This is the Multiple Choice Questions Part 1 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

### Start Practice Exam Test Questions Part I of the Series

**Choose the letter of the best answer in each questions.**

1. Evaluate:

- A. 1/5
- B. 2/5
- C. 3/5
- D. 4/5

2. Evaluate:

- A. Undefined
- B. 0
- C. Infinity
- D. 1/7

3. Evaluate:

- A. 0
- B. 1
- C. 8
- D. 16

4. Evaluate:

- A. 0
- B. 2
- C. 4
- D. 6

5. Evaluate:

- A. 0
- B. 1/2
- C. 2
- D. -1/2

6. Compute the following limit:

- A. 1
- B. 0
- C. 2
- D. Infinite

7. Evaluate:

- A. Undefined
- B. 3/5
- C. Infinity
- D. Zero

8. Evaluate:

- A. 24
- B. 26
- C. 28
- D. 30

9. Evaluate:

- A. e^2x
- B. e^(2/π)
- C. 0
- D. infinity

10. Differentiate y = e^x cos x^2

- A. – e^x sin x^2
- B. e^x (cos x^2 – 2x sin x^2)
- C. e^x cos x^2 – 2x sin x^2
- D. -2xe^x sin x

11. Differentiate y = sec (x2 + 2).

- A. 2x cos ( x^2 + 2)
- B. – cos (x^2 + 2) cot (x^2 + 2)
- C. 2x sec (x^2 + 2) tan (x^2 + 2)
- D. cos (x^2 + 2)

12. What is the derivative with respect to x of (x + 1)3 – x3?

- A. 3x + 6
- B. 3x – 3
- C. 6x – 3
- D. 6x + 3

13. Differentiate y = log 10 (x2 + 1)2

- A. 4x (x^2 + 1)
- B. 4xlog10e/x^2 + 1
- C. log e(x) (x^2 + 1)
- D. 2x(x^2 + 1)

14. Differentiate (x2 + 2)1/2

- A. [(x^2+2)^(1/2)]/2
- B. x/(x^2+2)^(1/2)
- C. 2x/(x^2+2)^(1/2)
- D. (x^2 + 2)^(3/2)

15. If y = (t^2 + 2)^2 and t = x^(1/2), determine dy/dx.

- A. 3/2
- B. (2x^2+2x)/3
- C. 2(x + 2)
- D. x^(5/2) + x^(1/2)

16. What is the first derivative of the expression (xy)^x = e^x?

- A. 0
- B. x/y
- C. [-y(1 + ln xy)] / x
- D. [-y(1-lnxy)/x^2]

17. Find the derivative with respect to x the function sqrt(2-3x^2)

- A. -2x^2/sqrt(2-3x^2)
- B. -3x/sqrt(2-3x^2)
- C. -3x^2/sqrt(2-3x^2)
- D. 3x/sqrt(2-3x^2)

18. Find y’ if y = arc sin cos x

- A. – 1
- B. -2
- C. 1
- D. 2

19. Find the derivative of arc cos 4x.

- A. -4/(1-16x^3)^(0.5)
- B. 4/(1-16x^3)^(0.5)
- C. -4/(1-4x^3)^(0.5)
- D. 4/(1-16x^3)^(0.5)

20. Find the derivative of [(x+1)^3]/x

- A. {[(x+1)^2]/x} – {[(x+3)^3]/x}
- B. {[4(x+1)^2]/x} – {[2(x+1)^3]/x}
- C. {[2(x+1)^2]/x} – {[(x+1)^3]/x^3}
- D. {[3(x+1)^2]/x} – {[(x+1)^3]/x^2}

21. Differentiate the equation: y = (x^2)/(x+1)

- A. (x^2+2x)/(x+1)^2
- B. 2x/(x+1)
- C. 2x
- D. (2x^2)/(x+1)

22. The derivative with respect to x of 2 cos^2 (x^2 + 2) is

- A. 2 sin (x^2 + 2) cos (x^2 + 2)
- B. -2 sin (x^2 + 2) cos (x^2 + 2)
- C. 8x sin (x^2 + 2) cos (x^2 + 2)
- D. -8x sin (x^2 + 2) cos (x^2 + 2)

