Exam 3 August 2012, questions and answers PDF

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AUGUST 2012 MATHEMATICS The equation y^2 = cx is the general equation of: A. y’ = 2y/x B. y’ = 2x/y C. y’ = y/2x D. y’ = x/2y A line segment joining two points on a circle is called: A. arc B. tangent C. sector D. chord Sand is pouring to form a conical pile such that its altitude is always twice it...

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REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

1. The equation y^2 = cx is the general equation of: A. y’ = 2y/x B. y’ = 2x/y C. y’ = y/2x D. y’ = x/2y 2. A line segment joining two points on a circle is called: A. arc B. tangent C. sector

D. chord

3. Sand is pouring to form a conical pile such that its altitude is always twice its radius. If the volume of a conical pile is increasing at the rate of 25 pi cu.ft/min, how fast is the radius is increasing when the radius is 5 feet? A. 0.5 ft/min B. 0.5 pi ft/min C. 5 ft/min D. 5 pi ft/min 4. Evaluate ʃ ʃ 2r²sin Ө dr dӨ, 0 > r >sin Ө, > Ө > pi/2 A. pi/2 B. pi/8 C. pi/24

D. pi/48

5. A shopkeeper offers a 25% discount on the marked price on an item. In order to now cost $ 48, what should the marked price be? A. $ 12 C. $ 60 B. $ 36 D. $ 64 6. An observer wishes to determine the height of a tower. He takes sights at the top of the tower from A to B, which are 50 ft. apart, at the same elevation on a direct line with the tower. The vertical angle at point A is 30 degrees and at point B is 40 degrees. What is the height of the tower? A. 85.60 ft B. 143.97 ft C. 110.29 ft D. 92.54 ft 7. A tangent to a conic is a line A. which is parallel to the normal B. which touches the conic at only one point C. which passed inside the conic D. all of the above 8. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axes. A. 3 B. 4 C. 5 D. 2 9. Find the general solution of (D² - D + 2)y = 0 A. y = e^x/2 (C1 sin sqrt. 7/2 x + C2 cos sqrt. 7/2 x) B. y = e^x/2 (C1 sin sqrt. 7/2 x - C2 cos sqrt. 7/2 x) C. y = e^x/2 (C1 cos sqrt. 7/2 x + C2 sin sqrt. 7/2 x) D. y = e^x/2 (C1 cos sqrt. 7/2 x - C2 sin sqrt. 7/2 x) 10. If 10 is subtracted from the opposite of a number, the difference is 5. What is the number? A. 5 B.15 C.-5 D. -15

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

11. If y = 5 – x, find x when y = 7 A. 12 B.-12

C. 2

D. -2

12. A ranch has a 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 13. Martin bought 3 pairs of shoes at P240 each pair and 3 pieces of t-shirts at P300 each. How much did he spent? A. P720 B. P900 C. P22,500 D. P 1,620 14. Find the standard equation of the circle with the center at (1,3) and tangent to the line 5x – 12y -8 =0. A. (x-1)2 + (y-3)2 = 8 C. (x-1)2 + (y-3)2 = 9 B. (x-1)2 + (y-3)2 = 12 D. (x-1)2 + (y-3)2 = 23 15. Find the volume of the solid formed by revolving the area bounded by the curve y2 = (x3)(1-x) in the first quadrant about x-axis. A. 0.137 B. 0.147 C. 0.157 D.0.167 16. In the pile of logs, each layer contains one more log than the layer above and the top contains just one log. If there are 105 logs in the pile, how many layers are there? A. 11 B. 12 C. 13 D. 14 17. A wall 8 feet high is 3.375 feet from a house. Find the shortest ladder that will reach from the ground to the house when leaning over the wall. A. 16.526 ft B. 15.625 ft C. 14.625 ft D. 17.525 ft 18. If f(x) = 10x + 1, then f(x+1) is equal to A. 10(10x ) B. 9(10x)

C. 1

D. 9(10x+1)

19. A particle moves on a straight line with a velocity v = (4 – 2t) 3 at time t. Find the distance traveled from t = 0 to t = 3. A. 32 B. 36 C. 34 D. 30 20. 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. 365.10 21. If the vertex of y = 2x2 + 4x + 5 will be shifted 3 units to the left and 2 units downward, what will be the new location of the vertex? A. (-2, 1) B. (-5, -1) C. (-3,1) D. (-4,1)

