Continuity And Differentiability Questions

We provide continuity and differentiability practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on continuity and differentiability skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

List of continuity and differentiability Questions

Question No Questions Class
1 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left(frac{mathbf{1}-boldsymbol{x}}{mathbf{1}+boldsymbol{x}}right)= )
A ( cdot frac{-1}{1+x^{2}} )
B ( cdot frac{1}{1+x^{2}} )
c. ( frac{1+x}{1-x} )
D. ( frac{2}{1+x^{2}} )
12
2 ( frac{d sin x^{2}}{d x} )
A. ( 2 x cos x^{2} )
B . ( 4 x cos x^{2} )
c. ( 2 x sin x^{2} )
D. ( -2 x sin x^{2} )
12
3 30.
Iff: R
R is a function defined by f (x) = [x]
(2x-1)
– Tt, where [x] denotes the greatest integer
COS
cos2
function, then fis.
[2012]
(a) continuous for every real x.
(b) discontinuous only at x = 0
(c) discontinuous only at non-zero integral values of x.
(d) continuous only at x =0.
12
4 ( left{begin{array}{ccc}text { Find } lim _{x rightarrow 0} f(x) & text { where } f(x)= \ x-1 & text { if } & x0end{array}right. ) 12
5 Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a
point ( ^{prime} c^{prime} ) in the indicated interval as
stated by the Lagrange’s mean value theorem:
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}+boldsymbol{2} ) on [-1,2]
12
6 (1+x,0 SX S2
(1983 – 2 Marks)
13 – x,2 5×53
Determine the form of g(x)=ff(x) and hence find the points
of discontinuity of g, if any
12
7 If ( u=tan ^{-1}left(frac{x^{2}+y^{2}}{x+y}right), ) then ( x frac{d u}{d x}+ )
( boldsymbol{y} frac{boldsymbol{d} boldsymbol{u}}{boldsymbol{d} boldsymbol{y}}= )
( mathbf{A} cdot sin 2 u )
B. ( frac{1}{2} sin 2 u )
c. ( frac{1}{3} sin 2 u )
D. ( 2 sin 2 u )
12
8 If ( int f(x) d x=frac{3}{55} sqrt[3]{tan ^{5} x}left(5 tan ^{2} x+right. )
11) ( +C ) then ( f(x) ) is equal to
This question has multiple correct options
A ( cdot sqrt[3]{sin ^{2} x cos ^{-14} x} )
B. ( sqrt[3]{tan ^{2} xleft(1+tan ^{2} xright)^{6}} )
c. ( sqrt[3]{cos ^{2} x sin ^{-14} x} )
D. ( frac{7}{3} sqrt[3]{sin ^{2} x cos ^{-14} x} )
12
9 ( operatorname{Let} mathbf{f}(mathbf{x})=mathbf{x}+tan ^{-1} mathbf{x}, mathbf{g}(mathbf{x})= )
( frac{x}{1+x^{2}}(x>0) ) Then
A ( cdot mathrm{f}(mathrm{x})0 )
B. ( f(x)>g(x), x>0 )
c. ( f(x)<g(x) ) in ( [1, infty) )
D. None of these
12
10 Evaluate ( int_{-6}^{0}|x+3| d x . ) What does this integral represent on the graph? 12
11 ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) in [-1,1] verify Rolle’s
theorem.
12
12 State if the given statement is True or False

Derivative of ( y=cos x ) with respect to ( x )
is ( sin x )
A. True
B. False

12
13 Differentiate: ( y=frac{x+2}{3 x+5} ) w.r.t ( x ) 12
14 Differentiate the following w.r.t. ( x: )
( sin left(tan ^{-1} e^{-x}right) )
12
15 Let ( boldsymbol{f}(boldsymbol{x}) ) be a differentiable function in
( [2,7] . ) If ( f(2)=3 ) and ( f^{prime}(x) leq 5 ) for all ( x )
in ( (2,7), ) then the maximum possible
value of ( f(x) ) at ( x=7 ) is
( A cdot 7 )
B. 15
c. 28
D. 14
12
16 ( boldsymbol{f}(boldsymbol{x})left{begin{aligned}=& frac{left|boldsymbol{x}^{2}-boldsymbol{x}right|}{boldsymbol{x}^{2}-boldsymbol{x}}, quad boldsymbol{x} neq boldsymbol{0}, boldsymbol{x} neq mathbf{1} \=& boldsymbol{x}=mathbf{0} \=-mathbf{1}, & boldsymbol{x}=mathbf{1} end{aligned} ) Discus right.
its continuity in ( 0<x leq 1 )
This question has multiple correct options
A. continuous at ( x=0 )
B. dis-continuous at ( x=1 )
c. dis-continuous at ( x=0 )
D. continuous at ( x=1 )
12
17 Assertion ( (A): f(x)=sin (pi[x]) ) is
differentiable every where [] is greatest
integer function
Reason ( (mathrm{R}): ) If ( mathbf{x}=mathbf{n} boldsymbol{pi} Rightarrow sin boldsymbol{x}=mathbf{0} forall mathbf{n} in )
( mathbf{Z} ) then
A. Both (A) and (R) are true and R is correct explanation for A
B. Both (A) and (R) are true and R is not correct explanation for
c. (A) is true (R) is false
D. (A) is false (R) is true
12
18 If ( boldsymbol{y}=tan ^{-1}left(frac{1+boldsymbol{x}^{2}}{1-boldsymbol{x}^{2}}right) ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? )
A. ( frac{2 x}{left(1+x^{4}right)} )
B. ( frac{-2}{left(1+x^{4}right)} )
c. ( frac{x}{left(1+x^{4}right)} )
D. none of these
12
19 Examine the continuity of:
[
begin{aligned}
boldsymbol{f}(boldsymbol{x}) &=boldsymbol{x}^{2}-boldsymbol{x}+boldsymbol{9} text { for } boldsymbol{x} leq mathbf{3} \
&=boldsymbol{4} boldsymbol{x}+mathbf{3} quad text { for } boldsymbol{x}>mathbf{3}, boldsymbol{a} boldsymbol{t} boldsymbol{x}=boldsymbol{3}
end{aligned}
]
12
20 Derivative of ( tan ^{3} theta ) with respect to ( sec ^{3} theta ) at ( theta=frac{pi}{3} ) is
A ( cdot frac{3}{2} )
B. ( frac{sqrt{3}}{2} )
( c cdot frac{1}{2} )
D. ( -frac{sqrt{3}}{2} )
12
21 If ( boldsymbol{y}=log _{10} boldsymbol{x}+log _{x} mathbf{1 0}+log _{x} boldsymbol{x}+ )
( log _{10} 10, ) then ( frac{d y}{d x}= )
A ( cdot frac{1}{x log _{e} 10}-frac{log _{e} 10}{xleft(log _{e} xright)^{2}} )
B. ( frac{1}{log _{e} 10}-frac{log _{e} 10}{xleft(log _{e} xright)^{2}} )
c. ( frac{1}{x log _{e} 10}-frac{log _{e} 10}{x^{2}left(log _{e} xright)^{2}} )
D. None of these
12
22 A function ( f(x) ) is defined by ( f(x)= )
( left{begin{array}{cl}frac{left[x^{2}-1right]}{x^{2}-1} & text { for } x^{2} neq 1 \ 0 & text { for } x^{2}=1end{array} ) Discuss the right.
contiuuity of ( f(x) ) at ( x=1 )
12
23 If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{a} cos ^{-1} boldsymbol{x}} ) then ( left(boldsymbol{1}-boldsymbol{x}^{2}right) boldsymbol{y}^{prime prime}-boldsymbol{x} boldsymbol{y}^{prime}= )
A . ( a y )
B. ( -a^{2} y )
c. ( -a y )
D ( cdot a^{2} y )
12
24 Differentiate: ( sin left(tan ^{-1} e^{x}right) ) 12
25 Identify whether ( f(x)=frac{x^{2}-4}{x-2} ) is
continuous at ( x=2 ) or not
12
26 Show that ( f(x)=frac{cos 3 x-cos 4 x}{x sin 2 x} ) for
( x neq 0, f(0)=frac{7}{4} ) is continuous at ( x= )
0
12
27 Differentiate w.r.t ( x )
( boldsymbol{y}=boldsymbol{x}^{2} sin 2 boldsymbol{x} )
12
28 Differentiate: ( mathbf{5}^{mathbf{5}^{5 x}}=boldsymbol{t} ) 12
29 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cc}frac{x^{2}-2 x-3}{x+1^{2}}, & text { when } x neq-1 \ k, & text { when } x=-1end{array}right. )
If ( f(x) ) is continuous at ( x=-1 ) then
( boldsymbol{k}=? )
A . 4
B. -4
c. -3
D.
12
30 If ( sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}}=4 ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{1} ) 12
31 Find the values of ( a ) and ( b ) so that the
function, ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{1-sin ^{2} x}{3 cos ^{2} x}, quad xpi / 2end{array}right. )
continuous
12
32 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=tan ^{-1}left(frac{mathbf{5} boldsymbol{x}+mathbf{1}}{mathbf{3}-boldsymbol{x}-mathbf{6} boldsymbol{x}^{2}}right) ) 12
33 ( f(x)=frac{x^{2}-16}{x-4}+a ) for ( x4
]
continuous at ( x=4, ) find ( a ) and ( b )
12
34 The velocity of a particle is given by ( v= )
( 12+3left(t+7 t^{2}right) . ) What is the
acceleration of the particle?
A ( .3+21 t )
B. ( 3+42 t )
( c .42 t )
D. ( 4 t )
12
35 ( operatorname{Let} sqrt{x}+sqrt{x+sqrt{x+ldots ldots infty}} ) then
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A. ( frac{1}{2 y-1} )
в. ( frac{x}{x+2 y} )
c. ( frac{1}{sqrt{1+4 x}} )
D. ( frac{y}{2 x+y} )
12
36 3.
The function f(x)=1+ sin xis (1986-2 Marks)
(a) continuous nowhere
(b) continuous everywhere
© differentiable nowhere
(d) not differentiable at x=0
(e) not differentiable at infinite number of points.
12
37 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of ( boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=mathbf{1 0 0} ) 12
38 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( log sqrt{frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}} )
12
39 Find the derivatives of ( x cos x ) 12
40 Find the derivative of ( x^{4}+4 ) 12
41 3.
(1994)
If y = (sin x)tanx, then is equal to
(a) (sin x)tan (1 + sec?x log sin x)
(b) tan x (sin x)tan x-1.cos x
(c) (sin x)tan x sec2x log sin x
(d) tan x (sin x)tan x-1
12
42 4.
If r2 + y2=1 then
(a) “-26”)2 + 1 = 0
(c) “+’)? – 1 = 0
(2000)
(b) yy”+(2+1=0
(d) yy”+26′)2+1 = 0
12
43 9.
3
If f(x + y) = f(x).f(y)Vx.y and f(5) = 2, f ‘(0) = 3, then
f'(5) is
[2002]
(a) o (6) 1 (c) 6 (d) 2
0
0*=2110-12 them
12
44 If ( x^{y}=e^{x-y}, ) then ( frac{d y}{d x} ) is equal to
A ( cdot frac{log x}{1+log x} )
B. ( frac{log x}{1-log x} )
c. ( frac{log x}{(1+log x)^{2}} )
D. ( frac{y log x}{x(1+log x)^{2}} )
12
45 [
f(x)=left{begin{array}{ll}
frac{1-sin ^{2} x}{3 cos ^{2} x}, & xfrac{pi}{2}
end{array}right.
]
then ( f(x) ) is continuous at ( x=frac{pi}{2} )
12
46 f ( boldsymbol{y}=tan (2 boldsymbol{x}+mathbf{3}) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
47 Assertion
Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}1+x & x<0 \ 1+[x]+sin x & 0 leq x leq pi / 2 \ 3 & x geq pi / 2end{array}right. )
is continuous on ( mathrm{R}-{1} )
Reason
The greatest integer function is
discontinuous at every integer.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
48 13. If x=2 cost-cos 2t, y=2 sin t-sin 2t, then at t =
I dy
(a) V2 +1 (b) V2+1
(d) None of these
2 (dy)²
12
49 If ( y=log left(frac{1-x^{2}}{1+x^{2}}right), ) then ( frac{d y}{d x} ) is equal
to
A ( cdot frac{-4 x}{1-x^{4}} )
в. ( frac{4 x^{3}}{1-x^{4}} )
c. ( frac{1}{4-x^{4}} )
D. ( frac{-4 x^{3}}{1-x^{4}} )
12
50 Let ( R ) be the set of all real numbers and
( boldsymbol{f}:[-mathbf{1}, mathbf{1}] rightarrow boldsymbol{R} ) be defined by ( quad boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{c}x sin frac{1}{x}, x neq 0 \ 0, x=0end{array} . ) Then right.
A. ( f ) satisfies the conditions of Rolle’s theorem on [-1,1]
B. ( f ) satisfies conditions of Lagrange’s Mean Value Theorem on [-1,1]
c. ( f ) satisfies the conditions of Rolle’s theorem on [0,1]
D. ( f ) satisfies the conditions of Lagrange’s Mean Value Theorem on [0,1
12
51 Assertion
Derivative of ( frac{x^{n}-a^{n}}{x-a} ) for some
constant ( n ) is
( frac{(n-1) x^{n}-n a x^{n-1}+a^{n}}{(x-a)^{2}} )
Reason ( frac{boldsymbol{d}}{boldsymbol{x}}left(frac{boldsymbol{u}}{boldsymbol{v}}right)=frac{boldsymbol{u}^{prime} boldsymbol{v}-boldsymbol{u} boldsymbol{v}^{prime}}{boldsymbol{v}^{2}} )
where ( u ) and ( v ) are two distinct
functions.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
52 If ( boldsymbol{x}=boldsymbol{y}(log boldsymbol{x} boldsymbol{y}) ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
53 If ( f(x)=sin x ) and ( g(x)=cos x ) then
( D *(f circ g) ) is equal to
( mathbf{A} cdot-sin 2(cos x) sin x )
B. – sin (cos ( x ) ) ( sin x )
c. ( -sin ^{2}(cos x) sin x )
D. ( -sin (cos x) sin ^{2} x )
12
54 28.
The values of p and q for which the function
120111
sin(p+1)x+sin x reo
x = 0 is continuous for all x in R, are
3/2
(b)
P
( p=34=
© p=29=
mp4=
) p=5.q=
(a
12
55 Differentiate the following functions
with respect to ( x ) :
( cos ^{-1}left{frac{x}{sqrt{x^{2}+a^{2}}}right} )
12
56 The solution of the differential equation ( left(frac{d y}{d x}right)^{2}-3 xleft(frac{d y}{d x}right)-2 y=8 )
A ( cdot y=2 x^{2}+4 )
B . ( y=2 x^{2}-4 )
( mathbf{c} cdot y=2 x+4 )
D. ( y=2 x-4 )
12
57 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, boldsymbol{i f} boldsymbol{x}^{frac{2}{3}}+boldsymbol{y}^{frac{boldsymbol{2}}{3}}=boldsymbol{a}^{frac{boldsymbol{2}}{3}} ) 12
58 The value of ( f(2) ) is
( A cdot 2 )
B. 4
( c .6 )
D. 8
12
59 Show that ( f(x)=|x-3| ) is continuous
but not differentiable at ( x=3 )
12
60 If ( 3 x^{2}+4 x y-7 y^{2}=0 ) Find ( (a) frac{d y}{d x} ) and
(b) ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} )
12
61 Suppose that ( f(0)=-3 ) and ( f^{prime}(x) leq 5 )
for all values of ( x ). Then the largest value
which ( f(2) ) can assume is ( ldots . )
A . 5
B. 6
( c cdot 7 )
D.
12
62 If ( y=frac{sin ^{2} x}{1+cot x}+frac{cos ^{2} x}{1+tan x} ) then ( y^{prime} ) is
equal to?
12
63 Differentiate with respect to ( x )
( boldsymbol{y}=sin 2 boldsymbol{x}-boldsymbol{4} boldsymbol{e}^{boldsymbol{3}} boldsymbol{x} )
12
64 ( operatorname{Let} y=sec left(frac{theta}{2}-1right) ) then find ( frac{d y}{d theta} ) 12
65 Find ( frac{d y}{d x} ) when ( x^{2}+y^{2}=c^{2} ) 12
66 Find the derivative of ( frac{2^{x} cot x}{sqrt{x}} )
A ( cdot frac{2^{x}}{sqrt{x}}left{log 2 cot x-csc ^{2} x-frac{cot x}{2 x}right} )
B. ( frac{2(x+1)}{sqrt{x}}left{log 2 cot x-csc ^{2} x-frac{cot x}{2 x}right} )
C ( frac{2^{x}}{sqrt{x}}left{log 2 cot x-csc ^{2} x-frac{cot x}{x^{2}}right} )
D. None of these
12
67 The difference of slopes of lines
represent by ( y^{2}-2 x y sec ^{2} alpha+ )
( left(3+tan ^{2} alpharight)left(tan ^{2} alpha-1right) x^{2}=0 ) is
( A cdot 3 )
B. 4
( c cdot 0 )
D.
12
68 f ( boldsymbol{y}=(mathbf{1}+boldsymbol{x})left(mathbf{1}+boldsymbol{x}^{2}right)left(mathbf{1}+boldsymbol{x}^{4}right) dots(mathbf{1}+ )
( left.x^{2^{n}}right), ) then ( frac{d y}{d x} ) at ( x=0 ) is
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. none of these
12
69 If ( x^{x}+x^{y}+y^{x}=a^{b}, ) then find ( frac{d y}{d x} ) 12
70 If the function ( u=f(x) ) is continuous at
the point ( x=a ) and the
function ( y=g(u) ) is continuous at the
point ( u=f(a), ) then the
composite function ( boldsymbol{y}=(boldsymbol{g} boldsymbol{o} boldsymbol{f})(boldsymbol{x})= )
( g(f(x)) ) is
A. continuous at the point ( x=f(a) )
B. continuous at the point ( x=a )
c. discontinuous at the point ( x=a )
D. continuous at the point ( x=g(a) )
12
71 If we apply the mean value theorem to
( f(x)=2 sin x+sin 2 x ) then ( c= )
This question has multiple correct options
A . ( pi )
в. ( pi / 4 )
c. ( pi / 2 )
D. ( pi / 3 )
12
72 19. Determine the values of x for which the following function
fails to be continuous or differentiable: (1997 – 5 Marks)
(1-x, xxl
f(x)= (1-x)(2-x), 1sx52 Justify your answer.
[3-x,
x>2
12
73 If ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{t}+frac{mathbf{1}}{boldsymbol{t}} ) and ( boldsymbol{x}^{4}+boldsymbol{y}^{4}=boldsymbol{t}^{2}+ )
( frac{1}{t^{2}} ) then ( frac{d y}{d x}= )
A ( cdot-frac{x}{y} )
в. ( frac{-y}{x} )
c. ( frac{x^{2}}{y^{2}} )
D. ( frac{y^{2}}{x^{2}} )
12
74 Solve:
( frac{x cos ^{-1} x}{sqrt{1-x^{2}}} d x )
12
75 Function ( f(x)=|x-2|-2|x-4| ) is
discontinous at:
A . ( x=2,4 )
B. ( x=2 )
C. No where
D. Except ( x=2 )
12
76 Solve ( : frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(sin boldsymbol{3} boldsymbol{x})=? ) 12
77 Find the derivative of the following functions(it is to be understood that
( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero
constants and ( m ) and ( n ) are integers):
( frac{a x+b}{c x+d} )
12
78 Find the intervals in which the following functions are increasing or decreasing ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}-mathbf{5} ) 12
79 Differentiate the function with respect
to ( boldsymbol{x} )
( f(x)=x^{2 / 3}+7 e^{x}-frac{5}{x}+7 tan x )
12
80 Assertion
Statement-1: ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{{boldsymbol{x}}} ) is
discontinuous for integral values of ( boldsymbol{x} ) where ( vartheta ) denotes the fractional part
function.
Reason
Statement-2: For integral values of ( boldsymbol{x} ) ( f(x) ) is not defined.
A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1
B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT the correct explanation for Statement-1
c. statement-1 is true, Statement-2 is false.
D. Statement-1 is false, Statement-2 is true.
12
81 If ( boldsymbol{y}=frac{1}{mathbf{3 – 4 x}} ) then ( boldsymbol{y}_{n}(1) ) equals
A .
B ( cdot(-1)^{n+1} n ! )
c. ( n ! 4^{n}(-1)^{n+1} )
D. None of these
12
82 Solve: ( frac{d}{d x} x sin ^{2} x ) 12
83 Find the derivative by first principle
( cos 5 x )
12
84 Differentiate ( sin boldsymbol{h}^{-1}left(frac{boldsymbol{x}}{mathbf{3}}right) ) with respect
to
( x ). Find out the solution of the
integration ( int frac{1}{left(x^{2}+9right)} d x ) Further find
out the value of the integral
( int frac{1}{left(x^{2}+49right)} d x ? )
12
85 The function ( f(x)=frac{tan left{pileft[x-frac{pi}{2}right]right}}{2+[x]^{2}} )
where ( [x] ) denotes the greatest integer
( leq x, ) is
A. continuous for all values of ( x )
B. Discontinuous at ( x=frac{pi}{2} )
c. Not differentiable for some values of ( x )
D. Discontinuous at ( x=-2 )
12
86 If the function ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{1-cos 4 x}{8 x^{2}}, x neq 0 \ k, x=0end{array} ) is continuous at right.
( boldsymbol{x}=mathbf{0} ) then ( boldsymbol{k}=? )
A . 1
B . 2
( c cdot frac{1}{2} )
D. ( frac{-1}{2} )
12
87 If ( y= )
( (x+sqrt{x^{2}-a^{2}})^{n} quad ) then ( quadleft(x^{2}-a^{2}right)left(frac{d y}{d x}right. )
A ( cdot n^{2} y )
B. ( -n^{2} y )
( c cdot n y^{2} )
D cdot ( n^{2} y^{2} )
12
88 If ( y=cos ^{-1}(sqrt{x}), ) then find ( frac{d y}{d x} ) using
first principle.
A ( cdot-frac{1}{sqrt{1-x}} )
в. ( frac{1}{sqrt{1-x}} )
c. ( -frac{1}{2 sqrt{x} sqrt{1-x}} )
D. ( frac{1}{2 sqrt{x} sqrt{1-x}} )
12
89 Given that, ( y=sin xleft(x^{2}right) e^{x} ), find ( y^{prime} ) at
( boldsymbol{x}=mathbf{0} )
12
90 Derivative of ( tan ^{-1}left(frac{x}{sqrt{1-x^{2}}}right) ) with
respect to ( sin ^{-1}left(3 x-4 x^{3}right) ) is
A ( cdot frac{1}{sqrt{1-x^{2}}} )
в. ( frac{3}{sqrt{1-x^{2}}} )
( c .3 )
D.
12
91 If ( x=a t^{2} ) and ( y=2 a t, ) then ( frac{d y}{d x} ) is equal
to
( mathbf{A} cdot t )
B.
( c . )
D. ( t^{2} )
12
92 If ( x sin y=3 sin y+4 cos y, ) then ( frac{d y}{d x}= )
( mathbf{A} cdot frac{-sin ^{2} y}{4} )
B. ( frac{sin ^{2} y}{4} )
( mathbf{c} cdot frac{-cos ^{2} y}{4} )
D. ( frac{cos ^{2} y}{4} )
12
93 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{y}=log left(boldsymbol{4} boldsymbol{x}-boldsymbol{x}^{5}right) ) 12
94 ff ( y=x^{2} cos x ) then ( y_{8}(0) ) is
A . 72
B . 56
( c cdot 0 )
D. – 56
12
95 Find the derivative of ( boldsymbol{y}=frac{1}{boldsymbol{x}}+frac{mathbf{1}}{boldsymbol{x}^{2}}+ )
( frac{mathbf{3}}{boldsymbol{x}^{mathbf{3}}} )
12
96 A function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) satisfies ( boldsymbol{f}(boldsymbol{x})= )
( boldsymbol{f}(boldsymbol{2} boldsymbol{a}-boldsymbol{x}) . ) Suppose ( boldsymbol{f}(boldsymbol{x}) ) is
differentiable at ( x=a ) then
This question has multiple correct options
A ( cdot f^{prime}(a)=0 )
B . ( f^{prime}left(a^{+}right)=-f^{prime}left(a^{-}right) )
c. ( f^{prime}left(a^{+}right)=f^{prime}left(a^{-}right)=0 )
D. None of these
12
97 Find the derivative of ( sin x ) with respect
to ( x ) from first principles.
12
98 3.
If f(x) = x” then the value of
20031
0
1
)
21
(b) 21
(1)
3!
(-1)”/”(1) :
n!
21-1
(a) I
(c)
(d) 0
12
99 If ( boldsymbol{x}^{m} cdot boldsymbol{y}^{n}=(boldsymbol{x}+boldsymbol{y})^{boldsymbol{m}+boldsymbol{n}}, ) then ( frac{d boldsymbol{y}}{d boldsymbol{x}} ) is :
A ( cdot frac{y}{2 x} )
в. ( frac{2 y}{x} )
( c cdot-frac{y}{x} )
D. ( frac{y}{x} )
12
100 Find the derivative of ( f(x) ) from the first
principles, where ( boldsymbol{f}(boldsymbol{x}) ) is
( sin x+cos x )
12
101 The values of ( f^{prime}(1) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
102 f ( sin theta+2 cos theta=1 ) then prove that
( 2 sin theta-cos theta=0 )
12
103 Discuss the continuity and differentiability of the function, ( boldsymbol{f}(boldsymbol{x})=left[begin{array}{cc}frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|} & |boldsymbol{x}|>mathbf{1} \ frac{boldsymbol{x}}{mathbf{1}-|boldsymbol{x}|} & |boldsymbol{x}| leq mathbf{1}end{array}right] ) 12
104 Derivative of ( left(tan ^{-1} xright)^{2} ) wrt to ( x ) 12
105 Assertion
Consider the polynomial function ( f(x)=frac{x^{7}}{7}-frac{x^{6}}{6}+frac{x^{5}}{5}-frac{x^{4}}{4}+frac{x^{3}}{3}- )
( frac{x^{2}}{2}+x . ) The equation ( f(x)=0 ) cannot
have two or more roots.
Reason
Rolle’s theorem is not applicable for ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) on any interval ( [boldsymbol{a}, boldsymbol{b}], ) where
( boldsymbol{a}, boldsymbol{b} boldsymbol{epsilon} boldsymbol{R} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
106 ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{sin (a+1) x+sin x}{x} text { if } x0end{array}right. )
( f(0)=c ) is continuous at ( x=0 ) then
A ( cdot a=frac{-3}{2}, c=frac{1}{2}, b neq 0 )
B cdot ( a=b=frac{1}{2}, c=0 )
C ( cdot a=b=frac{1}{2}, c=0 )
D ( cdot a=frac{1}{2} b neq 0 c=1 )
12
107 If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{x}-boldsymbol{3}, quad boldsymbol{x}<mathbf{0} \ boldsymbol{x}^{2}-mathbf{3} boldsymbol{x}+mathbf{2}, quad boldsymbol{x} geq mathbf{0}end{array} ) and right.
( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(|boldsymbol{x}|)+|boldsymbol{f}(boldsymbol{x})|, ) then ( boldsymbol{g}(boldsymbol{x}) ) is
This question has multiple correct options
A . continuous is ( R-{0} )
B. Continuous in ( R )
C . Differentiable in ( R-{0,1,2} )
D. Differentiable in ( R-{1,2} )
12
108 Differentiate the following functions
w.r.t. ( boldsymbol{x} )
( e^{operatorname{cosec}^{2} x} )
12
109 If ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}+boldsymbol{2} sqrt{boldsymbol{2} boldsymbol{x}-boldsymbol{4}}}+ )
( sqrt{x-2 sqrt{2 x-4}} ), then the value of
( mathbf{1 0} boldsymbol{f}^{prime}left(mathbf{1 0 2}^{+}right) ) is
A . -1
B.
c. 1
D. Does not exist
12
110 ( boldsymbol{g}(boldsymbol{x}+boldsymbol{y})=boldsymbol{g}(boldsymbol{x})+boldsymbol{g}(boldsymbol{y})+mathbf{3} boldsymbol{x} boldsymbol{y}(boldsymbol{x}+ )
( boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} ) and ( boldsymbol{g}^{prime}(mathbf{0})=-4 . ) For which of
the following values of ( x ) is ( sqrt{g(x)} ) not
defined?
A ( cdot[-2,0] )
в. ( [-2, infty] )
c. [-1,1]
D. none of these
12
111 Find the differential equation of the following. ( tan ^{-1}left(frac{1-cos x}{sin x}right) ) 12
112 ( f y^{prime}=-3 xleft(2-x^{2}right)^{frac{1}{2}} ) then find ( y^{prime prime} ) 12
113 Assertion
Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+mathbf{7} boldsymbol{x}+boldsymbol{4} ) be a polynomial
function, then ( boldsymbol{f}^{prime}(mathbf{2})=mathbf{1 1} )
Reason
A polynomial function is differentiable everywhere
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
114 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( log left(frac{x^{2}+x+1}{x^{2}-x+1}right) )
12
115 If ( f(x)=x sin x, ) then ( f^{prime}left(frac{pi}{2}right) ) is equal
to:
A.
B.
c. -1
D.
12
116 If ( y=A sin 5 x, ) then ( frac{d^{2} y}{d x^{2}}= )
begin{tabular}{l}
A. ( -25 y ) \
hline
end{tabular}
в. ( 25 y )
c. ( 5 y )
D. ( -5 y )
12
117 Differentiate with respect to ( x ) :
( boldsymbol{y}=boldsymbol{e}^{-mathbf{3} boldsymbol{x}}+sin mathbf{2} boldsymbol{x} )
12
118 If ( f(x) ) and ( g(x) ) are differentiable
functions for ( 0 leq x leq 1 ) such that
( boldsymbol{f}(mathbf{0})=mathbf{2}, boldsymbol{g}(mathbf{0})=mathbf{0}, boldsymbol{f}(mathbf{1})=mathbf{6}, boldsymbol{g}(mathbf{1})=mathbf{2} )
then in the interval ( (mathbf{0}, mathbf{1}) )
( mathbf{A} cdot f^{prime}(x)=0 ) for all ( x )
B . ( f^{prime}(x)=2 g^{prime}(x) ) for at least one ( x )
( mathbf{C} cdot f^{prime}(x)=2 g^{prime}(x) ) for at most one ( x )
D. None of these
12
119 1

2
2at
9. If x=
2 and y=-
1+1²
1+
dx
las 24
(c) a(t+1)
10 24
(d) a(t? – 1)
12
120 If ( boldsymbol{y}=log left[tan left(frac{boldsymbol{pi}}{boldsymbol{4}}+frac{boldsymbol{x}}{2}right)right] ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( A cdot sec x )
B. ( sin x )
c. ( operatorname{cosec} x )
D. ( sec frac{x}{2} )
12
121 ( lim _{x rightarrow 0} frac{tan ^{4} x-sin ^{4} x}{x^{4}}= )
A . 0
B.
