Derivatives & Their Application

7/28/2019 Derivatives & Their Application

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solving the MCQs

DERIVATIVES & THEIR APPLICATION

1) The curve y = x 5/1 has at (0,0) :A) a vertical tangent

B) a horizontal tangentC) an oblique tangent

D) no tangent.

2) If , y = a log x + bx2+ x has its extreme value at x = -1 and x =2, then

A) a = 2, b = -1B) a = 2, b = 1/2

C) a = 2, b = -1/2D) none of these.

3) The linea

x+

b

y= 1 touches the curve y = b axe / at the point

A) (a, b)B) (-a, -b)

C) (a, 0 )D) None of these

4) A cone of maximum volume is inscribed in a given sphere. Then the ratio of the

height of the cone to the diameter of the sphere isA) 2 : 3

B) 3 : 4C) 1 : 3

D) 1 : 4

5) The maximum value ofxlog

is

A) 1B) e

2

C) e

D)e1

6) The normal at the point (1,1) on the curve 2y = 3 -x2

is

A) x + y = 0

B) x + y +1 = 0C) x y + 1 = 0

D) x y = 0


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7)

The tangent to the curve y = e

2 X

at the point (0.1) meets x-axis atA) (0,0)B) (2,0)

C)

o,

2

1

D) None of these

8) If the slope of the normal to the curve x3= 8a

2y , a> 0 at a point in the first

quadrant is -3

2, then the point is

A) (a , -a )B) ( 2 a , a )

C) ( a, 2 a)D) ( -a , a )

9) The function f(x) = x 3 - 3x is

A) increasing in ( )1, U (1, ) decreasing in (-1,1)

B) decreasing in (- )1, U ,1 and increasing in (-1, 1)C) increasing in( 0, ) and decreasing in (-0, )

D) increasing in( 0, ) and decreasing in (-0, )

10) Tangents to the curve y = x3+3x at x = -1 and x = 1 are:A) ParallelB) intersecting obliquely but not at an angle of 45

0

C) intersecting at right anglesD) intersecting at an angle of 45

0

11) A stone is projected vertically upwards moves under the action of gravity alone and

its motion is described by x = 49t - 4.9t2. It is at a maximum height when:

A) t = 0

B) t = 5C) t = 10

D) none of these

12) The function f(x) = ax + b, is strictly decreasing for all x R if and only if :A) a = 0

B) a < 0C) a > 0

D) none of these

13) Tangents to the curve y = x3

at points (1,1) and (-1, -1) are

A) intersecting but not at right angles


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B) parallelC) perpendicular

D) none of these

14) The slope of the tangent to the cruve: x = a sin t, y = a

2

tanlogcost

t

at the point t is:A) tan t

B)ttan

1

C) tan tD) none of these

15) Let f(x) = x3- 6x

2+ 9x + 18, then f ( x) is strictly decreasing in:

A) (1, 3)B) (- , 1] [3, )C) [3, )D) (- , 1)

16) The equation of the tangent at the point t to the curve y= 4 ax at ( at2, 2at ) is

A) tx + y = 2at + at 3

B) tx + y = 2at

C) ty + x = at2

D) None of these

17. Let f (x) = x3+

2

3x

2+ 3x +3, then f (x) is:

A) a decreasing functionB) an increasing function

C) an odd functionD) an even function

18) The normal to a given curve is parallel to x - axis if:

A) 1dy

dx

B) 0dy

dx

C) 0dx

dy

D) 1dx

dy

19) The point on the curve y = x2, where slope of the tangent is equal

to the x coordinate of the point is

A) (-1,1)B) (0,0)


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C) (1, 0)D) (0,1)

20) The curve y - exy + x = 0 has a vertical tangent at the point

A)

(1,0)B) (1,1)C) ( 0,0)

D) none of these

21) The maximum value of f(x) = sinx + cosx is

A) 1B) 2

C)2

1

D) 2

22) If sum of two numbers is 3, then maximum value of the product of first and squareof second is:

A) 4B) 3

C) 2D) 1

23) The maximum value of the function f (x) =

x

x

1is

A) eB) (e)

1/e

C) (1/ e) e

D) none of these

24) If x>0 , xy = 1, minimum value of x + y isA) 2

B) -2C) 1

D) None of these

25) On uniform heating the side of a square sheet of metal is increasing at the rate of0-02 cm/sec. The rate at which the area is increasing when the side 10 cm long .

