1

400 KN 3m 300 KN 4m 200 KN 10 KN/m 15 KN/m 4m 6m

TECHNOLOGICAL INSTITUTE OF THE PHILIPPINES MANILA

PRE BOARD EXAM DAY 3

STUDENT NUMBER_______________________________________________________DATE________________ DIRECTIONS: PROVIDE THE SOLUTION TO SELECT THE CORRECT ANSWER.

1. Cynthia’s class number is a two digit number. The sum of the digits is 12. If the digits are reversed, the resulting number will be 15 more than twice her class number. What is her class number?

a. 39 b. 66 c. 48 d. 57

2 Find the orthogonal family to the family of circles, x2 _{+ y}2 _{= C}2

a. y = mx + b b. y = ln x c. y = mx d. NOTG

3. Simplify the following expression:

a. 2i/(2)1/2 _{b. 4i/(2)}1/2 _{c. (2)}1/2 _{/4i d. none of the given}

4. A 200 lb man hangs from the middle of a tightly stretched rope so that the angle between the rope and the horizontal direction is 

5

. Calculate the tension of the rope?

a. 1104 lb b. 1147 lb c. 1205 lb d. 1250 lb

5. The length of a steel wire is 30 ft with a load of 500 lb. If the maximum allowable stress is 20 ksi with a maximum allowable elongation of 0.20 inch. Calculate the diameter of the wire, E = 29 x 106 _psi

a. 0.50 in b. 0.36 in c. 0.95 in d. 0.20 inch

6. Determine the maximum shearing stress and elongation in a helical steel spring composed of 20 turns of 20-mm-diameter wire on a mean radius of 90 mm when the spring is supporting a load of 1.5 kN. G = 83 GPa.

a. 30 MPa b. 99.87 MPa c. 78.65 MPa d. NOTG

7. Six draws are made from an ordinary deck of playing cards, the card drawn being replaced and the deck thoroughly shuffled before the next draw is made. What is the probability of obtaining exactly 4 spades?

a. 135/4960 b. 153/4096 c 135/4690 d. 135/4096

8. Suppose that a corpse was discovered in a motel room at midnight and its temperature was 80F. The temperature of the room is kept constant at 60F. Two hours later the temperature of the corpse dropped to75F. Find the time of death. Note: The temperature of a live person is 98.6F.

a. 8:30 PM b. 7:30 PM c. 9:00 PM d. NOTG

9. The plate is acted on by 4 parallel forces. Three of which are shown. The fourth force P and its line of action are unknown. The resultant of this force system is the couple –vector CR _{= -1100i + 1500j KN-m. Determine P and its line of action (x,y)}

a. (3m, 4m) b. (4m, 3m) c. (3m, 3m) d. (4m, 2m)

Figure 10 Figure 9

10. Determine the resultant of the line load acting on the beam shown in the figure.

a. P = 105 KN b. P=75 KN c. P=90 KN d. P=45 KN

11. Tom’s retirement account in a company currently totals Php416,384. What perpetual income can Tom and his heirs receive per year if he retires now and the money is invested in an annuity earning 6% interest?

(A) Php20,980 (B) Php22,980 (C) Php24,980 (D) Php26,980

12. The equation of a circle on the y-axis and passes through the origin and the point (4,2) is

a. x2 _{+ y}2 _{- 10y = 0 c. x}2 _{+ y}2 _{- 10x = 0}

b. x2 _{+ y}2 _{- 10x + 10y = 0 d. x}2 _{+ y}2 _{+ 10y = 0}

13. Find the largest area of a rectangle with vertices at the origin of a cartesian coordinates system, on the x – axis, on the y – axis, and on the parabola y = 4 – x2_.

a. 16/3 b. 16/3½ _{c. 4(3)} - ½ _{d. NOTG}

14. Find the centroid of a hemisphere of radius R revolving about the x-axis.

a. 5/8 from center b. 2/5 from center c. 3/5 from center d. 3/8 from center

15. Any line through two points on a curve is called.

a. tangent line b. secant line c. slope d. normal line

16. The solution for dy/dx = xe –x _{when y(0) = 0}

a. y = - e–x _{(x+1) b. y = - e}–x _{(x+1) + 2 c. y = - e}–2x _{(x+1) + 1 d. NOTG}

17. Given the force vectors; A = 8i + 4j -2k and the position vectors B = 2j + 6k m and C =3i – 2j + 4k m, the dot product of A and B is

a. – 4N.m b. – 3.71 N.m c. – 4.5 N.m d. 7.3 N.m

18. Evaluate the definite integral

19. You are saving up for a big investment in six years. You estimate it will take Php14,500 to secure this investment. How much do you need to put into a savings account at the end of each year if the savings account earns 4%?