( c cdot frac{2}{3} )
D.
12
122 11.
ху
Let y be an implicit function of x defined by X-
it function of x defined by x2x2x4 coty
-1=0. Then y'(1) equals
(a) 1 (b) log 2 (C) -log 2 (d) -1
[20091
12
123 f the following function is continuous at ( x=frac{pi}{2}, ) then find ( a ) and ( b ) ( f(x)=left{begin{array}{l}frac{1-sin ^{2} x}{3 cos ^{2} x}, text { if } xfrac{pi}{2}end{array}right. ) 12
124 6. If y=x* sin x + int , then it will be
(a) 2x sin x + r cos + 3 tan x – 3x sec- x
tan” x
(b) 2x sin x + 3x sec? x – 3 tan x
(b) 2x sin x +
tan” x
x? cos x + (3 tan x – 3x sec? x)
tan” x
x? cos x – 2x sin x – (3 tan x – 3x sec? x)
tan? x
12
125 State true or false:
If ( u(x) ) and ( v(x) ) are differentiable functions such that ( frac{u}{v}(x)=7 ) ( frac{boldsymbol{u}^{prime}(boldsymbol{x})}{boldsymbol{v}^{prime}(boldsymbol{x})}=boldsymbol{p} ) and ( left(frac{boldsymbol{u}(boldsymbol{x})}{boldsymbol{v}(boldsymbol{x})}right)^{prime}=boldsymbol{q}, ) then
( frac{boldsymbol{p}+boldsymbol{q}}{boldsymbol{p}-boldsymbol{q}}=mathbf{1} )
A. True
B. False
12
126 Show that ( f(x)=e^{2 x} ) is increasing on ( R ) 12
127 Differentiate
( mathbf{3} boldsymbol{x}^{mathbf{1} / mathbf{3}}+frac{mathbf{6}}{mathbf{7}} boldsymbol{x}^{mathbf{7} / mathbf{6}}+mathbf{3} boldsymbol{x}^{mathbf{2} / mathbf{3}}+boldsymbol{C} )
12
128 If a function is everywhere continuous
and differentiable such that ( f^{prime}(x) geq 6 )
for all ( boldsymbol{x} epsilon[mathbf{2}, mathbf{4}] ) and ( boldsymbol{f}(mathbf{2})=-mathbf{4}, ) then
( mathbf{A} cdot f(4)<8 )
в. ( f(4) geq 8 )
c. ( f(4) geq 2 )
D. none of these
12
129 If ( f(x) ) is a polynomial function and
( boldsymbol{f}^{prime}(boldsymbol{x})>boldsymbol{f}(boldsymbol{x}), forall boldsymbol{x} geq 1 ) and ( boldsymbol{f}(1)=mathbf{0} )
then
A. ( f(x) geq 0, forall x geq 1 )
B. ( f(x)<0, forall x geq 1 )
c. ( f(x)=0, forall x geq 1 )
D. None of the above
12
130 Let ( f(x) ) be a real valued function not identically zero, such that
( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{n}right)=boldsymbol{f}(boldsymbol{x})+(boldsymbol{f}(boldsymbol{y}))^{n} quad forall boldsymbol{x}, boldsymbol{y} in )
( boldsymbol{R} )
where ( n in N(n neq 1) ) and ( f^{prime}(0) geq 0 . ) We
may get an explicit form of the function ( boldsymbol{f}(boldsymbol{x}) )
( int_{0}^{1} f(x) d x ) is equal to
A ( cdot frac{1}{2 n} )
B. ( 2 n )
( c cdot frac{1}{2} )
D. 2
12
131 ( f(x)=left{begin{array}{c}a sin frac{pi}{2}(x+1), x leq 0 \ frac{tan x-sin x}{x^{3}}, x>0end{array}right. )
continuous at ( x=0 . ) Find the value of
( boldsymbol{a} )
12
132 If ( f(x)=frac{e^{x^{2}}-cos x}{x^{2}}, ) for ( x neq 0, ) is
continuous at ( boldsymbol{x}=mathbf{0}, ) find ( boldsymbol{f}(mathbf{0}) )
12
133 Let ( y=sin ^{-1}(cos x) ) then find ( frac{d y}{d x} ) 12
134 If ( cos y=x cos (a+y), ) find ( frac{d y}{d x} ) 12
135 The value of ( K ) which the Function
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}
frac{tan 4 boldsymbol{x}}{tan 5 x}, & 0<x<frac{pi}{2} \
boldsymbol{k}+frac{2}{5}, & boldsymbol{x}=frac{pi}{2}
end{array}right.
]
continuous at ( x=frac{pi}{2}, ) is
12
136 Consider the piecewise defined
[
text { function }left{begin{array}{ll}
sqrt{-x}, & text { if } x4
end{array}right.
]
the answer which best describes the
continuity of this function-
A. the function is unbounded and therefore cannot be continuous
B. the function is right continuous at ( x=0 )
c. the function has a removable discontinuity at 0 and 4 but is continuous on the rest of the real line
D. the function is continuous on the entire real line
12
137 ff ( y=fleft(x^{2}+2right) ) and ( f^{prime}(3)=5, ) then ( frac{d y}{d x} )
at ( x=1 ) is
( mathbf{A} cdot mathbf{5} )
B . 25
c. 15
D. 10
12
138 If ( f(x)=frac{1+tan x}{1-tan x} ) then ( fleft(frac{pi}{6}right) ) 12
139 Differentiate:
( sin ^{2} boldsymbol{y}+cos boldsymbol{x} boldsymbol{y}=boldsymbol{K} )
12
140 ff ( boldsymbol{x}=boldsymbol{a} sin boldsymbol{theta}+boldsymbol{b} cos boldsymbol{theta}, boldsymbol{y}=boldsymbol{a} cos boldsymbol{theta} )
( -b sin theta )
then show that ( (a x+a y)^{2}+(b x-a y)^{2} )
( =left(a^{2}+b^{2}right)^{2} )
12
141 Differentiation of ( (2 x+3)^{6} ) with respect
to ( x ) is
A ( cdot 12(2 x+3)^{5} )
B. ( 6(2 x+3)^{5} )
c. ( 3(2 x+3)^{5} )
D. ( 6(2 x+3)^{6} )
12
142 Discuss the continuity of the following function at the indicated point(s):
( boldsymbol{f}(boldsymbol{x})left{begin{array}{l}|boldsymbol{x}| cos left(frac{1}{boldsymbol{x}}right), boldsymbol{x} neq mathbf{0} \ mathbf{0}, quad boldsymbol{x}=mathbf{0}end{array} quad, text { at } boldsymbol{x}=mathbf{0}right. )
12
143 State True or False.
If ( frac{e^{y}}{e^{x}}=x y, ) then ( y^{prime}=frac{2-log x}{(1-log x)^{2}} )
A. True
B. False
12
144 Examine the continuity of the following function at given points:
(i) ( boldsymbol{f}(boldsymbol{x})= )
[
begin{array}{ll}
frac{e^{5 x}-e^{2 x}}{sin 3 x}, & text { for } x neq 0 \
= & text { for } x neq 0
end{array}
]
12
145 greatest integer function and
(1993 – 1 Mark)
10. Let [:] denote the greatest integer
f(x) = [tan x], then:
(a) limo f(x) does not exist
(b) f(x) is continuous at x=0
© f(x) is not differentiable at x=0
(d) f'(0=1
12
146 If ( x sqrt{1+y}+y sqrt{1+x}=0 ) and ( x neq y )
show that ( frac{d y}{d x}=frac{-1}{(1+x)^{2}} )
12
147 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( sin boldsymbol{x}-boldsymbol{3} boldsymbol{x}=mathbf{5} boldsymbol{y} )
12
148 ( sqrt{1+left(frac{d^{2} y}{d x^{2}}right)^{3}}=left(2+frac{d y}{d x}right)^{1 / 3} )
Find it’s order and degree.
A .2,3
в. 2,9
( c cdot 2,6 )
D. 2,2
12
149 Differentiable ( log _{7}(log x) ) with respect
to ( boldsymbol{x} )
12
150 If ( S_{1} ) and ( S_{2} ) are respectively the sets of local minimum and local maximum
points of the functions, ( f(x)=9 x^{4}+ )
( 12 x^{3}-36 x^{2}+25, x in R, ) then
A ( . S_{1}={-2,1} ; S_{2}={0} )
B. ( S_{1}={-2,0} ; S_{2}={1} )
c. ( S_{1}={-2,} ; S_{2}={0,1} )
D. ( S_{1}={-1} ; S_{2}={0,2} )
12
151 If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cl}frac{1-sqrt{2} sin x}{pi-4 x}, & text { if } x neq frac{pi}{4} \ a, & text { if } x=frac{pi}{4}end{array}right. )
is continuous at ( frac{n}{4} ) then ( a= )
( A )
B. 2
c. 1
( 0 . frac{1}{1} )
12
152 Find the equation of the tangent to the curve ( y=frac{x-y}{(x-2)(x-3)} ) at the point
where it cuts the ( x ) -axis.
12
153 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{y}=boldsymbol{e}^{left(1+log _{e} boldsymbol{x}right)} )
12
154 If ( boldsymbol{x y}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}} ) then
This question has multiple correct options
A ( cdot frac{d y}{d x} ) doesn’t exist at ( x=0 )
B. ( frac{d y}{d x}=0 ) when ( x=1 )
C. ( frac{d y}{d x}=frac{1}{2} ) when ( x=0 )
D. none of these
12
155 Differentiate ( x^{sin x}+(sin x)^{cos x} ) w.r.t ( x ) 12
156 ( f(x)=frac{left(e^{k x}-1right)(sin k x)}{4 x^{2}}, x neq 0 )
( boldsymbol{f}(mathbf{0})=mathbf{9}, ) is continuous at ( boldsymbol{x}=mathbf{0}, ) then ( mathbf{k} )
( =? )
( mathbf{A} cdot pm 2 )
B. ±6
( c .pm 4 )
D. None of the above
12
157 Assertion
Derivative of ( 3 cot x+5 operatorname{cosec} x ) is
( -operatorname{cosec} x(3 operatorname{cosec} x+5 cot x) )
Reason
( boldsymbol{f}^{prime}(boldsymbol{a})=lim _{boldsymbol{h} rightarrow mathbf{0}} frac{boldsymbol{f}(boldsymbol{a}+boldsymbol{h})-boldsymbol{f}(boldsymbol{a})}{boldsymbol{h}} )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
158 ff ( y=log left(sqrt{x}+frac{1}{sqrt{x}}right), ) prove that
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}-mathbf{1}}{boldsymbol{2} boldsymbol{x}(boldsymbol{x}+mathbf{1})} )
12
159 If ( y=sqrt{frac{1-sin ^{-1} x}{1+sin ^{-1} x}} ) then ( y^{prime}(0) ) is equal
to
( mathbf{A} cdot mathbf{1} )
B. ( 1 / 2 )
( c cdot-1 )
D. ( sqrt{2} / 3 )
12
160 7.
Iff(x)= x-1, then on the interval [0, ] (1989-2 Marks)
(a) tan (x)] and 1/f(x) are both continuous
(b) tan f(x)] and 1/f(x) are both discontinuous
(c) tan [fx)) and s-‘(x) are both continuous
(d) tan fx)] is continuous but 1/f(x) is not.
12
161 Find the value ( : frac{d}{d x}left{cos x^{0}right}=? ) 12
162 If ( f(x)=cos ^{-1}left[frac{1-(log x)^{2}}{1+(log x)^{2}}right], ) then
( boldsymbol{f}^{prime}(boldsymbol{e})= )
( A cdot frac{1}{e} )
B. ( frac{2}{e^{2}} )
( c cdot frac{2}{e} )
D. None of these
12
163 16. Differentiation of x2 w.r.t. x is… 12
164 Discuss the countinuity of the following
function at ( boldsymbol{x}=mathbf{0} )
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{1-cos x}{x^{2}}, & x neq 0 \ frac{1}{2}, & x=0end{array}right. )
12
165 Find the value ( c ) in mean value theorem
for the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}+frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x} boldsymbol{epsilon}[mathbf{1}, boldsymbol{3}] )
12
166 ( f(x)=left{begin{array}{lc}frac{x^{2}}{a} quad ; & 0 leq x<1 \ -1 & 1 leq x<sqrt{2} \ frac{2 b^{2}-4 b}{x^{2}} & ; sqrt{2} leq x<inftyend{array}right. )
then find the value of ( a ) and ( b ) if ( f(x) ) is
continuous in ( [mathbf{0}, infty) . ) Find ( boldsymbol{a}+boldsymbol{b} )
12
167 0
1
21.
Let f be differentiable for all x. Iff(1) =-2 and f ‘(x) 2 2 for
x 6 [1, 6], then
[2005]
@ f6 28 (b) f(6<8 (c) f(6)<5 (d) f(6=5
12
168 Differentiate with respect to ( x ) :
( mathbf{3}^{e^{x}} )
12
169 If ( y sin x=x+y ) then ( left(frac{d y}{d x}right)_{x=0} ) equals
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. 2
12
170 Let ( f(x)=tan 2 x cdot tan 3 x cdot tan 5 x, ) then
( boldsymbol{f}^{prime}(boldsymbol{pi}) ) equals
A . 10
B. -10
( c .0 )
D.
12
171 Solution of differential equation ( x^{2}= ) ( 1+left(frac{x}{y}right)^{-1} frac{d y}{d x}+frac{left(frac{x}{y}right)^{-2}left(frac{d y}{d x}right)^{2}}{2 !}+ )
( frac{left(frac{x}{y}right)^{-3}left(frac{d y}{d x}right)^{3}}{3 !}+ldots )
12
172 For what value of ( a ), the function
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}
frac{1-cos 4 x}{boldsymbol{x}^{2}}, & boldsymbol{i f} quad boldsymbol{x}0
end{array}right.
]
continuous at ( boldsymbol{x}=mathbf{0} )
12
173 Let ( h(x) ) be differentiable for all ( x ) and
let ( boldsymbol{f}(boldsymbol{x})=left(boldsymbol{k} boldsymbol{x}+boldsymbol{e}^{boldsymbol{x}}right) boldsymbol{h}(boldsymbol{x}) ) where ( boldsymbol{k} ) is
some constant. If ( h(0)=5, h^{prime}(0)=-2 )
and ( f^{prime}(0)=18, ) then the value of ( k ) is equal to
( A cdot 3 )
B. 4
c.
D.
12
174 Find the derivative of ( boldsymbol{y}= ) ( tan ^{-1}left(frac{boldsymbol{a} boldsymbol{x}-boldsymbol{b}}{boldsymbol{b} boldsymbol{x}+boldsymbol{a}}right) ) 12
175 ( boldsymbol{g}(boldsymbol{x})=lim _{m rightarrow infty} frac{boldsymbol{x}^{m} boldsymbol{f}(boldsymbol{x})+boldsymbol{h}(boldsymbol{x})+boldsymbol{3}}{boldsymbol{2} boldsymbol{x}^{m}+boldsymbol{4} boldsymbol{x}+mathbf{1}} )
when ( x neq 1 ) and ( g(1)=e^{3} ) such that
( f(x), g(x) ) and ( h(x) ) are continuous function at ( boldsymbol{x}=mathbf{1} ) and ( boldsymbol{f}(mathbf{1})-boldsymbol{h}(mathbf{1})= )
( a(b-g(1)) ) then ( a+b ) is
12
176 If ( f(x)=frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}}, ) if ( x neq 2 )
[
=k, quad text { if } x=2
]
is continuous at ( x=0 ) then
( mathbf{A} cdot k=2 )
B. ( k=0 )
c. ( k=20 )
( mathbf{D} cdot k=7 )
12
177 ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{a}+boldsymbol{b}^{frac{1}{x}}}{boldsymbol{c}+boldsymbol{d}^{frac{1}{x}}}, boldsymbol{b}>1, boldsymbol{d}>1, boldsymbol{c} neq mathbf{0} )
( boldsymbol{f}(mathbf{0})=mathbf{1} ) is left continuous at ( boldsymbol{x}=mathbf{0} )
then
( mathbf{A} cdot a=0 )
B . ( a=2 c )
c. ( a=c )
D. ( a neq c )
12
178 ( lim _{x rightarrow 1} 2 x+1 ) 12
179 If ( f^{prime prime}(x)<0, forall x epsilon(a, b), ) and ( (c, f(c)) ) is
point of maxima, where ( c epsilon(a, b), ) then
( f^{prime}(c) ) is
A ( cdot frac{f(b)-f(a)}{b-a} )
в. ( left[frac{f(b)-f(a)}{b-a}right] )
c. ( 2left[frac{f(b)-f(a)}{b-a}right] )
D.
12
180 6. If f(x) = mx + c,fo) =f(0) = 1 then f(2)=
(a) 1 (6) 2 (c) 3 (d) – 3
12
181 The function ( y=f(x) ) is ?
A . odd
B. even
c. increasing
D. decreasing
12
182 Identify a possible graph for function
given by ( f(x)=-(x-2)^{3}+1 )
A. graph a
B. graph b
c. grpah c
D. grpah d
12
183 Differentiate: ( log (log x), x>1 ) 12
184 Diff: ( cos ^{-1}left(frac{2 x}{1+x^{2}}right) ) w.r.t. ( x ) 12
185 Let ( boldsymbol{f}:[mathbf{0}, mathbf{1}] rightarrow boldsymbol{R} ) be a continuous
function then the maximum value of ( int_{0}^{1} f(x) cdot x^{2} d x-int_{0}^{1} x cdot(f(x))^{2} d x ) for all
such function(s) is
( A cdot frac{1}{8} )
B. ( frac{1}{20} )
c. ( frac{1}{12} )
D. ( frac{1}{16} )
12
186 Given, ( boldsymbol{f}(boldsymbol{x})=-frac{boldsymbol{x}^{3}}{mathbf{3}}+boldsymbol{x}^{2} sin mathbf{1 . 5} boldsymbol{a}- )
( x sin a cdot sin 2 a-5 a r c sin left(a^{2}-8 a+17right) )
then
A. ( f(x) ) is not defined at ( x=sin 8 )
B. ( f^{prime}(sin 8)>0 )
c. ( f^{prime}(x) ) is not defined at ( x=sin 8 )
D. ( f^{prime}(sin 8)<0 )
12
187 If ( f(x)=(a x+b) cos x+(c x+d) sin x )
and ( f^{prime}(x)=x cos x, ) for all values of ( x in )
( R, ) then ( a, b, c, d ) are given by
A ( a=b=c=d )
( d )
в. 0,1,-1,0
c. 1,0,-1,0
D. 0,1,1,0
12
188 [
text { If } f(x)=frac{x^{3}+x^{2}-16 x+20}{(x-2)^{2}}, x neq 2
]
( =k, x=2 ) is continuous at ( x=2 )
find the value of k.
12
189 3.
There exist a function f(x), satisfying f(0) = 1,7
nction f (x), satisfying f(0) = 1, f'(0)=-1,
f(x) > 0 for allx, and
(1982 – 2 Marks)
(a) “(x) > 0 for all x
(b) -1<f"(x) <0 for all x
(c) -2 "(x) S-1 for all (d) F"(x)<-2 for all x
12
190 ( operatorname{Let} h(x)=min left{x, x^{2}right} ) for ( x in R ) Then
which of the following is correct
A. ( h ) is continuous for all ( x )
B. ( h ) is differentiable for all ( x )
C ( cdot h(x)=1 ) for all ( x>1 )
D. ( h ) is not a differentiable at 2 values of ( x )
12
191 If the function ( f ) is continuous at ( x=0 )
find ( boldsymbol{f}(mathbf{0}) )
where ( boldsymbol{f}(boldsymbol{x})=frac{cos 3 boldsymbol{x}-cos boldsymbol{x}}{boldsymbol{x}^{2}}, boldsymbol{x} neq mathbf{0} )
12
192 ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}-1) ) in the interval ( [mathbf{1}, mathbf{2}] ) if
( boldsymbol{f}^{prime}(boldsymbol{c})=boldsymbol{f}(1) . ) Find ( c )
12
193 If ( y=tan ^{-1}left(frac{1}{1+x+x^{2}}right)+ )
( tan ^{-1}left(frac{1}{x^{2}+3 x+2}right)+ )
( tan ^{-1}left(frac{1}{x^{2}+5 x+6}right)+ldots+ ) upto ( n )
terms then ( frac{d y}{d x} ) at ( x=0 ) and ( n=1 ) is
equal to
A ( frac{1}{2} d )
B. ( -frac{1}{2} )
( c )
( D )
12
194 Find the derivative of ( left(x^{2}+cos xright) )
( mathbf{A} cdot 2 x+sin x )
B. ( 2 x^{2}+sin x )
( mathbf{c} cdot 2 x-sin x )
D. ( 2 x-cos x )
12
195 If ( boldsymbol{y}=frac{boldsymbol{x}}{boldsymbol{x}+mathbf{1}}+frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}}, ) then ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) at ( boldsymbol{x}= )
1 is equal to
( A cdot frac{7}{4} )
B. ( frac{7}{8} )
( c cdot frac{1}{4} )
D. ( frac{-7}{8} )
E ( frac{-7}{4} )
12
196 If ( boldsymbol{y}=(mathbf{1}+boldsymbol{x})left(mathbf{1}+boldsymbol{x}^{mathbf{2}}right)left(mathbf{1}+boldsymbol{x}^{mathbf{4}}right) dots mathbf{.} )
( left.x^{2^{n}}right), ) then ( left(frac{d y}{d x}right)_{x=0}= )
( mathbf{A} cdot mathbf{0} )
в. ( frac{1}{2} )
( c cdot 1 )
( D )
12
197 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
( boldsymbol{y}=sin boldsymbol{x} . cos boldsymbol{x} )
12
198 Let ( x^{k}+y^{k}=a^{k},(a, k>0) ) and
( frac{d y}{d x}+left(frac{y}{x}right)^{1 / 3}=0, ) then ( k ) is :
A ( cdot frac{1}{3} )
B. ( frac{2}{3} )
( c cdot frac{4}{3} )
( D cdot 3 )
( overline{2} )
12
199 Assertion
Consider the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x} )
and ( boldsymbol{g}(boldsymbol{x})=-|boldsymbol{x}| )
The composite function ( boldsymbol{F}(boldsymbol{x})= )
( boldsymbol{f}(boldsymbol{g}(boldsymbol{x})) ) is not derivable at ( boldsymbol{x}=mathbf{0} )
Reason
( boldsymbol{f}^{prime}left(mathbf{0}^{+}right)=mathbf{2} ) and ( boldsymbol{f}^{prime}left(mathbf{0}^{-}right)=-mathbf{2} )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
200 If Rolle’s theorem is applicable to the
function, ( boldsymbol{f}(boldsymbol{x})=frac{ln boldsymbol{x}}{boldsymbol{x}} ) over the interval
( [a, b], ) where ( a, b in I^{+} ) then the value of
( a+b ) is
12
201 If ( y=tan ^{-1}left(frac{a cos x-b sin x}{b cos x+a sin x}right), ) then
( frac{d y}{operatorname{isequal}} ) to
( d x )
( A cdot 2 )
B. –
( c cdot a )
( D )
12
202 21. If y = sin x, then
will be …
12
203 Evaluate
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{e}^{boldsymbol{x}} boldsymbol{operatorname { s i n }} boldsymbol{x}}{left(boldsymbol{x}^{2}+boldsymbol{2}right)^{3}} )
12
204 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(tan ^{2} boldsymbol{a} boldsymbol{x}right) )
A ( cdot 2 a tan a x sec ^{2} a x )
B. ( -2 a tan a x sec ^{2} a x )
( mathbf{c} cdot a tan a x sec ^{2} a x )
D. ( 2 a cot a x sec ^{2} a x )
12
205 If for ( boldsymbol{x} inleft(mathbf{0}, frac{mathbf{1}}{mathbf{4}}right), ) the derivative
( tan ^{-1}left(frac{6 x sqrt{x}}{1-9 x^{3}}right) ) is ( sqrt{x} . g(x), ) then ( g(x) )
equals:
A ( cdot frac{3}{1+9 x^{3}} )
в. ( frac{9}{1+9 x^{3}} )
c. ( frac{3 x sqrt{x}}{1-9 x^{3}} )
D. ( frac{3 x}{1-9 x^{3}} )
12
206 Differentiate the following with respect
to ( boldsymbol{x} )
( cos ^{-1}[sqrt{frac{1+x}{2}}],-1<x<1 )
12
207 If ( boldsymbol{y}=sqrt{boldsymbol{x}+boldsymbol{y}}, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{1}{(2 y-1)} )
12
208 Find ( frac{d y}{d x}, ) if ( y=sqrt{1+sin 2 x} ) 12
209 Find the derivative of ( y=sqrt{x^{2}+1} ) 12
210 Which one of the following function is continuous everywhere in its domain but has at least one point where it is not
differentiable?
A ( . f(x)=x^{1 / 3} )
в. ( f(x)=frac{|x|}{x} )
( mathbf{c} cdot f(x)=e^{-x} )
D. ( f(x)=tan x )
12
211 If ( boldsymbol{y}=(boldsymbol{A}+boldsymbol{B} boldsymbol{x}) e^{m boldsymbol{x}}+(boldsymbol{m}-mathbf{1})^{-2} boldsymbol{e}^{boldsymbol{x}} )
( operatorname{then} frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}-2 boldsymbol{m} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{m}^{2} boldsymbol{y} ) is equal to:
A ( cdot e^{x} )
B. ( e^{m} )
( mathbf{c} cdot e^{-m x} )
D. ( e^{(1-m) x} )
12
212 Find the second derivative of the
function ( log x )
12
213 The solution set of ( f^{prime}(x)>g^{prime}(x) ) where
( f(x)=left(frac{1}{2}right) 5^{2 x+1} ) and ( g(x)=5^{x}+ )
( 4 x log 5 ) is
( A cdot(1, infty) )
B ( cdot(0,1) )
( c cdot[2, infty) )
D. ( (0, infty) )
12
214 If ( frac{cos ^{4} theta}{x}+frac{sin ^{4} theta}{y}=frac{1}{x+y} ) then ( frac{d y}{d x}= )
A . ( x y )
B. ( tan ^{2} theta )
( c cdot 0 )
D. ( left(x^{2}+y^{2}right) sec ^{2} theta )
12
215 ( boldsymbol{f}(boldsymbol{x})=mathbf{1}+frac{mathbf{1}}{boldsymbol{x}} ; boldsymbol{g}(boldsymbol{x})=frac{mathbf{1}}{mathbf{1}+boldsymbol{f}(boldsymbol{x})} Rightarrow )
( boldsymbol{g}^{prime}(mathbf{2})= )
( A cdot frac{1}{5} )
в. ( frac{1}{25} )
c. 5
D. ( frac{1}{16} )
12
216 If ( y=a^{frac{1}{2} log _{a} cos x}, ) find ( frac{d y}{d x} ) 12
217 ( f(x)=cot ^{-1}left(frac{x^{x}-x^{-x}}{2}right) ) then ( f^{1}(1)= )
( mathbf{A} cdot-log 2 )
B. ( log 2 )
( c cdot 1 )
D. –
12
218 ( frac{e^{1 / x}-e^{-1 / x}}{e^{1 / x}+e^{-1 / x}} ) check continuity at ( x= )
( mathbf{0}^{-} )
12
219 1-cos 4x
xco
14. Let f(x) = {a,
x=0
(1990 – 4 Marks)
>0
| 16+ √x – 4
Determine the value of a, if possible, so that the function is
continuous at x = 0
12
220 Solve ( : boldsymbol{y}=sin ^{-1}left(frac{1-boldsymbol{x}^{2}}{1+boldsymbol{x}^{2}}right), boldsymbol{0}<boldsymbol{x}<1 ) 12
221 Find the derivative of ( cos ^{2} x, ) by using
first principle of derivatives.
12
222 Find the derivative of
( x^{-4}left(3-4 x^{-5}right) )
12
223 Find the value of ( k, ) so that the function
( f(x) ) is continuous at the indicated
point ( left.boldsymbol{f}(boldsymbol{x}) begin{array}{l}=frac{8^{x}-2^{x}}{k^{x}-1} text { for } boldsymbol{x} neq boldsymbol{o} \ =boldsymbol{2} quad boldsymbol{x}=mathbf{0}end{array}right} boldsymbol{a} boldsymbol{t} boldsymbol{x}=0 )
12
224 If ( y=5^{x} x^{5}, ) then ( frac{d y}{d x} ) is
A ( cdot 5^{x}left(x^{5} log 5-5 x^{4}right) )
B . ( x^{5} log 5-5 x^{4} )
c. ( x^{5} log 5+5 x^{4} )
D. ( 5^{x}left(x^{5} log 5+5 x^{4}right) )
12
225 differentiate :
( boldsymbol{y} log boldsymbol{x} )
12
226 If ( y=frac{f(x)}{phi(x)} ) and ( z=frac{f^{prime}(x)}{phi^{prime}(x)}, ) then ( frac{f^{prime prime}}{f}- )
( frac{phi^{prime prime}}{phi}+frac{2(y-z)}{f phi}left(phi^{prime}right)^{2}= )
A ( cdot frac{d^{2} y}{d x^{2}} )
B. ( frac{1}{y} cdot frac{d^{2} y}{d x^{2}} )
c. ( y cdot frac{d^{2} y}{d x^{2}} )
D. None of these
12
227 Find the derivative of ( f(log x) ) with
respect to ( x ) where ( f(x)=log x )
12
228 Identify the graph of the polynomial function ( boldsymbol{f} )
( f(x)=x^{4}-2 x^{3}-x^{2}+2 x )
A. graph a
B. graph b
c. graph c
D. graoh d
12
229 Xis
01+ 1 x D
(1987-2 Marks)
The set of all points where the function f(x)
differentiable, is
(a) (-00,00)
(b) [0,00)
(c) (-0,0 (0,0) (d) (0,00)
(e) None
12
230 Say true or false.
The derivative of a constant function is
always non-zero.
A. True
B. False
12
231 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(cos ^{-1}left(4 boldsymbol{x}^{3}-mathbf{3} boldsymbol{x}right)right),=therefore frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{boldsymbol{m}}{sqrt{mathbf{1}-boldsymbol{x}^{k}}} . ) Find ( boldsymbol{k}-boldsymbol{m} ) ?