A) 0.4 cm 2 /cm

B) 0.2 cm 2 /sec

C) 4.0 cm 2 /sec

D) 40 cm 2 /sec


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26) Letf(x) = (x 2 -4) 31 , thenfhas a

A) local maxima at x = 0

B) local minima at x = 0C) point of inflexion at x = 0

D) None of these

27) The functionf(x) = 2 + 4x2+ 6x

4+ 8x

6 has

A) only one maximaB) only one minima

C) no maxima and minimaD) many maxima and minima

28) Letf(x) have second derivative at c such thatf' ( c) = 0

andf" (c) > 0, then c is a point ofA) inflexion

B) Local maximaC) local minima

D) None of these

29) The functionf(x) = 2x 3 - 3x 2 -12x + 4 has

A) no maxima and minimaB) two minima

C) two maximaD) one maxima and one minima

30) The sum of two non- zero numbers is 4. The minimum value of the sum of

their reciprocals isA) 0

B) 1C) 2

D) 3

31) On the interval 1.0 , the functionx 75125 x takes its maximumvalue at the point

A) 0

B)3

1

C)4

1

D)2

1


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32) The function f(x) = ex (x-1) (x-2) decreases in the interval

A) (- , -2)

B)

(-2, -1)C) (1, 2)D) (D)(2, + )

33) If the normal to the curve y =f(x) at the point (3,4) makes an angle 3 /4 with the

positive x-axis, thenf' (3) isA) -1

B) -2C) 2

D) 1

34)

The tangent to the curve y = x

3

6x

2

+ 9x + 4 , for 0

x

5

has maximum slope at x = ,A) 2

B) 3C) 5

D) 4

35) A stone thrown vertically upwards satisfies the equation s = 80t -16t2. The time

required to reach the maximum height is

A) 2B) 4

C)

3D) none of these

36) If the function f (x) = x3

+k

is maximum at x = 2 then k is :

A) 8

B) 16C) 24

D) 32

37) The minimum value of x log e x is equal to

A) e

B) -e

1

C) e

D)e

2


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38) The velocity m/sec of particles is proportional to the cube of the time.

If the velocity after 2 sec is 4 m/sec then is equal to:

A) t3

B)2

3t

C)3

3t

D)4

3t

39) The maximum value of f(x) =241 xx

x

on 1,1 is

A)4

1

B) -3

1

C)6

1

D)5

1

40) The maximum value of sin x (1 + cos x) will be at

A) x = 2/ B) x = 6/

C) x = /3

D) x = /4

41 ) The abscissae of the points of the curve y =x (x-2) (x -4), where tangents areparallel to x-axis, is obtained as

A) x = 23

2

B) x = 1

3

1

C) x = 23

1

D) x = 1

42 ) If the function f(x) = 2x ax93 112 22 xa , where a > 0 attains its

maximum and minimum at p and q respectively such theirp q2 then a equals:

A) 1

B) 2C) 4


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D) 3

43) A point an the parabola y2= 18x at which the oridinate increases at twice the

rate of the abscissa is:

A) (2,4)B) (2, -4)

C) (9,9)D) None of these

44) The function f(x) = xx decreases on the interval

A) (0, e)

B) 0.1)

C) (0, e1 )

D) None of these

45) The interval of increase of the function y =x ex + tan ( 7/ ) isA) (- )1,

B) (0, )

C) (- 0, )

D) (1, )

