Differentiation & Application Qns

1. d dx cos -1 x x x x − + − − 1 1 = (A) ₁+ x1 2 (B) − +1 1 _x2 (C) ₁+ x2 2 (D) − +2 1 _x2 2. d dx

(

x + 1x

)

2 = (A) 1 - 1₂ x (B) 1 + 1_x2 (C) 1 - 1_2x (D) None of these 3. d dx tan cos sin − +    1 _ 1 x x = (A) - 1₂ (B) 1 2 (C) - 1 (D) 1

4. If x = a(t - sint) & y = a(1 - cost), then

d dx =

(A) tan

( )

₂t (B) - tan

( )

₂t (C) cot

( )

₂t (D) - cot

( )

₂t 5. If y = xx_{, then} dy

dx =

(A) xx _{(1+logx) (B) x}x

( )

₁+ 1

x (C) (1 + log x) (D) None of these

6. If y = ex e x ex + + +...∞ , then dy dx = (A) _{1 −}y_y (B) _{1 − y}1 (C) _{1 +}y_y (D) _{y − 1}y 7. If xy _{= e}x - y _{, then} dy dx = (A) log x . [log (ex)] -2

(B) log x [og (ex)]2

(C) log x . (log x)2 (D) None of these 8. If y = sin -1 _{x 1}− +_{x x 1}−_x2   _{ ,} then dy dx = (A) − − + − 2 1 1 2 2 2 x x x x (B) − − − − 1 1 1 2 2 2 x x x (C) 1 1 1 2 2 2 −x + x x− (D) None of these

9. If y = A cos nx + B sin nx, then d y dx 2 2 is equal to : (A) n2 _{y (B) - y} (C) - n2 _{y (D) None of these}

10. The volume of a spherical balloon is increasing at the rate of 40 cubic centimetres per minute . The rate of change of the surface of the balloon at the instant when its radius is 8 cm is :

(A) 5₂ sq cm/min. (B) 5 sq cm/min. (C) 10 sq cm/min. (D) 20 sq cm/min.

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11. If y = 1 4 u4 _{, u =} 2 3 x 3 _{+ 5, then} dy dx = (A) 1₂₇ x2 _(2x3 ₊₁₅₎3 (B) 2 27 x (2x3 ₊₅₎ (C) 2 27 x (2x3 ₊₁₅₎3 (D) None of these

12. A stone thrown vertically upwards from the surface of the moon at a velocity of 24 m/sec. reaches a height of s = 24t - 0.8t2 _{metres after t sec.}

The acceleration due to gravity in m/sec2 _{at the surface of the moon is :}

(A) 0.8 (B) 1.6 (C) 2.4 (D) 4.9 13. If y = f 2 1 1 2 x x − +       & f′(x) = sin x2_, then dy dx = (A)

(

)

6 2 2 1 2 2 2 x x x − + + sin 2 1 1 2 2 x x − +    _ (B)

(

)

6 2 2 1 2 2 2 x x x − + + sin2 2 1 1 2 x x − +       (C)

(

)

− + + + 2 2 2 1 2 2 2 x x x sin 2 2 1 1 2 x x − +       (D)

(

)

− + + + 2 2 2 1 2 2 2 x x x sin 2 1 1 2 2 x x − +      

14. Differential co-efficient of, sec -1 1 2_{x −}2 1 w.r.t. 1− x at 2 x = 1 2 (A) 2 (B) 4 (C) 6 (D) 1

15. A body moves according to the formula v = 1 + t2_{, where v is the}

velocity at time t . The acceleration after 3 sec. will be (v in cm/sec.) : (A) 24 cm/sec2 _{(B) 12 cm/sec}2

(C) 6 cm/sec2 _{(D) None of these}

16. If 1−x2 + 1−y = a2 (x - y), then dy dx = (A) 1 1 2 2 − −xy (B) 1₁ 2 2 − −yx (C) x _y 2 2 1 1 − − (D) y _x 2 2 1 1 − − 17. If y = (x logx) log log x _{, then} dy

dx = (A) (xlogx)log log x

{

1

xlogx(logx+ log log )x

+ _ _    (log log ) log x x x 1

(B) (x logx)x log x _{log logx} 2 1

logx + x 



  (C) (x logx)x log x log logx

x 1 ₁ logx +     (D) None of these

18. If y = 1₁+₋tan_tanx_x , then dy dx = (A) 1 21₁−₊ tan_tanx_x . sec2

(

π

)

4 + x (B) 1₁−₊tan_tanx_{x . sec}2

(

π

)

