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MULTIPLE CHOICE QUESTIONS CHAPTER – 6 APPLICATIONS OF DERIVATIVES Q1. For the function f  x   x  (a) 1

1 , x 1,3 , the value of c for the mean value theorem is: x

(b)

3

(c) 2

(d) none of these

Q2. The value of c in Rolle ’s Theorem when f  x   2 x3  5x2  4 x  3 , x1 3,3 is: (c) 2

(b) 1 3

(a) 2

(d) 2 3

Q3. For the function f  x   x  x  2  , x1, 2 , the value of c for the mean value theorem is: (a) 1

(b) 1 2

(c) 2 3

(d) 3 2

Q4. The value of c in Rolle ’s Theorem when f  x   e x sin x , x0,   is: (a)  6

(b)  4

(c)  2

(d) 3 4

Q5. The cost function of a firm is C  3x 2  2 x  3 . Then the marginal cost, when x  3 is: (a) 10

(b) 20

(c) 5

(d) 25

Q6. The function f  x   tan x  x is: (a) Always increasing

(b) always decreasing

(c) not always decreasing

(d) sometimes increasing and sometimes decreasing

Q7. The function f  x   x3  6 x 2 15x 12 is: (a) Strictly decreasing on R (b) Increasing on   , 2 and decreasing in  2,  

(b) Strictly increasing on R (d) none of these

Q8. The function f  x   4  3x  3x 2  x3 is: (a) Decreasing on R

(b) Increasing on R

(c) Strictly increasing on R

(d) Strictly increasing on R

Q9. The function f  x  

x is: sin x

(a) Increasing in  0,1  1 (c) Increasing in  0,  and decreasing in  2

(b) Decreasing in  0,1 1   ,1 2 

(d) none of these

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Q10. The function f  x   x x is decreasing in the interval:  1 (b)  0,   e

(a)  0, e 

(c)  0,1

(d) none of these

Q11. The function f  x    x  x  3 is increasing in: 2

(a)  0,   Q12. The function f  x   (a)  1,1

(b)   , 0 

(c) 1,3

 3 (d)  0,    3,    2

x is increasing in: x 1 2

(b)  1,  

(c)   ,  1  1,  

(d) none of these

Q13. The function f  x   cot 1 x  x increases in the interval: (a)

1,  

(b)  1,  

(c)   ,  

(d)  0,  

Q14. If the function f  x   kx3  9 x 2  9 x  3 is monotonically increasing in every interval, then: (a) k  3

(b) k  3

(c) k  3

(d) k  3 0.

Q15. Side of an equilateral triangle expands at rate of 2 cm / sec. The rate of increase of its area when each side is 10 cm is (in cm2 /sec ): (a) 10 2

(b) 10 3

(c) 10

(d) 5

Q16. The radius of a sphere is changing t the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm (in cm2 /sec ) is: (a) 8

(b) 12

(c) 160

(d) 200

Q17. A cone whose height is always equal to its diameter is increasing in volume at the rate of 40cm3 / sec . At what rate is the radius increasing when its circular base area is 1m 2 ? (a) 1 mm/sec

(b) 0.001 cm/sec

(c) 2 mm/sec

(d) 0.002 cm/sec

Q18. A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 m3 / min . The rate at which the surface of the oil is rising (in m / min ) is: (a) 1

(b) 2

(c) 5

(d) 1.25

Q19. The coordinates of the point on the ellipse 16 x2  9 y 2  400 where the ordinate decrease at the same rate at which the abscissa increases, is:  16  (a)  3,   3

16   (b)  3,  3 

16   (c)  3,   3 

 1 (d)  3,   3

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Q20. If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to: (a) 1 unit

(b)

2 unit

1 unit 2

(c)

(d)

1 2 

unit

Q21. A man of height 6 ft walks at a uniform speed of 9 ft/sec from a lamp post fixed at 15 ft height. The length of his shadow is increasing at the rate of: (a) 15 ft/sec

(b) 9 ft/sec

(c) 6 ft/sec

(d) none of these

Q22. If there is an error of a% in measuring the edge of a cube, then percentage error in its surface is: (a) 2a%

(b)

a % 2

(c) 3a%

(d) none of these

Q23. In an error of k % is made in measuring radius of a sphere, then percentage error in its volume is: (a) k %

(b) 3k %

(c) 2k %

(d)

k % 3

Q24. A sphere of radius 100 mm shrinks to radius 98 mm, then approximate decrease in its volume is: (a) 12000 mm3

(b) 800 mm3

(c) 8000 mm3

(d) 120 mm3

(c) 2.01

(d) none of these

Q25. The approximate value of  33 is: 15

(a) 2.0125

(b) 2.1

Q26. The circumference of a circle is measured as 28 cm with an error of 0.01 cm. The percentage error in its area is: (a)

