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7/28/2019 Derivatives & Their Application
1/38
Note: Please read the Important Instructions and Disclaimer file before
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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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.
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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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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 .
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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.
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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.
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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.
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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
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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
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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)
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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
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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
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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
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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
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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