23. Find the second derivative of y by implicit differentiation from the equation 4x^2 + 8y^2 = 36.

- A. 64x^2
- B. (-9/4)y^3
- C. 32xy
- D. (-16/9)y^3

24. Find the partial derivatives with respect to x of the function xy^2 – 5y + 6.

- A. y^2 – 5
- B. y^2
- C. xy – 5y
- D. 2xy

25. Find the second derivative of x^3 – 5x^2 + x = 0.

- A. 10x – 5
- B. 6x – 10
- C. 3x + 10
- D. 3x^2 – 5x

26. Given the function f(x) = x to the 3rd power – 6x + 2. Find the first derivative at x = 2.

- A. 6
- B. 7
- C. 3x^2 – 5
- D. 8

27. Find the slope of the ellipse x^2 + 4y^2 – 10x – 16y + 5 = 0 at the point where y = 2 + 8^0.5 and x = 7.

- A. -0.1463
- B. -0.1538
- C. -0.1654
- D. -0.1768

28. If y = 4 cos x + sin 2x, what is the slope of the curve when x = 2 radians?

- A. -2.21
- B. -4.94
- C. -3.25
- D. 2.21

29. Find the slope of the tangent to the curve y = x^3 – 2x + 1 at x = 1.

- A. 1
- B. 1/2
- C. 1/3
- D. 1/4

30. Give the slope of the curve at the point (1, 1): y = [(x^3)/4] – (2x+1).

- A. 1/4
- B. -1/4
- C. 1 1/4
- D. -1 1/4

31. Find the slope of x^2y = 6 at the point (2, 2).

- A. 2
- B. -1
- C. -1/2
- D. -2

32. Find the slope of the curve x^2 + y^2 – 6x + 10y + 5 = 0.

- A. 1/5
- B. 2/5
- C. 1/4
- D. 2

33. Find the slope of the tangent to the curve y = -2x – x^2 + x^3 at (0, 2).

- A. 1
- B. 2
- C. 3
- D. 4

34. Find the coordinates of the vertex of the parabola y = x^2 – 4x + 1 by making use of the fact that at the vertex, the slope of the tangent is zero.

- A. (2, -3)
- B. (3, 2)
- C. (-1, -3)
- D. (-2, -3)

35. Find the equation of the normal to x^2 + y^2 = 5 at the point (2, 1).

- A. y = 2x
- B. x = 2y
- C. 2x + 3y = 3
- D. x + y = 1

36. What is the equation of the normal to the curve x^2 + y^2 = 25 at (4, 3)?

- A. 5x + 3y = 0
- B. 3x – 4y = 0
- C. 3x + 4y = 0
- D. 5x – 3y = 0

37. Locate the points of inflection of the curve y = f(x) = x^2e^x

- A. –2 ± sqrt (3)
- B. 2 ± sqrt (2)
- C. –2 ± sqrt (2)
- D. 2 ± sqrt (3)

38. In the curve 2 + 12x – x^3, find the critical points.

- A. (2, 18) & (-2, -14)
- B. (2, 18) & (2, -14)
- C. (-2, 18) & (2, -14)
- D. (-2, 18) & (-2, 14)

39. Find the radius of curvature of a parabola y^2 – 4x = 0 at point (4, 4).

- A. 22.36 units
- B. 25.78 units
- C. 20.33 units
- D. 15.42 units

40. Find the radius of curvature at any point in the curve y + ln cos x = 0.

- A. cos x
- B. 1.5707
- C. sec x
- D. 1

41. Find the minimum distance from the point (4, 2) to the parabola y2 = 8x.

- A. 4sqrt (3)
- B. 2sqrt (2)
- C. sqrt (3)
- D. 2sqrt (3)

42. The sum of the two positive numbers is 50. What are the numbers if their product is to be the largest possible.

- A. 24 & 26
- B. 28 & 22
- C. 25 & 25
- D. 20 & 30

43. A triangle has variable sides x, y, z subject to the constraint such that the perimeter is fixed to 18 cm. What is the maximum possible area for the triangle?

- A. 15.59 sq. cm
- B. 18.71 sq. cm
- C. 17.15 sq. cm
- D. 14.03 sq. cm

44. A farmer has enough money to build only 100 meter of fence. What are the dimensions of the field he can enclose the maximum area?

- A. 25 m x 25 m
- B. 15 m x 35 m
- C. 20 m x 30 m
- D. 22.5 m x 27.5 m

45. Find the minimum amount of tin sheet that can be made into a closed cylinder having a volume of 108 cu. inches in square inches.

- A. 125.50
- B. 127.50
- C. 129.50
- D. 123.50

46. A box is to be constructed from a piece of zinc 20 sq. in by cutting equal squares from each corner and turning up the zinc to form the side. What is the volume of the largest box that can be so constructed?

- A. 599.95 cu. in.
- B. 592.59 cu. in.
- C. 579.90 cu. in.
- D. 622.49 cu. in.

47. A poster is to contain 300 (cm square) of printed matter with margins of 10 cm at the top and bottom and 5 cm at each side. Find the overall dimensions if the total area of the poster is minimum.

- A. 27.76 cm, 47.8 cm
- B. 20.45 cm, 35.6 cm
- C. 22.24 cm, 44.5 cm
- D. 25.55 cm, 46.7 cm

48. A normal window is in the shape of a rectangle surmounted by a semi-circle. What is the ratio of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter?

- A. 1
- B. 1/2
- C. 2
- D. 2/3

49. Determine the diameter of a closed cylindrical tank having a volume of 11.3 cu. m to obtain minimum surface area.

- A. 1.22
- B. 1.64
- C. 2.44
- D. 2.68

50. The cost of fuel in running a locomotive is proportional to the square of the speed and is $25 per hour for a speed of 25 miles per hour. Other costs amount to $100 per hour, regardless of the speed. What is the speed which will make the cost per mile a minimum?

- A. 40
- B. 55
- C. 50
- D. 45

### Online Questions and Answers in Calculus Series

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

**Calculus MCQs**

**MCQs from Number 1 – 50**Answer key:

**PART I**

**MCQs from Number 51 – 100**Answer key:

**PART II**

**MCQs from Number 101 – 150**Answer key:

**PART III**

**MCQs from Number 151 – 200**Answer key:

**PART IV**

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