12
232 A curve passing through the point (1,1) is such that the intercept made by a tangent to it on ( x ) -axis is three times the
x co-ordinate of the point of tangency, then the equation of the curve is:
A ( cdot y=frac{1}{x^{2}} )
в. ( y=sqrt{x} )
c. ( y=frac{1}{sqrt{x}} )
D. none
12
233 If ( boldsymbol{y}=cos sqrt{boldsymbol{x}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
234 2. If y =
1
-. then
(a) (-a)
(e) (3 + a)2
(6) -(2-a)2
(d) -(z + a)2
12
235 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}{log (boldsymbol{x}+sqrt{boldsymbol{a}^{2}+boldsymbol{x}^{2}})}= )
A. ( frac{1}{(x+sqrt{a^{2}+x^{2}})} )
в. ( frac{x}{sqrt{a^{2}+x^{2}}} )
c. ( frac{1}{x(x+sqrt{a^{2}+x^{2}})} )
D. ( frac{1}{sqrt{a^{2}+x^{2}}} )
12
236 Find the derivative of the following functions(it is to be understood that
( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero
constants and ( m ) and ( n ) are integers):
os as ( 1+sin )
12
237 Differentiate wirh respect to
( sqrt{x+frac{1}{x}} )
12
238 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
if ( boldsymbol{y}=(sin boldsymbol{x})^{boldsymbol{x}}+sin ^{-1} sqrt{boldsymbol{x}} )
12
239 If ( boldsymbol{y}=frac{mathbf{1}}{mathbf{1}+boldsymbol{x}^{boldsymbol{beta}-boldsymbol{alpha}}+boldsymbol{x}^{boldsymbol{gamma}-boldsymbol{alpha}}}+ )
( frac{1}{1+x^{alpha-beta}+x^{gamma-beta}}+frac{1}{1+x^{alpha-gamma}+x^{beta-gamma}} )
then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{r}} ) is equal to
A . 0
B. 1
C ( cdot(a+beta+gamma) X^{alpha+beta+gamma-1} )
D. None of these
12
240 If ( f(x)=x^{frac{1}{x}} ) then ( f^{prime prime}(e) ) is equal to
B ( cdot e^{1 / e} )
( mathbf{c} cdot e^{1 /(e-2)} )
D. ( left.-e^{((1 / e)-3}right) )
12
241 Differentiate ( frac{x^{2}+1}{x} ) w.r.t ( x ) 12
242 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) where ( boldsymbol{x}^{mathbf{3}}+boldsymbol{y}^{mathbf{3}}+mathbf{3} boldsymbol{x} boldsymbol{y}=mathbf{7} ) 12
243 25.
Iff(x) = xa log x and f(0) = 0, then the value of a for which
Rolle’s theorem can be applied in [0, 1] is (20045
(a) -2 (6) -1 (c) o
(d) 1/2
12
244 Diffrentiate w.r.t ( x: )
( boldsymbol{y}=e^{2 x}(boldsymbol{a}+boldsymbol{b} boldsymbol{x}) )
12
245 Let y be an implicit function of ( mathbf{x} ) defined by ( mathbf{x}^{2 mathbf{x}}-mathbf{2} mathbf{x}^{mathbf{x}} cot boldsymbol{y}-mathbf{1}=mathbf{0} )
Then ( y^{prime}(1) ) equals
A . -1
B.
( c cdot log 2 )
D. ( -log 2 )
12
246 If ( y=sqrt{frac{1+tan x}{1-tan x}} ) then ( frac{d y}{d x} ) is equal to
A ( cdot frac{1}{2} sqrt{frac{1-tan x}{1+tan x}} sec ^{2}left(frac{pi}{4}+xright) )
в. ( sqrt{frac{1-tan x}{1+tan x}} sec ^{2}left(frac{pi}{4}+xright) )
c. ( frac{1}{2} sqrt{frac{1-tan x}{1+tan x}} sec left(frac{pi}{4}+xright) )
D. None of these
12
247 Find the derivative of ( operatorname{cosec}^{2} x, ) by using
first principle of derivatives?
12
248 The function ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}3 x-1, & text { if } x2end{array} ) continuous on right.
A ( cdot(-infty, 1) )
B. ( (2, infty) )
C ( .(-infty, 1) cup(2, infty) )
D. (1,2)
12
249 If the function ( mathbf{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{sin 3 x}{x} & (x neq 0) \ frac{k}{2} & (x=0)end{array} ) is continuous at right.
( x=0, ) then ( k ) is:
( A cdot 3 )
B. 6
( c cdot 9 )
( D )
12
250 Let ( boldsymbol{f}:[mathbf{1}, infty] rightarrow[mathbf{2}, infty] ) if ( boldsymbol{f}(mathbf{1})=mathbf{2} . ) be
differentiable function such that
( 6 int_{1}^{x} f(t) d t=3 x f(x)-x^{3} ) then the
value of ( boldsymbol{f}(mathbf{2}) ) is….
12
251 Differentiate w.r.t. ( boldsymbol{x} )
( boldsymbol{y}=boldsymbol{e}^{cos (boldsymbol{6} boldsymbol{x}-mathbf{1})} )
12
252 If ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) satisfies ( |boldsymbol{f}(boldsymbol{x})-boldsymbol{f}(boldsymbol{y})| leq )
( |x-y|^{3} ) and ( f(4)=192 ) then ( f(7) ) is
equal to
12
253 ( boldsymbol{g}(boldsymbol{x}+boldsymbol{y})=boldsymbol{g}(boldsymbol{x})+boldsymbol{g}(boldsymbol{y})+mathbf{3} boldsymbol{x} boldsymbol{y}(boldsymbol{x}+ )
( boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} ) and ( boldsymbol{g}^{prime}(mathbf{0})=-4 . ) The value of
( boldsymbol{g}^{prime}(1) ) is
A . 0
B.
c. -1
D. none of these
12
254 Show that the function ( f(x) ) defined as
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} cos frac{1}{x}, boldsymbol{x} neq mathbf{0},=mathbf{0}, boldsymbol{x}=mathbf{0} ) is
continuous at ( x=0 ) but not
differentiable at ( x=0 )
12
255 Let ( boldsymbol{f} ) be a function satisfying
( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x} ) and ( boldsymbol{y} )
and ( boldsymbol{f}(mathbf{0})=boldsymbol{f}^{prime}(mathbf{0})=mathbf{1} ) then
This question has multiple correct options
A. ( f ) is differentiable for all ( x )
B . ( f^{prime}(x)=f(x) )
( mathbf{c} cdot f(x)=e^{x} )
D. ( f ) is continuous for all ( x )
12
256 28. Let a + b = 4, where a 0 for all x, prove that so g(x) dx + So g(x)dx
dx
increases as (b-a) increases.
(1997- 5 Marks)
12
257 Differentiate w.r.t ( boldsymbol{x}: boldsymbol{x}^{boldsymbol{y}}+boldsymbol{y}^{boldsymbol{x}}=mathbf{1} ) 12
258 ff ( (x)=3 e^{x^{2}}, ) then ( f^{prime}(x)-2 x f(x)+ )
( frac{1}{3} f(0)-f^{prime}(0) ) is equal to
A.
B.
( mathrm{c} cdot frac{7}{3} mathrm{e}^{x} )
D. ex ( ^{x^{2}} )
12
259 f ( p^{2}=a^{2} cos ^{2} theta+b^{2} sin ^{2} theta ) then ( frac{d^{2} p}{d theta^{2}}+ )
( p ) is equal to ( (a neq b) )
A ( cdot frac{a^{2} b^{2}}{p^{4}} )
в. ( frac{a^{2} b^{2}}{p^{2}} )
c. ( frac{a b}{p} )
D. ( frac{a^{2} b^{2}}{p^{3}} )
12
260 5.
If x=ete
.
(20041
. x > 0, then
1+X
1-*
(b)
(d)
12
261 If ( y=tan ^{-1}left(frac{2^{x^{prime}}}{1+2^{2 x+1}}right) ) then ( frac{d y}{d x} ) at
( boldsymbol{x}=mathbf{0} ) is?
A ( cdot frac{1}{10} log 2 )
в. ( frac{1}{5} log 2 )
c. ( -frac{1}{10} log 2 )
D. ( log 2 )
12
262 Find the differentiation of ( sec left(tan ^{-1} xright) )
w.r.t. ( boldsymbol{x} )
12
263 Ify is a function of x and log (x + y) – 2xy=0, then the value
of y’ (O) is equal to
(2004S)
(a) 1 (b) -1 (c) 2
(d) O
12
264 ( frac{d^{2} x}{d y^{2}}=-frac{d^{2} y}{d x^{2}} cdotleft(frac{d x}{d y}right)^{3} ) 12
265 Find the value of ( f(2), ) so that the function ( boldsymbol{f}(boldsymbol{x})=frac{12 boldsymbol{x}-mathbf{2 4}}{(mathbf{4}+mathbf{2} boldsymbol{x})^{1 / 3}-mathbf{2}}, boldsymbol{x} neq mathbf{2} )
is continuous everywhere
12
266 If ( f(x) ) is continuous and ( fleft(frac{9}{2}right)=frac{2}{9} ) then ( lim _{x rightarrow 0} fleft(frac{1-cos 3 x}{x^{2}}right) ) is equal to:
A ( cdot frac{9}{2} )
в. ( frac{2}{9} )
( c cdot 0 )
D. ( frac{8}{9} )
12
267 ( y=frac{x^{2}}{1+x} ) Find ( frac{d y}{d x} ) 12
268 ff ( y=sin ^{-1} x, ) show that
( left(1-x^{2}right) cdot frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}=0 )
12
269 7.
[20021
fis defined in (-5, 5] as
f(x)=x ifx is rational
= -x ifx is irrational. Then
(a) f(x) is continuous at every x, except x = 0
(b) f(x) is discontinuous at every x, except x = 0
(c) f(x) is continuous everywhere
(d) f(x) is discontinuous everywhere
12
270 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left(frac{1-boldsymbol{x}}{mathbf{1}+boldsymbol{x}}right)= )
A ( cdot frac{2}{1+x^{2}} )
B ( cdot frac{-1}{1+x^{2}} )
c. ( frac{1}{1+x^{2}} )
D. ( frac{-2}{1+x^{2}} )
12
271 If ( y=frac{(1-x)^{2}}{x^{2}} ) where ( x neq )
( 0, ) then ( frac{d y}{d x} i s )
A ( cdot frac{2}{x^{2}}+frac{2}{x^{3}} )
B. ( -frac{2}{x^{3}}+frac{2}{x^{2}} )
( c cdot-frac{2}{x^{2}}+frac{2}{x^{3}} )
D. ( -frac{2}{x^{2}}-frac{2}{x^{3}} )
12
272 Differentiate ( y=sin b x^{2} ) w.r.t ( x ) 12
273 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( frac{e^{x} sin x}{left(x^{2}+2right)^{3}} )
12
274 Let ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon} boldsymbol{R} )
and suppose that ( f ) is differentiable at
0 and ( f^{prime}(0)=4 . ) If ( fleft(x_{0}right)=8 ) then
( f^{prime}left(x_{0}right) ) is equal to
12
275 Let ( f(x) ) be defined by ( f(x)= ) ( left{begin{array}{cl}sin 2 x & text { if } 0<x leq frac{pi}{6} \ a x+b & text { if } frac{pi}{6}<x leq 1end{array} . text { The values of } aright. )
and ( b ) such that ( f ) and ( f^{prime} ) are
continuous, are
A ( cdot a=1, b=frac{1}{sqrt{2}}+frac{pi}{6} )
в. ( a=frac{1}{sqrt{2}}, b=frac{1}{sqrt{2}} )
c. ( _{a=1, b}=frac{sqrt{3}}{2}-frac{pi}{6} )
D. None of these
12
276 The derivative of ( sin ^{-1} frac{2 x}{1+x^{2}} ) with respect to ( cos ^{-1} frac{1-x^{2}}{1+x^{2}} ) is
A . -1
B.
( c cdot 2 )
D. 4
12
277 Find the derivative of ( f(x) ) from the first
principle. ( sin x div cos x )
12
278 If ( (cos x)^{y}=(sin y)^{x} ) then ( frac{d y}{d x}= )
A ( cdot frac{log (sin y)+y tan x}{log (cos x)-x cot y} )
B. ( frac{log (sin y)-y tan x}{log (cos x)+cot y} )
c. ( log (sin y) )
D. ( frac{log (cos x)}{log (sin y)} )
12
279 Evaluate ( frac{d}{d x} 3^{log _{3} sqrt{x}}=dots dots dots )
( A cdot frac{1}{sqrt{x}} )
B. ( sqrt{x} )
c. ( frac{1}{2 sqrt{x}} )
D. ( -frac{1}{sqrt{x}} )
12
280 Prove that
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{rl}
frac{boldsymbol{x}^{2}-boldsymbol{2 5}}{boldsymbol{x}-mathbf{5}}, & boldsymbol{w} boldsymbol{h} boldsymbol{e} boldsymbol{n} quad boldsymbol{x} neq mathbf{5} \
boldsymbol{1 0}, boldsymbol{w h e n} & boldsymbol{x}=mathbf{5}
end{array}right.
]
continuous at ( boldsymbol{x}=mathbf{5} )
12
281 Evaluate ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{n} ) in ( [-mathbf{1}, mathbf{1}], boldsymbol{n} boldsymbol{epsilon} boldsymbol{Z}^{+} ) 12
282 ( operatorname{Let} f(x)=frac{1}{a x+b} ) then ( f^{prime prime}(0)= )
A ( cdot frac{2 a^{3}}{b^{2}} )
в. ( frac{2 a^{2}}{b^{3}} )
c. ( frac{2 a^{3}}{b^{3}} )
D. none of these
12
283 f ( y=tan ^{-1}left[frac{5 cos x-12 sin x}{12 cos x+5 sin x}right] )
then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot-2 )
D.
12
284 Find the derivative of ( e^{x}+e^{y}=e^{x+y} )
( mathbf{A} cdot-e^{x-y} y^{y} )
B . ( e^{x-y} )
( mathbf{c} cdot-e^{y-x} )
D. ( e^{y-x} )
12
285 Write the value of the derivative of
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-1|+|boldsymbol{x}-mathbf{3}| ) at ( boldsymbol{x}=mathbf{2} )
12
286 [
begin{aligned}
boldsymbol{f}(boldsymbol{x})=& frac{|boldsymbol{x}|}{boldsymbol{x}} boldsymbol{x} neq mathbf{0} \
mathbf{0} boldsymbol{x}=mathbf{0}
end{aligned}
]
Check whether ( f(x) ) is continous or not
at ( x=0 )
12
287 If ( y=cos ^{-1} cos x, ) then ( frac{d y}{d x} ) at ( x=frac{5 pi}{4} )
A . 1
в. – 1
c. ( frac{1}{sqrt{2}} )
D. ( frac{5 pi}{4} )
12
288 Discuss the continuity of
[
boldsymbol{f}(boldsymbol{x})=
]
( left{begin{array}{l}frac{sin 2 x}{sqrt{1-cos 2 x}} text { for } 0<x leq pi / 2 \ frac{cos }{pi-2 x} quad text { for } frac{pi}{2}<x<piend{array} ) at right.
( boldsymbol{x}=boldsymbol{pi} / 2 )
12
289 If ( f ) is a differentiable function at a point
‘a’ and ( f^{prime}(a) neq 0 ) then which of the
following is true.
A ( cdot f^{prime}(a)=lim _{h rightarrow 0} frac{f(a)-f(a-h)}{h} )
B. ( frac{1}{2} f^{prime}(a)=lim _{h rightarrow 0} frac{f(a+2 h)-f(a-h)}{2 h} )
c. ( f^{prime}(a)=lim _{h rightarrow 0} frac{f(a+2 h)-f(a)}{h} )
D. none of these
12
290 (0)

If y = y(x) and it follows
then y” (0)=
and it follows the relation x cos y + y cos x = Tt
(2005)
(b) 1 (C) T-1 (d)
(a) 1
12
291 Verify LMVT :
( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}} ) for ( boldsymbol{x}=[mathbf{1}, boldsymbol{3}] )
12
292 The value of ( f(0) ) so that the function ( f(x)=frac{sqrt{1+x}-sqrt[3]{1+x}}{x} )
becomes continuous, is equal to
A ( cdot frac{1}{6} )
B. ( frac{1}{4} )
( c cdot 2 )
D.
12
293 If ( f(x) ) is differentiable in ( [a, b] ) such that ( f(a)=2, f(b)=6, ) then there
exists at least one ( c, a<c<b, ) such
( operatorname{that}left(b^{3}-a^{3}right) f^{prime}(c)= )
A ( cdot c^{2} )
B . ( 2 c^{2} )
( c .-3 c^{2} )
D. ( 12 c^{2} )
12
294 f ( boldsymbol{y}=tan ^{-1}left(frac{sqrt{1+boldsymbol{a}^{2} boldsymbol{x}^{2}}-mathbf{1}}{boldsymbol{a} boldsymbol{x}}right), ) then
( left(1+a^{2} x^{2}right) y^{prime prime}+2 a^{2} x y^{1}= )
A . ( -2 a^{2} )
B ( cdot a^{2} )
( c cdot 2 a^{2} )
( D )
12
295 If ( e^{x}+e^{y}=e^{x+y}, ) find ( frac{d y}{d x} ) 12
296 Let ( boldsymbol{f}(boldsymbol{x})= )
( frac{boldsymbol{x}(mathbf{1}+boldsymbol{a} cos boldsymbol{x})-boldsymbol{b} sin boldsymbol{x}}{boldsymbol{x}^{3}}, boldsymbol{x} neq )
0 ( operatorname{and} f(0)=1, ) then values if ‘a’ and ‘b’
so that ‘f’ is continuous are
A ( cdot frac{5}{2}, frac{3}{2} )
B. ( frac{5}{2}, frac{-3}{2} )
c. ( -frac{5}{2}-frac{3}{2} )
D. ( frac{1}{2}-frac{3}{2} )
12
297 Given a function ‘g’ whcih has a
derivative ( g^{prime}(x) ) for every real ‘ ( x ) ‘ and
which satisfy ( g^{prime}(0)=2 ) and ( g(x+y)= )
( e^{y} cdot g(x)+e^{x} cdot g(y) ) for all ( x, y . ) Find ( g(x) )
( A cdot 2 x e^{x} )
В. ( x e^{x} )
c. ( x+e^{x} )
D. ( x-e^{x} )
12
298 If ( f(x)=int_{0}^{x} t(sin x-sin t) d t ) then?
A ( cdot f^{prime prime prime}(x)+f^{prime}(x)=cos x-2 x sin x )
B . ( f^{prime prime prime}(x)+f^{prime prime}(x)-f^{prime}(x)=cos x )
C. ( f^{prime prime prime}(x)-f^{prime prime}(x)=cos x-2 x sin x )
D. ( f^{prime prime prime}(x)+f^{prime prime}(x)=sin x )
12
299 Find differentiation of ( sec ^{-1} tan x ) 12
300 If ( frac{x+a}{2}=b cot ^{-1}(b ln y), b>0, ) then
value of ( boldsymbol{y} boldsymbol{y}^{prime prime}+boldsymbol{y} boldsymbol{y}^{prime} ln boldsymbol{y} ) equals
A ( cdot y^{prime} )
B . ( y^{prime} )
( c cdot 0 )
D.
12
301 If ( cos (x+y)=y sin x, ) then find ( frac{d y}{d x} ) 12
302 If ( f(x)=x^{n} ln x ) and ( f(0)=0 ) then
value of ( alpha ) for which Rolle’s Theorem can
be applied in ( [mathbf{0}, mathbf{1}] )
A . -2
B. –
c. 0
D.
12
303 Range of ( boldsymbol{y}=log _{frac{3}{4}}(boldsymbol{f}(boldsymbol{x})) )
( A cdot(-infty, 1] )
в. ( left[frac{3}{4}, inftyright) )
( c cdot(-infty, infty) )
D. ( R )
12
304 If ( y=e^{x}+sin x-4 x^{3}, ) find ( frac{d y}{d x} ) 12
305 If f(x)= x(VxVx+1), then (1985 – 2 Marks)
(a) f(x) is continuous but not differentiable at x=0
(b) f(x) is differentiable at x = 0
(c) f(x) is not differentiable at x=0
(d) none of these
12
306 If ( boldsymbol{y}=sqrt{frac{1-sin 2 x}{1+sin 2 x}}, ) then ( left(frac{d y}{d x}right)_{x=0}= )
This question has multiple correct options
A ( cdot frac{1}{2} )
B.
c. -2
D.
12
307 Let ( mathbf{f}: mathbf{R} rightarrow mathbf{R} ) be any function. Define
( mathbf{g}: mathbf{R} rightarrow mathbf{R} ) by ( boldsymbol{g}(boldsymbol{x})=|boldsymbol{f}(boldsymbol{x})| ) for all ( boldsymbol{x} )
Then ( g ) is
A. ( g ) may be bounded even if ( f ) is unbounded
B. one-one if fis one
c. continuous if ( f ) is continuous
D. differentiable if f is differentiable
12
308 If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{p}|sin boldsymbol{x}|+boldsymbol{q} e^{|boldsymbol{x}|}+boldsymbol{r}|boldsymbol{x}|^{3} ) and if
( f(x) ) is differentiable at ( x=0, ) then
A ( cdot p=q=r=0 )
B. ( p+q=0 ; r ) is any real number
c. ( q+r=0 ; p ) is any real number
D. ( r=0 ; p=0, q ) is any real number
12
309 Differentiate the following function with
respect to ( x )
( boldsymbol{x}^{5}left(boldsymbol{3}-boldsymbol{6} boldsymbol{x}^{-boldsymbol{9}}right) )
A. ( 15 x^{-4}+24 x^{-5} )
B ( cdot 15 x^{5}+24 x^{-5} )
( mathbf{c} cdot 15 x^{4}+24 x^{-4} )
D. ( 15 x^{4}+24 x^{-5} )
12
310 ( lim _{x rightarrow 0^{+}}left(left(x^{x^{x}}right)-x^{x}right) ) is
A. Equal to 0
B. Equal to 1
c. Equal to – 1
D. Non existent
12
311 Differentiate w.r.t. ( x )
( boldsymbol{y}=(cos boldsymbol{x})left(1-sin ^{2} boldsymbol{x}right) )
12
312 The radius of a sphere is changing at the rate of ( 0.1 mathrm{cm} / ) sec. The rate of
change of its surface area when the radius is ( 200 mathrm{cm}, ) is.
A ( cdot 8 pi c m^{2} / ) sec
В. ( 12 pi c m^{2} / )sec
c. ( 160 pi c m^{2} / ) sec
D. ( 200 pi c m^{2} / ) sec
12
313 If ( y=sin ^{-1}left(x^{2}right) ) then find ( frac{d y}{d x} ) using first
principle.
A. ( frac{2 x}{sqrt{1-x^{4}}} )
в. ( frac{2}{sqrt{1-x^{2}}} )
c. ( frac{x}{sqrt{1-x^{4}}} )
D. ( -frac{1}{sqrt{1-x^{4}}} )
12
314 Let ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{x} ) and ( boldsymbol{g}(boldsymbol{x})=[boldsymbol{x}+mathbf{2}] )
where [.] denotes the greatest integer function. Then, ( (g o f)^{prime}left(frac{pi}{2}right) ) is?
( mathbf{A} cdot mathbf{1} )
B.
c. -1
D. Does not exist
12
315 Find the value of ( p ) if following function ( boldsymbol{f}(boldsymbol{x})= )
[
left{begin{array}{ll}
frac{sqrt{1+p x}-sqrt{1-p x}}{x}, & text { if }-1 leq x< \
frac{2 x+2}{x-2}, & text { if } 0 leq x<1
end{array}right.
]
is continuous at ( x=0 )
12
316 Differentiate the following function with
respect to ( x ) ( 1+3 x )
12
317 If ( f(x)=frac{e^{x^{2}}-cos x}{x^{2}}, ) for ( x neq 0 ) is
continuous at ( boldsymbol{x}=mathbf{0}, ) then value of ( boldsymbol{f}(mathbf{0}) )
is
A ( cdot frac{2}{3} )
в. ( frac{5}{2} )
( c cdot 1 )
D.
12
318 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( sin ^{-1}left(2 x^{2}-1right), 0<x<1 )
12
319 8. If f(x)=va
– (2) 1 (1) 1
, then f (a) =
(co
(d) a
12
320 ( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{3} boldsymbol{x}+boldsymbol{2} )
Find ( frac{d y}{d x} ) if the given function is
continuous.
12
321 Differentiate the following w.r.t. ( x: ) ( e^{sin ^{-1} x} ) 12
322 The value of ( frac{boldsymbol{f}(boldsymbol{t})}{boldsymbol{f}^{prime}(boldsymbol{t})} cdot frac{boldsymbol{f}^{prime prime}(-boldsymbol{t})}{boldsymbol{f}^{prime}(-boldsymbol{t})}- )
( frac{f(-t)}{f^{prime}(-t)} cdot frac{f^{prime prime}(t)}{f^{prime}(t)} forall t epsilon R ) is equal to
A . -2
B. 2
c. -4
( D )
12
323 ( frac{boldsymbol{d}(sin boldsymbol{x})}{boldsymbol{d} boldsymbol{x}} )
( A cdot cos x )
B. ( sec x )
( c .-cos x )
D. – ( tan x )
12
324 Solve the following differential equation ( frac{d y}{d x}=3 x ) 12
325 What is the derivative of ( x^{3} ) with respect
to ( x^{2} ? )
A ( cdot 3 x^{2} )
в. ( frac{3 x}{2} )
c. ( x )
D. ( frac{3}{2} )
12
326 If ( f(x) ) satisfies the conditions of Rolle’s
theorem in [1,2] and ( f(x) ) is continuous in [1,2] then ( int_{1}^{2} f^{prime}(x) d x ) is equal to
A . 3
B. 0
( c .1 )
D. 2
12
327 If ( f(x)=(cos x+i sin x)(cos 3 x+ )
( i sin 3 x) ldots(cos (2 n-1) x+i sin (2 n- )
1) ( x ), then ( f^{prime prime}(x)= )
( mathbf{A} cdot n^{2} f(x) )
B . ( -n^{4} f(x) )
( mathbf{c} cdot-n^{2} f(x) )
D. ( n^{4} f(x) )
12
328 If the function ( f(x)=x^{3}-6 a x^{2}+5 x )
satisfies the conditions of Lagrange’s mean theorem for the interval [1,2] and the tangent to the curve ( y=f(x) ) at
( boldsymbol{x}=mathbf{7} / mathbf{4} ) is parallel to the chord joining
the points of intersection of the curve
with the ordinates ( x=1 ) and ( x=2 )
Then the value of a is?
( mathbf{A} cdot 35 / 16 )
B. ( 35 / 48 )
c. ( 7 / 16 )
D. ( 5 / 16 )
12
329 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(tan ^{-1} boldsymbol{x}right) )
A ( cdot frac{1}{1+x^{2}} )
B ( cdot frac{-1}{1+x^{2}} )
c. ( frac{-1}{1-x^{2}} )
D. ( frac{1}{1-x^{2}} )
12
330 Solve: ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=cos (boldsymbol{x}+boldsymbol{y}) ) 12
331 The function ( f(x)=sin ^{-1}(cos x) ) is
A. Discontinuous at ( x=0 )
B. continuous at ( x=0 )
C. differentiable at ( x=0 )
D. None of these
12
332 ( y=tan ^{-1}left(frac{1}{x}right) ) find ( frac{d y}{d x} ) 12
333 If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}-sin boldsymbol{t}), boldsymbol{y}=boldsymbol{a}(1+cos boldsymbol{t}) ) then
find ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} )
12
334 If ( y=sin left(log _{e} xright) ) prove that ( frac{d y}{d x}= )
( frac{sqrt{1-y^{2}}}{x} )
12
335 If ( x^{2}+2 x y+y^{3}=42, ) find ( frac{d y}{d x} ) 12
336 Differentiate ( cos ^{-1} x )
A ( cdot frac{-1}{sqrt{left(1-x^{2}right)}} )
B. ( frac{1}{sqrt{left(1-x^{2}right)}} )
c. ( frac{-1}{sqrt{left(1+x^{2}right)}} )
D. ( frac{1}{sqrt{left(1+x^{2}right)}} )
12
337 Differentiate w.r.t. ( mathbf{x} )
( f(x)=sqrt{sin (cos x)} )
12
338 Let ( boldsymbol{J}, boldsymbol{g}:lfloor-mathbf{1}, boldsymbol{2}rfloor rightarrow boldsymbol{K} ) be continuous
functions which are twice differential
on the interval ( (-1,2) . ) Let the values of
( f ) and ( g ) at the points -1,0 and 2 be as given in the following table:
( f(x)=left{begin{array}{l}3 ; x=-1 \ 6 ; x=0 quad text { and } g(x)= \ 0 ; x=2end{array}right. )
( left{begin{array}{l}0 ; x=-1 \ 1 ; x=0 \ 0 ; x=-1end{array}right. )
In each of the intervals (-1,0) and
( (0,2), ) the function ( (f-3 g)^{prime prime} ) never
vanishes. Then the correct statement(s)
is (are)
This question has multiple correct options
A ( cdot f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly three solutions in (-1,0)( cup(0,2) )
B. ( f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly one solutions in (-1,0)
C ( cdot f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly one solutions in (0,2)
D. ( f^{prime}(x)-3 g^{prime}(x)=0 ) has exactly two solutions in (-1,0) and exactly two solutions in (0,2)
12
339 2.
is
For a real number y, let [y] denotes the greatest integer less
than or equal to y: Then the function f(x) = –
tan(Te[x – 1)
1+[x]?
(1981 – 2 Marks)
(a) discontinuous at some x
(b) continuous at all x, but the derivative f'(x) does not
exist for somex
c) f'(x) exists for all x, but the second derivative f'(x)
does not exist for some x
(d) f'(x) exists for all x
12
340 From means value theorem ( boldsymbol{f}(boldsymbol{b})- )
( boldsymbol{f}(boldsymbol{a})=(boldsymbol{b}-boldsymbol{a}) boldsymbol{f}^{prime}left(boldsymbol{x}_{1}right) ; mathbf{0}<boldsymbol{a}<boldsymbol{x}_{1}<boldsymbol{b} ) if
( f(x)=frac{1}{x}, ) then ( x_{1}= )
A ( cdot sqrt{a b} )
в. ( frac{a+b}{2} )
c. ( frac{2 a b}{a+b} )
D. ( frac{b-a}{b+a} )
12
341 (a) –
(b) 1+2
(d) None of these
12
342 Derivative of ( (x+3)^{2}(x+4)^{3}(x+5)^{4} )
( boldsymbol{w} cdot boldsymbol{r} cdot operatorname{to} boldsymbol{x} ) is
A ( cdot(x+3)(x+4)(x+5)^{2}left(9 x^{2}+70 x+133right) )
B cdot ( (x+3)(x+4)^{2}(x+5)^{3}left(9 x^{2}+70 x+133right) )
C ( cdot(x+3)(x+4)^{2}(x+5)left(9 x^{2}-70 x-133right) )
D. none of these
12
343 ( f(x)=frac{1+e^{1 / x}}{1-e^{1 / x}}(x neq 0), f(0)=1, ) then
( f(x) ) is
A. left coninuous at ( x=0 )
B. right continuous at ( x=0 )
c. continuous at ( x=0 )
D. none
12
344 Differentiate the following w.r.t. ( x ) ( e^{x^{3}} ) 12
345 Let ( boldsymbol{f}(boldsymbol{x})= )
( (x-4)(x-5)(x-6)(x-7) ) then
A ( cdot f^{prime}(x)=0 ) has four roots
B. three roots of ( f^{prime}(x)=0 ) lie in (4,5)( cup(5,6) cup(6,7) )
C. the equation ( f^{prime}(x)=0 ) has only one root
D. three roots of ( f^{prime}(x)=0 ) lie in (3,4)( cup(4,5) cup(5,6) )
12
346 If ( boldsymbol{y}=boldsymbol{e}^{2 x}(boldsymbol{a} boldsymbol{x}+boldsymbol{b}), ) show that ( boldsymbol{y}_{2} )
( mathbf{4} boldsymbol{y}_{1}+mathbf{4} boldsymbol{y}=mathbf{0} )
12
347 A function ( f(x) ) defined as ( f(x)= ) ( left{begin{array}{ll}sin x, & x text { is rational } \ cos x, & x text { is irrational }end{array} ) is continuous right.
at
A ( cdot x=n pi+frac{pi}{4}, n in I )
В ( cdot x=n pi+frac{pi}{8}, n in I )
c. ( x=n pi+frac{pi}{6}, n in I )
D・ ( x=n pi+frac{pi}{3}, n in I )
12
348 The set of all points where ( boldsymbol{f}(boldsymbol{x})= )
( sqrt[3]{x^{2}|x|}-|x|-1 ) is not differentiable is
( A cdot{0} )
B ( cdot{-1,0,1} )
( mathbf{c} cdot{0,1} )
D. None of these
12
349 Find number of terms in ( left(1+x+x^{4}right)^{12} ) 12
350 ( lim _{boldsymbol{x} rightarrow infty} sum_{boldsymbol{r}=1}^{boldsymbol{n}} tan ^{-1}left(frac{2 r}{1-boldsymbol{r}^{2}+boldsymbol{r}^{4}}right) ) is equal to
( mathbf{A} cdot pi / 4 )
B. ( pi / 2 )
( c cdot frac{3 pi}{4} )
D. None of these
12
351 If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{m} sin ^{-1} boldsymbol{x}} )
Then ( left(1-x^{2}right)left(frac{d y}{d x}right)^{2}=A y^{2}, ) then ( A= )
?