4 + x

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(C) 1₂ 1 1 − + tantan x x . sec

(

π₄ + x

)

(D) None of these

19. If y secx + tanx + x2 _{y = 0, then} dy

dx is equal to : (A) 2 2 2 xy x y x x x x + + +

sec sec tan sec (B) - 2 2 2 xy x x x x x + + +

sec sec tan sec (C) - 2 2 2 xy x y x x x x + + +

sec sec tan sec

(D) None of these

20. If sin(xy) + x_{y = x}2 _{- y, then} dy

dx = (A)

[

]

y xy y xy xy xy y x 2 2 1 2 2 − − + − cos ( ) cos ( ) (B)

[

2 1

]

2 2 2 xy y xy xy xy y x − − + − cos ( ) cos ( ) (C) - y

[

xy y xy

]

xy xy y x 2 2 1 2 2 − − + − cos ( ) cos ( ) (D) None of these 21. _dxd cos− _ −₊ _       1 2 2 1 1 x x = (A) ₁+ x1 2 (B) - ₁+ x1 2 (C) - ₁+ x2 2 (D) ₁+ x2 2 22. If y = tan-1 x a x a 1 3 1 3 1 3 1 3 1 / / / / + −    _ , then dy dx = (A)

(

1

)

3_x2 3/ 1+_x2 3/ (B)

₍

a

₎

x x 3 2 3/ 1+ 2 3/ (C) -

₍

1

₎

3_x2 3/ 1+_x2 3/ (D) -

₍

a

₎

x x 3 2 3/ 1+ 2 3/ 23. If y = cot -1 1 1 + −       x x , then dy dx = (A) 1 1+ x2 (B) - ₁+ x1 2 (C) ₁+ x2 2 (D) - ₁+ x2 2 24. The function f(x) = x at x = 0, is (A) Continuous & non−differentiable (B) Discontinuous & differentiable (C) Discont. & non-differentiable (D) Continuous & differentiable 25. For which interval, the given function

f(x)=−2x2 −_9x2 −_{12x+1, is decreasing}

(A) (−2, ∞) (B) (-2, -1) (C) (−∞, -1)

(D) (−∞, -2) and (−1, ∞) 26. For which interval, the function,

x x

x

2 ₃

1 −

− satisfies all the conditions of Rolle’s theorem

(A) [0, 3] (B) [-3, 0]

(C) [1.5, 3] (D) for no interval 27. The abscissae of the points of the curve, y = x2 _{in the interval [-2, 2],}

where the slope of the tangents can be obtained bt mean value theorem for the interval [-2, 2], are :

(A) ± 2

3 (B) ± 3

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(C) ± 3

2 (D) 0

28. If x = secθ−cosθ & y = secn θ−_cosn θ

then : (A) (x2 _{+ 4)} dy dx      2 = n2 _(y2 _{+ 4)} (B) (x2 _{+ 4)} dy dx    _2 _{= x}2 _(y2 _{+ 4)} (C) (x2 _{+ 4)} dy dx    _2 _{= (y}2 _{+ 4)} (D) None of these 29. If xy _{= y}x_{, then} dy dx =

(A) y x_{( logy}( log_xy₋−_xy₎) (B) y x y y x y x x ( log ) ( log )

− − (C) ( log_{( log}x_y y_x −₋y_x)₎ (D) None of these

30. If y = x

( )

x

x

, then dy dx = (A) y [xx _{(log ex) . log x + x}x_]

(B) y [xx _{(log ex) . log x + x]}

(C) y [xx _{(log ex) . log x + x}x - 1_]

(D) y [xx _(log e x) . log x + x x - 1_] 31. If y = x2 _{+ x}log x _{, then} dy dx = (A) x x x x x 2 + log . log (B) x2 _{+ log x . x}log x (C) 2 2 x x x x x +    _{log .} log _ (D) None of these

32. If f(x + y) = f(x) . f(y) for all x & y and f(5) = 2, f′(0) = 3, then f′(5) will

(C) 6 (D) 8 33. If y = sec-1 x x + −    1_ 1 + sin-1 x x − +    1_ 1 then dy dx = (A) 0 (B) 1 (C) 2 (D) 3 34. f(x) = x2 _{- 27x + 5, is an increasing} function, when : (A) x < - 3 (B) x > 3 (C) x ≤ − 3 (D) x < 3 35. If y = (x2 − ₁₎m_{, then the (2m)}th_, differential co-efficient of y is : (A) m (B) (2m) ! (C) 2m (D) m! 36. If y = aemx _{+ be} −mx _{, then} d y dx 2 2 − m2_{y =} (A) m2 _(aemx − be −mx₎ (B) 1 (C) 0 (D) None of these