46) The number of points extremum of the functionf(x) = 3x bxx 234 64 for anyvalue of b is

A) 4B) 3

C) 1D) 2

47) The area of the triangle formed byt eh positive x- axis and the normal and the

tangent to the circlex 22 y = 4 at (1, 3 ) is

A) 2 3

B) 3

C) 4 3

D) 3

48) The critical points of the function f(x) = (x -2) 32 (2x + 1) areA) -1 and 2

B) 1C) 1 and -1/2

D) 1 and 2

49 ) Letf(x) =x xe2 then

A) maxf(x) = e1

B) max f(x) = 4 2e


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C) minf(x) = e 1 D) minf(x)> 0

50) A rectangle with perimeter 32 cm has greatest area if its length isA) 12

B) 10C) 8

D) 14

51 ) A particle is moving along the parabola y 2 = 4 (x + 2). As it is passes through the

point (7, 6) its y coordinate is increasing at the rate of 3 units per second. Therate at which x coordinate change at this instant is (in units/sec)

A) 4B) 6

C) 8D) 9

52 ) Let f(x) =

bax

x2

1

1

1

xfor

xfor

The coefficients a and b so thatfis continuous and differentiable at any point,are equal to

A) a = -1/2, b = 3/2B) a = 1/2, b = -3/2

C) a = 1, b = -1D) None of these

53 ) If y = log xe (x-2)2 x 0, 2, then y' (3) is equal to

A) 1/3

B) 2/3C) 4/3

D) None of these

54 ) The function offdefined by

F(x) =

x

x2sin

0

0

xfor

xforis

A) continuous and derivable at x = 0B) neither continuous nor desirable at x = 0

C) continuous but not desirable at x = 0D) None of these

55 ) . The derivative of tan 1

x

x 112

w.r.t tan 1

2

2

21

12

x

xxat x = 0 is

A) 1/4.

B) 1/8


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C) 1/2.D) 1

56 ) If y = sin 1

x

x

sincos1

sinsin

, then y (0) is

A) 1B) tan

C) (1/2) tan

D) sin 66.

57 ) If x = log tandy = t2

-1, then y (1) at t= 1 is

A) 2B) 4

C) 3D) None of these

58 ) If x = log t and y = t2 -1, then y" (1) at t = 1 is

A) 2B) 4

C) 3D) none of these

59 ) If x = sin 1 t and y = log (1 - t 2 ); then2

2

dx

yd2/1t is

A) -8/3B) 8/3

C) 3/4.D) -3/4

60 ) The function y = sin 1 x satisfies

A) (1 - x 2 ) y "' = xy"

B) (1 - x 2 ) y" = xy'

C) (1 - x 2 ) y" = x 2 y'

D) (1 - x 2 ) y' = 2xy'

61 ) If y = cos1(x

1) then y (-2) is equal to

A)32

1

B) -32

1

C)52

1


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D)52

1

62 ) . If x = sin 1 (t 2 )1 , y = cos 1 (2t) thendx

dyat t = 0 is

A) - 2

B) -2

1

C) 2

D)2

1

63 ) If tan 1 y y + x = 0 then2

2

dx

ydis equal to

A)