(A) Php2,185 (B) Php2,375 (C) Php2,415 (D) Php2,485

20. A man wants to fence a rectangular area as inexpensively as possible. If the fencing material cost Php 38 per meter and Php 1,520 is available for the project, how much area can be enclosed?

a. 100 m2 _{b. 105 m}2 _{c. 110 m}2 _{d. NOTG}

21. Find the smallest angle between the lines 2x + y = 0 and x + 3y + 4 = 0

a. 40° b. 60° c. 45° d. 30°

22. In preparing a tea, 600 g of water at C 

90

is poured into 200 g china pot whose specific heat capacity is 840J/kg C  and temperature of

C



20

. What is the final temperature of water? a. C  6 . 81 b. C  6 . 83 c. C  6 . 85 d. C  6 . 87

23. In a certain culture the rate of growth of bacteria is proportional to the amount present. Initially, 1000 bacteria are present and the amount doubles in 12 minutes. Determine the time it takes for the bacteria to multiply to 10,000.

a. 20 min b. 40 min c. 60 min d. NOTG

24. Find the equation of the hyperbola with vertices at ( ±2,0) and asymptotes y = ±3x

a. 4x2 _{– 36y}2 _{= 144 b. 36x}2 _{– 4y}2 _{= 144 c. 9x}2 _{– 4y}2 _{= 36 d. 4x}2 _{– 9y}2 _{= 36}

25. Certain manufactured articles are sold in boxes of 25 each. They are inspected by taking a random sample of 5 from each box and passing the box as satisfactory if no defective articles are found in the sample. Find the probability that a box will be passed if it contains 1 defective.

26. Assume that a 20-mm-diameter rivet joins the plates that are each 110 mm wide. The allowable stresses are 120 MPa for bearing in the plate material and 60 MPa for shearing of rivet. Determine (a) the minimum thickness of each plate; and (b) the largest average tensile stress in the plates.

a. 8.9 mm, 30 MPa b. 7.9 mm, 26.7 MPa c. 4.2 mm, 28.5 MPa d. NOTG

27. At each point (x,y) on a certain curve, the tangent line has slope 4x3 _{+ 2. Find the equation of the curve if it contains the point} (1,7)

a. x4 _{+ 2x + 3 b. x}4 _{+ 2x + 4 c. x}4 _{+ 2x + 6 d. NOTG}

28. 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

29. Find an approximation for the sum of all the terms of {16, -8, 4, -2, 1, -1/2, ¼, …}

a. 11 1/3 b. 10 2/3 c. 9 ¼ d. none of the given

30. A circle whose equation is X2 _{+ Y}2 _{+ 4X+ 6Y- 23 = 0 has a radius of}

a. 3 b. 4 c. 5 d. 6

31. A man 5.5 ft tall walks away from a lamppost 10 feet high at a rate of 8 feet per second. How fast does his shadow lengthen?

a. 88/9 fps b. 160/9 fps c. 9/88 fps d. 9/160 fps

32. A project requires 8 yearly investments. The initial investment at end of year one isPhp20,000 with a 5% negative gradient for the other 7 investments. Interest is 6%. The present cost of the total investment required is most nearly

(A) Php98,465 (B) Php104,355 (C) Php110,515 (D) Php132,000

33. The graph of a polar equation in the shape of a heart is known as

a. limacon with a dimple b. limacon c. cardiod d. limacon with inner loop

34. A tank can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the tank is full, it can be drained by pipe C in 4 hours. if the tank is initially empty and all three pipes are open, how many hours will it take to fill up the tank?

a. 4.55 hours b. 2.84 hours c. 3.53 hours d. none of the given

35. A rubber ball is dropped on a floor from a height of 5 m. If the coefficient of restitutions is 0.7, find the height of the which ball rebounds?

