b) 120 c) 5040 d) 4050 8. Water is pouring into a swimming pool. After t hours, there are t+ t1/2 gallons in the pool. At what rate is the water pouring into the pool when t = 9 hours? a. 7/6 gph b. 1 gph c. ½ gph d. 1/6 gph 9. The area enclosed by the ellipse x2/9 + y2/4 =1 is revolved about the line x=3. What is the volume generated? A. 355.3 B. 360.1 C.370.3 D.365.1 10. The total surface of two cubes is 150 sq. m, The total length of their edges is 84 cm. Find the edge of the larger cube. a. 4 b. 5.5 c. 6.4 d. 7.2 11. Glycerin has an index of reaction of 1.47. Find the focal length in glycerin of a lens made of flint glass(n= 1.63) whose focal length in air is 10 cm. a. 11 cm b. 23 cm c. 39 cm d. 58 cm 12. Find the integrating factor for the equation (x2+y2 – x)dx –ydy = 0 a. 1/(x+y) b. y/(x+y) c. 1/(x2+y2) d. 1/x2 e. 1/y2 13. The equation of a circle on the y-axis and passes through the origin and the point (4,2) is a. x2 + y2 - 10y = 0 c. x2 + y2 - 10x = 0 2 2 b. x + y - 10x + 10y = 0 d. x2 + y2 + 10y = 0 14. Two forces PQ and PR of magnitudes 5.0 kg and 8.0 kg, respectively, acts at a point P. The direction of PQ is N20ºE, and the direction of PR is N65ºE. Approximate the magnitude and direction of the resultant PS. a) 11.2 kg, N42oE b) 12.1 kg, N48oE c) 14.1 kgN73oE d) 15.1 kg.N27oE 15. A particle moves according to the following function of time: x(t) = 3sint and y(t) = 4 cost, what is the resultant velocity at t= pi? a. 3 b. 4 c. 9 d. 25 16. Find the volume generated by revolving about the x – axis the area bounded by the line x + 2y – 2 and the x and y axes.
17. Twelve persons are to sit at a round table. Two particular people insist on sitting opposite each other. Find the number of ways the twelve can be seated. a) 2638800 b) 6238800 c) 3826800 d) 3628800 18. Given part of an arithmetic sequence ( -312, ___, ___, -174) and t50 = 1160, what is the value of the 26th term? a. 48 b. 56 c. 67 d. 74 19. A square and a regular hexagon have the same perimeter. If the area of the square is 2.25, what is the area of the hexagon? a. 2.250 b. 2.598 c. 2.838 d. 3.464 20. A steel beam in a bridge extends 25 m across a small stream. What is the change in length from the winter, when the temperature is 20 C, to the summer when it is 37 C. The coefficient of linear expansion of steel is 12x10-6/ C . a. 17mm b. 19 mm c. 21mm d. 23 mm 21. At a temperature of 80°C, a steel tire 12 mm thick and 90 mm wide that is to be shrunk onto a locomotive driving wheel 2 m in diameter just fits over the wheel, which is at a temperature of 25°C. Determine the contact pressure between the tire and wheel after the assembly cools to 25°C. Neglect the deformation of the wheel caused by the pressure of the tire. Assume α = 11.7 x 10 –6 m /(m·°C) and E = 200 Gpa. a. 1.54 MPa b. 1.96 MPa c. 2.24 MPa d. 2.86 MPa
F 8 FT Figure 24 Figure 22 Figure 23
4 FT
W
22. A rigid horizontal bar of negligible mass is connected to two rods as shown in the figure above. If the system is initially stressfree. Calculate the temperature change that will cause a tensile stress of 90 MPa in the brass rod. Assume that both rods are subjected to the change in temperature. a. 35.5˚C b. 42.08˚C c. 56.01˚C d. 60.2˚C 23. A tube 2 mm thick has the shape shown in the figure. Find the shearing stress caused by a torque of 600 N·m. a. 78.4 MPa b. 81.92 MPa c. 83.67 MPa d. 84.35 MPa 24. A cylindrical tank is set at rest as shown. What force, F, should be applied horizontally at point C to raise the tank? a. 25.4 lbf b. 57.7 lbf c. 66.7 lbf d. 100 lbf 25. It is known that y(x) passes through the points (0,2) and (1,4). Solve for y(x) if the second derivative is: (d2y/dx2) = 1 a. Y= X2 + 3X + 4 b. 2Y= X2 + 3X + 2 c. 6Y= 3X2 + 4X + 1 d. NOTG 26. If x increases uniformly at the rate of 0.001 feet per second, at what rate is the expression (1+x)3 increasing when x becomes 9 feet? a. 0.001 cfs b. 0.3 cfs c. 0.003 cfs d. 1.003 cfs 27. An archeologists discovered some ancient objects which had organic remains. He found that 60% of the original amount of C – 14 remains and the disappearance of C-14 obeys the first-order decay law, give an estimation for the age of the objects given the half life of C14 is approximately 5600 years, determine the age of the fossil. a. 4127. 