5

2 )1(2

y

y

B)5

21

y

y

C)4

2 )1(2

y

y

D)5

2 )1(2

y

y

64 ) If F (x) =

x

x

x

x

xx

6

3

2

2

0

1 2

32

then F' (x) is equal to

A) 6x3

B) x 23 6x C) 3x

D) 6x2

65 ) If y =x

x

cot1

sin 2

+

x

tan1

cos2

then y (x) is equal to

A) cos 2xB) cos 2x

C) cos x2

D) cos x3


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66 ) The derivative of sin

n

n

x

x2

21

1

1with respect to x n2 is

A) - nn x31

B)nn x3

1

C)nn x 2

1

D) -nn x 2

1

67 ) If y = cos1

41

sin5cos4 xx, then

dx

dyis equal to

A) 0

B) 1C) -1

D) None of these

68 ) If y = log cos

2

tan1

xx eethen y (0) is equal to

A) e + e1

B) e - e1

C)2

1

ee

D) None of these

69 ) If the function y = log1

1satisfies the relation xy + 1 =f (y) thenf (y) is equal

toA) y

B) y 12

C) ey

D) ey

70 ) If y =2

1

1

sin

x

x

satisfies the relation (1-x 2 )y xy = k then the value of k is

A) 1B) 0

C) -1


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D) 2

71 ) If sin y = x sin (a + y) anddx

dy=

a

A

cos21 2 then the value of A is

A) 2

B) cos aC) sin a

D) None of these

72 ) If x = a cos 2t, y = b sin2t then

2

2

dx

ydis equal to

A) tb

a2cos

B) 0C) 1

D) t

a

b2sin

73 ) If f(x) = tan 1 )sin1/()sin1( xx , 0 ,2/x thenf ( )6/ is

A) 1/4.B) -1/2

C) 1/4D) 1/2

74 ) If 2x + 2y = 2 yx , then the value ofdx

dyat x = y = 1 is

A) 0

B) -1C) 1

D) 2

75 ) (dx

dcos

-1x + sin

-1x) is

A)2

B) 0

C)21

2x

D) none of these.


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76 ) The derivatives ofelogx

isA) (logx) elogx

B) elogx

C)

(logx) e

logx-x

D) 1.

77 ) Iff(x) = log(x+ 12 x ), then f ' (x) equals

A) 12 x

B)12 x

x

C) 1+12 x

x

D)1

12 x

78 ) If y = sec-1

dx

dythen

x

x

x

x,

1

11sin

1

1

A) 1B) 0

C)1

1

x

x

D)1

1

x

79 ) Iff(x) =x

x

2

4thenf'(0) is

A) 0

B) 1

C) does not existD) none of these.


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80 ) The differential co-efficient off(x) = log (logx) is

A)

x

x

log

B) (x logx)-1

C)xlog

D) x logx

81 ) If isdx

dythenyxayx ),(11 22

A)2

2

1

1

x

y

B)2

2

1

1

y

x

C) 21 x

D) 21 y

82 ) If y =

ac

a

ccb

c

bba

b

a

xx

xx

xx

, then dx

dy

A) 0

B) 1C) a + b + c

D) none of these

83 ) If y = tan-1 dx

dythen

x

x,

1

1

is equal to

A) 211

B) -21

1

C) x1tan4

D) tan-1

x


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84 )2

1

1

1)(cos

xx

dx

d

where

A) -1


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D)t

1

90 ) If yx = 4, thendy

dxat y =1 is

A) 3B) -1C) -3

D) none of these.

91 ) Differential co-efficient of log 10x w.r.t log x 10 is

A)2

2

)10(log

)(logx

B)2

2

10

)10(log

)(log x

C)2

2

)10(log

)10(logx

D)2

2

)(log

)10(log

x

92 ) Ifx = at 2y = 2at, then 2

2

dx

yd

A) -2

1

t

B)32

1

at

C)3

1

t

D) -32

1

at

93 ) If y = axn+1

+bx-n

, thenx2

2

2

dx

yd

A) n (n-1) y

B) n(n+1) yC) nyD) n2y


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19/38

100 ) If y =

0,

,0,0

0,

xx

x

xx

then at x = 0, y is:

A) not continuous

B)

continuous but not differentiableC) differentiable

D) None of these.

101 ) If (x) is a polynomial of degree m ( 1), then which of the following is the not true

A)n

n

dx

yd= 0, for all n > m

B) is a derivable at all x RC) is continuous at all x RD) none of these.

102 ) Let (x) =

0,

0,

2

2

xx

xxthen:

A) (x) is not derivable at x = 0B) (x) is derivable at x = 0

C) (x) is not continuous at x = 0D) (x) is continuous but not derivable at x = 0.