a. 2.45 m b. 2.75 m c. 3.25 m d. 3.5 m

36. A revolving beacon is located 1000 ft. from a straight sea wall and rotates at the constant rate of 2 rpm. How fast does the beam of light cast by the beacon sweep along the sea wall at the nearest point to the beacon?

a. 262 fps b. 210 fps c. 186 fps d. NOTG

37. Compute the derivative of the given function at the indicated point. Y = (x3 _{– x + 1)/(x}3 _{+ 1) at x = 2}

a. 4/28 b. 5/27 c. 2/19 d. NOTG

38. As shown in Fig. P-269, there is a gap between the aluminum bar and the rigid slab that is supported by two copper bars. At 10°C, Δ = 0.18 mm. Neglecting the mass of the slab, calculate the stress in each rod when the temperature in the assembly is increased to 95°C. For each copper bar, A = 500 mm2_{, E = 120 GPa, and α = 16.8 µm/(m·°C). For the aluminum bar, A = 400} mm2_{, E = 70 GPa, and α = 23.1 µm/(m·°C).}

a. Stress copper = 30.35 MPa, Stress Al = 12.14 MPa b. Stress copper = 12.14 MPa, Stress Al = 30.35 Mpa

c. Stress copper = 35.30 MPa, Stress Al = 14.12 MPa d. Stress copper = 20.35 MPa, Stress Al = 22.14 MPa 39. A 6-in.-long bronze tube, with closed ends, is 3 in. in diameter with a wall thickness of 0.10 in. With no internal pressure, the tube just fits between two rigid end walls. Calculate the longitudinal and tangential stresses for an internal pressure of 6000 psi. Assume ν = 1/3 and E = 12 × 106 _psi.

a. tangential = 40000 psi, longitudinal = 20000 psi b. tangential = 90000 psi, longitudinal = 30000 psi

c. tangential = 80000 psi, longitudinal = 50000 psi d. tangential = 50000 psi, longitudinal = 30000 psi

40. Two charges, one of + C

 5 . 0

and the other of C

 2 . 0

, attract each other with a force of 100 N. How far apart are they?

a. 3 mm b. 4 mm c. 30 mm d. 40 mm

41. A particular moon rock is analyzed and it is found to contain equal amounts of Potassium-19 and Argon-18, which are denoted as , respectively. is a radioactive isotope that can decay to . Note that for potassium decay and that the yield of is about 11% (i.e. about 1 out of every 9 atoms decays to ). Assuming that the initial amount of argon was zero and that all of the argon is the result of potassium decay, estimate the age of the moon rock.

a. 2 billion years b. 4.3 billion years c. 5.3 billion years d. NOTG

42. In how many ways can 3 red balls, 4 black balls, and 2green balls be arranged adjacent to each other in a shelf if one does not distinguish between balls of the same color?

a. 1260 b. 1344 c. 1458 d. NOTG

43. Towns A and B are on a north-south line and 34.6 miles apart. An observer on the east of that line finds that the direction of A and B are N37o _{20’ W and S 52}o _{40’ W. How far is the farther town from the observer?}

a. 12.0 miles b. 21.0 miles c. 27.5 mile d. 25.7 miles 44. 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 = 3x}2 _{− 2x + 3 c. y = 2x}2 _{− 2x + 2 d. NOTG}

45. 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

46. The direction of the resultant force from the x-axis for the three forces shown in the figure is a. cosθ = - 0.302 b. cosθ = - 0.864 c. cosθ = 0.403 d. cosθ = 0.252

47. For the machine part shown in the figure, replace the applied load of 150 kN acting at point A by 2 horizontal forces, one acting at point B, the other at point C.

a. – 18 KN b. 450 KN c. 150 KN d. 300 KN.

