MCQs in Calculus Part II

This is the Multiple Choice Questions Part 2 of the Series in Calculus topic in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

Multiple Choice Questions Topic Outline

• MCQs in Complex Variables | MCQs in Derivatives and Applications | MCQs in Integration and Applications | MCQs in Transcendental Functions | MCQs in Partial Derivatives | MCQs in Indeterminate forms | MCQs in Multiple Integrals | MCQs in Differential Equations | MCQs in Maxima/Minima and Time Rates

Online Questions and Answers in Calculus Series

Following is the list of multiple choice questions in this brand new series:

Calculus MCQs

PART 1: MCQs from Number 1 – 50                        Answer key: PART I

PART 2: MCQs from Number 51 – 100                        Answer key: PART II

PART 3: MCQs from Number 101 – 150                        Answer key: PART III

PART 4: MCQs from Number 151 – 200                        Answer key: PART IV

Continue Practice Exam Test Questions Part II of the Series

Choose the letter of the best answer in each questions.

51. The cost C of a product is a function of the quantity x of the product: C(x) = x^2 – 4000x + 50. Find the quantity for which the cost is minimum.

• A. 1000
• B. 1500
• C. 2000
• D. 3000

52. An open top rectangular tank with square bases is to have a volume of 10 cu. m. The materials for its bottom are to cost P 15 per square meter and that for the sides, P 6 per square meter. Find the most economical dimensions for the tank.

• A. 1.5m x 1.5m x 4.4m
• B. 2m x 2m x 2.5m
• C. 4m x 4m x 0.6m
• D. 3m x 3m x 1.1m

53. What is the maximum profit when the profit-versus-production function is as given below? P is profit and x is unit of production.

P = 200,00 – x – [1.1/(x+1)]^8

• A. 285,000
• B. 200,000
• C. 250,000
• D. 300,000

54. A boatman is at A which is 4.5 km from the nearest point B on a straight shore BM. He wishes to reach in minimum time a point C situated on the shore 9 km from B. How far from C should he land if he can row at the rate of 6 kph and can walk at the rate of 7.5 kph?

• A. 4.15 km
• B. 3.0 km
• C. 3.25 km
• D. 4.0 km

55. 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

56. The cost per hour of running a motor boat is proportional to the cube of the speed. At what speed will the boat run against a current of 8 km/hr in order to go a given distance most economically?

• A. 10 kph
• B. 13 kph
• C. 11 kph
• D. 12 kph

57. Given a cone of diameter x and altitude of h. What percent is the volume of the largest cylinder which can be inscribed in the cone to the volume of the cone?

• A. 44%
• B. 46 %
• C. 56%
• D. 65%

58. At any distance x from the source of light, the intensity of illumination varies directly as the intensity of the source and inversely as the square of x. Suppose that there is a light at A, and another at B, the one at B having an intensity 8 times that of A. The distance AB is 4 m. At what point from A on the line AB will the intensity of illumination be least?

• A. 2.15 m
• B. 1.33 m
• C. 1.50 m
• D. 1.92 m

59. A wall “h” meters high is 2 m away from the building. The shortest ladder that can reach the building with one end resting on the ground outside the wall is 6 m. How high is the wall in meters?

• A. 2.34
• B. 2.24
• C. 2.44
• D. 2.14

60. The coordinates (x,y) in feet of a moving particle P are given by x = cost – 1 and y = 2 sin t + 1, where t is the time in seconds. At what extreme rates in fps is P moving along the curve?

• A. 3 and 2
• B. 3 and 1
• C. 2 and 0.5
• D. 2 and 1

61. A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum?

• A. 3.41 m
• B. 3.51 m
• C. 3.71 m
• D. 4.41 m

62. A man walks across a bridge at the rate of 5 fps as a boat passes directly beneath him at 10 fps. If the bridge is 10 feet above the boat, how fast are the man and the boat separating 1 second later?

• A. 8 fps
• B. 8.25 fps
• C. 8.33 fps
• D. 8.67 fps

63. An LRT train 6 m above the ground crosses a street at 9 m/s at the instant that a car approaching at a speed of 4 m/s is 12 m up the street. Find the rate of the LRT train and the car separation one second later.