103 ) The function (x) = (x a) sina

1for x a and (a) = 0 is :

A) derivable at x = aB) not continuous at x =a

C) continuous but not derivable at x = aD) none of these.

104 ) The derivative of an even function is :

A) an even functionB) an odd function

C) non- negativeD) none of these.

105 ) Derivative of tan-1

x

x 121 with respect to tan

-1x is :

A)21

1

B)2

121

x

x

C) 1

D)2

1.


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106 ) If y = sin -1

2

2

1

1

x

x, then

dx

dyequals :

A)21

2

B)2

1

2

x

C)22

1

D)22

2

.

107 )If x = sec - cos ,y = secn - cos

n , then

2

dx

dyis:

A)4

)4(2

22

x

yn

B)

2

22 4

x

yn

C) n4

42

2

y

D)

2

x

ny 4.

108 ) If (x) = ex

g(x) , g (0) = 2, g (0) = 1, then (0) is:

A) 1B) 3

C) 2D) 0.

109 ) If (x) = x tan-1

x, then (1) is:

A)42

1

B)42

1

C)42

1

D) none of these.


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110 ) If xy = exy, thendx

dyis equal to:

A) 2log1 x

y

B) 2log1 x

x

C) 2log1

log

x

x

D) none of these.

111 ) .If y = sec-1

1

1

x

x

+ sin-1

1

1

x

x,then

dx

dyis:

A) 1

B)1

1

x

x

C)1

1

x

D) 0.

112 ) .The differential coefficient of tan-1

xx

xx

sincos

cossinw.r.t. x is:

A) 0

B)2

1

C) 1

D) none of these.

113 ) The differential coefficient of log tan x is :

A) sec 2xB) 2 cosec 2x

C) 2 sec3

x

D) 2 cosec3 x.

114 )The differential coefficient of (x) =log (log x) is:

A)x

x

log

B) xx log -1

C)xlog

D) x log x.


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115 ) The derivative of sin-1

x w.r.t. cos-1

21 x is:

A)21

1

x

B)

1C) cos-1

x

D) tan-1

21 x

x

.

116 ) If 2x

2y

= 2x+y

, thendn

dyis equal to:

A)yx

yx

22

22

B)yx

yx

21

22

C)x-y

x

y

21

12

D)y

xyx

2

22

117 )dx

d(tan

-1(sec x +tan x )) is equal to :

A) 0

B) sec x- tan x

C)2

1

D) 2.

118 ) If y = ....... xxx to , thendx

dyis equal to :

A) 1

B)xy

1

C) xy 2

1

D)12

1

y.


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119 ) If y = ....sinsinsin xxx to , then the value ofdx

dyis equal

A)1

sin

Y

x

B) 1

sin

Y

x

C)12

cos

y

x

D)12

cos

y

x.

120 ) .The derivative of sec-1

12

12x

w.r.t. 21 x at x =2

1is:

A) 2B) 4

C) 1D) -2

121 ) If y = log ,tanx the value ofdx

dyat x = /4, is given by :

A) 1

B) 0

C)2

1

D) .

122 ) The differentiable coefficient of x6 w.r.t. x3 is:A) 6x

5

B) 3x2

C) 2x3

D) X3.

123 ) If (x) = log e2(log x), then (e) is :A) 0

B)e2

1

C)2

e

D)e

2.


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124 ) If (x) = cot-1

(cos 2x)1/2

, then ' ( 6/ ) is :

A)3

1

B)3

2

C)3

2

D) -3

2.

125 ) If y = a (1 + cos t) and x = a (t sin t ), thendx

dyis equal to :

A) tan2

t

B) -tan2t

C) -cot2

t

D) none of these.

126 ) Let (x) =2

2

1 x

x

, x 0, then the derivative of (x) w.r.t. x is :

A) 221

2

x

x

B) 221

1

x

C)2

22

1

x

D) 222

1

x.