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

22. A coat of paint of thickness 0.01 inch is applied to the faces of a cube whose edge is 10 inches, thereby producing a slightly larger cube. Estimate the number of cubic inches of paint used. A. 4 B. 6 C. 3 D. 5

23. Find the mass of lamina in the given region and density function: π D [ (x , y) ] , 0≤ x ≤ , 0 ≤ y ≤ cos x∧ρ=7 x 2 A. 2

B. 3

C. 4

D. 5

24. Find the area of the region bounded by the curves y = x2 – 4x and x + y = 0 A. 4.5 B. 5.5 C. 6 D. 5 25. A conic section whose eccentricity is less than one is known as: A. circle B. parabola C. hyperbola

D. ellipse

26. The plate number of a vehicle consists of 5-alphanumeric sequence is arranged such that the first 2 characters are alphabet and the remaining 3 are digits. How many arrangements are possible if the first character is a vowel and repetitions are not allowed? A. 90 B. 900 C. 9,000 D. 90,000 27. The axis of the hyperbola, which is parallel to its directrices, is known as: A. conjugate axis B. transverse axis C. major axis D. minor axis 28. The minute hand of a clock is 8 units long. What is the distance traveled by the tip of the minute hand in 75 minutes. A. 10pi B. 20pi C. 25pi D. 40pi 29. Find k so that A = (3, -2) and B = (1, k) are perpendicular. A. 2 B. 3 C. 1/2

D. 3/2

30. The probability of a defect of a collection of bolts is 5%. If a man picks 2 bolts, what is the probability that does not pick 2 defective bolts? A. 0.950 B. 0.9975 C. 0.0025 D. 0.9025 31. If f(x) = A. -7

1 ,(f·g)’*(1) = 6 and g’(1) = -1, then g(1) = x−2 B. -5 C. 5

D. 7

32. 3 randomly chosen senior high school students were administered a drug test. Each student was evaluated as positive to the drug test (P) or negative to the drug test (N). Assume the possible combinations of the 3 students drug test

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

evaluation as PPP, PPN, PNP, NPP, PNN, NPN, NNP, NNN. Assuming each possible combination is equally likely, what is the probability that at least 1 student gets a negative result? A. 1/8 B. 1/2 C. 7/8 D. ¼ 33. The tangent line to the function h(x) at (6, -1) intercepts the y-axis at y = 4. Find h’ (6). A. -1/6 B. -2/3 C. -4/5 D. -5/6 34. The cable of a suspension bridge hangs in the form if a parabola when the load is uniformly distributed horizontally. The distance between two towers is 150m, the points of the cable on the towers are 22 m above the roadway, and the lowest point on the cables is 7 m above the roadway. Find the vertical distance to the cable from a point in the roadway15 m from the foot of a tower. A. 16.6 m B. 9.6 m C. 12.8 m D. 18.8 m 35. In how many ways different orders may 5 persons be seated in a row? A. 80 B. 100 C. 120 D. 160 36. The symbol “/” used in division is called. A. modulus B. minus

C. solidus

D. obelus

37. Find the area of one loop r2 = 16 sin 2theta. A. 16 B. 8

C. 4

D. 6

38. Find the centroid of the upper half of the circle x2 + y2 = 9. C. (0, 5/pi) A. (0, 3/pi) B. (0, 4/ pi¿

D.(0, 6/pi)

39. In polar coordinate system, the distance from a point to the pole is known as A. polar angle C. radius vector B. x-coordinate D.y-coordinate 40. The number that is subtracted in subtraction. A. minuend C. dividend B. subtrahend D. quotient 41. In how many ways can a person choose 1 or more of a 4 electrical appliances? A. 12 B. 13 C. 14 D. 15 42. The surface area of a spherical segment. A. lune B. Zone

C. Wedge

D. sector

43. A particle has a position vector (2cos2t, 1+3sint). What is the speed of the particle at time t = pi/4? A. 1.879 B. 4.5 C. 5.427 D. 7.245

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

44. If the equation is unchanged by the substitution of –x for x, its curve is symmetric with respect to the A. y-axis C. origin B. x-axis D. line 45 degrees with the axis 45. Find the number of sides of a regular polygon if each interior angle measures 108 degrees. A. 7 B. 8 C.5 D. 6 46. The integer part of common logarithm is called the________. A. radicand B. root C. characteristic