1 14

(b) 0.01

(c)

1 7

(d) none of these

Q27. The equation of normal to the curve y  x  sin x cos x at x   2 is: (a) x  2

(b) x  

(c) x    0

(d) 2x  

Q28. The point on the curve y  x 2  3x  2 where tangent is perpendicular to y  x is: (a)

1 2,1 4 

(b) 1 4,1 2 

(c)  4, 2 

(d) 1,1

Q29. The point on the curve y 2  x where tangent makes 450 angle with x-axis is: (a)

 0, 0 

(b)  2,16 

(c)  3,9 

(d) none of these

Q30. The angle between the curves y 2  x and x 2  y at 1,1 is: (a) tan 1  4 3

(b) tan 1  3 4 

(c) 900

(d) 450

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Q31. The equation of tangent at those points where the curve y  x 2  3x  2 meets x-axis are: (a) x  y  2  0 , x  y 1  0

(b) x  y  1  0 , x  y  2  0

(c) x  y 1  0 , x  y  0

(d) x  y  0 , x  y  0

Q32. At what point the slope of the tangent to the curve x2  y 2  2 x  3 is zero? (a)  3, 0  ,  1,0 

(b)  3, 0  , 1, 2 

(c)  1, 0  , 1, 2 

(d) 1, 2  , 1,  2 

Q33. If the curve ay  x 2  7 and x3  y cut each other at 900 at 1,1 , then value of a is: (a) 1

(b) 6

(c) 6

(d) 0

Q34. The equation of normal x  a cos3  , y  a sin 3  at the point    4 is: (a) x  0

(b) y  0

(c) x  y

(d) x  y  a

Q35. The angle of intersection of the parabolas y 2  4ax and x 2  4ay at the origin is: (a)  6

(b)  3

(c)  2

(d)  4

(c) 1,  2 

(d)  1, 2 

Q36. The line y  x 1 touches y 2  4 x at the point: (a) 1, 2 

(b)  2,1

Q37. The tangent to the curve y  e2x at the point  0,1 meets x  axis at: (a)  0,1

 1  (b)   , 0   2 

(c)  2, 0 

(d)  0, 2 

Q38. The curves y  4 x2  2 x  8 and y  x3  x 13 touch each other at the point: (a)  3, 23

(b)  23,  3

Q39. The maximum value of f  x   (a)

1 e

(b)

Q40. The minimum value of x 2  (a) 0

(c)  34,3

(d)  3,34 

(c) e

(d) 1

(c) 50

(d) 75

log x is: x

2 e 250 is: x

(b) 25

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Q41. The maximum value of f  x    x  2  x  3 is: 2

(a)

7 3

(b) 3

(c)

4 27

(d) 0

Q42. The least value of f  x   e x  e x is: (a) 2

(b) 0

Q43. For all real values of x , the minimum value of y 

(a) 0

(d) can’t be determine

(c) 2

(b) 1

1 x  x2 is: 1 x  x2

(c) 3

(d)

1 3

Q44. The maximum value of y  sin x.cos x is: (a)

1 4

(b)

1 2

(c)

(d) 2 2

2

Q45. If the function f  x   x3  ax 2  bx 1 is maximum at x  0 and minimum at x 1 , then: 2 (a) a  , b  0 3

Q46. If y 

3 (b) a   , b  0 2

(c) a  0, b 

3 2

(d) none of these

ax  b has a turning point at P  2,  1 . The value of a and b so that y is maximum  x 1 x  4 

at P is: (a) a  0 , b  1

(b) a  1 , b  0

(c) a  1 , b  2

(d) a  2 , b  1

Q47. The smallest value of polynomial 3x4  8x3 12 x2  48x 1 in 1, 4 is: (a) 49

(c) 59

(b) 59

(d) 257

Q48. The function f  x   2 x3  3x 2 12 x  4 , has: (a) Two points of local maximum

(b) Two points of local minimum

(c) one maxima and one minima

(d) no maxima or minima

Q49. The sum of two non-zero numbers is 8, the minimum value of the sum of their reciprocals is: (a) 1 4

(b) 1 2

(c) 1 8

(d) none of these

Q50. The point on the curve x 2  2 y which is nearest to the point  0,5  is:



(a) 2 2 , 4





(b) 2 2 , 0



(c)  0, 0 

(d)  2, 2 

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ANSWERS 1. b

2. a

3. d

4. d

5. b

6. a

7. b

8. a

9. a

10. b

11. d

12. a

13. c

14. c

15. b

16. c

17. d

18. a

19. a

20. d

21. c

22. a

23. b

24. c

25. a

26. a

27. d

28. b

29. b

30. b 31. b

32. d

33. c

34. c

35. c

36. a

37. b

38. d

39. a

40. d

41. c

42. c

43. d

44. b 45. b

46. b

47. c

48. c

49. b

50. a

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