• A. 3.64 m/s
• B. 3.94 m/s
• C. 4.24 m/s
• D. 4.46 m/s

64. Water is flowing into a conical cistern at the rate of 8 m^3/min. If the height of the inverted cone is 12m and the radius of its circular opening is 6 m. How fast is the water level rising when the water is 4 m deep?

• A. 0.64 m/min
• B. 0.56 m/min
• C. 0.75 m/min
• D. 0.45 m/min

65. Water is pouring into a conical vessel 15 cm deep and having a radius of 3.75 cm across the top. If the rate at which the water rises is 2 cm/sec, how fast is the water flowing into the conical vessel when the water is 4 cm deep?

• A. 2.37 m^3/sec
• B. 5.73 m^3/sec
• C. 6.28 m^3/sec
• D. 4.57 m^3/sec

66. Water is pouring into a swimming pool. After t hours, there are t = sqrt(t) gallons in the pool. At what rate is the water pouring into the pool when t = 9 hours?

• A. 7/6 gph
• B. 8/7 gph
• C. 6/5 gph
• D. 5/4 gph

67. A helicopter is rising vertically from the ground at a constant rate of 4.5 meters per second. When it is 75 m off the ground, a jeep passed beneath the helicopter traveling in a straight line t a constant rate of 80 kph. Determine how fast the distance between them changing after 1 second.

• A. 12.34 m/s
• B. 11.10 m/s
• C. 10.32 m/s
• D. 9.85 m/s

68. A balloon is released from the ground 100 meters from an observer. The balloon rises directly upward at the rate of 4 meters per second. How fast is the balloon receding from the observer 10 seconds later?

• A. 1.68 m/sec
• B. 1.36 m/sec
• C. 1.55 m/sec
• D. 1.49 m/sec

69. A balloon is rising vertically over a point A on the ground at the rate of 15 ft./sec. A point B on the ground level with and 30 ft. from A. When the balloon is 40 ft. from A, at what rate is its distance from B changing?

• A. 13 ft./s
• B. 15 ft./s
• C. 12 ft./s
• D. 10 ft./s

70. Car A moves due East at 30 kph at the same instant car B is moving S 30° E, with a speed of 60 kph. The distance from A to B is 30 pm. Find how fast is the distance between them separating after one hour.

• A. 36 kph
• B. 38 kph
• C. 40 kph
• D. 45 kph

71. A car starting at 12:00 noon travels west at a speed of 30 kph. Another car starting from the same point at 2:00 P.M. travels north at 45 kph. Find how (in kph) fast the two are separating at 4:00 P.M.

• A. 49
• B. 51
• C. 53
• D. 55

72. Two railroad tracks are perpendicular to each other. At 12:00 P.M. there is a train at each track approaching the crossing at 50 kph, one being 100 km and the other 150 km away from the crossing. How fast in kph is the distance between the two trains changing at 4:00 P.M.?

• A. 67.08
• B. 68.08
• C. 69.08
• D. 70.08

73. Water is running into a hemispherical bowl having a radius of 10 cm at a constant rate of 3 cm^3/min. When the water is x cm. deep, the water level is rising at the rate of .0149 cm/min. What is the value of x?

• A. 3
• B. 2
• C. 4
• D. 5

74. 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

75. A standard cell has an emf “E” of 1.2 volts. If the resistance “R” of the circuit is increasing at the rate of 0.03 ohm/sec, at what rate is the current “I” changing at the instant when the resistance is 6 ohms? Assume Ohm’s law E=IR.

• A. -0.002 amp/sec
• B. 0.004 amp/sec
• C. -0.001 amp/sec
• D. 0.003 amp/sec

76. What is the integral of (3t – 1)^3 dt?

• A. (1/12)(3t - 1)^4 + C
• B. (1/12)(3t – 4)^4 + C
• C. (1/4)(3t – 1)^4 + C
• D. (1/4)(3t – 1)^3 + C

77. Evaluate the integral of dx / (x + 2) from -6 to -10.

• A. 2^(1/2)
• B. 1/2
• C. ln 3
• D. ln 2

78. Integrate x cos (2x^2 + 7) dx.

• A. (1/4) sin(2x^2 + 7) + C
• B. (1/4)cos(2x^2 + 7) + C
• C. (sin θ) / ((4(2x^2 + 7)) + C
• D. sin (2x^2 + 7) + C