01 yrs. b.4205.6 yrs. c. 4569.5 yrs. d. NOTG 28. Divide the number 60 into two parts so that the product (P) of one part with the square of the other part is a maximum. a. 30, 30 b. 25, 35 c. 50, 10 d. 40, 20 29. How many different committees of 5 can be selected from 15 Juniors and 10 Seniors if it must contain at least 3 Juniors? a) 20475 b) 37128 c) 24075 d) 73128 30.There are 8 baseball teams in a league. How many games will be played if each team plays each of the other teams 40 times? a) 280 b) 560 c) 320 d) 1120 31. A rifle with a muzzle velocity of 200m/s is fired with its barrel horizontal at a height of 1.5 m above the ground. How far away from the rifle does the bullet strike the ground? a. 100 m b. 110 m c. 120 m d. 130 m 32. Given : dy1/dx = (2/13) (1+ 5/2 x - 3/2 – ¾ k), What is the value of k such that y1 is perpendicular to the curve y2 = 2x at (1,1) a. 3 b. 6 c. 7 d. 10 33. If it is known that y = 1 when x =1, what is the constant of integration for the following integral? y(x) = ∫( e2x- 2x) dx a. e2+2 b. ½ e2x + 1 c. ½ e2 + 2 d. NOTG
34. If a cube has an edge of length 2, what is the distance from any vertex to the center of the cube? a. (2) 1/2/2 b. (3)1/2 c. 2(2)1/2 d. 2(3)1/2 35. A parabola has its focus at (7, -4) and directrix Y = 2. Find the equation a. X2 + 12X – 14Y + 61 = 0 b. X2 + 14X – 12Y + 61 = 0 c. X2 – 12X + 14 Y + 61 = 0 d. X2 – 14X + 12 Y + 61 = 0 36. In how many ways can one make a selection of 4 black balls, 5 red balls, and 2 white balls from a box containing 8 black balls, 7 red balls and 5 white balls? a) 14700 b) 17400 c) 47100 d) 10740 37. A certain ball always rebounds to 4/5 of the distance of the previous fall. Suppose it is dropped from a height of 400 meters, what is the total distance it has traveled up and down when it hits the ground for the fourth time? a. 534.8 meters b. 895.8 meters c. 1235.5 meters d. 1961.6 meters 38. The rigid platform in the figure has negligible mass and rests on two steel bars, each 250.00 mmm long. The center bar is aluminum and 249.90 mm long. Compute the stress in the steel bar after the center load P=400 kN has been applied. For each steel bar, the area is 1200 mm2 and E = 200 Gpa. For the Al bar, the area is 2400 mm2 and E=70 GPa. a. 22.5 MPa b. 25.4 MPa c. 32.1 MPa d. NOTG Figure 40.
Figure 38.
8 ft.
2P
P Figure 39
8 ft. 8 ft.
39. Determine the force in member CD. a. P/12 b. P/3 c. 5P/12 d.5P/4 40. Determine the vertical force per unit width at F necessary to resist the wind load of 0.5 psi on DEF. a. 216 psi b. 432 psi c. 648 psi c. 864 psi 41. The angle of elevation of a hot air balloon, climbing vertically, changes from 25 degrees at 10:00 am to 60 degrees at 10:02 am. The point of observation of the angle of elevation is situated 300 meters away from the take off point. What is the upward speed, assumed constant, of the balloon? a. 2.25 m/s b. 2.76 m/s c. 3.16 m/s d. 3.89 m/s e. 4.104 m/s 42. Find the area of the shaded region between y= 6x – 1 and y = ¼ x + 3, bounded by x = 0 and the intersection point. 43. Find the value of x for which the tangent to y = 4x – x2 is parallel to the x – axis. a. 2 b. – 1 c. 1 d–2 44. A mothball loses mass by evaporation at a rate that is proportional to the surface area. If half the mass is lost in 100 days, how long will it take the radius to decrease to half its initial value? a. 243 days b. 255 days c. 234 days d. 275 days e. NOTG 45. Vector A has magnitude 6 units and is in the direction of the +x-axis. Vector B has a magnitude 4 units and lies in the xy-plane, making an angle of 30˚ with the + x-axis. Find the vector product A x B. a. 9 b. 10 c. 11 d. 12 46. A bag contains six white, five red, and four balls. If six balls are drawn, find the probability that there will be two balls of each color. a) 180/1001 b) 108/1001 c) 180/1100 d) 108/1100 47. An elevator system in an office building can either be refurbished or replaced. Refurbishing the elevators will cost Php55,000 and extend the life of the elevators another 20 years. Salvage value at the end of 20 years will be Php11,000. Annual maintenance costs will be Php1,000 per year. The current salvage value of the elevators is Php32,000. Replacing the elevator system will cost Php140,000, with an expected life of 50 years. Maintenance costs will be Php400 per year. Salvage value at the end of 50 years will be Php28,000. Interest is 4%. The annual cost for refurbishing the elevators is: (A) Php4,680 (B) Php5,930 (C) Php7,030 (D) Php8,240 48. The annual cost for replacing the elevators is: (A) Php5,840 (B) Php6,340 (C) Php6,540 (D) Php6,740 49. An inlet pipe can fill up the tank in 6 hours while an outlet pipe can empty the tank in 9 hours. One day, the tank was being filled. When it was 1/3 full, a boy opened the outlet pipe. How long did it take to fill the rest of the tank?