47. The constant “e” is named in honor of: A. Euler B. Eigen

C. Euclid

D. mantissa

D. Einstein

48. A man rows upstream and back in 12 hours. If the rate of the current is 1.5 kph and that of the man in still water is 4 kph, what was time spent downstream? A. 1.75 hrs B. 2.75 hrs. C. 3.75 hrs D. 4.75 hrs 49. The probability that A can solve a given problem is 4/5, that B can solve it is 2/3, and that C can solve it is 3/7. If all three try, compute the probability that the problem will be solved. A. 101/105 B. 102/105 C. 103/105 D. 104/105 50. A steel ball at 110 deg C cools in 8 min to 90 deg c in a room at 30 deg C. Find the temperature of the ball after 20 minutes. A. 58.97 °C B. 68.97 °C C. 78.97 °C D. 88.97 °C 51. A freight train starts from Los Angeles and head for Chicago at 40 mph. Two hours later passenger train leaves the same station for Chicago traveling at 60 mph. How long will it be before the passenger train overtakes the freight train? A. 3 hrs B. 4 hrs C. 5 hrs D. 6 hrs 52. Given the triangle ABC in which A = 30 deg 30 min, b = 100 m and c = 200 m. Find the length of the side a. A. 124.64 m B. 142.24 m C. 130.50 m D. 103.00 53. Lines that intersect in a point are called______. A. Skew lines B. Intersecting lines C. Agonic lines D. Coincident lines 54. Find the average rate of change of the area of a square with respect to its side x as x changes from 4 to 7. A. 14 B. 6 C. 17 D. 11

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

55. If the distance x from the point of departure at time t is defined by the equation x = -16t2 + 5000t + 5000, what is the initial velocity A. 20000 B. 5000 C. 0 D. 3000 56. What conic section is represented by 2x2 + y2 – 8x + 4y = 16? A. parabola B. ellipse C. hyperbola

D. circle

57. If 9 ounces of cereal will feed 2 adults or 3 children, then 90 ounces of cereal, eaten at the same rate, will feed 8 adults and how many children? A. 8 B. 12 C.15 D. 18 58. Mary is twice as old as Helen. If 8 is subtracted from Helen’s age and 4 is added to Mary’s age, Mary will then be four times as old as Helen. How old is Helen now? A. 24 B. 36 C. 18 D. 16 59. A point on the curve where the second the derivative of a function is equal to zero is called. A. maxima B. minima C. point of inflection D. point of intersection 60. Find the area of the triangle whose sides are 25, 39, and 40. A. 46 B. 684 C. 486

D. 864

61. A/An_______triangle is a triangle having three unequal sides. A. oblique B. scalene C. equilateral D. isosceles 62. Find the length of the arc of 6xy = x4 + 3 from x = 1 to x = 2. A. 1.34 B. 1.63 C. 1.42

D. 1.78

63. Give the degree measure of angle 3pi/5 radians. A. 108 B. 120 C. 105

D. 136

64. What do you call a radical expressing an irrational number? A. surd B. radix C. complex number

D. index

65. Find the derivative of the function f(x) = (2x – 3x)2. A. 2x - 4 B. 2x - 3 C. 6x - 8

D.

8x

-12

66. What is the length of the line with a slope of 4/3 from a point (6, 4) to the yaxis? A. 10 B. 25 C. 50 D. 75 67. The inclination of the line determine by the points (4, 0) and (5 A. 30 degrees B. 45 degrees C. 60 degrees

√ 3 ) is D. 90 degrees

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

68. A sequence of numbers where the succeeding term is greater than the preceding term is called: A. dissonant resonance C. Isometric series B. convergent series D.divergent series 69. Find the value of x for which y = 4 + 3x – 3x3 will have a maximum value. A. 0 B. -3 C. -2 D. 1 70. How many cubic meters is 500 gallons of liquid? A. 4.8927 B. 3.0927 C. 2.8927

D. 1.8927

71. A certain radioactive substance has a half-life of 3 years. If 10 grams are present initially, how much of the substance remains after 9 years? A. 1.50 grams B. 1.25 grams C. 2.50 grams D. 1.75 grams