A . ( m )
B. ( -m )
( mathrm{c} cdot m^{2} )
D. ( -m^{2} )
12
352 If ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) satisfies the property ( (boldsymbol{x}- )
( boldsymbol{y}) boldsymbol{f}(boldsymbol{x}+boldsymbol{y})-(boldsymbol{x}+boldsymbol{y}) boldsymbol{f}(boldsymbol{x}-boldsymbol{y})= )
( 4 x yleft(x^{2}-y^{2}right), f(1)=1, ) then the
number of real roots of ( boldsymbol{f}(boldsymbol{x})=mathbf{4} ) will be
A . 1
B . 2
( c cdot 3 )
D.
12
353 ( operatorname{Let} F(x)=f(x) g(x) h(x) ) for all real ( x )
where ( f(x), g(x) ) and ( h(x) ) are differentiable functions. At some point
( boldsymbol{x}_{0}, boldsymbol{F}^{prime}left(boldsymbol{x}_{0}right)=mathbf{2 1} boldsymbol{F}left(boldsymbol{x}_{0}right), boldsymbol{f}^{prime}left(boldsymbol{x}_{0}right)= )
( 4 fleft(x_{0}right), g^{prime}left(x_{0}right)=-7 gleft(x_{0}right) ) and
( boldsymbol{h}^{prime}left(boldsymbol{x}_{0}right)=boldsymbol{k h}left(boldsymbol{x}_{0}right) . ) Then ( boldsymbol{k} ) is equal to
12
354 Differentiate the following functions
with respect to ( boldsymbol{x} )
( boldsymbol{x} sin 2 boldsymbol{x}+mathbf{5}^{boldsymbol{x}}+boldsymbol{k}^{boldsymbol{k}}+left(tan ^{2} boldsymbol{x}right)^{2} )
12
355 The graph of ( f(x) ) is given below. Based
on this graph determine where the
function is discontinuous.
12
356 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cc}2 a-x, & text { if }-a<x<a \ 3 x-2 a, & text { if } a leq xend{array} . ) Then right.
which of the following is true?
A. ( f(x) ) is discontinuous at ( x=a )
B. ( f(x) ) is not differentiable at ( x=a )
c. ( f(x) ) is differentiable at ( x geq a )
D. ( f(x) ) is continuous at all ( x<a )
12
357 If ( a x+b y^{2}=cos y, ) then find ( frac{d y}{d x} ) 12
358 Domain of the function is ( boldsymbol{y}= )
( frac{sqrt{cos x-1 / 2}}{sqrt{6+35 x-6 x^{2}}} )
( mathbf{A} cdot therefore D_{1} cap D_{2}=[0, pi / 6] cup[5 pi / 3,6] )
B ( ldots D_{1} cap D_{2}=[0, pi / 3] cup[5 pi / 3,6] )
( mathrm{c} cdot therefore D_{1} cap D_{2}=[0, pi / 3] cup[5 pi / 6,6] )
D. ( therefore D_{1} cap D_{2}=[0, pi / 2] cup[5 pi / 3,6] )
12
359 Use Rolle’s theorem to prove that equation ( a x^{2}+b x=frac{a}{3}+frac{b}{2} ) has a root
between 0 and 1
12
360 Find the derivative with respect to ( x ) of
the function
( left(log _{cos x} sin xright)left(log _{sin x} cos xright)^{-1}+ )
( sin ^{-1} frac{2 x}{1+x^{2}} ) at ( x=frac{pi}{4} )
A ( cdot gleft(frac{4}{pi^{2}+16}-frac{1}{log 2}right) )
B. ( -8left(frac{4}{(pi+4)^{2}}-frac{1}{log 2}right) )
( ^{mathbf{c}} cdot_{8}left(frac{4}{pi^{2}+16}+frac{1}{log 2}right) )
D ( cdot gleft(frac{4}{(pi+4)^{2}}-frac{1}{log 2}right) )
12
361 if ( x_{1}, x_{2}, x_{3} dots x_{n} ) denote the values of ( x )
where ( f(x) ) vanishes such that ( x_{1}> )
( x_{2}>x_{3} ldots x_{n}, ) then ( lim _{n rightarrow infty} sum_{r=1}^{n} frac{r}{x_{r}} ) is equal to
A . -8
B. -4
( c .-2 )
D. –
12
362 will be
4. If y = sin(x2), then
(a) 2t cos(x2)
(c) 4×2 sin (+2)
(b) 2 cos (+2) – 4t sin (12)
(d) 2 cos (12)
12
363 Let ( boldsymbol{f}(boldsymbol{x})=frac{2}{pi} operatorname{cosec}^{-1} frac{x+1}{2} ) Then
( mathbf{A} cdot lim _{x rightarrow 1^{+}} f(x)=0 )
( mathbf{B} cdot lim _{x rightarrow 1} f(x)=frac{pi}{2} )
( mathbf{C} cdot lim _{x rightarrow-3^{-}} f(x)=-1 )
( mathbf{D} cdot lim _{x rightarrow-3} f(x)=1 )
E ( cdot lim _{x rightarrow-3} f(x)=-frac{pi}{2} )
12
364 ( f_{n}(x)=e^{f_{n-1}(x)} ) for all ( n epsilon N ) and
( f_{0}(x)=x, ) then ( frac{d}{d x}left{f_{n}(x)right} ) is
This question has multiple correct options
A ( cdot f_{n}(x) frac{d}{d x}left{f_{n-1}(x)right} )
В ( cdot f_{n}(x) f_{n-1}(x) )
( mathbf{c} cdot f_{n}(x) f_{n-1}(x) cdots f_{2}(x) cdot f_{1}(x) )
D. none of these
12
365 If ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y}+boldsymbol{z})=boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y}) cdot boldsymbol{f}(boldsymbol{z}) ) for all
( boldsymbol{x}, boldsymbol{y}, boldsymbol{z} ) and ( boldsymbol{f}(boldsymbol{2})=boldsymbol{4}, boldsymbol{f}^{prime}(boldsymbol{0})=boldsymbol{3}, ) then
( f^{prime}(2) ) equals
A . 12
B. 9
c. 16
D. 6
12
366 If the Rolle’s theorem holds for the
function ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{mathbf{3}}+boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x} ) in
interval [-1,1] for the point ( c=frac{1}{2}, ) then
find the value of ( 2 a+b ? )
A
B. –
( c cdot 2 )
( D cdot-2 )
12
367 ( operatorname{Let} f(x)=left(frac{tan left(frac{pi}{4}-xright)}{cot 2 x}right)left(x neq frac{pi}{4}right) )
The value which should be assigned to at ( frac{pi}{4} ) so that it is continuous
everywhere, is
A ( cdot frac{1}{2} )
B.
( c cdot 2 )
D. None of these
12
368 ff ( y=tan ^{-1} frac{1}{x^{2}+x+1}+ )
( tan ^{-1} frac{1}{x^{2}+3 x+3}+ )
( tan ^{-1} frac{1}{x^{2}+5 x+7}+ldots . ) to ( n ) terms
then
A ( cdot frac{d y}{d x}=frac{1}{1+(x+n)^{2}}-frac{1}{1+x^{2}} )
B. ( frac{d y}{d x}=frac{1}{(x+n)^{2}}-frac{1}{1+x^{2}} )
c. ( frac{d y}{d x}=frac{1}{1+(x+n)^{2}}+frac{1}{1+x^{2}} )
D. None of these
12
369 Let ( boldsymbol{f} ) be an increasing function on ( [boldsymbol{a}, boldsymbol{b}] ) and ( g ) be a decreasing function on ( [a, b] )
then on ( [a, b] )
This question has multiple correct options
A. fog is a decreasing function.
B. gof is an increasing function.
c. ( f o g ) is an increasing function.
D. None of these
12
370 If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
371 If ( f(x)=frac{1}{x-1}, ) then determine the
number of points of discontinuity of
( boldsymbol{f}[boldsymbol{f}{boldsymbol{f}(boldsymbol{x})}] )
12
372 ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{sqrt{1+boldsymbol{p} boldsymbol{x}}-sqrt{mathbf{1}-boldsymbol{p} boldsymbol{x}}}{boldsymbol{x}} \ frac{mathbf{2} boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{2}}, quad mathbf{0} leq boldsymbol{x} leq mathbf{1}end{array} quad, quad-mathbf{1} leq boldsymbol{x}<right. )
0 is continuous in the interval [-1,1]
then ( p ) equals-
A . -1
B. ( -frac{1}{2} )
( c cdot frac{1}{2} )
( D )
12
373 If ( y=sqrt{frac{sec x-1}{sec x+1}} ) then ( frac{d y}{d x}= )
A. ( frac{1}{2} sec ^{2} frac{x}{2} )
B. ( sec ^{2} frac{x}{2} )
c. ( frac{1}{2} tan frac{x}{2} )
D. ( tan frac{x}{2} )
12
374 ( y=6 x^{3}+3 x^{2}+4 x+5 )
Find the value of ( frac{d y}{d x} ? )
12
375 f ( boldsymbol{y}=log _{7}(log boldsymbol{x}) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
376 ( operatorname{Let} f(x)=left(2-frac{x}{a}right)^{tan left(frac{pi x}{2 a}right)}, x neq a )
The value which should be assigned to ( f ) at ( x=a ) so that it is continuous
everywhere is
A ( -frac{2}{pi} )
B . ( e^{-2 / pi} )
( c cdot 2 )
( mathbf{D} cdot e^{2 / pi} )
12
377 Let ( f ) be a function which is continuous
and differentiable for all real ( x ). If
( boldsymbol{f}(mathbf{2})=-mathbf{4} ) and ( boldsymbol{f}^{prime}(boldsymbol{x}) geq mathbf{6} ) for all ( boldsymbol{x} in )
( [2,4], ) then
( mathbf{A} cdot f(4)<8 )
B ( cdot f(4) geq 8 )
( mathbf{c} cdot f(4) geq 12 )
D. none of these
12
378 Find ( frac{d y}{d x}, ) if ( x+y=sin (x-y) )
A ( cdot frac{cos (x-y)-1}{cos (x-y)+1} )
B ( cdot frac{cos (x-y)+1}{cos (x-y)-1} )
C ( frac{cos (x+y)+1}{cos (x-y)-1} )
D. ( frac{cos (x+y)-1}{cos (x-y)+1} )
12
379 Illustration 2.35 If f(x) = x cos x, find f”(x). 12
380 If ( boldsymbol{y}=sec ^{-1}left(frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right)+sin ^{-1}left(frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right) )
then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
12
381 Differentiate the following expression
( boldsymbol{w} cdot boldsymbol{r} cdot boldsymbol{t} cdot boldsymbol{x} )
( boldsymbol{y}=csc ^{2}left(boldsymbol{x}^{2}right) )
12
382 Differentiate the given function w.r.t. ( x )
( log (log x), x>1 )
12
383 ( operatorname{Let} g(x)=lim _{n rightarrow infty} frac{x^{n} f(x)+h(x)+1}{2 x^{n}+3 x+3}, x^{1} 1 )
and ( g(1)=lim _{x rightarrow 1} frac{sin ^{2}left(pi cdot 2^{x}right)}{ln left(sec left(pi cdot 2^{x}right)right)} ) be a
continuous function at ( x=1, ) find the
value of ( 4 g(1)+2 f(1)-h(1) . ) Assume
that ( f(x) ) and ( h(x) ) are continuous at
( boldsymbol{x}=mathbf{1} )
12
384 6.
The function for
The function
In(1+ ax) – In(1-bx) is not defined
(x)=
x
-0. The value which should be assigned tofat x
that it is continuous at x=0, is
(1983 – 1 Mark)
(a) a-b
(b) a + b
(C) In a – In b
(d) none of these
12
385 If ( boldsymbol{x}^{m}+boldsymbol{y}^{m}=mathbf{1} ) such that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=-frac{boldsymbol{x}}{boldsymbol{y}} )
then what should be the value of ( m ? )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. None of the above
12
386 If ( f(x)=sqrt{1-e^{-x^{2}}}, ) then at ( x=0, f(x) )
is
A. differentiable as well as continuous
B. continuous but not differentiable
c. differentiable but not continuous
D. neither differetiable nor continuous
12
387 If ( f(x)=log left(frac{x^{2}+a b}{x(a+b)}right), ) then the
value of ( ^{prime} C^{prime} ) for which ( f^{prime}(c)=0 ) in ( [a, b] )
( mathbf{A} cdot C=pm sqrt{frac{a}{b}} )
B. ( C=pm sqrt{a b} )
( ^{c} cdot c=pm sqrt{frac{b}{a}} )
D. none of these
12
388 Find the derivatives of the following functions at the indicated points.
( boldsymbol{f}(boldsymbol{x})=sin 4 boldsymbol{x} cos mathbf{4} boldsymbol{x}, boldsymbol{f}^{prime}(boldsymbol{pi} / mathbf{3})=? )
12
389 ( boldsymbol{y}=tan ^{-1}left[frac{log left(frac{boldsymbol{e}}{boldsymbol{x}^{2}}right)}{log left(boldsymbol{e} boldsymbol{x}^{2}right)}right]+ )
( tan ^{-1}left(frac{3+2 log x}{1-6 log x}right), ) then ( frac{d^{2} y}{d x^{2}}= )
( A )
в.
( c cdot 0 )
D. –
12
390 Examine the applicability of Mean Value
Theorem for the following function. ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] ) for ( boldsymbol{x} boldsymbol{epsilon}[boldsymbol{2}, boldsymbol{2}] )
12
391 The value of ( c ) in the lagranges mean
value theorem for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}, boldsymbol{a}= )
( 1, h=frac{1}{2} ) is
A ( cdot frac{1}{3} )
в. ( sqrt{frac{19}{56}} )
c. ( sqrt{frac{19}{3}}+2 )
D. ( sqrt{frac{19}{3}}-2 )
12
392 Consider the functions defined
implicitly by the equation ( y^{3}-3 y+ )
( x=0 ) on various intervals in the real
line. If ( boldsymbol{x} epsilon(-infty,-2) cup(2, infty), ) the
equation implicitly defines a unique real valued differentiable function ( y= )
( f(x) . ) If ( x epsilon(-2,-2) ) the equation
implicitly defines a unique real valued
differentiable function ( boldsymbol{y}=boldsymbol{g}(boldsymbol{x}) )
satisfying ( boldsymbol{g}=boldsymbol{g}(mathbf{0})=mathbf{0} )
If ( f(-10 sqrt{2})=2 sqrt{2} ) then
( f^{prime prime}(-10 sqrt{2})= )
A ( cdot frac{4 sqrt{2}}{7^{3} cdot 3^{2}} )
B. ( -frac{4 sqrt{2}}{7^{3} cdot 3^{2}} )
c. ( frac{4 sqrt{2}}{7^{3} cdot 3^{3}} )
D. ( frac{4 sqrt{2}}{7 cdot 3} )
12
393 In the function ( f(x)=a x^{3}+b x^{2}+ )
( 11 x-6 ) satisfies condition of rolle’s
therorem in [1,3] and ( f^{prime}left(2+frac{1}{3}right)=0 ) then value of ( a ) and ( b ) are respectively
A. 1,-6
B. -1,6
c. -2,1
D. ( -1, frac{1}{2} )
12
394 differentiate ( e^{-2 tan ^{-1} x^{2}} ) 12
395 ff ( f(x)=e^{x} g(x) )
( boldsymbol{g}(mathbf{0})=mathbf{1}, boldsymbol{g}^{prime}(mathbf{0})=mathbf{3}, ) then ( boldsymbol{f}^{prime}(mathbf{0}) ) is
( mathbf{A} cdot mathbf{0} )
B. 4
( c cdot 3 )
D.
12
396 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}=sin boldsymbol{y} ) 12
397 Illustration 2.25 If y= 12++3]|2x* + 1], then find dy
Illustration 2.25
If y =
31|2x
+ 1), then find
12
398 Differentiate with respect to ( x ) :
( log left(cos x^{2}right) )
12
399 Find if limit of function exists as ( x )
tends to zero
( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{lr}frac{mathbf{1}}{boldsymbol{x}}-mathbf{1} & boldsymbol{x} neq mathbf{0} \ boldsymbol{e} boldsymbol{x}+mathbf{1} \ mathbf{0} & boldsymbol{x}=mathbf{0}end{array}right. )
12
400 Let ( f(x) ) be a real valued function not identically zero, such that
( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{n}right)=boldsymbol{f}(boldsymbol{x})+(boldsymbol{f}(boldsymbol{y}))^{n} quad forall boldsymbol{x}, boldsymbol{y} in )
( boldsymbol{R} )
where ( n in N(n neq 1) ) and ( f^{prime}(0) geq 0 . ) We
may get an explicit form of the function
( boldsymbol{f}(boldsymbol{x}) )
The value of ( f(5) ) is :
A . 6
B. 3
( c cdot 5 n )
D. 5
12
401 26. If y=x2 sin x, then
will be …
12
402 If ( boldsymbol{y}=tan boldsymbol{x}+cot boldsymbol{x} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
403 Differentiate the following
function with respect to ( x )
( tan h^{-1}(3 x+1) )
12
404 Let ( f, g ) and ( h ) are differentiable functions. If ( boldsymbol{f}(mathbf{0})=mathbf{1} ; boldsymbol{g}(mathbf{0})=mathbf{2} ; boldsymbol{h}(mathbf{0})= )
3 and the derivative of their pair wise
products at ( x=0 ) are ( (f g)^{prime}(0)= )
( mathbf{6} ;(boldsymbol{g} boldsymbol{h})^{prime}(mathbf{0})=mathbf{4} ) and ( (boldsymbol{h} boldsymbol{f})^{prime}(mathbf{0})=mathbf{5} ) then
compute the value of ( (f g h)^{prime}(0) )
A . 12
B . 15
c. 16
D. None of these
12
405 Examine if Mean value Theorem applies
to ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{3} boldsymbol{x}^{2}-mathbf{5} boldsymbol{x} ) in the interval
[1,2]. If it does, then find the intermediate point whose existence is asserted by theorem.
A. Mean Value theorem is applicable and intermediate points are ( c=-3.55,1.55 )
B. Mean Value theorem is not applicable
c. Mean Value theorem is applicable and intermediate points are ( c=3.55,-1.55 )
D. none of these
12
406 Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be defined by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cl}boldsymbol{alpha}+frac{sin [boldsymbol{x}]}{boldsymbol{x}} & boldsymbol{i f} boldsymbol{x}>0 \ boldsymbol{2} & boldsymbol{i f} boldsymbol{x}=mathbf{0} \ boldsymbol{beta}+left[frac{sin boldsymbol{x}-boldsymbol{x}}{boldsymbol{x}^{3}}right] & boldsymbol{i} boldsymbol{f} boldsymbol{x}<0end{array}right. )
where ( [x] ) denotes the integral part of ( y ) If ( f ) is continuous at ( x=0, ) then ( beta-alpha= )
( A )
B. 1
( c cdot 0 )
( D )
12
407 If ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{a}^{boldsymbol{x}}, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x} log _{e} boldsymbol{a}-boldsymbol{y}}{boldsymbol{x} log _{e} boldsymbol{x}} ) 12
408 If ( boldsymbol{y}=sin ^{-1}left(frac{mathbf{5} boldsymbol{x}+mathbf{1 2} sqrt{mathbf{1}-boldsymbol{x}^{2}}}{mathbf{1 3}}right), ) then
( frac{d y}{d x} ) is equal to
A. ( -frac{1}{sqrt{1-x^{2}}} )
B. ( frac{1}{sqrt{1-x^{2}}} )
c. ( frac{3}{sqrt{1-x^{2}}} )
D. ( -frac{x}{sqrt{1-x^{2}}} )
12
409 f ( boldsymbol{y}=log [boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}}], ) show that
( left(x^{2}+a^{2}right) frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}=0 )
12
410 If ( mathbf{a}+mathbf{b}+mathbf{c}=mathbf{0}, ) then the equation
( 3 a x^{2}+2 b x+c=0 ) has at least one
root in
This question has multiple correct options
A ( .(1,2) )
в. (0,1)
c. (-1,1)
D. (2,3)
12
411 If ( a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+ )
( c=0 ) then ( frac{d y}{d x}= )
A. ( -left(frac{a x+h y+g}{h x+b y+f}right) )
в. ( -left(frac{a x+h y+g}{b x+h y+f}right) )
c. ( -left(frac{h x+b y+f}{a x+h y+g}right) )
D. ( -left(frac{h x+b y+f}{h x+a y+g}right) )
12
412 If the function ( f(x)=frac{log x-1}{x-e}, ) for
( x neq e ) is continuous at ( x=e, ) then find
( boldsymbol{f}(boldsymbol{e}) )
12
413 If ( boldsymbol{y}=|cos boldsymbol{x}|+|sin boldsymbol{x}|, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}= )
( frac{2 pi}{3} ) is
A ( cdot frac{1}{2}(sqrt{3}+1) )
B. ( 2(sqrt{3}-1) )
c. ( frac{1}{2}(sqrt{3}-1) )
D. none of these
12
414 7.
„Osx<1
Let f(x) =
(1983 – 2 Marks)
*+2,15×52
Discuss the continuity of f,f' and f" on [0,2].
12
415 If ( f(x) ) is differentiable everywhere, then
( |boldsymbol{f}(boldsymbol{x})|^{2} ) is differentiable everywhere.
Enter ( 1 text { if true or } 0 text { otherwise }) )
12
416 Dfferentiate w.r.t ( x ) :
( tan ^{2} 7 x )
12
417 Find whether the following function is
differentiable at ( x=1,2: ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}boldsymbol{x}, & boldsymbol{x} leq mathbf{1} \ boldsymbol{2}-boldsymbol{x}, & boldsymbol{1} leq boldsymbol{x} leq mathbf{2} \ -boldsymbol{2}+boldsymbol{3} boldsymbol{x}-boldsymbol{x}^{2}, & boldsymbol{x}>boldsymbol{2}end{array}right. )
12
418 A derivable function ( boldsymbol{f}: boldsymbol{R}^{+} rightarrow boldsymbol{R} )
satisfies the condition ( f(x)-f(y) geq )
( ln frac{x}{y}+x-y ; forall x, y in R^{+} . ) If ( g ) denotes
the derivative of ( f ) then the value of the ( operatorname{sum} sum_{n=1}^{100} gleft(frac{1}{n}right) ) is ( 1030 k . ) Find the
value of ( k )
12
419 4.
st integer less than or equal to x. If
(1986-2 Marks)
Let [x] denote the greatest integer less than
$x)=[r sin tx], then f(x) is
a) continuous at r=0 (b) continuous in
(c) differentiable at x=1 (d) differentiable in (1,1)
(e) none of these
12
420 Verify Rolle’s Theorem for the function
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}-boldsymbol{8}, boldsymbol{x} in[-boldsymbol{4}, boldsymbol{2}] )
12
421 Differentiate the following function with
respect to ( x ) ( left(1+x^{2}right) cos x )
12
422 If ( left(1+x^{2}right) y_{1}=x(1-y), y(0)=frac{4}{3} )
then ( y(sqrt{8})-frac{1}{9} ) is
12
423 ( f(x)=left{begin{array}{lr}frac{2^{x+2}-16}{4^{x}-16}, text { if } x neq 2 \ k, & text { if } x=2end{array}right. )
continuous at ( x=2, ) find ( k )
12
424 Differentiate the following function with
respect to ( x )
( x^{2} e^{x} ) log ( x )
A ( cdot x e^{x}(x log x+2 log x) )
B. ( x e^{x}(1+2 log x) )
C ( cdot x e^{x}(1+x log x) )
D. ( x e^{x}(1+x log x+2 log x) )
12
425 Suppose that ( boldsymbol{f} ) is a differentiable
function with the property that
( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})+boldsymbol{x} boldsymbol{y} ) and
( lim _{h rightarrow 0} frac{1}{h} f(h)=3 ) then
A. ( f ) is a linear function
B ( cdot f(x)=3 x+x^{2} )
c. ( f(x)=3 x+frac{x^{2}}{2} )
D. None of these
12
426 Let ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}-mathbf{1}}+ )
( sqrt{x+24-10 sqrt{x-1}} ; 1<x<26 ) be
real valued function. Then ( f^{prime}(x) ) for ( 1< )
( boldsymbol{x}<26 ) is
( A cdot 0 )
в. ( frac{1}{sqrt{x-1}} )
c. ( 2 sqrt{x-1}-5 )
D. none of these
12
427 Illustration 2.30
Find the derivative of y = sin(x+ – 4).
a
2
1
1
12
428 If ( f(x)=log _{x^{2}}left(log _{e} xright), ) then ( f^{prime}(x) ) at
( boldsymbol{x}=boldsymbol{e} ) is
A . 1
B.
c. ( frac{1}{2 e} )
D.
12
429 Let ( f(x) ) be defined on ( [0, pi] ) by ( f(x)= ) ( left{begin{array}{ll}x+a sqrt{2} sin x & , 0 leq x leq pi / 4 \ 2 x cot x+b & , frac{pi}{4}<x leq frac{pi}{2} . text { If } f \ a cos 2 x-b sin x & , frac{pi}{2}<x<piend{array}right. )
is continuous on ( [0, pi] ) then
This question has multiple correct options
A ( a=frac{pi}{6} )
в. ( b=-frac{pi}{12} )
c. ( a=frac{pi}{6} ) and ( b=-frac{pi}{12} )
D・ ( a=frac{pi}{3} ) and ( b=-frac{pi}{12} )
12
430 Testify the mean value theorem in the interval ( [boldsymbol{a}, boldsymbol{b}], boldsymbol{f}(boldsymbol{x})=frac{1}{4 boldsymbol{x}-1} ) where
( a=1 ) and ( b=4 )
12
431 Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a
point ( ^{prime} c^{prime} ) in the indicated interval as
stated by the Lagrange’s mean value theorem:
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{2} boldsymbol{x}^{2}-boldsymbol{x}+boldsymbol{3} ) on ( [mathbf{0}, mathbf{1}] )
12
432 Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) is a function which is
defined by ( f(x)=max left{x, x^{3}right} . ) The set of all points on which ( f(x) ) is not differentiable
is
A ( cdot{-1,1} )
the
B. {-1,0}
( c cdot(0,1) )
D. {-1,0,1}
12
433 The value of ( c ) in Rolle’s theorem for the
function ( boldsymbol{f}(boldsymbol{x})=cos frac{boldsymbol{x}}{2} ) on ( [boldsymbol{pi}, boldsymbol{3} boldsymbol{pi}] ) is
( A cdot 0 )
B. ( 2 pi )
c. ( frac{pi}{2} )
D. ( frac{3 pi}{2} )
12
434 The function
( f(x)=frac{cos x-sin x}{cos 2 x} ) is not defined at
( x=frac{pi}{4} . ) The value of ( fleft(frac{pi}{4}right) ) so that ( f(x) )
is continuous everywhere, is
( A )
B.
( c cdot sqrt{2} )
D. ( frac{1}{sqrt{2}} )
12
435 Find the derivative of ( y=frac{2 x}{1-x^{2}} ) 12
436 Find the value of ( x ) for which the
derivative of the function ( f(x)= ) ( 20 cos 3 x+12 cos 5 x-15 cos 4 x )
is equal to zero?
12
437 If ( f(x)=x^{2}-x+5, x>frac{1}{2}, ) and ( g(x) ) is
its inverse function, then ( g^{prime}(7) ) equals:
A. ( -frac{1}{3} )
в. ( frac{1}{13} )
c. ( frac{1}{3} )
D. ( -frac{1}{13} )
12
438 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of the following implicit
functions ( : boldsymbol{y}^{3}-boldsymbol{3} boldsymbol{y}^{2} boldsymbol{x}=boldsymbol{x}^{3}+boldsymbol{3} boldsymbol{x}^{2} boldsymbol{y} )
12
439 Draw the graph of function ( boldsymbol{f}(boldsymbol{x})= )
( |x| / x . ) Is ( f(x) ) defined at ( x=0 ? ) Does the
limit of ( f(x) ) exist when ( x rightarrow 0 ? )
12
440 Evaluate ( : int(tan x-cot x)^{2} d x ) 12
441 If ( f^{1}(x)=sin (log x) ) and ( y= )
( fleft(frac{2 x+3}{3-2 x}right), ) then ( frac{d y}{d x} ) equals
A. ( frac{12}{(3-2 x)^{2}} )
B. ( sin left[log left(frac{2 x+3}{3-2 x}right)right] )
c. ( frac{12}{(3-2 x)^{2}} sin left[log left(frac{2 x+3}{3-2 x}right)right] )
D. ( frac{12}{(3-2 x)^{2}} cos left[log left(frac{2 x+3}{3-2 x}right)right] )
12
442 The value of ( c ) in Lagrange’s theorem for
the function ( f(x)=|x| ) in the interval
[-1,1] is
A. 0
B. ( 1 / 2 )
c. ( -1 / 2 )
D. non-existent in the interval
12
443 If ( s=sqrt{t^{2}+1}, ) then ( frac{d^{2} s}{d t^{2}} ) is equal to
A ( cdot frac{1}{s} )
в. ( frac{1}{s^{2}} )
c. ( frac{1}{s^{3}} )
D. ( frac{1}{s^{4}} )
12
444 ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}3 x-8 & text { if } x leq 5 \ 2 k & text { if } x>5end{array}right. )
continuous, find ( k )
( A cdot frac{2}{7} )
B. 3 ( overline{7} )
( c cdot frac{4}{7} )
D.