a.8 hours b. 10 hours c.12 hours d. 13 hours 50. The sum of the base and altitude of an isosceles triangle is 36 cm. Find the altitude of the triangle if its area is to be a maximum. a. 16 cm b. 17 cm c. 18 cm d. 9 cm 51. Find the equation of the parabola whose focus is at the origin and whose directrix is the line y = -4. a. x2 = 4y+4 b. x2 = 4y-4 c. x2 = 8y +16 d. x2 = 8y + 8 52. As part of a charity fund-raising drive, a Chicago marathon runner with mass 50.0 kg runs up the stairs to the top of the 443-m tall Sears Tower, the tallest building in the US. In order to lift herself to the top in 15.0 minutes, what must be her average power output in Watts. a. 116 W b. 240 W c. 320 W d. 455 W 53. Locate the centroid of the plane area bounded by the equation y2=4x; x=1 and the x – axis on the first quadrant. A.(3/4,3/5) B.(3/5,3/4) C.(3/5,3/5) D.(3/4,2/3) 54. An airplane is approaching point A along a straight line and at a constant altitude h. At 10:00 am, the angle of elevation of the airplane is 20o and at 10:01 it is 60o. What is the altitude h of the airplane if the speed of the airplane is constant and equal to 600 miles/hour? (round answer to 2 decimal places). a. 4.1 miles b. 4.6 miles c. 5.2 miles d. 5.5 miles 55. A mothball loses mass by evaporation at a rate that is proportional to the surface area. If half the mass is lost in 100 days, how long will it take the radius to decrease to half its initial value? a. 243 days b. 255 days c. 234 days d. 275 days e. NOTG 56. Find the area bounded by y2-3x+3=0 and x=4. A. 10 B. 11 C. 12 D. 13 57. Bill decides to start a 401(k) investment account beginning next year with an initial investment of Php500. His plan is to make annual investments which increase by Php100 each year. If Bill earns 10% on his investment, his 401(k) account will be worth how much in 15 years? (A) Php30,820 (B) Php31,760 (C) Php32,660 (D) Php33,520 58. The lines y = ±(5/12)x are asymptotes of a hyperbola whose foci are 26 units apart. Which CANNOT be a vertex of the hyperbola? a. (-12, 0) b. (0, 5) c. (0, -12) d. (12, 0) 59. Given y1 = 4x+3 and y2 = x2+C, find C such that y2 is tangent to y1. a. 2 b. 5 c. 7 d. 9 60. Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine the no. of infected students after 6 days if it is observed that 50 students are infected after 4 days. Assumed that nobody leaves the campus throughout the duration of the disease. a. 276 students b. 305 students c. 348 students d. 487 students 61. At 7:00 AM, Mr. Castro left home for work driving at a rate of 40kph. 15 minutes later, his wife discovered that Mr. Castro left his wallet at home so Mrs. Castro pursue him on her car at a rate of 50 kph. At what time did Mrs. Castro catch up with her husband? a. 8:00 AM b. 8:15 AM c. 8:30 AM d. 8:45 AM 62. In an arithmetic sequence, t23 = 519 and t71 = 1503, what is t114? a. 3824.5 b. 2458.3 c. 2384.5 d. none of the given 63. A lift station sewage pump initially costs Php20,000. Annual maintenance costs are Php300. The pump salvage value is 10 percent of the initial cost in 20 years. Using 4% interest, the annual cost of the pump is most nearly: (A) Php1,200 (B) Php1,705 (C) Php1,772 (D) Php1,840 64. The total surface of two cubes is 150 sq. m, The total length of their edges is 84 cm. Find the edge of the larger cube. a. 4 b. 5.5 c. 6.4 d. 7.2 65. At each point (x,y) on a certain curve, the tangent line has slope 4x 3 + 2. Find the equation of the curve if it contains the point (1,7) a. x4 + 2x + 3 b. x4 + 2x + 4 c. x4 + 2x + 6 d. NOTG 66. What is the standard equation of the locus of points in the Cartesian plane, the absolute difference of whose distances from the points (±4,0) is 4? a. (x2/9)-(y2/16) =1 b. (x2/4)-(y2/12) =1 c. (x2/12)-(y2/16) =1 d. (x2/12)-(y2/9) =1. 67. Gabriel made three investments totaling P220,000.00 with interest rates at 4%, 5%, and 8%. The total annual income from the three investments was P13,500.00. The income from the 8% investment was P2,500.00 more than the combined income from the other two investments. How much was invested at 5% rate? a. 50000 b. 60000 c. 70000 d. 100000 68. An investment option is available with continuous compounding at 5% interest. If you invest Php8,000 now, how much interest income will you earn if you cash out in 3.5 years?