72. A statement of the truth of which is admitted without proof is called: A. an axiom B. a postulate C. a theorem D. a corollary 73. A rectangular trough is 8 feet long, 2 feet across the top and 4 feet deep. If water flows in at a rate of 2 ft 3/min, how fast is the surface rising when the water is 1 ft deep? A. ¼ ft/min B. ½ ft.min C. 1/8 ft/min D. 1/6 ft/min 74. Find the point(s) on the graph of y = x 2 at which the tangent line is parallel to the line y = 6x -1. A. (3, 17) B. (3, 9) C. (1, 2) D. (2, 4) 75. How many petals are three in the rose curve r = 3 cos 5theta? A. 5 B. 10 C. 15

D. 6

76. Find the acute angle between the vectors z1 = 3 – 4i and z2 = -4 + 3i. A. 17 deg 17 min C. 15 deg 15 min B. 16 deg 16 min D. 18 deg 18 min 77. If z1 =1 – i and z2 = -2 + 4i evaluate z12 + 2z1 – 3. A. -1 + 4i B. 1 - 4i C. -1 – 4i

D.

1

+

4i

78. A motorboat moves in the direction N 40 deg E for 3 hours at 20 mph. How far north does it travel? A. 58 mi B. 60 mi C. 46 mi D. 32 mi 79. Find the value of 4 sinh(pi i/3) A. 2i (sqrt. of 3) B. 4i (sqrt. of 3)

C. i (sqrt. of 3)

D. 3i (sqrt. of 3)

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

80. Find the upper quartile in the set (0, 1, 3, 4) A. 0.5 B. 0.25 C. 2

D. 3.5

81. In debate on two issues among 32 people, 16 agreed with the first issue, 10 agreed with the second issue and of these 7 agreed with both. What is the probability of selecting a person at random who did not agree with either issue? A. 1/32 B. 13/32 C. 3/8 D. 3/10 82. From the top of the lighthouse, 120 m above the sea, the angle of depression of a boat is 15 degrees. How far is the boat from the lighthouse? A. 448 m B. 428 m C. 458 m D. 498 m 83. The cross section of a certain trough are inverted isosceles triangles with height 6 ft and base 4 ft. Suppose the trough contains water to a depth of 3 ft. Find the total fluid force on one end. A. 187.2 lb B. 178.2 lb C. 192.4 lb D. 129.4 lb 84. Two lines are not coplanar. A. Parallel lines B. Skew lines C. Secant lines 85. Find the inverse Laplace transform of A. 2 e-3t

D. Straight lines

−2 . s−3

B. 2e3t

C. 3e-2t

D. 3e2t

86. Find the length of the latus rectum of the curve rcos2 theta – 4cos theta = 16sin theta. A. 4 B. 16 C. 12 D. 18 87. A quadrilateral with no pair of parallel sides. A. Trapezoid B. Trapezium C. Rhombus

D. Rhomboid

88. Find the equation of the line tangent to the curve y = x 3 – 6x2 + 5x + 2 at its point of inflection. A. 7x – y B. -7x + y = 0 C. 7x +y = 10 D. -7x – y = 10 89. Find the area of the polygon with vertices at 2 + 3i, 3 + i, -2 – 4i, -1 + 2i. A. 47/5 B. 47/2 C. 45/2 D.45/4 90. Find the radius of curvature of y = x3 at x =1. A. 5.27 B. 4.27

C. 6.27

D.

7.27

91. Determine the probability of throwing a total of 8 in a single throw with two dice, each of whose faces is numbered from 1 to 6. A. 1/3 B. 1/18 C. 5/36 D. 2/9

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

92. Find the distance between the point (3, 2, -1) and the plane 7x – 6y + 6z + 8 = 0. A. 1 B. 2 C. 3 D. 4 93. How many parallelograms are formed by a set of 4 parallel lines intersecting another set of 7 parallel lines? A. 123 B. 124 C. 125 D. 126 94. The graphical representation of the cumulative frequency distribution in a set of statistical data is called: A. Ogive B. Histogram C. Frequency polyhedron D. mass diagram 95. Find the area bounded by the curve defined by the equation x 2 = 8y and its latus rectum. A. 11/3 B. 32/3 C. 16/3 D. 22/3 4 96. Evaluate lim ( i z +3 z ²−10 i ) z→2i