12
445 The function ( y=frac{2-x^{2}}{x^{4}} ) takes on
equal values at the end-points of the interval ( [-1,1] . ) Is Rolle’s theorem valid in
this interval?
12
446 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( sin ^{-1}left{frac{sin x+cos x}{sqrt{2}}right}, frac{pi}{4}<x<frac{3 pi}{4} )
12
447 [
begin{array}{rlr}
text { If } boldsymbol{f}(boldsymbol{x}) & =frac{boldsymbol{x}^{2}-mathbf{9}}{boldsymbol{x}-mathbf{3}}+boldsymbol{alpha}, text { for } boldsymbol{x}>mathbf{3} \
& =mathbf{5}, & text { for } boldsymbol{x}=mathbf{3} \
& =mathbf{2} boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}+boldsymbol{beta}, & text { for } boldsymbol{x}<mathbf{3}
end{array}
]
is continuous at ( x=3, ) find ( alpha ) and ( beta )
12
448 If ( x^{4}+7 x^{2} y^{2}+9 y^{4}=24 x y^{3}, ) then
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A ( cdot frac{x}{y} )
B. ( underline{y} )
c. ( -frac{x}{y} )
D. ( -frac{y}{x} )
12
449 Differentiate ( boldsymbol{y}=sin ^{-1}left(frac{mathbf{2}^{x+1} mathbf{3}^{x}}{1+mathbf{3 6}^{x}}right) ) 12
450 If ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) forall boldsymbol{x}, boldsymbol{y} ) and
( boldsymbol{f}(mathbf{5})=mathbf{2}, boldsymbol{f}^{prime}(mathbf{0})=mathbf{3} ; ) then ( boldsymbol{f}^{prime}(mathbf{5}) ) is equal
to-
A .2
B. 4
( c cdot 6 )
D. 8
12
451 ( operatorname{Let} f(x)=sin frac{1}{x}, x neq 0 . ) Then ( f(x) ) can
be continuous at ( 4 x=0 )
A. If ( f(0)=1 )
B. If ( f(0)=0 )
c. If ( f(0)=-1 )
D. For no definite value of ( f(0) )
12
452 If ( a x^{2}+2 x y+b y^{2}=0 ) then find ( frac{d y}{d x} ) 12
453 Find the value of ( k ) is continuous at ( x= )
where
2
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}frac{boldsymbol{k} cos boldsymbol{x}}{boldsymbol{pi}-mathbf{2} boldsymbol{x}}, text { if } boldsymbol{x} neq frac{boldsymbol{pi}}{mathbf{2}} \ boldsymbol{3}, quad text { if } boldsymbol{x}=frac{boldsymbol{pi}}{2}end{array}right. )
12
454 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{x} sin boldsymbol{y}+boldsymbol{y} sin boldsymbol{x}=mathbf{0} ) 12
455 Sketch the graph ( y=|x+3| . ) Evaluate ( int_{-6}^{0}|x+3| d x . ) What does this integral represent on the graph? 12
456 Differentiate the following function with
respect to ( x )
( boldsymbol{x}^{-4}left(boldsymbol{3}-boldsymbol{4} boldsymbol{x}^{-boldsymbol{5}}right) )
A . ( -12 x^{-6}+36 x^{-10} )
B. ( -12 x^{-5}+36 x^{-11} )
c. ( -12 x^{-5}+36 x^{-10} )
D. ( -12 x^{5}+36 x^{-10} )
12
457 If ( boldsymbol{f}(boldsymbol{x})=sqrt{1-sin 2 boldsymbol{x}}, ) then ( boldsymbol{f}^{prime}(boldsymbol{x}) ) is
equal to:
This question has multiple correct options
A. ( -(cos x+sin x) ), for ( x in(pi / 4, pi / 2) )
B. ( (cos x+sin x) ), for ( x in(0, pi / 4) )
c. ( -(cos x+sin x) ), for ( x in(0, pi / 4) )
D. None of these
12
458 Consider the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2 boldsymbol{x}-mathbf{1}, & boldsymbol{0} leq boldsymbol{x}<mathbf{2} \ boldsymbol{x}+boldsymbol{a} & boldsymbol{2} leq boldsymbol{x} leq mathbf{4} \ boldsymbol{3} boldsymbol{x}+boldsymbol{b} & boldsymbol{4}<boldsymbol{x} leq mathbf{6}end{array}right. )
(i) Find ( f(2-) ) and ( f(2+) )
(ii) Find ( a ) if ( f ) is continuous at ( x=2 )
(iii) Find ( b ) if ( f ) is continuous on [0,6]
12
459 If ( y=frac{x+c}{1+x^{2}}, ) then the value of ( x y )
where ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=mathbf{0} ) is
A ( cdot frac{1}{2} )
B. ( frac{3}{4} )
( c cdot frac{5}{4} )
D. None of these
12
460 By Rolles theorem for ( f(x)=(x- )
( a)^{m}(x-b)^{n} ) on ( [a, b] ; m, n ) being
positive integer. Find the value of ( c ) which lies between ( a )
& b.
A ( cdot c=frac{m b+n a}{m+n} )
в. ( c=frac{m b-n a}{m+n} )
c. ( _{c}=frac{n b+m a}{m+n} )
D. ( c=frac{n b-m a}{m+n} )
12
461 The function ( f ) is defined as ( f(x)= ) ( left{begin{array}{ll}x^{2}+a x+b, & text { if } 0 leq x<2 \ 3 x+2, & text { if } 2 leq x leq 4, text { If } f text { is } \ 2 a x+5 b, & text { if } 4<x leq 8end{array}right. )
continuous in [0,8] find the values of ( a )
and ( b )
12
462 Find the derivative of ( boldsymbol{y}=(boldsymbol{x}+ )
1) ( (x+2)^{2} )
12
463 If ( sqrt{1-x^{2}}+sqrt{1-y^{2}}=a(x-y) )
prove that ( frac{d y}{d x}=sqrt{frac{1-y^{2}}{1-x^{2}}} )
12
464 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: sin boldsymbol{x}+mathbf{7} boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{5} ) 12
465 If ( y=sqrt{x log _{e} x, text { then } frac{d y}{d x} text { at } x=e text { is }} )
( A cdot frac{1}{e} )
B. ( frac{1}{sqrt{e}} )
( c cdot sqrt{e} )
D. None of these
12
466 Verify that ( y=a e^{-x} ) is a solution of ( frac{d^{2} y}{d x^{2}}=frac{1}{y}left(frac{d y}{d x}right)^{2} ) 12
467 Assertion
Statement ( -1 f(x)=|x| cos x ) is not
differentiable at ( mathbf{x}=mathbf{0} )
Reason
Statement – 2 Every absolute value functions are not differentiable.
A. Statement-1 is True, Statement-2 is True
Statement- 2 is a correct explanation for
Statement-1.
B. Statement-1 is True, Statement-2 is True
Statement-2 is NOT a correct explanation
for Statement-
c. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True
12
468 Let ( boldsymbol{y}=sin ^{-1} boldsymbol{x}, ) then find ( left(mathbf{1}-boldsymbol{x}^{2}right) boldsymbol{y}_{2} )
( boldsymbol{x} boldsymbol{y}_{1} )
Where ( y_{1} ) and ( y_{2} ) denote the first and
second order derivatives respectively.
( mathbf{A} cdot mathbf{1} )
в. – 1
c. 0
D.
12
469 Differentiate with respect to ( x ) :
( frac{x^{2}+2}{sqrt{cos x}} )
12
470 If ( y=e^{sqrt{x}}+e^{-sqrt{x}} ) then ( frac{d y}{d x} ) equals
This question has multiple correct options
A ( cdot frac{e^{sqrt{x}}-e^{-sqrt{x}}}{2 sqrt{x}} )
( frac{e^{sqrt{x}}-e^{-sqrt{x}}}{2 x} )
c. ( frac{1}{2 sqrt{x}} sqrt{y^{2}-4} )
D. ( frac{1}{2 sqrt{x}} sqrt{y^{2}+4} )
12
471 If ( f(x)=|cos 2 x| ) then ( f^{prime}left(frac{pi}{4}+0right) ) is
equal to
A .2
B.
c. -2
D. none of these
12
472 If ( f ) be a continuous function on ( [mathbf{0}, mathbf{1}] )
differentiable in (0,1) such that
( f(1)=0, ) then there exists some ( c in )
(0,1) such that
A ( cdot c f^{prime}(c)-f(c)=0 )
B. ( f^{prime}(c)+c f(c)=0 )
c. ( f^{prime}(c)-c f(c)=0 )
D. ( c f^{prime}(c)+f(c)=0 )
12
473 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{2} boldsymbol{y}^{2}+boldsymbol{6} boldsymbol{x}=mathbf{5} )
12
474 Find the slope of tangent to the curve ( y=3 x^{2}-6 ) at the point on it whose ( x )
coordinate is 2
12
475 If ( boldsymbol{u}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{h} boldsymbol{x} boldsymbol{y}+boldsymbol{b} boldsymbol{y}^{2} )
( operatorname{then} x frac{partial^{2} u}{partial x partial y}+y frac{partial^{2} u}{partial y^{2}}=? )
( mathbf{A} cdot 2(h x+b y) )
B. ( 2(h x-b y) )
c. ( 2(b x+h y) )
D. ( 2(b x-h y) )
12
476 Differentiate ( boldsymbol{y}=log (log sqrt{boldsymbol{x}}) ) 12
477 Explain Mean Value Theorem 12
478 15.
A function f:R → R satisfies the equation f (x + y)=f(x)f)
for all x, y in Randf(x) #0 for anyx in R. Let the function be
differentiable at x=0 and f'()=2. Show that f'(x)=2f(x) for
all x in R. Hence, determine f(x). (1990 – 4 Marks)
12
479 ( mathbf{f} boldsymbol{y}=tan ^{-1}left[frac{sqrt{mathbf{1}+boldsymbol{x}^{2}}-mathbf{1}}{boldsymbol{x}}right], ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ? ) 12
480 If ( f(x)=x cdotleft(frac{a^{1 / x}-a^{-1 / x}}{a^{1 / x}+a^{-1 / x}}right), x neq )
( mathbf{0}(boldsymbol{a}>mathbf{0},) boldsymbol{f}(mathbf{0})=mathbf{0} ) then
A . fis differentiable at ( x=0 )
B. fis not differentiable at ( x=0 )
c. ( f ) is not continuous at ( x=0 )
D. None of these
12
481 26. Let [x] be the greatest integer less than or equals to x. Then,
at which of the following point(s) the function
f(x)= x cos(To(x+[x])) is discontinuous? (JEE Adv. 2017)
(a) x=-1
(b) x=0
(c) X=1
(d) x=2
12
482 Differentiate with respect to ( x ) :
( frac{2^{x} cos x}{left(x^{2}+3right)^{2}} )
12
483 A function ( f ) is defined as follows:
( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}1 & text { for }-infty< \ 1+sin x & text { for } 0 leq x<frac{pi}{2} \ 2+left(x-frac{pi}{2}right)^{2} & text { for } frac{pi}{2} leq x<+inftyend{array}right. )
Discuss the continunity and
differentiability at ( boldsymbol{x}=mathbf{0} & boldsymbol{x}=boldsymbol{pi} / mathbf{2} )
This question has multiple correct options
A. continuous but not differentiable at ( x=0 )
B. differentiable and continuous at ( x=pi / 2 )
c. neither continuous but nor differentiable at ( x=0 )
D. continuous but not differentiable at ( x=pi / 2 )
12
484 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{cot ^{-1} frac{sqrt{1+boldsymbol{x}}-sqrt{1-boldsymbol{x}}}{sqrt{1+boldsymbol{x}}+sqrt{1-boldsymbol{x}}}right}= )
( A cdot frac{1}{sqrt{1-x^{2}}} )
B ( cdot frac{-1}{2 sqrt{1-x^{2}}} )
( mathbf{C} cdot frac{1}{1+x^{2}} )
D. ( frac{-1}{2left(1+x^{2}right)} )
12
485 A balloon which always remains spherical, has a variable diameter ( frac{3}{2}(2 x+3) . ) The rate of change of volume with respect to ( x ) will be
A ( cdot frac{27 pi}{8}(2 x-3)^{2} )
в. ( frac{27 pi}{8}(2 x+3)^{2} )
c. ( frac{27 pi}{8}(3 x-2)^{2} )
D. ( frac{8}{27 pi}(2 x+3)^{2} )
12
486 Let ( mathbf{g}(mathbf{x})=log (mathbf{f}(mathbf{x})) ) where ( mathbf{f}(mathbf{x}) ) is a
twice differentiable positive function on ( (0, infty) ) such that ( f(x+1)=x f(x) )
Then, for ( mathbf{N}=mathbf{1}, mathbf{2}, mathbf{3}, dots mathbf{g}^{prime prime}left(mathbf{N}+frac{mathbf{1}}{mathbf{2}}right)- )
( mathrm{g}^{prime prime}left(frac{1}{2}right)= )
A ( cdot-4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}-1)^{2}}right) )
B. ( 4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}-1)^{2}}right) )
c. ( -4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}+1)^{2}}right) )
D ( 4left(1+frac{1}{9}+frac{1}{25}+ldots+frac{1}{(2 mathrm{N}+1)^{2}}right) )
12
487 Find ( frac{d y}{d x}, ) when ( y=x^{x}-2^{sin x} ) 12
488 If ( x^{2}-2 x^{2} y^{2}+5 x+y-5=0 ) and
( boldsymbol{y}(1)=1, ) then
This question has multiple correct options
A ( cdot y^{prime}(1)=1 )
в. ( y^{prime prime}(1)=-frac{4}{3} )
c. ( quad y^{prime prime}(1)=-frac{22}{3} )
D. ( y^{prime}(1)=frac{2}{3} )
12
489 Let ( boldsymbol{f}(boldsymbol{x} boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y}) ) for all ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} )
f ( f^{prime}(1)=2 ) and ( f(4)=4, ) then ( f^{prime}(4) )
equal to
( mathbf{A} cdot mathbf{4} )
B.
c. ( frac{1}{2} )
D. 8
12
490 If ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{1+boldsymbol{x}^{2}+boldsymbol{x}^{4}}{1+boldsymbol{x}+boldsymbol{x}^{2}}right)=boldsymbol{a} boldsymbol{x}+ )
( b, ) then ( (a, b)= )
A. (-1,2)
B. (-2,1)
c. (2,-1)
D. (1,2)
12
491 If the graphs of ( y=f(x) ) and ( y=g(x) )
intersect in coincident points the ( lambda ) can take values:
This question has multiple correct options
A . 3
B. 1
( c cdot-1 )
D.
12
492 If ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}+mathbf{1}|(|boldsymbol{x}|+|boldsymbol{x}-mathbf{1}|) ) then at
what points the function is/are not differentiable at in the interval [-2,2] This question has multiple correct options
A . -1
B.
c. 1
D. ( 1 / 2 )
12
493 Differentiate: ( boldsymbol{x}^{mathbf{1 0 0}}+sin boldsymbol{x}-mathbf{1} )
A ( cdot 100 x^{99}-cos x )
( x )
B. ( 100 x^{99}+cos x )
c. ( x^{99}+cos x )
D. ( 100 x^{99}+sin x )
12
494 If ( y=frac{x}{|n| c x mid} ) (where ( c ) is an arbitrary
constant) is the general solution of the differential equation ( frac{d y}{d x}=frac{y}{x}+phileft(frac{x}{y}right) )
then the function ( phileft(frac{x}{y}right) )
A ( cdot frac{x^{2}}{y^{2}} )
в. ( -frac{x^{2}}{y^{2}} )
c. ( frac{y^{2}}{x^{2}} )
D. ( -frac{y^{2}}{x^{2}} )
12
495 Solve ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin boldsymbol{theta}), boldsymbol{y}=boldsymbol{a}(mathbf{1}+cos boldsymbol{theta}) )
find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ? )
12
496 If ( mathbf{y}=mathbf{b} cos log left(frac{boldsymbol{x}}{boldsymbol{n}}right)^{boldsymbol{n}}, boldsymbol{t h e n} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A ( cdot frac{-n b sin log (x)^{n}}{x} )
B. ( n b sin log left(frac{x}{n}right)^{n} )
c. ( _{-n b sin log }left(frac{x}{n}right)^{n} )
D. None of these
12
497 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left(frac{cos boldsymbol{x}}{mathbf{1}+sin boldsymbol{x}}right) )
A. ( -frac{1}{2} )
B. ( -frac{1}{4} )
( c cdot-frac{1}{8} )
D. 3
12
498 Let ( boldsymbol{f} ) be differential for all ( boldsymbol{x} ). If ( boldsymbol{f}(mathbf{1})= )
-2 and ( f^{prime}(x) geq 2 ) for ( x epsilon[1,6], ) then ( ? )
A. ( f(6)=5 )
В. ( f(6)<5 )
C. ( f(6)<8 )
D. ( f(6) geq 5 )
is ( 5(6) geq 5 )
12
499 Value of ( c ) of Lagranges mean theorem for
[
boldsymbol{f}(boldsymbol{x})=mathbf{2}+boldsymbol{x}^{mathbf{3}} text { if } boldsymbol{x} leq mathbf{1}
]
( =3 x ) if ( x>1 ) on [-1,2] is
( A cdot pm frac{sqrt{5}}{3} )
B. ( pm frac{sqrt{3}}{2} )
( c cdot pm frac{sqrt{2}}{5} )
( D cdot pm frac{3}{sqrt{5}} )
12
500 Find the value of ( k ) so that
( boldsymbol{f}(boldsymbol{x})left{begin{array}{ll}boldsymbol{k} boldsymbol{x}+mathbf{1} & boldsymbol{i} boldsymbol{f} boldsymbol{x} leq boldsymbol{pi} \ cos boldsymbol{x} & boldsymbol{i} boldsymbol{f} boldsymbol{x}>piend{array}right. )
continuous at ( boldsymbol{x}=boldsymbol{pi} )
12
501 If ( f(x) ) is continuous and ( fleft(frac{9}{2}right)=frac{2}{9} ) ( operatorname{then} lim _{x rightarrow 0} fleft(frac{1-cos 3 x}{x^{2}}right) ) is equal to 12
502 If the p.d.f of a continuous random variable ( boldsymbol{x} ) is ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}k x^{2}(1-x) & , 0<x<1 \ 0 & , text { otherwise }end{array}right. )
then value of ( k ) is
A . 12
B. 10
c. -12
D.
12
503 Find the derivative of the following functions (it is to be understood that
( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers) ( sin (x+a) )
12
504 Let ( mathbf{f}=left{begin{array}{ll}mathbf{a x}^{2}+mathbf{1} & text { for } mathbf{x}>mathbf{1} \ mathbf{x}+mathbf{a} & text { for } mathbf{x} leq mathbf{1}end{array} text { then } mathbf{f}right. )
is derivable at ( x=1 ) if
( mathbf{A} cdot mathbf{a}=0 )
B. ( a=frac{1}{2} )
( mathbf{c} cdot mathbf{a}=1 )
( mathbf{D} cdot mathbf{a}=mathbf{2} )
12
505 State Rolle’s theorem. 12
506 Let ( y=2^{x}+x^{2}+2 ) then find ( frac{d y}{d x} ) 12
507 Differentiate the following w.r.t ( x: frac{e^{x}}{sin x} ) 12
508 Say true or false.
Derivative of ( x^{n} ) is ( n x^{n-1} )
A. True
B. False
12
509 Is Rolle’s theorem valid for the function
( boldsymbol{y}=boldsymbol{x}^{3}+boldsymbol{4} boldsymbol{x}^{2}-boldsymbol{7} boldsymbol{x}-mathbf{1 0} ) in the
interval [-1,2]
12
510 Prove cos is continuous on R. 12
511 If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}+boldsymbol{e}^{-boldsymbol{x}}+log boldsymbol{x}^{2}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
512 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} cot left(frac{x}{2}right)= ) 12
513 f ( boldsymbol{y}=tan ^{-1} frac{boldsymbol{x}}{mathbf{1}+sqrt{mathbf{1}-boldsymbol{x}^{2}}}+ )
( sin left(2 tan ^{-1} sqrt{frac{1-x}{1+x}}right), ) then find ( frac{d y}{d x} ) for
( boldsymbol{x} in(-1,1) )
12
514 ( left{begin{array}{cl}text { Evaluate } lim _{x rightarrow 2^{+}} f(x), text { where } f(x)= \ (x-[x], quad x2end{array}right. ) 12
515 If ( 5 f(x)+3 fleft(frac{1}{x}right)=x+2 ) and ( y= )
( boldsymbol{x} boldsymbol{f}(boldsymbol{x}) )
( operatorname{then} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{n}} ) at ( boldsymbol{x}=mathbf{1} )
A . 14
в. ( frac{7}{8} )
c. 1
D.
12
516 ( f(x)=frac{a sin x-b x+c x^{2}+x^{3}}{2 x^{2} ell n(1+x)-2 x^{3}+x^{4}} )
when ( x neq 0 ) and ( f(x) ) is continuous at
( boldsymbol{x}=mathbf{0}, ) find value of ( mathbf{2 0 0} times boldsymbol{f}(mathbf{0}) )
12
517 Differentiate the functions with respect
to ( x )
( sec (tan (sqrt{x})) )
12
518 Find ( frac{d y}{d x}=sin ^{-1} x ) 12
519 Discuss the continuity of the function defined by ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{5}| ) 12
520 Differentiate the following functions
with respect to ( x ) ( sin ^{-1}left{frac{x+sqrt{1-x^{2}}}{sqrt{2}}right},-1<x<1 )
12
521 Answer the following question in one
word or one sentence or as per exact
requirement of the question. If ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}|+|boldsymbol{x}-mathbf{1}|, ) write the value of
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(boldsymbol{f}(boldsymbol{x})) )
12
522 If ( sin ^{-1} x+sin ^{-1} y=frac{pi}{2}, ) then ( frac{d y}{d x} ) is
equal to
A ( cdot frac{x}{y} )
в. ( -frac{x}{y} )
c. ( frac{y}{x} )
D. ( -frac{y}{x} )
12
523 Find derivative of ( (boldsymbol{a} boldsymbol{x}+boldsymbol{b})^{n}(boldsymbol{c} boldsymbol{x}+boldsymbol{d})^{boldsymbol{n}} ) 12
524 Differentiate the following with respect
to ( x )
( cos ^{-1} 2 x sqrt{1-x^{2}}, frac{1}{sqrt{2}}<x<1 )
12
525 For every twice differentiable function f:R
2,2W
(JEE Adv. 2018)
(0) +(f'(o))2 = 85, which of the following statements)
is (are) TRUE?
a) There exist r,seR, wherer00
(d)
There exists a € (-4,4) such that f(a)+f”(a) = 0
and f'(a)+0
12
526 Let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}]+[-boldsymbol{x}] . ) Then for any
integer ( n ) and ( k in R-I )
This question has multiple correct options
A ( cdot lim _{x rightarrow n} f(x) ) exists
B. ( lim _{x rightarrow k} f(x) ) exists
c. f is continuous at ( x=n )
D. f is continuous at ( x=k )
12
527 The function
( boldsymbol{f}(boldsymbol{x})left{begin{array}{l}frac{sin sqrt[3]{boldsymbol{x}} log (1+boldsymbol{3} boldsymbol{x})}{left(tan ^{-1} sqrt{boldsymbol{x}}right)^{2}left(boldsymbol{e}^{boldsymbol{5}} sqrt[3]{boldsymbol{x}}-mathbf{1}right)} quad, boldsymbol{x} neq mathbf{0} \ boldsymbol{a}, quad boldsymbol{x}=mathbf{0}end{array}right. )
is continuous at ( boldsymbol{x}=mathbf{0}, ) if
( mathbf{A} cdot a=0 )
B. ( a=frac{5}{3} )
( mathbf{c} cdot a=2 )
D. ( a=frac{3}{5} )
12
528 ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) and ( boldsymbol{f}(boldsymbol{x})= )
( frac{boldsymbol{x}left(boldsymbol{x}^{4}+mathbf{1}right)(boldsymbol{x}+mathbf{1})+boldsymbol{x}^{4}+mathbf{2}}{boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1}}, operatorname{then} boldsymbol{f}(boldsymbol{x}) ) is
A. one-one ito
B. many-one onto
c. one-one onto
D. many-one into
12
529 If ( y=sqrt{frac{1-x}{1+x}} ) then ( frac{d y}{d x} ) equals-
A ( cdot frac{y}{1-x^{2}} )
в. ( frac{y}{x^{2}-1} )
c. ( frac{y}{1+x^{2}} )
D. ( frac{y}{y^{2}-1} )
12
530 14. The function f (x) = (x2 – 1) x2-3x+2 +cos ( x is
NOT differentiable at
(1999-2 Marks)
(a) -1 (6) 0 (c) 1 (d) 2
12
531 Differentiate
( frac{x^{4}}{4}-frac{x^{-3}}{3}-frac{2}{x}+C )
12
532 The left hand derivative of ( f(x)= )
( [x] sin pi x ) at ( x=k, k ) is an integer, is
( mathbf{A} cdot(-1)^{k}(k-1) pi )
B cdot ( (-1)^{k-1}(k-1) pi )
( mathbf{c} cdot(-1)^{k} k pi )
D. ( (-1)^{k-1} k pi )
12
533 Let ( S ) be the set of all functions ( f: )
( [0,1] rightarrow R, ) which are continuous on
[0,1] and differentiable on ( (0,1) . ) Then
for every ( f ) in ( S, ) there exists ( a c in(0,1) )
depending on ( f, ) such that
( ^{text {A } cdot frac{f(1)-f(c)}{1-c}=f^{prime}(c)} )
B cdot ( |f(c)-f(1)|<(1-c)left|f^{prime}(c)right| )
c. ( |f(c)+f(1)|<(1+c)left|f^{prime}(c)right| )
D ( cdot|f(c)-f(1)|<mid f^{prime}(c) )
12
534 If ( sin y=x sin (a+y), ) then show that
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{sin ^{2}(boldsymbol{a}+boldsymbol{y})}{sin boldsymbol{a}} )
12
535 ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2})(boldsymbol{x}-boldsymbol{3}), boldsymbol{x} in[mathbf{0}, boldsymbol{4}] )
find ( ^{prime} c^{prime} ) if ( L M V T ) can be applied.
12
536 If ( boldsymbol{y}=tan ^{2}left(log boldsymbol{x}^{3}right), ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
537 Show that ( f, ) given by ( f(x)= ) ( frac{boldsymbol{x}-|boldsymbol{x}|}{boldsymbol{x}}(boldsymbol{x} neq mathbf{0}), ) is continuous on ( mathrm{R}[mathbf{0}] ) 12
538 Find the derivative of ( sqrt{tan x} ) with respect to ( x ) using the first principle.
A ( cdot frac{sec ^{2} x}{2 sqrt{tan x}} )
в. ( frac{s e c x}{2 sqrt{tan x}} )
( ^{mathrm{c}} cdot frac{sec ^{2} x}{sqrt{tan x}} )
D. ( frac{sec ^{2} x}{2 sqrt{tan } x} )
12
539 ( operatorname{Let} f^{prime}(x)=e^{x^{2}} ) and ( f(0)=10 . ) If ( A< )
( f(1)<B ) can be concluded from the
mean value theorm, then the largest value of ( (boldsymbol{A}-boldsymbol{B}) ) equals
( A )
B. ( 1-e )
( mathbf{c} cdot e-1 )
D. ( 1+e )
12
540 If ( x^{2}+y^{2}=t-frac{1}{t} ) and ( x^{4}+y^{4}=t^{2}+ )
( frac{1}{t^{2}}, ) then prove that ( frac{d y}{d x}=frac{1}{x^{3} y} )
12
541 Illustration 2.20
If y= –
= (x)-1/2, then find dyldx.
12
542 If ( mathbf{y}=mathbf{c e}^{x /(x-a)}, ) then ( frac{mathbf{d y}}{mathbf{d x}} ) equals
( A cdot a(x-a) )
B. ( -frac{text { ay }}{(x-a)^{2}} )
( c cdot a^{2}(x-a)^{2} )
D.
12
543 Let ( (a-b cos y)(a+b cos x)=a^{2}-b^{2} )
and ( frac{d y}{d x}=frac{sin x f(y))}{(a+b cos x)^{2}} cdot ) If ( a^{2}-b^{2}= )
( 192, ) then ( f(pi / 2) )
12
544 If ( f(x)=sec (3 x), ) then ( f^{prime}left(frac{3 pi}{4}right)= )
A. ( -3 sqrt{2} )
B. ( -frac{3 sqrt{2}}{2} )
( c cdot frac{3}{2} )
D. ( frac{3 sqrt{2}}{2} )
E. ( 3 sqrt{2} )
12
545 If ( f(x)=left|x^{2}-4 x+3right|, ) then ( f^{prime}(x) ) is
A. ( 2 x-4 ) for ( 1<x<3 )
B. ( 4-2 x ) for ( 1<x<3 )
c. ( 2 x-4 ) for ( 1 leq x leq 3 )
D. ( 4-2 x )
12
546 If ( y=log _{10}(sin x), ) then ( frac{d y}{d x} ) equals to:
A ( cdot sin x log _{10} e )
B. ( cos x log _{10} e )
C. ( cot x log _{10} e )
D. ( cot x )
12
547 Show that ( boldsymbol{f}(boldsymbol{x})= )
( begin{array}{ll}frac{sin 3 x}{tan 2 x}, & text { if } x0end{array} )
12
548 Find ( frac{d y}{d x}, x=aleft(cos t+log tan frac{t}{2}right), y= )
( a sin t )
12
549 The derivative of the function ( f(x)= ) ( sqrt{x^{2}-2 x+1} ) in the interval [0,2] is
A . -1
B.
( c cdot 0 )
D. does not exist
12
550 Differentiate with respect to ( x e^{x} x^{5} )
A ( cdot 5 e^{x} x^{4}+e^{x} x^{5} )
B. ( 4 e^{x} x^{5}+e^{x} x^{5} )
( mathbf{c} cdot 5 e^{x} x^{4}+e^{x} x^{4} )
D. ( 4 e^{x} x^{5}+e^{x} x^{4} )
12
551 The derivative of ( f(tan x) ) with respect
( operatorname{tog}(sec x) ) at ( quad x=frac{pi}{4}, ) where ( f^{prime}(1)= )
( mathbf{2} ; quad boldsymbol{g}^{prime}(sqrt{mathbf{2}})=mathbf{4} ) is
A ( cdot frac{1}{sqrt{2}} )
B. ( sqrt{2} )
c. 1
D.