37. The rate of change of x2 +16 w.r.t. x x − 1 at x = 3, will be : (A) - 24 5 (B) 245 (C) 12 5 (D) - 125 38. If y = f 5 1 10 2 3 x x + −    _ & f′(x) = cos x, then dy dx = (A) cos 5 1 10 2 3 x x + −       d dx 5 1 10 2 3 x x + −      

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(B) 5 1 10 2 3 x x + − cos 105 2 13 x x + − (C) cos 5 1 10 2 3 x x + − (D) None of these 39. Consider f(x) = x x x x 2 0 0 0 , , ≠ =    

(A) f(x) is discontinuous everywhere (B) f(x) is continuous everywhere (C) f′(x) exists in (-1, 1)

(D) f′(x) exists in (-2, 2)

40. 36 factorize into two factors in such a way that sum of factors is minimum, then the factors are :

(A) 2, 18 (B) 9, 4

(C) 3, 12 (D) None of these 41. If f(x) = 2x3 − 3x2 − 12x_{+5 and}

x ∈ [-2, 4], then the maximum value of function is at the following value of x . (A) 2 (B) - 1 (C) - 2 (D) 4 42. If y2 _{= p(x) is a polynomial of degree} three then, 2 _dxd y d y dx 2 2 2 .       = (A) p′′′(x)+p′(x) (B) p′′(x) . p′′′(x) (C) p(x) . p′′′(x) (D) Constant 43. The ratio of height of a cone having

maximum volume which can be inscribed in a sphere with the diameter of sphere, is : (A) 2/3 (B) 1/3 (C) 3/4 (D) 1/4 44. If f(x) = sin x − x 2 is increasing function, then : (A) 0 < x < π 3 (B) -π 3 < x < 0 (C) -π 3 < x < π 3 (D) x = π 2 45. The function y = 2x3 _{− 9x}2 _{+ 12x − 6}

is monotonic decreasing when : (A) 1 < x < 2 (B) x > 2

(C) x < 1 (D) None of these 46. Rolle’s theorem is not applicable to

the function f(x) = x defined on [-1, 1], because :

(A) f is not continuous on [-1, 1] (B) f is not differentiable on (-1, 1) (C) f (-1) ≠ f(1)

(D) f(-1) = f(1) ≠ 0

47. The slope of tangent to the curve, x = t2 _{+ 3t - 8, y = 2t}2 _{- 2t - 5 at the} point (2, -1) is : (A) 22 7 (B) 6 7 (C) -6 (D) None of these 48. If f(x) = 1 1 0 0 ₂ + < ≤ <     sin , , x x x π , then f′(0) = (A) 1 (B) 0

(C) ∞ (D) Does not exist 49. Let f′(x) be continuous at x = 0 and

f′′(0) = 4 . The value of, Limit x → 0 2 3 2 4 2 f x f x f x x ( )− ( )+ ( ) _{is :} (A) 11 (B) 2 (C) 12 (D) None of these 50. At the point x = 1, the given function

f(x) = x x x x 3 ₁ 1 1 1 − − < < ∞ − ∞ < ≤    , , is :

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(A) Continuous & differentiable (B) Condinuous & not differentiable (C) Discontinuous & differentiable (D) Discont. & not differentiable 51. Let [x] denotes the greatest integer

less than or equal to x . If, f(x) = [x sin πx], then f(x) is : (A) Continuous at x = 0 (B) Continuous in (-1, 0) (C) Differentiable in (-1, 1) (D) All the above

52. If f(x) = kx3 − _{9 x}2 _{+ 9x + 3 is}

monotonically increasing in each interval, then :

(A) k < 3 (B) k ≤ 3

(C) k > 3 (D) None of these 53. If y = a log x + bx2 _{+ x has its}

extremum values at x = -1 & x = 2, then : (A) a = 2, b = −1 (B) a = 2, b = − 1₂ (C) a = −2, b = (D) None of these 1₂ 54. If f(x) = e x x x x 1 0 0 − ≤ >     , , , then : (A) f(x) is differentiable at x = 0 (B) f(x) is continuous at x = 0 (C) f(x) is differentiable at x = 1 (D) f(x) is continuous at x = 1 55. If f(2) = 4, f′(2) = 1, then Limit x → 2 x f f x x ( )2 2 ( ) 2 − − = (A) 1 (B) 2 (C) 3 (D) -2