A. -12 +6i

B. 12 - 6i

97. Naperian logarithm have a base of A. 3.1416 B. 2.171828

C. 12 +6i

D. -12 – 6i

C. 10

D. 2.71828

98. If an aviator flies around the world at a distance 2km above the equator, how many more km will he travel than a person who travels along the equator? A. 12.6 km B. 16.2 km C. 15.8 km D. 18.5 km 99. Find the volume of a spherical whose central angle is pi/5 radians on a sphere of radius 6 cm. A. 90.48 cu. cm B. 86.40 cu. cm C. 78.46 cu. cm D. 62.48 cu. cm 100. What is the coefficient of the (x -1) 3 term in the Taylor series expansion of f(x) = lnx expanded about x = 1? A. 1/6 B. 1/4 C. 1/3 D. 1/2

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

SOLUTIONS: 1. Solution: y 2=cx c=

y2 x

Differentiate: 0=x(2 y y ' )− y 2 ¿/ x 2 y 2=2 xyy ' y'=

y2 = y /2 x 2 xy

2. D. chord 3. Solution: h=2r , r=5 ft 1 2 1 Vcone= πr ² h= π r 2 (2 r ) = πr ³ 3 3 3

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

2 dr 25 ft ³= π ,3 πr ² dt 3 25 π =2 π (5)²

dr dt

25 π dr =0.5 ft /min = dt 2 π (25) 4.

Solution: π 2 sin θ

∫ ∫ 2 r ²sin θ cos ² θ drdθ 0

0 π 2 sin θ

¿∫ ∫ 2 r ² dr sin θ cos ² θ dθ 0

0

π 2

sin θ

2 ¿∫ r ² ∫ sin θ cos ² θ dθ 0 0 3

=

π 2

∫ 23 (sin θ)³ sinθ cos ² θ dθ 0

π 2

2 ¿ ∫ sin 3 0

4

θ cos ² θ dθ

5. Solution: 48=( 1−0.25 ) X

x= 6.

48 =$ 64 0.75

Solution: β=180−40=140°

α=180−30−140 =10°

¿

[

]

2 ( 3 )( 1) (1) π π = 3 (6)( 4 )(2 ) 2 48

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

50 x ;x =143.969621 = sinθ sin 30 h=143.969621sin ( 40 )=92.54 ft

7. B. which touches the conic at only one point 8. Solution: 2 x −3 ( 0)+6=0

x=

−6 =−3 2

2 ( 0)−3 y +6 =0 6 y= =2 3 1 A= ( 3 )( 2) =3 sq . units 2 9. Solution:

( D2−D +2) y=0 m ²−m+2=0

( ) m−

1 2 7 + =0 4 2

√ √

7 −7 1 m− = = i 2 2 4 1 7 m= + √ i 2 2 y = e Ax (C1 cosBx +C2 sinBx) x

y=e2 (C 1 cos 10.

√7 x +C 2 cos √ 7 x) 2 2

Solution: x - 10 = 5 Opposite of x – 10 = 5 15 – 10 = 5

2

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

∴−5

11. Solution: y = 5 – x, find x when y = 7 7=5–x x = -7 + 5 = −2 12. Solution x

Cattle → Horses →

y

x 9 = ; x= y +80 y 5 y=

5x 9

Substitute: 5x + 80=180 9 y=180−80=100

x + y =180+100 =280 13. Solution: 3(240) + 3(300) =

P1,620

14. Solution: 5x -12y – 8 = 0, center of the circle C (1,3) d=r=

5 ( 1) −12( 3 )−8

|√ 5² + 12² |

(x – h)² + (y – r)² = r 2 2 ( x−1) + ( y−3) =9

15.

Solution: y 2=( x 3 ) ( x−1 )

=3

REGISTERED ELECTRICAL ENGINEERS PRE-BOARD EXAMINATION AUGUST 2012 MATHEMATICS

y ²=( x 3 −x 4 ) a=1

LR=4 1

π ∫ ( x 3−x 4 ) dx=0.157 0

16. Solution: Sn=

n [ a + (n−1 ) d ] 2 1

a1=1 a2=2 Sn=105

n=1 105=

n [ 2( 1 ) +( n−1 ) (1) ] 2

∴n=14 layers

17. Solution: 2

2

2

L3 =h 3 + x 3 2

2

2

L3 =8 3 +3.375 3

∴ L=15.625 ft

18. Solution: if f ( x )=10 x +1,then f ( x+1 )−f ( x ) =? let x =1 f ( 1) =10 1+1=11 f ( 1+1 )=10

1+ 1

+ 1=101
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