M0IITU14 - Differentiation & Application Qns

7/30/2019 M0IITU14 - Differentiation & Application Qns

1/8

1

QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

1. ddx

cos -1 x xx x

+

1

1=

(A)1

1 2+ x (B)+

1

1 2x

(C)2

1 2+ x (D)+

2

1 2x

2.d

dx( )x x+ 1

2

=

(A) 1 - 12x

(B) 1 + 12x

(C) 1 - 12x

(D) None of these

3.d

dx tan

cossin

+

1

1x

x=

(A) - 12

(B)

12

(C) - 1 (D) 1

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

then

d

dx=

(A) tan ( )t2 (B) - tan ( )t2

(C) cot ( )t2 (D) - cot ( )t2

5. If y = xx, thendy

dx=

(A) xx (1+log x) (B) xx ( )1 1+ x(C) (1 + log x) (D) None of these

6. If y = ex e

x ex

++ + .......

, thendy

dx=

(A)y

y1 (B)1

1 y

(C)

y

y1 + (D)y

y 1

7. If xy = ex - y , thendy

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 x x x1 1 2 + ,

then dy

dx=

(A)

+

2

11

22 2x

x x x

(B)

1

11

22 2x x x

(C) 11

122 2

+x x x

(D) None of these

9. If y = A cos nx + B sin nx, thend 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.

Differentiation & Application of Derivative


2/8

2

QUEST TUTORIALS


11. If y =

1

4

u4 , u = 23

x3 + 5, then dydx

=

(A) 127

x2 (2x3 +15)3

(B)

227

x (2x3 +5)

(C)

227

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 heightof 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

12

x

x

+

& f (x) = sin x2,

thendydx

=

(A) ( )

6 2 2

1

2

22

x x

x

+

+sin 2 112

2xx

+

(B)( )

6 2 2

1

2

22

x x

x

+

+sin2

2 1

12x

x

+

(C)( )

+ +

+

2 2 2

1

2

2 2

x x

xsin2

2 1

12x

x

+

(D)

( )

+ +

+

2 2 2

1

2

2 2

x x

xsin

2 1

12

2x

x

+

14. Differential co-efficient of,

sec -1 12 12x

w.r.t. 1 2 x at x =1

2

is :

(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/sec2

(C) 6 cm/sec2 (D) None of these

16. If 1 12 2

+ x y = a(x - y), then

dy

dx

=

(A)

1

1

2

2

x

y

(B)1

1

2

2

y

x

(C)x

y

2

2

1

1

(D)

y

x

2

2

1

1

17. If y = (x logx) log log x , thendydx

=

(A) (xlog x)log log x

{ 1x x x xlog (log loglog )+ +

(log log )log

xx x

1

(B) (x logx)x log x log logx 2 1logx x

+

(C) (x logx)x log xlog logx

x

1 1

logx

+

(D) None of these

18. If y =11

+

tantan

xx

, thendydx

=

(A)

12

1

1

+

tan

tan

x

x

. sec2 ( )4 + x

(B)11

+

tantan

xx . sec

2 ( )4 + x


3/8

3

QUEST TUTORIALS


(C) 12

1

1

+

tan

tan

x

x

. sec ( )4 + x(D) None of these

19. If y secx + tanx + x2 y = 0, thendydx

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) + xy = x2 - y, then

dydx

=

(A)

[ ]y xy y xy

xy xy y x

2 12

2 2

+

cos( )

cos( )

(B)[ ]2 12

2 2

xy y xy

xy xy y x

+

cos( )

cos( )

(C) -[ ]y xy y xyxy xy y x

2 12

2 2

+

cos( )

cos( )

(D) None of these

21. ddx

cos

+

12

2

1

1

x

x

=

(A)1

1 2+ x (B) - 1

1 2+ x

(C) -

2

1 2+ x (D)2

1 2+ x

22. If y = tan-1x a

x a

1 3 1 3

1 3 1 31

/ /

/ /

+

, then

dydx

=

(A)

( )

13 12 3 2 3x x/ /+

(B) ( )a

x x3 12 3 2 3/ /+

(C) -( )1

3 12 3 2 3x x/ /+

(D) -( )a

x x3 12 3 2 3/ /+

23. If y = cot -111

+

xx

, thendydx

=

(A)

11 2+ x

(B) -1

1 2+ x

(C)2

1 2+ x (D) -2

1 2+ x

24. The function f(x) = x at x = 0, is(A) Continuous & nondifferentiable(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 xx

2 31

satisfies all the conditions of

Rolles 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) 23

(B) 3


4/8

4

QUEST TUTORIALS


(C) 32 (D) 0

28. If x = seccos & y = secn cosn then :

(A) (x2 + 4)

dy

dx

2

= n2 (y2 + 4)

(B) (x2 + 4)dydx

2

= x2 (y2 + 4)

(C) (x2 + 4)dy

dx

2

= (y2 + 4)

(D) None of these

29. If xy = yx, thendy

dx=

(A)y x y y

y x x( log )

( log )

(B)y x y yx y x x

( log )( log )

(C)( log )( log )x y yy x x

(D) None of these

30. If y =

( )x

xx

, then

dy

dx =(A) y [xx (log ex) . log x + xx]

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

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

(D) y [xx (loge

x) . log x + xx - 1]

31. If y = x2 + xlog x , then

dydx

=

(A)

x x xx

x2 + log . log

(B) x2 + log x . x log x

(C)2 2x 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) willbe :

(A) 2 (B) 4

(C) 6 (D) 8

33. If y = sec-1xx

+

11

+ sin-1xx

+

11

then

dydx

=

(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 < 335. If y = (x 2 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

m2y =

(A) m2 (aemx be mx)(B) 1 (C) 0

(D) None of these

37. The rate of change of x2

16+ w.r.t.

xx 1

at x = 3, will be :

(A) - 245

(B) 245

(C) 125

(D) - 125

38. If y = f5 1

10 32x

x

+

& f(x) = cos x,

thendydx

=

(A) cos5 1

10 32x

x

+

ddx

5 1

10 32

x

x

+


5/8

5

QUEST TUTORIALS


(B) 5 110 32xx+ cos 5 110 32xx

+

(C) cos5 1

10 32x

x

+

(D) None of these

39. Consider f(x) =xx

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 andx [- 2, 4], then the maximum valueof 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 ddx

yd y

dx

22

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 thediameter of sphere, is :

(A) 2/3 (B) 1/3

(C) 3/4 (D) 1/4

44. If f(x) = sin x x2

is increasing

function, then :

(A) 0 < x

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M0IITU14 - Differentiation & Application Qns