48. A nominal rate of 9.5% has an equivalent effective rate of 9.84%. What is the compounding period? a) monthly b) bi-monthly c) quarterly d) semi- annually

49. 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. 3.54 b. 3 c. 4 d. none of the given

50. What is the probability that 5 cards dealt from a well shuffled deck , 3 will be hearts and 2 spades?

a. 0.00633 b. 0.00858 c. 0.011 d. 0.0153

51. Evaluate the definite integral:

52. A total of 0.8 kg of water at C 

20

is placed in 1 kW electric kettle. How long a time is needed to raise the temperature of water to C  100 . Figure 39 Figure 38 150KN 80 mm Figure 47 Figure 46. 40 mm 40 mm 80 mm

a. 3.5 min b. 4.5 min c. 5.5 min d. 6.5 min 53. Evaluate the integral

54. A steel shaft 3 ft long that has a diameter of 4 in is subjected to a torque of 15 kip·ft. Determine the maximum shearing stress. Use G = 12 × 106 _psi.

a. 15.3 ksi b. 18.3 ksi c. 14.3 ksi d. NOTA

55. A flanged bolt coupling consists of 8 10 – mm diameter steel bolts on a bolt circle 400 mmm in diameter, and 6 10 mm diameter steel bolts on a concentric bolt circle 300 mm in diameter. What torque can be applied without exceeding a shearing stress of 60 Mpa in the bolts.

a. T = 10.7 kN.m b. 15.7 kN.m c. 20.5 kN.m d. NOTA

56. The 36 m cable shown in Figure weigh 1.5 KN/m. determine the sag H and the maximum tension in the cable. a. T = 44.3 KN, H = 8.67 m b. T = 34.3 KN, H = 8.77 m

c. T = 74.3 KN, H = 5.77 m d. T = 94.3 KN, H = 6.77 m

57. Miles is camping in Glacier National Park. In the midst of a glacier canyon, he makes a loud holler. He hears an echo 1.22 seconds later. The air temperature is 20 degrees C. How far away are the canyon walls?

a. 310 m b. 209 m c. 107 m d. 236 m

58. You purchase 1,000 shares of stock at Php15.00 per share. The stock pays quarterly dividends of Php125 for two years at which time you sell the stock at the trading price of Php16.50 per share. The yearly return on your investment is most nearly:

(A) 2% (B) 4% (C) 6% (D) 8%

59.What is the distance between the center of the circles X2 _{+ Y}2 _{+ 2X + 4Y – 3 = 0 and X}2 _{+ Y}2_{- 8X – 6Y + 7 = 0}

a. 7.07 b. 7.77 c. 8.07 d. 8.77

60. Calculate the distance when the moment is zero.

a. 3.25 m b. 2.55 m c. 2.154 m d. 1.75 m

61. A company produces transmission gears used by several farm tractor manufacturers. The base cost of operation is Php596,700 per year. The cost of manufacturing is Php18.40 per gear. If the company sells the gears at an average price of Php37.90 each, how many gears must be sold each year to break even?

(A) 15,740 (B) 18,400 (C) 30,600 (D) 32,430

62. In a triangle ABC, angle B = 45o _{and angle A= 60}o_{. What is the ratio of the sides AB to AC?} a. 1.37 b. 1.22 c. 1.48 d. 0.73

63. An elevator system for a 20-yr old high-rise office building cost Php400,000 when first installed. The system was designed to last 30 years, with salvage at 10% of initial cost. Depreciation is straight line. The allowable depreciation charge per year is:

(A) Php12,000 (B) Php13,333 (C) Php18,000 (D) Php20,000

64. From the mountain 1780 feet high, the angle of depression of a point on the nearer shore of a river is 4840’ and of a point directly across on the opposite side is 2220”. What is the width of the river between the two points.

a) 2767 feet b) 4332.9 feet c) 1565 feet d) 3381 feet

65. The daily cost to a small firm of producing x units of a product is C(x) = 2002 + 120 x – 5x2 + 1/3 x3 dollars. The market price of 1 unit is $264. What should be the daily output x in order to maximize the daily profit.

a. 10 units b. 18 units c. 36 units d. NOTG

66. Evaluate the indefinite integral



dx

Figure 56 Figure 55

67. It is a quantity that when applied to a member, tends to deform the member by causing a rotation of one part of the member relative to another.

a. couple b. torque c. axial force d. reaction

68.The set of all possible outcomes in an experiment.

a. sample space b. event c. sample point d. population

69. Future value less the present value

a. discount b. rate discount c. discount rate d. profit

70. It is the worth of the property as recorded in the books of account of the company.