56. The function f(x) = max{(1 - x), (1 + x), 2}, x ∈ (−∞, ∞), is :

(B) Differentiable at all points (C) Differentiable at all points except

at x = 1 and x = -1

(D) Continuous at all points except at x = 1 and x = -1, where it is discontinuous 57. The function f(x) =

n x n e x ( ) ( ) π + + is : (A) Increasing on [0, ∞) (B) Decreasing on [0, ∞) (C) Decreasing on [0, π/e) and

Increasing on [π/e, ∞) (D) Increasing on [0, π/e) and

Decreasing on [π/e, ∞)

58. The number of points at which the function, f(x) = x - 0.5 + x - 1 + tanx does not have a derivative in the interval (0, 2) is :

(A) 1 (B) 2 (C) 3 (D) 4 59. The function f(x) = tan x - x

(A) Always increases (B) Always decreases (C) Never decreases

(D) Sometimes increases and some times decreases

60. On the interval [0, 1], the function x25 _{(1 - x)}75 _{takes its maximum value}

at the point : (A) 0 (B) 1/2 (C) 1/3 (D) 1/4 61. If x2_{ey + 2xye}x _{+ 13 = 0, then} dy dx = (A) 2 2 1 2 xe y x x xe y x y x − −       + + +( )

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(B) 2 2 1 2 xe y x x xe x y y x − −       + + + ( ) (C) - 2 2 1 2 xe y x x xe x y x y − −       + + + ( ) (D) None of these

62. y = (tan )x (tan )xtanx , then at x = π 4 , the value of dy dx = (A) 0 (B) 1 (C) 2 (D) None of these 63. Let g(x) be the inverse of the function

f(x) & f′(x) = 1 1+ x2 . Then g′(x) is equal to : (A)

(

1

)

1+ g x( )3 (B)

(

)

1 1+ f x( )3 (C) 1 + (g(x))3 _{(D) 1 + (f(x)}3 64. If x = t2_{, y = t}3_{, then} d y dx 2 2 = (A) 3₂ (B) 3 4 t (C) _{2 t}3 (D) 3₂t 65. d y dx 20 20 (2 cosx cos3x) = (A) 220 _{(cos2x - 2}20 _{cos 4x)}

(B) 220 _{(cos2x + 2}20 _{cos 4x)}

(C) 220 _{(sin2x + 2}20 _{sin 4x)}

(D) 220 _{(sin2x - 2}20 _{sin 4x)}

66. Which of the following is not true ? (A) Every differentiable function is

continuous

(B) If derivative of a function is zero at all points, then the function is constant

(C) If a function has maxima or minima at a point, then the

function is differentiable at that point and its derivative is zero (D) If a function is constant, then its

derivative is zero at all points 67. If f(x) = x5 _{- 20x}3 _{+ 240x, then f(x)}

satisfies which of the following . (A) It is monotonically decreasing

everywhere (B) It is monotonically decreasing only in (0, ∞) (C) It is monotonically increasing everywhere (D) It is monotonically increasing only in (−∞, 0)

68. A ladder 5m in length is resting against vertical wall . The bottom of the ladder is pulled along the ground away from the wall at the rate of 1.5 m/sec. The length of the highest point of the ladder when the foot of the ladder is 4.0 m away from the wall decreases at the rate of :

(A) 2 m/sec. (B) 3 m/sec. (C) 2.5 m/sec. (D) 1.5 m/sec. 69. From mean value theorem,

f(b) − f(a) = (b-a) f′(x₁) ; a < x₁ < b if f(x) = 1_x , then x₁ = (A) a b (B) a+₂b (C) 2a b a+b (D) b a b a − +

70. If sum of two numbers is 3, then maximum value of the product of first and the square of second is :

(A) 4 (B) 3

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(C) 2 (D) 1 ANSWERS 1. D 2. A 3. A 4. C 5. A 6. A 7. A 8. C 9. A 10. C 11. C 12.B 13. D 14. B 15. C 16. B 17. A 18.A 19. C 20. A 21. D 22. A 23. B 24.A 25. D 26. D 27. A 28. A 29. B 30.C 31. C 32. C 33. A 34. B 35. B 36.C 37. D 38. A 39. B 40. D 41. D 42.C 43. B 44. C 45. A 46. B 47. B 48.D 49. C 50. B 51. D 52. C 53. B 54.D 55. B 56. AC57. B 58. C 59. A 60.D 61. C 62. C 63. C 64. B 65. B 66.C 67. C 68. A 69. A 70. A

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