12
552 ( x=t cos t, y=t+sin t . ) Then ( frac{d^{2} x}{d y^{2}} ) at
( t=frac{pi}{2} ) is
A ( cdot frac{pi+4}{2} )
в. ( -frac{pi+4}{2} )
c. -2
D. none of these
12
553 Find:
( frac{d y}{d x}=sin (x+y)+cos (x+y) )
12
554 If for all ( x, y ) the function ( f ) is defined by
( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})+boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y})=1 ) and
( boldsymbol{f}(boldsymbol{x})>0 . ) When ( boldsymbol{f}(boldsymbol{x}) ) is differentiable
( boldsymbol{f}^{prime}(boldsymbol{x})= )
A . -1
B. 1
( c .0 )
D. cannot be determined
12
555 Let S (x) be defined in the interval [-2,2] such that
f(x) =1-1-2sxso
Xx-10<x2
and g(x)=f(xD) + f(x)!
Test the differentiability of g(x) in (-2,2). (1986-5 Marks)
12
556 Differentiate w.r.t ( x, ) the following function:
( log sqrt{frac{1+cos x}{1-cos x}} )
12
557 Consider the function ( y=|x-1|+ )
( |x-2| ) in the interval [0,3] and discuss
the continuity and differentiability of
the function in this interval.
This question has multiple correct options
A. continuous everywhere
B. differentiable everywhere except at ( x=1 ) and ( x=2 )
c. differentiable everywhere
D. continuous everywhere except at ( x=1 ) and ( x=2 )
12
558 Find ( frac{d y}{d x} ) if
(a) ( x^{3}+2 x^{2} y+3 x y^{2}+4 y^{3}=5 )
(b) ( x=2 cos ^{3} theta, y=2 sin ^{3} theta )
(c) ( y=sin ^{-1}(2 x sqrt{1-x^{2}}) ;-1 leq x leq 1 )
12
559 Differentiate with respect to ( x ) :
( e^{tan ^{-1} sqrt{x}} )
12
560 Differentiate the following functions
with respect to ( x: ) ( tan ^{-1}left(frac{a+b x}{b-a x}right) )
12
561 If ( f(x)=frac{1}{2} x-1, ) then on the interval
( [mathbf{0}, boldsymbol{pi}] )
A ( cdot tan (f(x)) ) and ( frac{1}{f(x)} ) are continuous
в.
( tan (f(x)) ) and ( frac{1}{f(x)} ) are discontinuous
c. ( tan (f(x)) ) is continuous but ( frac{1}{f(x)} ) is discontinuous
D ( cdot tan (f(x)) ) is discontinuous but ( frac{1}{f(x)} ) is continuous
12
562 If ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}} ) then
( frac{d y}{d x}=frac{log x}{(1-log x)^{2}} )
A . True
B. False
12
563 Let ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) and ( boldsymbol{f}(boldsymbol{x})= )
( 1+(sin 2 x) g(x) ) where ( g(x) ) is
continuous, then ( f^{prime}(x) ) equals
A ( . f(x) g(0) )
B ( .2 f(x) g(0) )
D. ( 2 f(0) )
12
564 Let ( f(x) ) be a real valued function not
identically zero, such that
( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{n}right)=boldsymbol{f}(boldsymbol{x})+(boldsymbol{f}(boldsymbol{y}))^{n} quad forall boldsymbol{x}, boldsymbol{y} in )
( boldsymbol{R} )
where ( n in N(n neq 1) ) and ( f^{prime}(0) geq 0 . ) We
may get an explicit form of the function ( boldsymbol{f}(boldsymbol{x}) )
The value of ( $ $ f^{prime}(0) $ $ f^{prime}(0) ) is :
A.
B. ( n )
c. ( n+1 )
D. 2
12
565 COMO
10. Let f (x) be a continuous and g(x) be a discontinuous
function. prove that f(x) + g(x) is a discontinuous function.
(1987-2 Marks)
12
566 If ( x=frac{1-t^{2}}{1+t^{2}} ) and ( y=frac{2 t}{1+t^{2}} ) at then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? )
A ( cdot frac{-1}{x^{3}} )
в. ( frac{y}{x} )
c. ( frac{-x}{y} )
D. ( frac{x}{y} )
12
567 What is the nature of the graph:
( boldsymbol{y}=boldsymbol{6} e^{-4 boldsymbol{x}} )
A. monotonically increasing
B. monotonically decreasing
c. Increasing then decreasing
D. decreasing then increasing
12
568 Differentiate with respect to ( x )
( boldsymbol{y}=log (1+sin boldsymbol{x}) )
12
569 Find ‘c’ of Lagrange’s mean-value
theorem for
(i) ( f(x)=left(x^{3}-3 x^{2}+2 xright) ) on ( left[0, frac{1}{2}right] )
(ii) ( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{2 5}-boldsymbol{x}^{2}} ) on ( [mathbf{0}, mathbf{5}] )
(iii) ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}+boldsymbol{2}} ) on ( [boldsymbol{4}, boldsymbol{6}] )
12
570 Differentiate ( sin ^{2} 3 x cdot tan ^{3} 2 x ) 12
571 The function ( mathbf{f}(mathbf{x})=frac{cos boldsymbol{x}-sin boldsymbol{x}}{cos mathbf{2} boldsymbol{x}} ) is not
defined at ( x=frac{pi}{4} ) The value of ( fleft(frac{pi}{4}right) ) so that ( mathbf{f}(mathbf{x}) ) is continuous at ( boldsymbol{x}=frac{boldsymbol{pi}}{mathbf{4}} ) is
A ( cdot frac{1}{sqrt{2}} )
B. ( sqrt{2} )
( c cdot-sqrt{2} )
D.
12
572 What is the derivative of ( |x-1| ) at ( x= )
( mathbf{2} ? )
A . -1
B. 0
c. 1
D. Derivative does not exist
12
573 If ( y ) is a function of ( x, ) then ( frac{d^{2} y}{d x^{2}}+ ) ( y frac{d y}{d x}=0 . ) If ( x ) is a function of ( y, ) then the
equation becomes
A ( cdot frac{d^{2} x}{d y^{2}}+x frac{d x}{d y}=0 )
в. ( frac{d^{2} x}{d y^{2}}+yleft(frac{d x}{d y}right)^{3}=0 )
c. ( frac{d^{2} x}{d y^{2}}-yleft(frac{d x}{d y}right)^{2}=0 )
D ( cdot frac{d^{2} x}{d y^{2}}-xleft(frac{d x}{d y}right)^{2}=0 )
12
574 Differentiate
( 2 x^{3 / 2}+2 x^{5 / 2}+C )
A ( cdot frac{d y}{d x}=sqrt{x}(3+5 x) )
B. ( frac{d y}{d x}=sqrt{x}(3-5 x) )
c. ( frac{d y}{d x}=-sqrt{x}(3+5 x) )
D. None of these
12
575 Find the derivative of the following functions (it is to be understood that
( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero
constants and ( m ) and ( n ) are integers)
( (a x+b)^{n}(c x+d)^{m} )
12
576 If ( e^{y}(x+1)=1 ) show that ( frac{d y}{d x}=-e^{y} ) 12
577 If ( boldsymbol{x}+boldsymbol{y}=tan ^{-1} boldsymbol{y} ) and ( boldsymbol{y}^{prime prime}=boldsymbol{f}(boldsymbol{y}) boldsymbol{y}^{prime} ) then
( boldsymbol{f}(boldsymbol{y})= )
A ( cdot frac{1}{yleft(1+y^{2}right)} )
В. ( frac{3}{yleft(1+y^{2}right)} )
c. ( frac{2}{yleft(1+y^{2}right)} )
D. ( frac{-2}{yleft(1+y^{2}right)} )
12
578 If ( boldsymbol{f}(boldsymbol{x})=min left(|boldsymbol{x}|^{2}-mathbf{5}|boldsymbol{x}|, mathbf{1}right) ) then ( boldsymbol{f}(boldsymbol{x}) )
is non differentiable at ( lambda ) points, then
( lambda+13 ) equals
12
579 f ( boldsymbol{y}=boldsymbol{a} cos (log boldsymbol{x})-boldsymbol{b} sin (log boldsymbol{x}), ) then
the value of ( x^{2} frac{d^{2} y}{d x^{2}}+x frac{d y}{d x}+y ) is
( mathbf{A} cdot mathbf{0} )
B. 1
c. 2
D. 3
12
580 If ( x=A cos 4 t+B sin 4 t, ) then ( frac{d^{2} x}{d t^{2}}= )
( A )
B . ( -16 x )
( c .15 x )
D. ( 16 x )
E . ( -15 x )
12
581 If ( y=log _{x^{2}+4}left(7 x^{2}-5 x+1right), ) then
( frac{d y}{d x}= )
A ( cdot frac{1}{log _{e}left(x^{2}+4right)}left(frac{14 x-5}{7 x^{2}-5 x+1}-frac{2 x y}{x^{2}+4}right) )
В ( cdot frac{1}{log _{e}left(x^{2}+4right)}left(frac{14 x-5}{7 x^{2}-5 x+1}+frac{2 x y}{x^{2}+4}right) )
c. ( -frac{1}{log _{e}left(x^{2}+4right)}left(frac{14 x-5}{7 x^{2}-5 x+1}-frac{2 x y}{x^{2}+4}right) )
D. None of these
12
582 A value of c which the conclusion of
Mean Value Theorem holds for the
function ( boldsymbol{f}(boldsymbol{x})=log _{e} boldsymbol{x} ) on the interval
[1,3] is
( mathbf{A} cdot 2 log _{3} e )
B – ( frac{1}{2} log _{e} 3 )
( c cdot log _{3} )
( mathrm{D} cdot log _{e} 3 )
12
583 If ( x=sec theta-cos theta ) and ( y=sec ^{3} theta- )
( sec ^{3} theta-cos ^{3} theta, ) then the value of
( left(frac{d y}{d x}right)^{2} ) at ( x=0 )
A . 0
B. 2
( c cdot 4 )
D.
12
584 If ( f(x)=frac{1-tan x}{1-sqrt{2} sin x}, ) for ( x neq )
( frac{pi}{4} ) is continous ( boldsymbol{a t} quad boldsymbol{x}= )
( frac{pi}{4}, quad ) find ( quad fleft(frac{pi}{4}right) )
12
585 Find the value of ( k ) for which ( f(x)= ) ( left{begin{array}{l}frac{1-cos 4 x}{8 x^{2}}, text { when } x neq 0 \ k, quad text { when } x=0end{array}right. )
continuous at ( boldsymbol{x}=mathbf{0} )
12
586 ( operatorname{Let} g(x)=frac{f(x)}{x+1} ) where ( f(x) ) is
differentiable on [0,5] such that
( boldsymbol{f}(mathbf{0})=mathbf{4}, boldsymbol{f}(mathbf{5})=-1 . ) There exists ( boldsymbol{c} in )
(0,5) such that ( g^{prime}(c) ) is ?
( A cdot-frac{1}{6} )
B. ( frac{1}{6} )
c. ( -frac{5}{6} )
D. –
12
587 ( frac{boldsymbol{d}}{d x}left(e^{tan x}right) )
( mathbf{A} cdot e^{tan x} cdot sec ^{2} x )
B ( cdot e^{cot x} cdot sec ^{2} x )
( mathbf{C} cdot e^{cos x} cdot sec ^{2} x )
D ( cdot e^{sin x} cdot sec ^{2} x )
12
588 Find ( frac{d y}{d x} ) of ( x y+y^{2}=tan x+y ) 12
589 ff ( y=e^{x}(sin x+cos x), ) then prove that
( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}-boldsymbol{2} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{2} boldsymbol{y}=mathbf{0} )
12
590 If ( f(x) ) is continuous in [0,1] and ( fleft(frac{1}{3}right)=1 ) then ( lim _{n rightarrow infty} fleft(frac{n}{sqrt{9 n^{2}+1}}right) ) is
( mathbf{A} cdot mathbf{1} )
B.
c. ( frac{1}{3} )
D. none of these
12
591 Assertion Derivative of ( (boldsymbol{p} boldsymbol{x}+boldsymbol{q})left(frac{boldsymbol{r}}{boldsymbol{x}}+boldsymbol{s}right) ) is ( boldsymbol{p s}+ )
( frac{boldsymbol{q} boldsymbol{r}}{boldsymbol{x}^{2}} )
Reason
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(boldsymbol{u} boldsymbol{v})=boldsymbol{u}^{prime} boldsymbol{v}+boldsymbol{u} boldsymbol{v}^{prime} )
where ( u ) and ( v ) are two distinct
functions.
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
592 (a) Differentiate ( boldsymbol{y}=cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right) ) with
respect to ( boldsymbol{x}, mathbf{0}<boldsymbol{x}<mathbf{1} )
(b) Differentiate ( x^{x}-2^{sin x} ) with respect
to ( x )
12
593 If ( x^{y}-y^{x}=1, ) then the value of ( frac{d y}{d x} ) is : 12
594 If ( boldsymbol{x}^{p} cdot boldsymbol{y}^{boldsymbol{q}}=(boldsymbol{x}+boldsymbol{y})^{boldsymbol{p}+boldsymbol{q}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? )
A. ( frac{y}{x} )
B. ( -frac{y}{x} frac{y}{x} )
c. ( frac{x}{y} )
D. ( -frac{x}{y} )
12
595 If ( x^{3}-y^{3}+3 x y^{2}-3 x^{2} y+1=0, ) then
( operatorname{at}(0,1) frac{d y}{d x} ) equals
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 2 )
D.
12
596 ff ( x=a t^{4}, y=b t^{3}, ) then find ( frac{d y}{d x} ) 12
597 If ( f(x)=cos ^{-1}left{frac{1-left(log _{e} xright)^{2}}{1+left(log _{e} xright)^{2}}right} ), then
( boldsymbol{f}^{prime}(boldsymbol{e}) )
A. Does not exist
B.
c.
D. Is equal to 1
12
598 ( * boldsymbol{f}(boldsymbol{x})= )
( frac{1-sin x}{(pi-2 x)^{2}} cdot frac{log sin x}{log left(1+pi^{2}-4 pi x+4 x^{2}right)}, x )
( pi / 2 ) The assigned to function at ( x= )
( pi / 2 ) in order that it may be continuous
at ( x=pi / 2 ) is ( -frac{1}{m} . ) Find ( m )
12
599 ( operatorname{Let} f(x)=left{begin{array}{ll}2 x+3 & ,-3 leq x<-2 \ x+1, & -2 leq x<0 \ x+2, & 0 leq x leq 1end{array}right. )
At what points the function is/are not differentiable in the interval (-3,1)
This question has multiple correct options
A . -2
B.
( c .1 )
D. ( 1 / 2 )
12
600 f ( f(x)=sqrt{1+cos ^{2}left(x^{2}right)}, ) then ( f^{prime}(x) ) is? 12
601 Find ( frac{d y}{d x} ) of ( a x+b y^{2}=cos y ) 12
602 Differentiate w.r.t ( x )
( boldsymbol{y}=log left(boldsymbol{4} boldsymbol{e}^{boldsymbol{3} boldsymbol{x}}right) )
12
603 ( boldsymbol{y}=sin ^{1}left[frac{boldsymbol{2} boldsymbol{x}}{mathbf{1}+boldsymbol{x}^{2}}right], boldsymbol{t h e n} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
604 If ( x sin y=sin (y+a) ) and ( frac{d y}{d x}= )
( frac{A}{1+x^{2}-2 x cos a} ) then the value of ( A ) is
( A cdot 2 )
B. ( cos a )
( c .-sin a )
D. – –
12
605 [
operatorname{Let} f(x)=x+frac{1}{2 x+frac{1}{2 x+frac{1}{2 x+ldots . . infty}}}
]
Compute the value of ( boldsymbol{f}(mathbf{1 0 0}) cdot boldsymbol{f}^{prime}(mathbf{1 0 0}) )
12
606 Solve the following differential equation ( frac{d y}{d x}=x^{2} ) 12
607 If ( y=sqrt{sin x+y} ) then ( frac{d y}{d x} ) equals to
A ( cdot frac{cos x}{2 y-1} )
B. ( frac{cos }{1-2 y} )
c. ( frac{sin x}{1-2 y} )
D. ( frac{sin x}{2 y-1} )
12
608 Find the derivation of ( sqrt{tan x} ) with respect to x using first principle. 12
609 Differentiate the following functions
with respect to ( x: ) ( mathbf{f} boldsymbol{y}=mathbf{s e c}^{-1}left(frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right)+ )
( sin ^{-1}left(frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right), boldsymbol{x}>mathbf{0} . ) Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
12
610 ( f(x)=frac{sin 2 x+1}{sin x-cos x} ) is discontinuous at
( boldsymbol{x}= )
A ( cdot frac{pi}{4} )
в. ( frac{pi}{3} )
c.
D.
12
611 ( mathbf{f} boldsymbol{y}=sin left{tan ^{-1} sqrt{left(frac{1-x}{1+x}right)}right} ) prove that
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{- x}}{sqrt{mathbf{1 – x}^{2}}} )
12
612 If ( f ) is a real-valued differentiable
function satisfying ( |boldsymbol{f}(boldsymbol{x})-boldsymbol{f}(boldsymbol{y})| leq )
( (x-y)^{2}, quad x, y in R ) and ( f(0)=0, ) then
( boldsymbol{f}(mathbf{1}) ) equals
( mathbf{A} cdot mathbf{1} )
B. 2
c. 0
D. –
12
613 Differentiate the following function with
respect to ( x )
( x^{3} sin x )
12
614 Verify lagrange’s mean value theorem
for the function
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{3} ) where ( boldsymbol{x} in[boldsymbol{4}, boldsymbol{6}] )
12
615 If ( y=x tan y, ) then ( frac{d y}{d x} ) is equal to
A. ( frac{tan y}{x-x^{2}-y^{2}} )
в. ( frac{y}{x-x^{2}-y^{2}} )
c. ( frac{tan y}{y-x} )
D. ( frac{tan x}{x-y^{2}} )
12
616 Differentiate w.r.t ( boldsymbol{x} )
( e^{operatorname{cosec}^{2} x} )
12
617 25. If|cs
and f(x) is a differentiable function at x = 0 given
+ x
bsin -1
<x<0
by f(x) = {
x=0
2x 12 – 1
0<x< 1 / 2
Find the value of a' and prove that 64 b2=4-c2
(2004 – 4 Marks)
12
618 The value of ( c ) in Lagrange mean
value theorem for ( f(x)=log (sin x) ) in ( left[frac{pi}{6}, frac{5 pi}{6}right] ) is
This question has multiple correct options
A ( cdot frac{pi}{4} )
в. ( frac{pi}{2} )
c. ( frac{2 pi}{3} )
D. ( frac{3 pi}{4} )
12
619 ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{r}frac{boldsymbol{K} cos boldsymbol{x}}{boldsymbol{pi}-mathbf{2} boldsymbol{x}} ; boldsymbol{x} neq frac{boldsymbol{pi}}{mathbf{2}} \ boldsymbol{5} ; boldsymbol{x}=frac{boldsymbol{pi}}{2}end{array}right. )
Find the value of ( K ) so that the function is continuous at the point ( boldsymbol{x}=frac{boldsymbol{pi}}{mathbf{2}} )
12
620 Left hand derivative and right hand
derivative of a function ( f(x) ) at a point ( x=a ) are defined as
( f^{prime}left(a^{-}right)=lim _{h rightarrow 0^{+}} frac{f(a)-f(a-h)}{h}= )
( lim _{h rightarrow 0^{-}} frac{f(a)-f(a-h)}{h}= )
( lim _{x rightarrow a^{+}} frac{f(a)-f(x)}{a-x} ) respectively
Let ( f ) be a twice differentiable function.
We also know that derivative of an even
function is odd function and derivative
of an odd function is even function.
If ( f ) is even, which of the following is
Right hand derivative of ( boldsymbol{f}^{prime} ) at ( boldsymbol{x}=boldsymbol{a} )
A. ( lim _{h rightarrow 0^{-}} frac{f^{prime}(a)+f^{prime}(-a+h)}{h} )
B. ( lim _{h rightarrow+} frac{f^{prime}(a)+f^{prime}(-a-h)}{h} )
c. ( lim _{h rightarrow 0} frac{-f^{prime}(a)+f^{prime}(-a+h)}{-h} )
D. ( lim _{h rightarrow 0^{+}} frac{f^{prime}(a)+f^{prime}(-a+h)}{h} )
12
621 1. Differentiation of sin(x?) w.r.t. x is
(a) cos(x2)
(b) 2x cos(x2)
(c) x2 cos(x2)
(d) -cos(2x)
dy
12
622 Differentiate the following functions
with respect to ( boldsymbol{x} )
[
mathbf{f} boldsymbol{y}=cot ^{-1}left{frac{sqrt{mathbf{1}+sin boldsymbol{x}}+sqrt{mathbf{1}-sin boldsymbol{x}}}{sqrt{mathbf{1}+sin boldsymbol{x}}-sqrt{mathbf{1}-sin boldsymbol{x}}}right}
]
( 0<x<frac{pi}{2}, ) show that ( frac{d y}{d x} ) is
independent of ( boldsymbol{x} )
12
623 Find the derivative of the following functions: ( 3 cot x+5 cos e c x ) 12
624 Find the derivative of the following
function.
( log (log x) )
12
625 Let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] ) and ( boldsymbol{g}(boldsymbol{x})= )
( left{begin{array}{cc}mathbf{0}, & boldsymbol{x} in boldsymbol{Z} \ boldsymbol{x}^{2}, & boldsymbol{x} in boldsymbol{R}-boldsymbol{Z}end{array} . ) Then which of the right.
following is not true ([.] represents the greatest integer function)? This question has multiple correct options
A ( cdot lim _{x rightarrow 1} g(x) ) exists but ( g(x) ) is not continuous at ( x=1 )
B. ( lim _{x rightarrow 1} f(x) ) does not exist and ( f(x) ) is not continuous at
[
x=1
]
c. ( g o f ) is a continuous function.
D. gof is a discontinuous function.
12
626 If ( y^{2}+b^{2}=2 x y, ) then ( frac{d y}{d x} ) equals
This question has multiple correct options
A ( cdot frac{1}{x y-b^{2}} )
в. ( frac{y}{y-x} )
c. ( frac{x y-b^{2}}{(y-x)^{2}} )
( frac{x y-b^{2}}{y} )
12
627 Discuss the continuity of the following function at the indicated point(s):
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}
frac{boldsymbol{e}^{boldsymbol{x}}-mathbf{1}}{log (mathbf{1}+mathbf{2} boldsymbol{x})}, boldsymbol{i} boldsymbol{f} boldsymbol{x} neq mathbf{0} \
boldsymbol{7}, boldsymbol{i} boldsymbol{f} quad boldsymbol{x}=mathbf{0}
end{array}right.
]
( boldsymbol{x}=mathbf{0} )
12
628 If ( x^{2}+2 x y+2 y^{2}=1, ) then ( frac{d y}{d x} ) at the
point where ( y=1 ) is equal to:
A .
B. 2
( c cdot-1 )
D. –
( E )
12
629 Verify Rolle’s theorem for the following function: ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+mathbf{1 0} ) on ( [mathbf{0}, boldsymbol{4}] ) 12
630 Let ( f(x) ) satisfy the requirements of Lagrange’s mean value theorem in [0,1] ( boldsymbol{f}(mathbf{0})=mathbf{0} ) and ( boldsymbol{f}^{prime}(boldsymbol{x}) leq mathbf{1}-boldsymbol{x}, forall boldsymbol{x} boldsymbol{epsilon}(mathbf{0}, mathbf{1}) )
then
A. ( f(x) geq x )
в. ( |f(x)| geq 1 )
C ( . f(x) leq x(1-x) )
D. ( f(x) leq frac{1}{4} )
12
631 d?
dv2 equals
(2007-3 marks)
(d
12
632 Find ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}, ) where ( boldsymbol{y}=log left(frac{boldsymbol{x}^{2}}{boldsymbol{e}^{2}}right) ) 12
633 Let ( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{mathbf{3}}+mathbf{3} boldsymbol{x} forall boldsymbol{x} in boldsymbol{R}, ) then
equation of tangent for ( y=f^{-1}(x) ) at
( boldsymbol{x}=mathbf{5} ) will be
A ( cdot 9 y-x=4 )
В. ( 9 y-4 x=-19 )
c. ( 49 y-9 x=4 )
D. ( 9 y-2 x=-1 )
12
634 9.
The following functions are continuous on (0,7).
(1991 – 2 Marks)
(a) tan x
o
0<****
V
x sin x,
sin(+*), <x<*
12
635 Solve:
( lim _{x rightarrow 3} frac{left(x^{frac{1}{3}}+3 sqrt{3}right)left(x^{frac{1}{3}}-3 sqrt{3}right)}{x-3} )
12
636 For instantaneous speed, the distance
traveled by the object and the time taken are both equal to zero.
A. True
B. False
12
637 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}} ) 12
638 ( boldsymbol{x}^{frac{1}{2}}+mathbf{1}=boldsymbol{t} )
differentiate w.r.t. ( mathbf{x} )
( mathbf{A} cdot frac{d t}{d x}=frac{1}{2 sqrt{x}} )
B. ( frac{d t}{d x}=frac{1}{4 sqrt{x}} )
( mathbf{C} cdot frac{d t}{d x}=frac{1}{8 sqrt{x}} )
D. ( frac{d t}{d x}=frac{1}{16 sqrt{x}} )
12
639 If ( x sqrt{1+y}+y sqrt{1+x}=0, ) for ( -1< )
( x<1, ) prove that ( frac{d y}{d x}=frac{1}{(1+x)^{2}} )
12
640 Differentiate w.r.t ( mathbf{x} ) ( boldsymbol{y}=boldsymbol{x}^{2} ln (sqrt{frac{boldsymbol{x}^{2}+mathbf{9}}{boldsymbol{x}^{2}+boldsymbol{4}}} ) 12
641 If ( boldsymbol{y}=sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}+sqrt{boldsymbol{x}+sqrt{boldsymbol{y}+ldots infty}}} )
then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A ( cdot frac{y^{2}-x}{2 y^{3}-2 x y-1} )
В. ( frac{x^{2}-x}{2 x^{3}-2 x y-1} )
C. ( frac{x^{2}-x}{2 x^{3}-2 x y^{2}-1} )
D. None of these
12
642 Find ( k, ) if the given function is
continuous at ( x=2 ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}mathbf{3} boldsymbol{x}-mathbf{4} text { for } mathbf{0} leq boldsymbol{x} leq mathbf{2} \ mathbf{2} boldsymbol{x}+boldsymbol{k} text { for } mathbf{2} leq boldsymbol{x} leq mathbf{4}end{array}right. )
12
643 If ( f(x) ) is continuous at ( x=c ) and ( g(x) )
is continuous at ( x=f(c) ) then which of
the following is/are continuous at ( x=c )
( ? )
A ( cdot(f(x)-g(x)) * f(x) )
в. ( f(g(x)) )
c. ( f(f(x)) )
D. None
12
644 If ( y^{2}=a x^{2}+b x+c, ) then ( y^{3} frac{d^{2} y}{d x^{2}} ) is
A . a constant
B. a function of x only
c. a function of y only
D. a function of ( x ) and ( y )
12
645 Show that ( 3+frac{1}{1+} frac{1}{6+1+} frac{1}{6+} dots= )
( 3left(1+frac{1}{3+2+3+2+cdots}right) )
12
646 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=log left(frac{1-boldsymbol{x}^{2}}{1+boldsymbol{x}^{2}}right) ) 12
647 Let ( f ) be differentiable for all ( mathbf{x} . ) If ( boldsymbol{f}(mathbf{1})= )
-2 and ( f^{prime}(x) geq 2 ) for all ( x epsilon[1,6] ) then
A ( cdot f(6)<8 )
B. ( f(6) geq 8 )
( mathbf{c} cdot f(6) geq 5 )
D. ( f(6) leq 5 )
12
648 Is every continuous function
differentiable?
12
649 If ( x^{2}+y^{2}=4, ) then ( y frac{d y}{d x}+x ) is equal to
( A cdot 4 )
B. 0
c. 1
D. –
12
650 Examine the continuity of the following function at the point ( boldsymbol{x}=-frac{1}{2} ) ( f(x)=left{begin{array}{ll}frac{4 x^{2}-1}{2 x+1} & x neq-frac{1}{2} \ -2, & x=-frac{1}{2}end{array}right. ) 12
651 Differentiate w.r.t. ( boldsymbol{x} )
( boldsymbol{y}=boldsymbol{e}^{boldsymbol{3} boldsymbol{x}} sin boldsymbol{4} boldsymbol{x} )
12
652 If ( g(X)=frac{x}{[X]} ) for ( X>2 ) then ( lim _{x rightarrow 2^{+}} )
( frac{boldsymbol{g}(boldsymbol{X})-boldsymbol{g}(boldsymbol{2})}{boldsymbol{X}-boldsymbol{2}}= )
A . –
B.
c. ( frac{1}{2} )
D.
12
653 Find the derivative of the following functions:
(i) ( tan x cos x )
(ii) ( sec x )
12
654 ( y=e^{x^{2}} ) the value of ( frac{d y}{d x} ) is ( m x e^{x^{2}} . ) Find
( m )
12
655 Form the differential equation from the following primitive, where constant is
arbitrary.
( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} )
A ( cdot frac{d y}{d x}=0 )
B. ( frac{d^{2} y}{d x^{2}}=0 )
c. ( frac{d^{3} y}{d x^{3}}=0 )
D. None of these
12
656 Check the continuity of the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{|boldsymbol{x}|}{boldsymbol{x}}, & boldsymbol{x} neq mathbf{0} \ mathbf{0}, & boldsymbol{x}=mathbf{0}end{array} quad mathbf{a t} boldsymbol{x}=mathbf{0}right. ) 12
657 Is the function defined by ( f(x)=|x|, ) a
continuous function?
12
658 If ( f(x)= ) ( left{begin{array}{cc}(mathbf{1}+|sin boldsymbol{x}|)^{frac{a}{|sin |}} & ;-frac{pi}{6}<boldsymbol{x}<mathbf{0} \ boldsymbol{b} & ; boldsymbol{x}=mathbf{0} \ boldsymbol{e}^{left(frac{tan 2 x}{tan 3 x}right)} & ; mathbf{0}<boldsymbol{x}<frac{pi}{mathbf{6}}end{array}right. )
is a continuous function on ( left(-frac{pi}{6}, frac{pi}{6}right) )
then
A ( cdot a=frac{2}{3}, b=e^{2} )
в. ( a=frac{1}{3}, b=e^{1 / 3} )
c. ( _{a=frac{2}{3}, b=e^{2 / 3}} )
D. ( a=e^{2 / 3}, b=frac{2}{3} )
12
659 If ( 2^{x}-2^{y}=2^{x+y} ) then ( frac{d y}{d x}= )
( mathbf{A} cdot 2^{y-x} )
B . ( 2^{y / x} )
( mathbf{c} cdot-2^{y / x} )
D. ( 2^{x / y} )
12
660 Differentiate the following function with
respect to ( x )
( frac{boldsymbol{x}^{n}}{sin boldsymbol{x}} )
12
661 Determine the values of ( a, b, c ) for which
[
left{begin{array}{ll}
f(x)=frac{sin (a+1) x+sin x}{x} & text { for } x
end{array}right.