127 )dxd

xxec

21cos

21 is :

A) -21

2

, x 0

B)21

2

,x 0

C)

222

11

12

xx

x

, x 1, 0.

D) None of these.


25/38

128 )If y = log axx , thendx

dyis :

A)axx

1

B)axx 2

1

C)axx

1

D) None of these.

129 )dx

d

)22log(

2

222

2axx

aax

xis:

A)axx 2

1

B) 22 ax

C)22

1

ax

D) None of these.

130 ) If y = ....coscoscos 222 xxx to , thendx

dyis :

A) 12sin 2

yx

x

B)12

sin2 2

y

xx

C)12

sin

y

x

D) None of these.


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27/38

135 ) If y = x xx ...

, then xdx

dyequals:

A)2

)log1(

y

xyx

B))log1(

2

xyx

y

C)xy

y

log1

2

D) None of these.

136 ) If y = (sin x)tan x

,then

dx

dyis equal to:

A) tan x (sin x)tan x 1

B) (sin x)tan x

(1 + sec2

x log sin x)C) tan x(sin x)

tan x - 1cos x

D) (sin x)tan x

sec2

x. log sin x.

137 ) If x = a sin ,y = a (1 + sin ), thendx

dyat =

3

is :

A)3

1

B) 3

C)32

3

D)3

32 .

138 ) If y =

X

x

11 , then

dx

dyequals:

A)

X

xx

1

xx 1

11

1log

B)

X

x

11

xx 1

111log

C)

X

x

11

x

11log

D)

X

xx

1

1)1log(

x

xx .


28/38

139 ) If y = cex/ (x - a),

theny

1dx

dyis equal to :

A) a (x-a)2

B) -2)( ax

ay

C) a2

(x- a)2

D) None of these.

140 ) If (x) =3e2

x, then (x) -2x (x) +

3

1(0) (0)is equal to:

A) 0

B) 1

C)2

3

7 xe

D) None of these.

141 ) If x = 2 cos t cos 2t, y = 2 sin t sin 2t, then the value of2

2

dx

ydat t = /2 is:

A) 3/2

B) -3/2

C) 5/2D) -5/2.

142 ) If y = e-x

(A cos x + B sin x),then y satisfies:

A)2

2

dx

yd+

dx

dy2= 0

B)2

2

dx

yd-

dx

dy2+ 2y =0

C)2

2

dx

yd+

dx

dy2+ 2y = 0

D)2

2

dx

yd+2y = 0.


29/38

143 ) If y = log (sin x),then2

2

dx

ydis:

A) -cosec2 xB) Sec

2x

C) cosec x x cot x

D) Sec x tan x..

144 ) If y = a sin mx + b cos mx,,then2

2

dx

ydis :

A) -m2

y

B) myC) m

2y

D) None of these.

145 ) If y = a emx

+ b e-mx

, then y 2 is:

A) -m2y

B) -my1

C) m2y

D) None of these.

146 ) If y2 = ax2 + b, then2

2

dx

ydis:

A)3y

ab

B) 3ab

C)2y

ab

D) None of these.

147 ) If x =2

2

1

1

t

t

and y =

2121

2121

tt

tt

,then the value of

2

2

dx

ydat t = 0 is given by:

A) -1

B) 1C) 0

D)2

1.


30/38

148 ) If yx + xy = c, then2

2

dx

ydis equal to:

A)2

2

c

B) c

2

C) -2

2

c

D) None of these.

149 ) If y = x + ex

, then2

2

dx

ydis:

A) 21

1

xe

B) - 21 x

x

e

e

C) - 31 x

x

e

e

D) ex.

150 ) If y = aex

+ be2x

, then:

A)2

2

dx

yd+ 3

dx

dy+ 2y = 0

B)2

2

dxyd + 3

dxdy - 2y = 0

C)2

2

dx

yd- 3

dx

dy-2y = 0

D)2

2

dx

yd- 3

dx

dy+ 2y = 0 .