79. Integrate: (7x^3 + 4x^2) dx.

• A. (7x^3)/3 + (4x^2)/2 + C
• B. (7x^4)/4 + (4x^2)/5 + C
• C. (7x^4)/4 + (4x^3)/3 + C
• D. (7x^4) - (4x)/2 + C

80. What is the integral of sin^5 x cos^3 x dx if the lower limit is zero and the upper limit is π/2?

• A. 0.0203
• B. 0.0307
• C. 0.0417
• D. 0.0543

81. What is the integral of sin^5 x dx if the lower limit is 0 and the upper limit is π/2?

• A. 0.233
• B. 0.333
• C. 0.433
• D. 0.533

82. Find the integral of 12 sin^5 x cos^5 x dx if the lower limit = 0 and upper limit = π/2.

• A. 0.2
• B. 0.3
• C. 0.4
• D. 0.5

83. Evaluate the integral of sin^6 x dx from 0 to π/2.

• A. π/32
• B. 2π/17
• C. 3π/32
• D. 5π/32

84. Evaluate the integral of x(x – 5)^12 dx from 5 to 6.

• A. 0.456
• B. 0.556
• C. 0.656
• D. 0.756

85. Evaluate the integral of (xdx) / ((x + 1)^8) from 0 to 1.

• A. 0.011
• B. 0.022
• C. 0.033
• D. 0.044

86. Evaluate the integral of (cos3A) dA from 0 to π/6.

• A. 27π/363
• B. 35π/768
• C. 23π/765
• D. 12π/81

87. Integrate 1/ (3x + 4) with respect to x and evaluate from x = 0 and x = 2.

• A. 0.278
• B. 0.336
• C. 0.252
• D. 0.305

88. Evaluate the integral of cos^2 ydy.

• A. y/2 + (sin 2y)/4 + C
• B. y + 2cosy + C
• C. y/4 + (sin 2y)/4 +C
• D. y + sin 2y + C

89. Integrate the square root of (1 – cosx) dx.

• A. -2*sqrt(2) cos(x/2) + C
• B. -2*sqrt(2) cos x +C
• C. 2*sqrt(2) cos(x/2) + C
• D. -2*sqrt(2) cos x +C

90. Evaluate the integral of cos x dx limits from π/4 to π/2

• A. 0.423
• B. 0.293
• C. 0.923
• D. 0.329

91. Evaluate the integral of ln x dx, the limits are 1 and e.

• A. 0
• B. 1
• C. 2
• D. 3

92. Evaluate the integral of (2log₁₀edx)/x from 1 to 10.

• A. 2.0
• B. 49.7
• C. 3.0
• D. 5.12

93. What is the integral of cos 2x e^(sin 2x) dx?

• A. (e^(sin 2x))/2 + C
• B. -(e^(sin 2x))/2 + C
• C. -e^(sin 2x)/2 + C
• D. e^(sin 2x)/2 + C

94. The integral of cos x with respect to x is

• A. sin x + C
• B. sec x + C
• C. –sin x + C
• D. csc x + C

95. Find the integral of [(e^x – 1] divided by [e^x +1] dx

• A. ln (e exp x – 1) square + x + C
• B. ln (e exp x + 1) - x + C
• C. ln (e exp x – 1) + x + C
• D. ln (e exp x + 1) square - x + C

96. Evaluate the double integral of r sin u dr du, the limits of r are 0 and cos u and the limits of u are 0 and pi.

• A. 1
• B. 1/2
• C. 0
• D. 1/3

97. Evaluate the integral of (3x^2 + 9y^2) dx dy if the interior limits has an upper limit of y and a lower limit of 0, and whose outer limit has an upper limit of 2 and a lower limit of 0.

• A. 10
• B. 20
• C. 30
• D. 40

98. Evaluate integral of zdz r^2dr sinu du, the limits of z are from 0 to 2, the limits of r are from 0 to 1, and the limits of u are from 0 to π/2.

• A. 2/3
• B. 4/3
• C. 1/3
• D. 5/3

99. Find the area of the region bounded by y^2 = 8x and y = 2x.

• A. 1.22 sq. units
• B. 1.33 sq. units
• C. 1.44 sq. units
• D. 1.55 sq. units

100. What is the area bounded by the curve x^2 = -9y and the line y+ + 1 = 0?

• A. 3 sq. units
• B. 4 sq. units
• C. 5 sq. units
• D. 6 sq. units

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