]
is continuous at ( boldsymbol{x}=mathbf{0} )
12
662 Differentiate the following w.r.t.x:
( 5^{x} cdot sec ^{-1} 2 x )
12
663 The function ( boldsymbol{f}:(boldsymbol{R}-mathbf{0}) rightarrow mathbf{R} ) given by
( f(x)=frac{1}{x}-frac{2}{e^{2 x}-1} ) can be made
continuous at ( x=0 ) by defining ( f(0) ) as
A . 2
B. – –
c. 0
( D )
12
664 ffunction ( f(x) ) is continuous in interval ( [-2,2], ) find the value of ( (a+b) )
where ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}frac{sin a x}{x}-2, & text { for }-2 leq x<0 \ 2 x+1, & text { for } 0 leq x leq 1 \ 2 b sqrt{x^{2}+3}-1, & text { for } 1<x leq 2end{array}right. )
12
665 If ( f(x) ) is differentiable in the interval
( [2,5], ) where ( f(2)=frac{1}{5} ) and ( f(5)=frac{1}{2} )
then there exists a number ( c, 2<c< )
5 for which ( f^{prime}(c)= )
A ( cdot frac{1}{2} )
B. ( frac{1}{5} )
( c cdot frac{1}{10} )
D. None
12
666 Find the value ( f(0) ) so that the function
( boldsymbol{f}(boldsymbol{x})=frac{1}{x}-frac{2}{e^{2 x}-1}, boldsymbol{x} neq 0 ) is continuous
at ( x=0 & ) examine the differentiability
of ( f(x) ) at ( x=0 )
A ( cdot f(0)=0, ) differentiable at ( x=0 )
B. ( f(0)=0, ) not differentiable at ( x=0 )
( mathrm{c} . f(0)=1, ) differentiable at ( x=0 )
D. ( f(0)=1, ) not differentiable at ( x=0 )
12
667 ( frac{d}{d x}left(tan ^{-1} frac{cos x-sin x}{cos x+sin x}right) )
( A )
B. –
( c cdot 1 )
D.
12
668 Find the derivative of ( frac{tan ^{-1} x}{1+tan ^{-1} x} ) w.r.t.
( tan ^{-1} x )
A ( cdot frac{1}{sec ^{-1} x} )
в. ( frac{1}{left(tan ^{-1} xright)^{2}} )
c. ( frac{1}{1+tan ^{2} x} )
D. ( frac{1}{left(1+tan ^{-1} xright)^{2}} )
12
669 If ( int frac{sin x}{sin (x-alpha)} d x=A x+ )
( B log sin (x-alpha)+c ) then find the value
of ( (boldsymbol{A}, boldsymbol{B}) )
12
670 The function ( f(x)=[x], ) at ( x=5 ) is:
A. Ieft continuous
B. right continuous
c. continuous
D. cannot be determined
12
671 Differentiate ( tan ^{-1}left(frac{sin x}{1+cos x}right) ) w.r.t. 12
672 If the function ( mathbf{f}(boldsymbol{x})= ) ( left{begin{array}{ll}frac{2^{x+2}-16}{4^{x}-16} & text { for } x neq 2 \ mathbf{A} & x=2end{array} ) is continuous right.
at ( boldsymbol{x}=mathbf{2}, ) then ( mathbf{A}= )
( A cdot 2 )
B.
( c cdot frac{1}{4} )
D.
12
673 Derivative of an odd function.
A. May be even or may be odd
B. Is always odd
C. Is always even
D. None of these
12
674 The function ( f(x)=sin ^{-1}(cos x) ) is :
A. Discontinuous at ( x=0 )
B. Continuous at ( x=0 )
c. Differentiable at ( x=0 )
D. None of these
12
675 Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function such that ( boldsymbol{f}left(frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2}}right)=frac{boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})}{2} ) for all ( mathbf{x}, mathbf{y} )
and ( f(0)=3 ) and ( f^{prime}(0)=3 . ) Then
A . ( f(x) / x ) is continuous on ( mathbb{R} )
B. ( f(x) ) is continuous on ( R )
c. ( f(x) ) is bounded on ( R )
D. none of these
12
676 Find the derivative of
( f(x)=left(x^{2}-5right)left(x^{3}-2 x+3right) )
12
677 If ( y=a^{frac{1}{2} log _{a} cos x} . ) Find ( frac{d x}{d} ) 12
678 If ( y=x^{2} tan x, ) find ( frac{d y}{d x} ) 12
679 If ( y=e^{x} sin x, ) then find ( frac{d y}{d x} )
A ( cdot e^{x}(sin x+cos x) )
B . ( e^{x}(sin x-cos x) )
( mathbf{c} cdot e^{x} sin x )
D. None of these
12
680 Discuss the continuity of the function ( f ) defined by ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{boldsymbol{x}}, boldsymbol{x} neq mathbf{0} ) 12
681 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[sin ^{2}left(cot ^{-1} sqrt{frac{boldsymbol{1}+boldsymbol{x}}{1-boldsymbol{x}}}right)right]=? ) 12
682 Find ( frac{d y}{d x}, ) if ( y=log left(sqrt{x}-frac{1}{sqrt{x}}right) ) 12
683 If ( y=log (log x)+2 sin x, ) find ( frac{d y}{d x} ) 12
684 Show that ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{2}|+|boldsymbol{x}-mathbf{3}| ) is
not differentiable at ( x=2 )
12
685 20. If y = 2 sin x, then dyldt will be… 12
686 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[log _{e}left{left(boldsymbol{e}^{boldsymbol{x}}+boldsymbol{2}right)+right.right. )
( sqrt{boldsymbol{e}^{2 boldsymbol{x}}+boldsymbol{4} boldsymbol{e}^{boldsymbol{x}}+boldsymbol{5}}}]= )
A. ( frac{1}{sqrt{e^{2 x}+4 e^{x}+5}} ) C. ( frac{e^{x}}{sqrt{e^{2 x}+4 e^{x}+5}} )
D. ( frac{e^{x}}{sqrt{e^{2 x}+4 e^{x}+3}} )
( sqrt{e^{2 x}+4 e^{x}+3} )
12
687 If ( g ) is inverse function of ( f ) where
( f(x)=int_{0}^{pi} frac{1}{sqrt{1+t^{2}}} d t quad ) and
( int gleft(g^{prime}(x)right)^{2} d x=frac{left[1+(g(x))^{alpha}right]^{beta}}{gamma}+c )
Then the value of ( alpha beta gamma ) is equal to [where ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} in boldsymbol{R}] )
A . 9
B. 6
( c .3 )
D.
12
688 Differentiate: ( y=sin (2 x+3) ) w.r.t ( x ) 12
689 Suppose that ( f ) is differentiable for all
( boldsymbol{x} in boldsymbol{R} ) and that ( boldsymbol{f}^{prime}(boldsymbol{x}) leq 2 ) for all ( boldsymbol{x} . ) If
( f(1)=2 ) and ( f(4)=8, ) then ( f(2) ) has
the value equal to
A . 3
B. 4
( c cdot 6 )
D. 8
12
690 The value of ( c ) in Lagrange’s theorm for
the function ( f(x)=log sin x ) in the
interval ( [boldsymbol{pi} / mathbf{6}, mathbf{5} boldsymbol{pi} / mathbf{6}] ) is
( mathbf{A} cdot pi / 4 )
в. ( pi / 2 )
( mathrm{c} cdot 2 pi / 3 )
D. none of these
12
691 Assertion
Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a function defined by
( boldsymbol{f}(boldsymbol{x})=max left{boldsymbol{x}, boldsymbol{x}^{3}right} . ) Then, ( boldsymbol{f}(boldsymbol{x}) ) is not
differentiable at ( boldsymbol{x}=-mathbf{1}, mathbf{0}, mathbf{1} )
Reason
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{x}, boldsymbol{x} leq-mathbf{1} \ boldsymbol{x}^{3},-mathbf{1}<boldsymbol{x} leq mathbf{0} \ boldsymbol{x}, mathbf{0}mathbf{1}end{array}right. )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
692 Let ( mathbf{f}:(-mathbf{1}, mathbf{1}) rightarrow mathbf{R} ) be a differentiable
function with ( mathbf{f}(mathbf{0})=-mathbf{1} ) and ( mathbf{f}^{prime}(mathbf{0})=mathbf{1} )
Let ( mathbf{g}(mathbf{x})=[mathbf{f}(mathbf{2 f}(mathbf{x})+mathbf{2})]^{2} . ) Then ( mathbf{g}^{prime}(mathbf{0})= )
A . -4
B.
( c cdot-2 )
D. 4
12
693 In
[0,1] Lagrange’s Mean Value theorem is NOT applicable to
A ( cdot(mathrm{x})=left{begin{array}{ll}frac{1}{2}-mathrm{x}, & mathrm{x}<frac{1}{2} \ left(frac{1}{2}-mathrm{x}right)^{2} & mathrm{x} geq frac{1}{2}end{array}right. )
B. ( f(x)=left{begin{array}{ll}frac{sin x}{x}, & x neq 0 \ 1, & x=0end{array}right. )
c. ( f(x)=x|x| )
D. ( f(x)=|x| )
12
694 If ( f(x) ) is a polynomial in ( x, ) then
the second derivative of ( fleft(e^{x}right) ) at ( x=1 )
is
A ( cdot e f^{prime prime}(e)+f^{prime}(e) )
B . ( left(f^{prime prime}(e)+f^{prime}(e)right) e^{2} )
c. ( e^{2} f^{prime prime}(e) )
D. ( left(f^{prime prime}(e) e+f^{prime}(e)right) e )
12
695 The law of the mean can also be put in
the form
( mathbf{A} cdot f(a+h)=f(a)-h f^{prime}(a+q h) 0<q<1 )
B ( cdot f(a+h)=f(a)+h f^{prime}(a+q h) 0<q<1 )
( mathbf{c} cdot f(a+h)=f(a)+h f^{prime}(a-q h) 0<q<1 )
D. ( f(a+h)=f(a)-h f^{prime}(a-q h) 0<q<1 )
12
696 If ( f(x), phi(x), varphi(x) ) are continuous
on ( [a, b] ) and differentiable on
( (a, b) exists c epsilon(a, b), ) then
( left|begin{array}{lll}boldsymbol{f}(boldsymbol{a}) & boldsymbol{phi}(boldsymbol{a}) & boldsymbol{varphi}(boldsymbol{a}) \ boldsymbol{f}(boldsymbol{b}) & boldsymbol{phi}(boldsymbol{b}) & boldsymbol{varphi}(boldsymbol{b}) \ boldsymbol{f}^{prime}(boldsymbol{c}) & boldsymbol{phi}^{prime}(boldsymbol{c}) & boldsymbol{varphi}^{prime}(boldsymbol{c})end{array}right|= )
( A cdot f^{prime}(c) )
B. ( phi^{prime}(c) )
c. ( varphi^{prime}(c) )
D.
12
697 If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}-boldsymbol{2})(boldsymbol{x}-boldsymbol{4}), mathbf{1} leq boldsymbol{x} leq mathbf{4} )
then a number satisfying the conditions of the mean value theorem is
( mathbf{A} cdot mathbf{1} )
B. ( frac{5}{2} )
( c .3 )
D. ( frac{7}{2} )
12
698 If ( f(x)=x^{3} ) and ( g(x)=x^{3}-4 x ) in
( -2 leq x leq 2, ) then consider the
statements:
(a) ( f(x) ) and ( g(x) ) satisfy mean value theorem.
(b) ( f(x) ) and ( g(x) ) both satisfy Rolle’s
theorem.
(c) Only ( g(x) ) satisfies Rolle’s theorem. Of these statements
A . (a) alone is correct.
B. (a) and (c) are correct
c. (a) and (b) are correct
D. None is correct
12
699 Examine the continuity of the function
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}
frac{log boldsymbol{x}-log mathbf{7}}{boldsymbol{x}-mathbf{7}} & text { for } boldsymbol{x} neq mathbf{7} \
mathbf{7}, & text { for } boldsymbol{x}=mathbf{7}
end{array}right.
]
at ( x=7 )
12
700 Applying Lagranges’s MeanValue Theorem for a suitable function ( f(x) ) in
( [0, h], ) we have ( f(h)+h f^{prime}(theta h), 0<theta< )
1. Then for ( f(x)=cos x, ) the value of
( lim _{h rightarrow 0^{+}} theta ) is
A .
B. 0
( c cdot frac{1}{2} )
D.
12
701 Suppose that a function ( f ) satisfies the following conditions for all real values of
( x ) and ( y )
( (i) f(x+y)=f(x) cdot f(y) )
( (i i) f(x)=1+x g(x), ) where
( lim _{x rightarrow 0} g(x)=1 . ) The value of ( log f(8) ) is
12
702 The function ( frac{|boldsymbol{x}|}{boldsymbol{x}^{2}+mathbf{2} boldsymbol{x}}, boldsymbol{x} neq mathbf{0} ) and
( boldsymbol{f}(mathbf{0})=mathbf{0} ) is not continuous at ( boldsymbol{x}=mathbf{0} )
because-
A ( cdot lim _{x rightarrow 0} f(x) neq f(0) )
B. ( lim _{x rightarrow 0^{+}} f(x) ) does not exist
c. ( lim _{x rightarrow 0^{-}} f(x) ) does not exist
D. ( lim _{x rightarrow 0} f(x) ) does not exist
12
703 If ( f(x) ) is a differentiable function in the
interval ( (0, infty) ) such that ( f(1)= ) 1 ( operatorname{and} lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1, ) for eacch
( boldsymbol{x}>0, ) then ( boldsymbol{f}(mathbf{3} / 2) ) is equal to:
( mathbf{A} cdot frac{13}{6} )
B . ( frac{23}{18} )
c. ( frac{25}{9} )
D. ( frac{31}{18} )
12
704 Discuss the applicability of Lagrange’s
mean value theorem for the function
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) on [-1,1]
12
705 if ( y=5 x^{2}+8 x ) find ( frac{d y}{d x} )
A. ( 10 x+8 )
B. ( 5 x+8 )
c. ( 10 x^{2}+8 x )
D. none of these
12
706 Solve ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} sin ^{boldsymbol{n}} boldsymbol{x} ) 12
707 ( frac{dleft(x^{n}right)}{d x}=? )
A ( cdot n x^{n-1} )
В . ( n x^{n} )
c. ( (n-1) x^{n-1} )
D. ( (n-1) x^{n} )
12
708 The function ( f: R / 0 rightarrow R ) given by ( f(x)= ) ( frac{1}{x}-frac{2}{e^{2 x}-1} ) can be made continuous at
( x=0 ) by
defining ( f(0) ) as
A .
B.
( c cdot 2 )
D. –
12
709 The number of continuous functions on
R which satisfy ( (f(x))^{2}=x^{2} ) for all ( x in )
( boldsymbol{R} ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 4 )
D. 8
12
710 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ccc}x & text { if } & x text { is rational } \ 2-x & text { if } & x text { is irrational }end{array} ) Then right.
fof ( (x) ) is continuous
A. everywhere
B. no where
c. at all irrational ( x )
D. at all rational ( x )
12
711 If ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{20^{x}+3^{x}-6^{x}-10^{x}}{1-cos 8 x} ; text { for } x neq 0 \ left(frac{k}{16}right) log left(frac{10}{3}right) cdot log 2 ; text { for } x=0end{array}right. )
continous at ( x=0, ) then the value of ( k )
is
( A cdot sin ^{2} 30^{circ} )
B. ( log _{3}left(frac{1}{2}right) )
[
3
]
( c cdot sqrt[3]{1} )
D. ( frac{log 2^{2}}{3} )
12
712 Differentiate each of the functions with
respect to ( ^{prime} boldsymbol{x}^{prime} )
( frac{a x+b}{c x+d} )
12
713 Verify Rolle’s theoremlquad for the
function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{-boldsymbol{x}} sin boldsymbol{x}, boldsymbol{x} boldsymbol{epsilon}[mathbf{0}, boldsymbol{pi}] )
12
714 If ( boldsymbol{f}(boldsymbol{x})= )
( lim _{p rightarrow infty} frac{boldsymbol{x}^{p} boldsymbol{g}(boldsymbol{x})+boldsymbol{h}(boldsymbol{x})+mathbf{7}}{mathbf{7} boldsymbol{x}^{boldsymbol{p}}+mathbf{3} boldsymbol{x}+mathbf{1}} ; boldsymbol{x} neq mathbf{1} ) and
( boldsymbol{f}(mathbf{1})=mathbf{7}, boldsymbol{f}(boldsymbol{x}), boldsymbol{g}(boldsymbol{x}) ) and ( boldsymbol{h}(boldsymbol{x}) ) are all
continuous functions at ( x=1 . ) Then
which of the following statement(s) is/are correct
This question has multiple correct options
A. ( g(1)+h(1)=70 )
B. ( g(1)-h(1)=28 )
D. ( g(1)-h(1)=-28 )
12
715 Illustration 2.19 If y =

= x-10, then find dyldx.
12
716 Answer the following question in one
word or one sentence or as per exact
requirement of the question. f ( frac{pi}{2}<x<pi, ) then find ( frac{d}{d x}(sqrt{frac{1+cos 2 x}{2}}) )
12
717 Show that the function ( |x| ) is not
differentiable at ( x=0 )
12
718 Identify the graph of the polynomial
function ( boldsymbol{f} )
( f(x)=x^{3}-2 x^{2}-x+2 )
A. graph a
B. graph b
c. graph c
D. graph d
12
719 If ( x=t^{3}+t+5 & y=sin t ) then
( frac{d^{2} y}{d x^{2}}= )
A. ( frac{left(3 t^{2}+1right) sin t+6 t cos t}{left(3 t^{2}+1right)^{3}} )
B. ( frac{left(3 t^{2}+1right) sin t+6 t cos t}{left(3 t^{2}+1right)^{2}} )
c. ( frac{left(3 t^{2}+1right) sin t+6 t cos t}{left(3 t^{2}+1right)^{2}} )
D. ( frac{c o s t}{3 t^{2}+1} )
12
720 The function ( f(x)=sin ^{-1}(cos x) ) is?
This question has multiple correct options
A. Discontinuous at ( x=0 )
B. Continuous at ( x=0 )
C. Differentiable at ( x=0 )
D. None of these
12
721 Show that ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(cos boldsymbol{h}^{-1} boldsymbol{x}right)=frac{mathbf{1}}{sqrt{boldsymbol{x}^{2}-mathbf{1}}} ) 12
722 If ( x=y sqrt{1-y^{2}}, ) then ( frac{d y}{d x} ) equals to? 12
723 If ( y=cos ^{3} x, ) then ( frac{d y}{d x}= ) 12
724 Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a
point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem:
( boldsymbol{f}(boldsymbol{x})=sqrt{mathbf{2 5}-boldsymbol{x}^{2}} ) on ( [-mathbf{3}, boldsymbol{4}] )
12
725 Find the derivative of
( frac{2}{x+1}-frac{x^{2}}{3 x-1} )
12
726 9.
Letf:R
Letfir
→ Rbe a differentiable function and (1)
able function and f(1) =4. Then
f(
2t
the value of lim
(1990 – 2 Marks)
dt is
X-1
X1
(a) 8 f'(l)
(b) 4 f'(1) (c) 2 f'(1)
(d) f'(1)
12
727 The set of points where f(x) = x is differentiable is
(a) (-0,0) (0,00) (b) (-00,-1) (-1,00)
(c) (-00,00) (d) (0,00)
[2006]
12
728 Let ( lim _{x rightarrow a} f(x) ) exists but it is not equal to
( f(a) . ) Then ( f(x) ) is discontinuous at ( x=a )
and a is called a removable
discontinuity. If ( lim _{x rightarrow a^{-}} f(x)= )
land ( lim _{x rightarrow a^{+}} f(x)=m ) exist but ( l neq )
( m . ) Then a is called a jump
discontinuity. If one of the limits (left
hand limit or right hand limit ) does not
exist, then a is called an infinite
discontinuity. ( operatorname{Let} f(x)=left{begin{array}{cl}x^{2}+|x|, & x-5end{array} . ) Then right.
( boldsymbol{x}=-mathbf{5} ) is
A. a point of discontinuity
B. a jump discontinuity
c. a removable discontinuity
D. an infinite discontinuity
12
729 Given that ( prod_{n=1}^{n} cos frac{x}{2^{n}}=frac{sin x}{2^{n} sin left(frac{x}{2^{n}}right)} )
and ( boldsymbol{f}(boldsymbol{x})= )
[
left{begin{array}{c}
lim _{n rightarrow infty} sum_{n=1}^{n} frac{1}{2^{n}} tan left(frac{x}{2^{n}}right), quad x in(0, \
frac{2}{pi}
end{array}right.
]
Then which one of the following is true?
A. ( f(x) ) has non-removable discontinuity of finite type at
[
x=frac{pi}{2}
]
B.
( f(x) ) has removable discontinuity at ( x=frac{pi}{2} )
C ( quad f(x) ) is continuous at ( x=frac{pi}{2} )
D. ( f(x) ) has non-removable discontinuity of infinite type at ( x=frac{pi}{2} )
12
730 The value of ( p ) for which the function
[
left{begin{array}{ccc}
boldsymbol{f}(boldsymbol{x})= & & \
& left(boldsymbol{4}^{boldsymbol{x}}-mathbf{1}right)^{mathbf{3}} & \
hline multirow{2}{*} {sin frac{boldsymbol{x}}{boldsymbol{p}} log left(1+frac{boldsymbol{x}^{2}}{mathbf{3}}right)} & ; & boldsymbol{x} neq mathbf{0} \
& mathbf{1 2}(log mathbf{4})^{mathbf{3}} & ; boldsymbol{x}=mathbf{0}
end{array}right.
]
continuous at ( boldsymbol{x}=mathbf{0}, ) is
A . 4
B. 2
( c .3 )
( D )
12
731 The function ( mathbf{f}(boldsymbol{x})=frac{|boldsymbol{x}|}{boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{0} ) is
A. Ieft continuous
B. right continuous
c. continuous
D. Discontinuous
12
732 T applicable
(2003)
In [0,1] Lagranges Mean Value theorem is NOT appli
to
V


X
X
(a) f(x)= (1 – x
xz 2
ΛΙ
sin x
,
X0
x=0
(b) f(x) = { x
T 1,
(C) f(x) = x/x/
(d) f(x) = |x|
12
733 Let ( boldsymbol{f}:[mathbf{2}, mathbf{7}] rightarrow[mathbf{0}, infty] ) be a continuous
and differentiable function. Then, the
value of ( (boldsymbol{f}(mathbf{7})- )
( f(2)) frac{(f(7))^{2}+(f(2))^{2}+f(2) cdot f(7)}{3}, ) is
(where ( c epsilon(2,7)) )
( mathbf{A} cdot 3 f^{2}(c) f^{prime}(c) )
B. ( 5 f^{2}(c) . f(c) )
c. ( 5 f^{2}(c) cdot f^{prime}(c) )
D. none of these
12
734 Find the values of ( a ) and ( b ) such that the
function defined by
( f(x)=left{begin{array}{l}5, text { if } x leq 2 \ a x+b, text { if } 2<x<10 text { is a } \ 21, text { if } x geq 10end{array}right. )
continuous function.
12
735 If ( y=tan ^{-1}left(frac{2^{x}}{1+2^{2 x+1}}right), ) then ( frac{d y}{d x} ) at
( boldsymbol{x}=mathbf{0} ) is ( ? )
( A cdot-frac{1}{5} )
B. 2
( c cdot epsilon 2 )
D. none of these
12
736 Solve-
( cos x^{3} cdot sin ^{2}left(x^{5}right) )
12
737 Prove that
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(cot boldsymbol{h}^{-1} boldsymbol{x}right)=frac{-mathbf{1}}{left(boldsymbol{x}^{2}-mathbf{1}right)} )
12
738 Let ( f ) be a function defined for all ( boldsymbol{x} boldsymbol{epsilon} boldsymbol{R} )
If ( f ) is differentiable and ( fleft(x^{3}right)=x^{5} ) for
all ( boldsymbol{x} boldsymbol{epsilon} boldsymbol{R}(boldsymbol{x} neq boldsymbol{0}) ) then the value of
( boldsymbol{f}^{prime}(mathbf{2 7}) ) is
A . 15
B. 45
c. 0
D. None
12
739 ( left(3 x^{4}-x^{3}+4right)^{5 / 2} ) differentiate w.r.t ( x ) 12
740 Show that between any two roots of the
equation ( e^{x} cos x=1 ) there exists
atleast one root of ( e^{x} sin x-1=0 ) by continuity and differentiability.
12
741 If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{a} boldsymbol{x}^{2}-boldsymbol{b}, & boldsymbol{i} boldsymbol{f} quad|boldsymbol{x}|<-1 \ -frac{1}{|boldsymbol{x}|} boldsymbol{i} boldsymbol{f} & |boldsymbol{x}| geq-1end{array}right. )
differential at ( x=1 . ) Find the values of
( a ) and ( b )
A ( . a=1 / 2 ; b=3 / 2 )
В. ( a=1 / 2 ; b=-3 / 2 )
c. ( a=-1 / 2 ; b=3 / 2 )
12
742 f ( y=x^{x}+(sin x)^{cot x} ). find ( frac{d y}{d x} ) 12
743 Differentiate the following functions
w.r.t. ( x )
( e^{log (log x)} )
12
744 Find the derivative of the following functions from first principle:
( sin (x+1) )
12
745 Differentiate the given function w.r.t. ( x ) ( frac{cos x}{log x}, x>0 ) 12
746 ( frac{1+tan ^{2} x}{1-tan ^{2} x} d x ) is equal to
A ( cdot log frac{1-tan x}{1+tan x}+c )
B. ( log frac{1+tan x}{1-tan x}+c )
c. ( frac{1}{2} log frac{1-tan x}{1+tan x}+c )
D. ( frac{1}{2} log frac{1+tan x}{1-tan x}+c )
12
747 Let ( f(x)=1+|sin x| . ) Then
This question has multiple correct options
( mathbf{A} cdot f(x) ) is continuous nowhere
B . ( f(x) ) is continuous everywhere
( mathbf{C} cdot f(x) ) is differentiable nowhere
D. ( f^{prime} )
(0) does not exist
12
748 Verify Rolle’s Theorem for the function
( f(x)=x(x-1)^{2} ) in the interval [0,1]
12
749 Evaluate :
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{x}+boldsymbol{y}+mathbf{1}}{boldsymbol{2} boldsymbol{x}+boldsymbol{2} boldsymbol{y}+boldsymbol{3}} )
12
750 Assertion
Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be any function. Define
( boldsymbol{g}: boldsymbol{R} rightarrow boldsymbol{R} ) by ( boldsymbol{g}(boldsymbol{x})=|boldsymbol{f}(boldsymbol{x})| ) for all ( boldsymbol{x} )
Then, ( g ) is continuous is ( boldsymbol{f} ) is
continuous.
Reason
Composition of two continuous functions is a continuous function
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
751 ( boldsymbol{g}(boldsymbol{x})=left{begin{array}{ll}mathbf{1} & boldsymbol{x} leq-boldsymbol{2} \ frac{mathbf{1}}{mathbf{2}} boldsymbol{x} & -boldsymbol{2}<boldsymbol{x}<mathbf{4} text { .then } \ sqrt{boldsymbol{x}} & , boldsymbol{x} geq mathbf{4}end{array}right. )
A. ( g ) is a continuous function
B. all the discontinuities are removable discontinuities
c. all the discontinuities are jump
D. all the discontinuities are infinitt
12
752 ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}^{2}+boldsymbol{2} boldsymbol{x}+boldsymbol{4} ) find ( boldsymbol{y} ) in terms of ( boldsymbol{x} ) 12
753 If ( boldsymbol{f}(boldsymbol{x})= )
( frac{1 . cos x+5 cos 3 x+cos 5 x}{cos 6 x+6 cos 4 x+15 cos 2 x+10} ) then
( boldsymbol{f}(mathbf{0})+boldsymbol{f}^{prime}(mathbf{0})+boldsymbol{f}^{prime prime}(mathbf{0})= )
A ( cdot frac{1}{2} )
B.
( c cdot-frac{1}{2} )
( D )
12
754 If ( y=sqrt{frac{1-x}{1+x}}, ) then ( frac{d y}{d x} ) equals
A ( cdot frac{y}{1-x^{2}} )
в. ( frac{y}{x^{2}-1} )
c. ( frac{y}{1+x^{2}} )
D. ( frac{y}{y^{2}-1} )
12
755 f ( x=tan left(frac{1}{a} log yright), ) prove that ( (1+ )
( left.x^{2}right) frac{d^{2} y}{d x^{2}}+2 x frac{d y}{d x}-a 0 )
12
756 ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cl}frac{tan [x]-[x] tan 1}{x} & ; boldsymbol{x} neq 0 \ boldsymbol{0} & ; boldsymbol{x}=0end{array}, ) then right.
( boldsymbol{f}^{prime}left(mathbf{0}^{-}right) ) is [where ( [boldsymbol{x}] ) denotes integer part
of ( boldsymbol{x} )
A . 0
B.
( c cdot-1 )
D. Does not exist
12
757 If ( boldsymbol{y}= )
( tan ^{-1}left(cot left(frac{pi}{2}-xright)right) ) then ( frac{d y}{d x}= )
( A )
B. – –
( c cdot 0 )
D.