151 ) If y = (x + 21 x )n , then (1 +x2)2

2

dx

yd+x

dx

dyis :

A) n2y

B) n2yC) yD) 2x2 y.


31/38

152 ) If x = ey+e......y

, x > 0, thendx

dyis:

A)x

1

B) x

1

C)x1

D)x1

.

153 ) If y is a function of x and log (x + y ) 2xy = 0, then y (0) is equal to :

A) 1B) -1

C) 2

D) 0.

154) If 3 sin ( xy) & 4 cos (xy) = 5 thendx

dyis

A)2

2

x

y

B))sin(4cos3

)cos(4sin3

xyxy

xyxy

C) 3 cos xy 4 sin xy

D) x

y

155) If xy = ex y

then

A)dx

dydoesnt exist at x = 1

B)dx

dy= 0 when x = 1

C)dx

dy=

2

1when x = e

D) None of these.

156 ) The derivative of the sin x3

with respect to cos x2

A) Co + x3

B) co + x3C) tan x

3

D) tan x3


32/38

157 ) If y = sec -1

1

1

x

x+ sin -1

2

1

x

xthen

dx

dyis equal to

A) 0B) x + 1

C)

1D) -1

158 ) The function y = cos-1

x satisfies

A) ( 1- x2 ) y" = xy"B) ( 1- x2 ) y" = - 2xy'

C) ( 1- x2 ) y" = x2 y'D) ( 1- x2 ) y" = 2 xy'

159 ) If y = tan-1

2log(

2/log

ex

xe+ tan-1

x

x

log61

log23, then

2

2

dx

ydis

A) ( A ) 2B) ( b ) 1

C) ( c ) 0D) ( d ) -1

160 ) The derivative of tan-1

x

x 12

1 w.r.t.tan

-1

2

2

21

12

x

xxat x = 0 is

A) 1/4

B) 1/8C) 1/2

D) 1

161 ) If A = 4 r2 , the value of

dr

dAwhen r = 3 is

A) 8

B) 24

C) 36

D) 16

162 ) If x = sin-1

t and y = log (1 t2

);then

2

1tdx

dyis

A) -3

2

B)3

4

C)3

2

D)3

4


33/38

163) If x = a sin , y = a cos1 then2

2

dx

ydat =

3 is

A) 2/a

B) 2/aC) 4/aD) 4/

164) The maximum value of

F(x) = xx

0,log

is

A)e

2

B)

e

1

C) e

D) e165) Forf (x)= 3 sinx+3 cosx, the point

6

x is

A) local minimum

B) local maximumC) point of inflexion

D) none of these

166) The maximum value of sinx + cosx is

A) 2

B) - 2

C) 3

D) 2

167) The maximum value off(x) =x1/x

is

A)e

2

B) eC) e1/e

D)e

1

168) The point on the curve y2

= 4x which is nearest to the point (2,1) is

A) (1, -2)B) (-2,1)

C) (1, 2 )2

D) (1,2)


34/38

169) The volume of a ball is increasing at the rate of 4 c.c/sec. The rate of increase of

the radius when the volume is 288 c.c is

A)6

1cm/sec

B)

36

1

cm/sec

C)9

1cm/sec

D)24

1cm/sec

170) The speed v of the particle moving along a straight line is given by a + bv2=x

2,

where x is its distance from the origin. The acceleration of the particle is

A)b

x

B)abx

C) ab xD) ax

171) If the area of an expanding circular region increases at a constant rate with respect

to time, then the rate of increase of the perimeter with respect to timeA) varies inversely as the radius

B) varies directly at the radiusC) remains constant

D) varies directly as square of the radius

172) The point at which the tangent to the curve y = 2x2 x +1 is parallel to y =

3x+9 is

A) (-2,1)B) (3,9)

C) (1,2)D) (2,1)

173) The height of the cylinder of maximum volume increasing in a sphere of radius a is