(A) Php1,172 (B) Php1,261 (C) Php1,490 (D) Php1,530 69. A conic has the lines y = -3 and y = 5 as directrices and the point (1, 26) as a focus. What are the coordinates of the vertex closest to the given focus? a. (1,11) b. (2, 11) c. (3, 8) d. NOTG 70. Find the particular solution of y''' = 0 given that: y(0) = 3, y'(1) = 4, y''(2) = 6 a. y = 2x2 − 3x + 3 b. y = 3x2 − 2x + 3 c. y = 2x2 − 2x + 2 d. NOTG 71. A passenger weighing 60 kg rides a ferris wheel with a radius of 8.0 m. The wheel makes one revolution in 10 s. Calculate the force the seat exerts on the passenger at the top of the circle. a. 398 N b. 498 N c. 588 N d. 778 N 72. A ship leaves port at 1 pm traveling north at the speed of 30 miles/hour. At 3 pm, the ship adjusts its course 20 degrees eastward. How far is the ship from the port at 4pm? a. 55 miles b. 68 miles c. 73 miles d. 89 miles 73. You are considering investing in a 5-yr CD(certificate of deposit) with an annual yield of 6.5% and monthly compounding. If you invest Php5,000, your effective interest earned is most nearly: (A) 6.5% (B) 6.6% (C) 6.7% (D) 6.8% 74. A plane cuts through a cube in such a way that the plane passes through three of the cube’s vertices, no two of which lie on the same edge. If the edge of the cube has a length of 1, what is the surface area of the smaller fragments of the cube? a. 2.4 b. 2.7 c. 3.1 d. 3.7 e. 4.3 75. You throw a 0.145 kg baseball straight up in the air, giving it an initial upward velocity of magnitude 20 m/ s. Use the conservation of energy to find how high it goes, ignoring air resistance. a. 15.2 m b. 17.8 m c. 20.4 m d. 21.9 m 76. A man draws a card from each of 2 separate decks. What is the probability that the card from the first deck is an ace and the one from the second deck a face card? a) 14/169 b) 2/169 c) 16/169 d) 3/169 77. A tank is initially filled with 100 gal of salt solution containing 1 lb of salt per gallon. Fresh brine containing 2 lb of salt per gallon runs into the tank at the rate of 5 gal/min, and the mixture , assumed to be kept uniform by stirring runs out at the same rate. Determine how long it will take for this amount to reach 150 lb. a. 10.2 min b. 12.6 min c. 13.9 min d. 14.5 min 100lb f
30 lbf/ft
60º
Figure 80
2000 lbf
1” Figure 78
2”
1” 8 ft
Figure 79
10 ft
30º
2” 8 ft
30º
78. The beam is supported by a cable and is loaded as shown. What is the tension in the cable? The beam is supported by a cable and is loaded as shown. What is the tension in the cable? a. 200 lbf b. 250 lbf c. 430 lbf d. 500 lbf 79. Determine the moment of inertia around the centroidal axis of the beam. a. 416 in4 b. 650 in4 c. 735 in4 d. 1067 in4 80. A scissor jack is used to raise a car. If the jack supports a weight of 2000 lbf, determine the force in member BE. a. 0 lbf b. 500(3)1/2 lbf c. 1000(3)1/2 lbf d. 2000(3)1/2 lbf