A) 3

2a

B)2

3a

C)3

2a

D)3

a


35/38

174 ) If y = xx , thendx

dyis equal to

A ) x 4/1

B ) x 4/1

C ) 4/1

4

3x

D ) 4/1

4

3 x

175 ) If y = (sin x)x

thendx

dy=

A ) x sin xx 1

B ) x log sin x

C ) sinx x )cotsin(log xxx

D ) x xx sinlog

176 ) . If y = tan )tan(sec1 xx then.dx

dy=

A ) 1

B )2

1

C )2

D ) 0

177 ) If x y = 7 yx .thendx

dy=

A )xx

yx

7log

7log

B )xx

xy

7log

7log

C )x

yx

7log

)7(log

D )xx

yx

7log

7log

178 ) Differentiating log ( sec x + tan x ) w.r.t. sec x we get,

A ) sec x ( sec x + tan x)

B ) cot xC ) tan x


36/38

D ) sec x tan x

179 ) The derivative of sin-1

21

2

x

xw. r. t cos

2

21

1

1

x

xis ,

A ) 1

B ) 0C ) -1

D ) 2

180 ) . If y = sec x + tan x then.2

2

dx

yd=

A ) y sec x

B ) xy sec2

C ) y cos x

D ) xy cos2

181 ) y = log xsin ,5dx

dy=

A )x

x

sin

cos

B )2

1cot x

C ) 2

1

tan x

D ) xx cossin

182 ) . If sin (x + y) = log (x + y) thendx

dy=

A ) 0B ) -2

C ) 1D ) -1

183 ) The perimeter of a rectangle is 100 cm. The lengths of its sides to give maximumArea are :

A ) 25 25B ) 30 20C ) 22 28D ) 24 26


37/38

184 ) The length seg AB is 12 cm. The point P on seg AB such that AP2

+ BP2

is minimum

A ) AP = 5 , BP = 7

B )

AP = 6 , BP = 6C ) AP = 8 , BP = 4D ) AP = 9 , BP = 3

Derivatives &their applications

Sr.no. answer

1 A

2 C

3 D

4 A

5 D

6 D

7 C

8 B

9 A

10 A

11 B

12 B

13 C

14 B

15 A

16 C

17 B

18 B

19 B

20 A

21 D

22 A

23 B

24 A

25 A

26 B

27 B

28 C

29 D

30 B

31 C

32 C

33 D

34 B

35 D

36 B

37 B

38 B

39 C

40 C

41 A

42 B

43 D

44 C

45 C

46 C

47 A

48 D

49 B

50 C

51 D52 A

53 B

54 A

55 A

56 D

57 B

58 B

59 A

60 D

61 A

62 B

63 A

64 D

65 B

66 A

67 B

68 D

69 C

70 A

71 C

72 B

73 D

74 B

75 D

76 D

77 D

78 D

79 C

80 B

81 A

82 A

83 A

84 A

85 A

86 D

87 B

88 D

89 D90 C

91 A

92 D

93 B

94 B

95 A

96 C

97 B

98 D

99 D

100 B

101 D

102 B

103 C

104 B

105 D

106 A

107 A

108 B

109 A

110 C

111 D


38/38

112 C

113 B

114 B

115 B

116 C

117 C

118 D

119 D

120 B

121 A

122 C

123 B

124 C

125 C

126 A

127 B

128 B

129 B130 B

131 C

132 D

133 C

134 C

135 C

136 B

137 A

138 B

139 B

140 B

141 B

142 C

143 A

144 A

145 C

146 A

147 D

148 A

149 C

150 D

151 A

152 C

153 A

154 D155 B

156 B

157 A

158 B

159 C

160 A

161 B

162 A

163 C

164 B

165 A

166 A

167 C

168 D

169 C

170 A

171 A

172 C

173 C

174 D

175 C

176 B

177 A

178 B

179 A180 B

181 B

182 D

183 A

184 C

Derivatives & Their Application

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