12
758 If ( boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=tan boldsymbol{x}+boldsymbol{y}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is equa
to
A ( frac{sec ^{2} x-y}{(x+2 y-1)} )
B. ( frac{cos ^{2} x+y}{(x+2 y-1)} )
C ( frac{sec ^{2} x-y}{(2 x+y-1)} )
D. ( frac{cos ^{2} x+y}{(2 x+2 y-1)} )
12
759 If ( x^{y}=e^{x-y}, ) then show that ( frac{d y}{d x}= ) ( frac{log boldsymbol{x}}{(1+log boldsymbol{x})^{2}} ) 12
760 If ( y=left(tan ^{-1} xright)^{2}, ) show that ( left(x^{2}+right. )
1) ( ^{2} y_{2}+2 xleft(x^{2}+1right) y_{1}=2 )
12
761 ( boldsymbol{f}(boldsymbol{x})=mathbf{1} /left(mathbf{1}-boldsymbol{e}^{-mathbf{1} / boldsymbol{x}}right), boldsymbol{x} neq mathbf{0}, boldsymbol{f}(mathbf{0})= )
0 at ( x=0 ) Is function continuous at
( x=0 ? )
12
762 Differentiate with respect to ( x ) :
( log _{7}(2 x-3) )
12
763 Let ( y=e^{frac{1}{x}} ) then find ( frac{d^{2} y}{d x^{2}} ) 12
764 ( operatorname{Let} int_{0}^{x}left(frac{b t cos 4 t-a sin 4 t}{t^{2}}right) d t= )
( frac{a sin 4 x}{x} ) then ( a ) and ( b ) are given by
A ( cdot a=frac{1}{4}, b=1 )
В. ( a=2, b=2 )
c. ( a=-1, b=4 )
D. ( a=2, b=4 )
12
765 If ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}frac{sqrt{1+p x}-sqrt{1-p x}}{x} & -1 leq x<0 \ frac{2 x+1}{x-2} & 0 leq x leq 1end{array}right. )
continuous in the interval ( [-1,1], ) then
( boldsymbol{p}= )
A . –
B. ( frac{-1}{2} )
( c cdot frac{1}{2} )
( D )
12
766 The function ( f(x)=|x| ) at ( x=0 ) is:
A. continuous but non-differentiable
B. discontinuous and differentiable
c. discontinuous and non-differentiable
D. continuous and differentiable
12
767 Examine the continuity of the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}|boldsymbol{x}| cos frac{1}{x} & , text { if } boldsymbol{x} neq mathbf{0} \ boldsymbol{0} & , text { if } boldsymbol{x}=mathbf{0}end{array} text { at } boldsymbol{x}=right. )
( mathbf{n} )
12
768 If the function ( left{begin{array}{cc}frac{k cos x}{(pi-2 x)}, & text { when } x neq frac{pi}{2} \ 3, & x=frac{pi}{2}end{array}right. )
be continue at ( x=frac{pi}{2}, ) then the value of
( k ) is
A . 3
в. -3
c. -5
D. 6
12
769 If ( boldsymbol{y}=|cos boldsymbol{x}|+|sin boldsymbol{x}|, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}= )
( frac{2 pi}{3} ) is
A ( cdot frac{1-sqrt{3}}{2} )
B. 0
c. ( frac{1}{2}(sqrt{3}-1) )
D. None of these
12
770 Find the slope of the tangent to the curve ( y=x^{3}-x ) at ( x=2 ) 12
771 If ( y=tan ^{-1} frac{cos x+sin x}{cos x-sin x}, ) then find ( frac{d y}{d x} ) 12
772 If ( y=frac{sin x+cos x}{sin x-cos x} ) find ( frac{d y}{d x} ) at ( x=frac{pi}{4} ) 12
773 If ( x=a(cos theta+sin theta) ) and ( y= )
( a(sin theta-cos theta), ) then find ( frac{d^{2} y}{d x^{2}} )
12
774 Differentiate the following functions
with respect to ( boldsymbol{x} ) ff ( y=sin ^{-1}left(frac{2 x}{1+x^{2}}right)+ )
( sec ^{-1}left(frac{1+x^{2}}{1-x^{2}}right), 0<x<1, ) prove that
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{4}}{boldsymbol{1}+boldsymbol{x}^{2}} )
12
775 If ( boldsymbol{f}(boldsymbol{x})=cos boldsymbol{pi}(|boldsymbol{x}|+[boldsymbol{x}]), ) then ( boldsymbol{f}(boldsymbol{x}) )
is/are (where [.] denotes greatest
integer function) This question has multiple correct options
A . continuous at ( x=frac{1}{2} )
B. continuous at ( x=0 )
c. Differentiable in (2,4)
D. Differentiable in (0,1)
12
776 If ( y=frac{1}{2}left(sin ^{-1} xright)^{2}, ) then find ( (1- )
( left.boldsymbol{x}^{2}right) boldsymbol{y}_{2}-boldsymbol{x} boldsymbol{y}_{1} )
Where ( y_{1} ) and ( y_{2} ) denote first and second
derivatives of ( y ) respectively.
A . -1
B. 0
c. 1
D. 2
12
777 If ( boldsymbol{y}=tan ^{-1} sqrt{frac{1-sin x}{1+sin x}}, ) then the value
of ( frac{d y}{d x} ) at ( x=frac{pi}{6} ) is
A. ( -frac{1}{2} )
B.
c. 1
D. –
12
778 If ( f(x)=frac{x^{2}-10 x+25}{x^{2}-7 x+10} ) for ( x neq 5 ) and ( f ) is
continuous ( a t x=5, ) then ( f(5) ) has the
value equal to-
A .
B. 5
c. 10
D. 25
12
779 Differentiate w.r.t. ( x )
( boldsymbol{y}=sin left(5 x^{3}+2 xright) )
12
780 If ( boldsymbol{x} sqrt{boldsymbol{y}}+boldsymbol{y} sqrt{boldsymbol{x}}=1, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) equals
( mathbf{A} cdot-frac{y+2 sqrt{x y}}{x+2 sqrt{x y}} )
B. ( -sqrt{frac{x}{y}}left(frac{y+2 sqrt{x y}}{x+2 sqrt{x y}}right) )
( mathbf{c} cdot-sqrt{frac{y}{x}}left(frac{y+2 sqrt{x y}}{x+2 sqrt{x y}}right) )
D. None of these
12
781 The differential coefficient of ( f(sin x) )
with respect to ( x ) where ( f(x)=log x ) is:
A . ( tan x )
B. ( cot x )
c. ( f(cos x) )
D.
12
782 Find the derivative ( : cot x ) 12
783 The points where the function ( f(x)= )
( [boldsymbol{x}]+|mathbf{1}-boldsymbol{x}|,-mathbf{1} leq boldsymbol{x} leq mathbf{3}, ) where [
denotes the greatest integer function, is not differentiable are
A. ( x=-1,0,1,2,3 )
B. ( x=-1,0,2 )
c. ( x=0,1,2,3 )
D. ( x=-1,0,1,2 )
12
784 f ( y=cos 2 x cos 3 x, ) then ( y_{n} ) is equal to
Where, ( y_{n} ) denotes the ( n^{t h} ) derivative of
( boldsymbol{y} )
A ( cdot 6^{n} cos left(2 x+frac{n pi}{2}right) cos left(3 x+frac{n pi}{2}right) )
B. ( frac{1}{2}left[5^{text {n }} cos left(frac{text { n } pi}{2}+5 xright)+cos left(frac{text { n } pi}{2}+xright)right] )
c. ( frac{1}{2}left[5^{mathrm{n}} sin left(5 x+frac{mathrm{n} pi}{2}right)+sin left(x+frac{pi}{2}right)right] )
D.
12
785 If ( boldsymbol{f}(boldsymbol{x}) ) defined by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{left|x^{2}-xright|}{x^{2}-x}, x neq 0,1 \ 1, quad x=0 quad text { then } f(x) text { is } \ -1, quad x=1end{array}right. )
continuous for all
( A )
B. ( x ) except at ( x=0 )
c. ( x ) except at ( x=1 )
D. ( x ) except at ( x=0 ) and ( x=1 )
12
786 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for
( boldsymbol{y}=left(sin 20 x+a^{2 x}+10right) )
12
787 Differentiate the following function with
respect to ( x ) ( frac{1}{sin x} )
12
788 Differentiate with respect to ( x ) :
( sin ^{-1}left(frac{2^{x+1}}{1+4^{x}}right) )
12
789 Let ( f(x) ) be a continuous function
whose range is ( [2,6.5] . ) If ( h(x)= )
( left[frac{cos x+f(x)}{lambda}right], lambda in N, ) be continuous
where [.] denotes the greatest integer function, then the least value of ( lambda ) is
( mathbf{A} cdot mathbf{6} )
B. 7
( c cdot 8 )
D. None of these
12
790 Examine whether the given function ( f(x) ) is continuous at ( x=3 )
[
begin{array}{c}
boldsymbol{f}(boldsymbol{x})= \
frac{boldsymbol{x}^{4}-boldsymbol{8} boldsymbol{x}}{sqrt{boldsymbol{x}^{2}+mathbf{5}}-mathbf{3}}, quad text { for } quad boldsymbol{x} neq boldsymbol{3} \
boldsymbol{3} quad text { for } quad boldsymbol{x}=mathbf{3}
end{array}
]
[
boldsymbol{f}(boldsymbol{x})=mathbf{3}
]
12
791 Find the diffrentiation of ( x sin x ) 12
792 1.
Let f: R → R, g: R → R and h: R → R be
differentiable functions such that f(x)=x3 + 3x +2, g(f(x))=
x and h (g(g(x)))=x for all x e R. Then
(JEE Adv. 2016)
(a) f(2)= 15
(6) h'(1)=666
(d) h(g(3))=36
(c)
h(0)=16
12
793 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{1-sin ^{3} x}{3 cos ^{2} x}, quad text { if } quad xfrac{pi}{2}end{array}end{array}right. )
continuous at ( x=frac{pi}{2}, ) find ( a ) and ( b )
12
794 If ( boldsymbol{f}(boldsymbol{x})=sqrt{1}+sqrt{boldsymbol{x}}, boldsymbol{x}>0, ) then ( boldsymbol{f}(boldsymbol{x}) )
( f^{prime}(x) ) is equal to
A ( cdot frac{1}{2 sqrt{x}} )
B. ( frac{1}{2} )
c. ( frac{1}{4 sqrt{x}} )
D. ( frac{2 sqrt{x}+1}{4 sqrt{x}} )
12
795 ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be such that
( |boldsymbol{f}(boldsymbol{x})-boldsymbol{f}(boldsymbol{y})|^{2} leq|boldsymbol{x}-boldsymbol{y}|^{3} ) for all ( boldsymbol{x}, boldsymbol{y} in )
( R ) then the value of ( f^{prime}(x) ) is
A ( cdot f(x) )
B. constant possibly different from zero
c. ( (f(x))^{2} )
D.
12
796 If ( tan (x+y)+tan (x-y)=1 ), then
find ( frac{d y}{d x} )
12
797 Trace the curve ( boldsymbol{y}=boldsymbol{x}^{3} ) 12
798 ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}+boldsymbol{a} & ; quad boldsymbol{x}<0 \ |boldsymbol{x}-1| & ; quad boldsymbol{x} geq 0end{array}right. )
( boldsymbol{g}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}+mathbf{1} & ; text { if } boldsymbol{x}0), ) then
A ( . a=2, b=0 )
В. ( a=2, b=1 )
c. ( a=1, b=0 )
D. ( a=1, b=1 )
12
799 The function
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{r}
boldsymbol{x}^{2}-boldsymbol{a} boldsymbol{x}+mathbf{3}, boldsymbol{x} text { is ratio } \
boldsymbol{2}-boldsymbol{x}, quad boldsymbol{x} text { is irratio }
end{array}right.
]
is continuous at exactly two points then
the possible values of ‘ ( a ) ‘ are
( A cdot(2, infty) )
В ( cdot(-infty, 3) )
c. ( (-infty,-1) cup(3, infty) )
D. ( R )
12
800 If ( x^{m} cdot y^{n}=(x+y)^{m+n}, ) then ( frac{d y}{d x} ) is ( ? )
A. ( frac{y}{x} )
в. ( frac{x+y}{x y} )
c. ( x y )
D. ( frac{x}{y} )
12
801 Statement I: The function ( f(x) ) in the
figure is differentiable at ( x=a )
Statement II: The function ( f(x) )
continuous at ( x=a )
A. Both Statement I and Statement II are true and the
Statement II is the correct explanation of the Statement I
B. Both Statement I and Statement II are true and the Statement II is not the correct explanation of the Statement I
c. Statement l is true but Statement II is false
Statement I is false but Statement II is true
12
802 If ( f(x)=(x-1)(x-2) ) and interval
given was ( (0,4), ) find ‘c’ using Langrange’s mean value theorem.
A ( cdot 2+frac{2}{sqrt{3}} )
B. ( 2-frac{2}{sqrt{3}} )
c. 2
D. Both A and B
12
803 is derivable and has a continuous
derivative at ( boldsymbol{x}=mathbf{0} )
A ( . m in(1, infty) )
B. ( m in[2, infty) )
c. ( m in(2, infty) )
D. ( m in(-infty, 2) )
12
804 ff ( f(x)=1+x+x^{2}+ldots ldots+x^{1000}, ) then
( boldsymbol{f}^{prime}(-1)=ldots . . )
A . -50
B . -500
c. -100
D. 500500
12
805 If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}-sin boldsymbol{t}), boldsymbol{y}=boldsymbol{a}(boldsymbol{1}-cos boldsymbol{t}) )
find ( boldsymbol{d} boldsymbol{y} / boldsymbol{d} boldsymbol{x} ). at ( boldsymbol{t}=boldsymbol{pi} )
12
806 If ( boldsymbol{y}=sec sqrt{boldsymbol{a}+boldsymbol{b} boldsymbol{x}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) equals-
A ( cdot frac{b}{b sqrt{a+b x}} sec sqrt{a+b x} tan sqrt{a+b x} )
в. ( frac{b}{2 sqrt{a+b x}} sec sqrt{a+b x} tan sqrt{a+b x} )
c. ( 2 b sqrt{a+b x} sec sqrt{a+b x} tan sqrt{a+b x} )
D. None of these
12
807 If ( f(x)=frac{tan x}{sqrt{1+tan ^{2} x}}, lim _{x rightarrow(pi / 2)^{-}} f(x)= )
( boldsymbol{a} ) and ( lim _{boldsymbol{x} rightarrow(boldsymbol{pi} / mathbf{2})^{+}} boldsymbol{f}(boldsymbol{x})=boldsymbol{b} ) then
( mathbf{A} cdot a=b )
B . ( a=1+b )
( mathbf{c} cdot a+b=0 )
D. ( a+b=2 )
12
808 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{x}^{10}+mathbf{1 0}^{boldsymbol{x}}+mathbf{1 0} boldsymbol{x}+mathbf{1 0} ) 12
809 Check the continuity of ( f ) given by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ccc}left(x^{2}-9right) /left(x^{2}-2 x-3right) & text { if } & 0<x< \ 1.5 & text { if } & x=varepsilonend{array}right. )
and ( x neq 3 ) at the point 3
12
810 If ( y=ln left(x^{e^{x}}right) ) find ( frac{d y}{d x} ) 12
811 Differentiate the given function w.r.t. ( x )
( boldsymbol{y}=log left(cos e^{x}right) )
12
812 The solution of differential equation
( boldsymbol{y} boldsymbol{d} boldsymbol{x}+left(boldsymbol{x}-boldsymbol{y}^{2}right) boldsymbol{d} boldsymbol{y}=mathbf{0} )
( mathbf{A} cdot e^{frac{y}{x}}=sin x+c )
B. ( y=c x log x )
c. ( x=frac{y^{2}}{3}+frac{c}{y} )
D. ( cos left(frac{y-2}{x}right)=a )
12
813 If Rolle’s theorem is applicable to the function ( f(x)=frac{ln x}{x},(x>0) ) over the interval ( [a, b] ) where ( a epsilon I, b epsilon I, ) then the value of ( a^{2}+b^{2} ) can be
A . 20
B . 25
c. 45
D. 10
12
814 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left[frac{sqrt{mathbf{1}+sin boldsymbol{x}}-sqrt{mathbf{1}-sin boldsymbol{x}}}{sqrt{mathbf{1}+sin boldsymbol{x}}-sqrt{mathbf{1}-sin boldsymbol{x}}}right]= )
( A )
B. ( -frac{1}{2} )
( c cdot frac{1}{2} )
( D )
12
815 Draw a graph of the function y=[x] +1-, -18×3.
Determine the points, if any, where this function is not
differentiable.
(1989- 4 Marks)
12
816 Find the derivative of the following
function from first principle:
( -x )
12
817 The set of all points of differentiability of the function ( mathbf{f}(mathbf{x})=frac{sqrt{mathbf{x}+mathbf{1}}-mathbf{x}}{sqrt{mathbf{x}}} ) for ( mathbf{x} )
( neq 0 ) and ( mathrm{f}(0)=0 ) is
( mathbf{A} cdot(-infty, infty) )
B. ( [0, infty) )
( c cdot(0, infty) )
D. ( (-infty, infty) sim{0} )
12
818 For the function ( f(x)=frac{x^{100}}{100}+frac{x^{99}}{99}+ )
( +frac{x^{2}}{2}+x+1, f^{prime}(1)= )
( mathbf{A} cdot x^{100} )
в. 100
( c .10 )
D. None of these
12
819 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for the following
i) ( y=tan ^{-1}left(frac{3 x-x^{3}}{1-3 x^{2}}right),-frac{1}{sqrt{3}}<x< )
( frac{1}{sqrt{3}} )
ii) ( y=sin ^{-1}left(frac{1-x}{1+x}right), 0<x<1 )
12
820 f ( y=log sqrt{frac{1+tan x}{1-tan x}}, ) prove that ( frac{d y}{d x}= )
( sec 2 x )
12
821 If ( lim _{x rightarrow c} frac{f(x)-f(c)}{x-c} ) exists finitely, then
( mathbf{A} cdot lim _{x rightarrow c} f(x)=f(c) )
B. ( lim _{x rightarrow c} f^{prime}(x)=0 )
( mathbf{C} cdot lim _{x rightarrow c} f(x) ) does not exist
D. ( lim _{x rightarrow c} f(x) ) may or may not exist
12
822 Find derivative of ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} cos boldsymbol{x} ) 12
823 If ( f(x) ) be such that ( f(x)=max )
( left{|2-x|, 2-x^{3}right}, ) then
This question has multiple correct options
A continuous ( forall x in R )
[
f(x) text { is continuous } x in R
]
B. ( f(x) ) is differentiable ( forall x in R )
c. ( f(x) ) is non-differentiable at one point only
D. ( f(x) ) is non-differentiable at 4 points only
12
824 Find the differential of
( mathbf{y}=left(sin ^{-1} mathbf{x}right)^{2}+mathbf{A} cos ^{-1} mathbf{x}+mathbf{B} ) where
( A, B ) are arbitrary constants
12
825 If ( x y=tan ^{-1}(x y)+cot ^{-1}(x y), ) then ( frac{d y}{d x} )
is equal to
A ( cdot frac{y}{x} )
B. ( -frac{y}{x} )
c. ( frac{x}{y} )
D. ( -frac{x}{y} )
12
826 If ( boldsymbol{x}=operatorname{cost} ) and ( boldsymbol{y}=sin 4 boldsymbol{t} ) then ( (1- )
( left.boldsymbol{x}^{2}right) boldsymbol{y}_{2}-boldsymbol{x} boldsymbol{y}_{1}= )
A . ( 4 y )
в. ( -4 y )
c. ( 16 y )
D. ( -16 y )
12
827 Differentiate with respect to ( x ) :
( log left{cot left(frac{pi}{4}+frac{x}{2}right)right} )
12
828 Differentiate ( frac{x}{sin x} ) with respect to ( x )
A ( frac{sin x+x cos x}{sin ^{2} x} )
B. ( frac{sin x-x cos x}{sin ^{2} x} )
( frac{cos x+x cos x}{sin x} )
D. ( frac{cos x-x sin x}{sin x} )
12
829 For every pair of continuous functions
( boldsymbol{f}, boldsymbol{g}:[mathbf{0}, mathbf{1}] rightarrow boldsymbol{R} ) such that max
( {boldsymbol{f}(boldsymbol{x}): boldsymbol{x} in[mathbf{0}, mathbf{1}]}= )
( max {g(x): x in[0,1]}, ) the correct
statement (s) is (are)
A ( cdot(f(c))^{2}+3 f(c)=(g(c))^{2}+3 g(c) ) for some ( c in[0,1] )
B ( cdot(f(c))^{2}+f(c)=(g(c))^{2}+3 g(c) ) for some ( c in[0,1] )
( (f(c))^{2}+3 f(c)=(g(c))^{2}+g(c) ) for some ( c in[0,1] )
D ( cdot(f(c))^{2}=(g(c))^{2} ) for some ( c in[0,1] )
12
830 Find derivative of ( tan ^{-1} frac{cos x-sin x}{cos x+sin x} )
w.r.t. ( boldsymbol{x} )
A . -1
B.
c. 1
D.
12
831 Differential coefficient of ( log (sin x) ) with
respect to ( x ) is:
A . ( cot x )
B. ( operatorname{cosec} x )
( c cdot tan x )
D. sec ( x )
12
832 If ( y=sqrt{frac{1+sin x}{1-sin x}} ) then ( frac{d y}{d x}=? )
A ( cdot frac{1}{2} sec ^{2}left(frac{pi}{4}-frac{x}{2}right) )
B ( cdot frac{1}{2} csc ^{2}left(frac{pi}{4}-frac{x}{2}right) )
( ^{mathrm{c}} cdot frac{1}{2} csc left(frac{pi}{4}-frac{x}{2}right) cot left(frac{pi}{4}-frac{x}{2}right) )
D. none of these
12
833 f ( boldsymbol{y}=log (log sin boldsymbol{x}), ) then evaluate ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
834 If ( lim _{n rightarrow infty} frac{1^{a}+2^{a}+3^{a}+dots+n^{a}}{n^{a+1}}=frac{1}{5} )
( (w h e r e a>-1) ) then the value of ( a ) is
A .2
B. 3
( c cdot 4 )
D. 5
12
835 Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a
point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem:
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-mathbf{1} ) on ( [mathbf{2}, boldsymbol{3}] )
12
836 If ( y=tan ^{-1} sqrt{frac{1-cos x}{1+cos x}}, ) then for ( 0< )
( boldsymbol{x}<frac{boldsymbol{pi}}{mathbf{2}}, frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( mathbf{A} cdot 2 sec ^{2}(x / 2) )
B ( cdot frac{1}{2} sec ^{2}(x / 2) )
( c cdot frac{1}{2} )
D. ( -frac{1}{2} sec ^{2}(x / 2) )
12
837 Examine the applicability of Mean Value Theorem for the following function. ( f(x)=x^{2}-1 ) for ( x epsilon[1,2] ) 12
838 If ( f(x y)=f(x) . f(y) forall x, y in R ) If the
function is continuous at one point ( boldsymbol{x}= )
( a, ) then ( f(x) ) is:
A. continuous for all ( x in R-{0} )
B. continuous forall ( x in R )
c. discontinuous on
D. continuous at ( x=0 )
12
839 Differentiate with respective to ( x ) ( log (sec x+tan x) ) 12
840 If ( y ) and ( z ) are the functions of ( x ) and if ( boldsymbol{y}^{2}+boldsymbol{z}^{2}=boldsymbol{lambda}^{2}, ) then ( boldsymbol{y} frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{boldsymbol{y}}{boldsymbol{lambda}}right)+ )
( frac{d}{d x}left(frac{z^{2}}{lambda}right) ) is equal to
A ( cdot frac{z}{lambda} frac{d z}{d x} )
B. ( frac{z}{lambda} frac{d x}{d z} )
c. ( frac{lambda}{z} frac{d z}{d x} )
D. None of these
12
841 For ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{2} cdot sin boldsymbol{x}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
842 If the function ( boldsymbol{f}(boldsymbol{x})= )
( frac{boldsymbol{x}^{2}-(boldsymbol{A}+mathbf{2}) boldsymbol{x}+boldsymbol{A}}{boldsymbol{x}-mathbf{2}}, ) for ( boldsymbol{x} neq mathbf{2} ) and
( f(2)=2, ) is continuous at ( x=2, ) then
find the value of ( boldsymbol{A} ) ?
12
843 If ( f ) and ( g ) are differentiable functions
then ( D *(f g) ) is equal to
A. ( f D * g+g D * f )
В. ( D * f D * g )
C ( cdot f^{2} D * g+g^{2} D * f )
D ( cdot f(D * g)^{2}+g(D * f)^{2} )
12
844 Solve: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(operatorname{cosec} boldsymbol{x})=? ) 12
845 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{4}+boldsymbol{3} boldsymbol{x}+1,[-boldsymbol{2},-1] ) then
( A cdot f ) has exactly two zeros in [-2,-1]
B. f has exactly one zero in [-2,-1
C ( cdot ) f has at least one zero in [-2,-1]
D. f has no zero in [-2,-1]
12
846 ( operatorname{Let} f(x)=left{begin{array}{cc}1 /|x| & text { for }|x| geq 1 \ a x^{2}+b & text { for }|x|<1end{array} ) The right.
coefficients a and b so that fis
continuous and differentiable at any
point, are equal to
A. ( a=-1 / 2, b=3 / 2 )
в. ( a=1 / 2, b=-3 / 2 )
c. ( a=1, b=-1 )
D. none of these
12
847 For some constants ( a ) and ( b ) find the
derivative of
( left(a x^{2}+bright)^{2} )
12
848 Find the derivative of ( sec ^{-1}left(frac{x+1}{x-1}right)+ )
( sin ^{-1}left(frac{x-1}{x+1}right) )
A .
B.
( c cdot-1 )
D. ( frac{x+1}{x-1} )
12
849 If ( boldsymbol{x}^{boldsymbol{y}}=boldsymbol{e}^{boldsymbol{x}-boldsymbol{y}}, ) then find ( frac{d boldsymbol{y}}{d boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{1} ) 12
850 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{e}^{sin sqrt{tan boldsymbol{x}}} ) 12
851 Differentiate w.r.t. ( boldsymbol{x} )
( frac{boldsymbol{x}}{boldsymbol{y}^{3}}=mathbf{1} )
12
852 Find the second order derivatives of
( tan ^{-1} x )
12
853 Find ( frac{d y}{d x} ) if ( 3 x+4 y=9 ) 12
854 Find the values of ( a ) and ( b ) such that the
function defined by ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ccc}5, & text { if } & x leq 2 \ a x+b, & text { if } & 2<x<10 text { is } \ 21, & text { if } & x geq 10end{array}right. )
continuous function.
12
855 Show that the function ( boldsymbol{f}(boldsymbol{x})= )
( left[cos left(x^{10}+1right)right]^{1 / 3}, x in R ) is a continuous
function.
12
856 A non zero polynomial with real coefficients has the property that
( boldsymbol{g}(boldsymbol{x})=boldsymbol{g}^{prime}(boldsymbol{x}) cdot boldsymbol{g}^{prime prime}(boldsymbol{x}) cdot ) Let the leading
coefficient of ( g(x) ) be ( a ). Then ( 36 a= )
( A cdot 6 )
B. 4
( c .3 )
D.
12
857 If ( int_{0}^{y} frac{1}{sqrt{1+9 u^{2}}} d u=u, ) then ( frac{d^{2} y}{d u^{2}} ) is
( mathbf{A} cdot sqrt{1+9 y^{2}} )
В. ( frac{1}{sqrt{1+9 y^{2}}} )
c. ( 9 y )
D. ( 9 y^{2} )
12
858 Differentiate the function with respect
to ( x )
( 2 sqrt{cot left(x^{2}right)} )
12
859 If ( y=ln sqrt{frac{1-sin x}{1+sin x}} ) then ( frac{d y}{d x} ) equals-
( A cdot sec x )
B. – sec ( x )
( c cdot csc x )
( mathbf{D} cdot sec x tan x )
12
860 If ( boldsymbol{f}(boldsymbol{x})=sqrt{1+cos ^{2}left(boldsymbol{x}^{2}right)}, ) then
( f^{prime}left(frac{sqrt{pi}}{2}right) ) equal to
A ( cdot frac{sqrt{pi}}{6} )
в. ( -sqrt{frac{pi}{6}} )
c. ( frac{1}{sqrt{6}} )
D. ( frac{pi}{sqrt{6}} )
12
861 f ( x sin (a+y)=sin y, ) then ( y^{prime}=? ) 12
862 Solve:
( lim _{x rightarrow 0} frac{sin ^{3} x^{2}}{x^{6}} )
( mathbf{A} cdot mathbf{1} )
B.
( c cdot-1 )
D. ( infty )
12
863 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=log left(frac{1+boldsymbol{x}}{1-boldsymbol{x}}right) ) 12
864 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}= )
( sin ^{-1}left(frac{boldsymbol{6} boldsymbol{x}-boldsymbol{4} sqrt{boldsymbol{1}-boldsymbol{4} boldsymbol{x}^{2}}}{mathbf{5}}right) )
12
865 If ( y=frac{sin ^{-1} x}{sqrt{1-x^{2}}}, ) prove that ( (1- )
( left.x^{2}right) frac{d y}{d x}=(x y+1) )
12
866 Assertion ( f(x)=sin ^{2} x+sin ^{2}left(x+frac{pi}{3}right)+ )
( cos x cos left(x+frac{pi}{3}right) ) then ( f^{prime}(x)=0 )
Reason Derivative of a constant function is
always zero
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
867 Differentiate ( frac{x^{2}-1}{x} ) w.r.t ( x ) 12
868 ( frac{boldsymbol{d}^{2} boldsymbol{x}}{boldsymbol{d} boldsymbol{y}^{2}} ) equals
( mathbf{A} cdotleft(frac{d^{2} y}{d x^{2}}right)^{-1} )
B ( cdot-left(frac{d^{2} y}{d x^{2}}right)^{-1}left(frac{d y}{d x}right)^{-3} )
C ( cdotleft(frac{d^{2} y}{d x^{2}}right)^{-1}left(frac{d y}{d x}right)^{-2} )
D ( cdotleft(-frac{d^{2} y}{d x^{2}}right)^{-1}left(frac{d y}{d x}right)^{-2} )
12
869 If ( y=x sin y, ) then prove that ( frac{d y}{d x}= ) ( frac{boldsymbol{y}}{boldsymbol{x}(1-boldsymbol{x} cos boldsymbol{y})} ) 12
870 If ( f^{prime}(3)=2 ) then
( lim _{h rightarrow 0} frac{fleft(3+h^{2}right)-fleft(3-h^{2}right)}{2 h^{2}} ) is
( A )
B. 2
c. 37
( D cdot frac{1}{1} )
12
871 The function ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}2 x^{2}-1, & text { if } 1 leq x leq 4 \ 151-30 x, & text { if } 4<x leq 5end{array} ) is not right.
suitable to apply Rolle's theorem, since
B. ( f(1) neq f(5) )
c. ( f(x) ) is continuous only at ( x=4 )
D. ( f(x) ) is not differentiable in (4,5)
E. ( f(x) ) is not differentiable at ( x=4 )
12
872 ( boldsymbol{f}(boldsymbol{x})=sqrt{log _{1 / 2}left(frac{mathbf{5} boldsymbol{x}-boldsymbol{x}^{2}}{mathbf{4}}right)} ) 12
873 (d) For any illu
Letf:(0,7) →
be a twice differentiable function such tha:
f(x) sint-f(t)sinx = sina x for all x € (0,7).
lim
I-r
then which of the following statement(s)
12
(JEE Adv. 2018)
is (are) TRUE?
(6) f(x) <* _ xfor allx e (0,7)
© There exists a e (0, Tt) such that f'(a)=0
12
874 Differentiation gives us the instantaneous rate of change of one variable with respect to another
A. True
B. False
12
875 If ( f(x)=(x)^{frac{1}{x-1}} ) for ( x neq 1 ) and ( f ) is
continuous at ( mathbf{x}=1 ) then ( mathbf{f}(mathbf{1})= )
( A )
B . e-
( c cdot e^{-2} )
D. e ( ^{2} )
12
876 The value of ( c ) in Lagrange’s theorem for
the function ( |x| ) in the interval [-1,1] is
( mathbf{A} cdot mathbf{0} )
в. ( 1 / 2 )
c. ( -1 / 2 )
D. non existent in the interval
12
877 The function ( boldsymbol{f}(boldsymbol{x})= )
( frac{log (1+a x)-log (1-b x)}{x} ) is not
defined at ( x=0 . ) The value which
should be assigned to ( f ) at ( x=0 ) so that
it is continuous there, is
A ( . a-b )
B. ( a+b )
( mathbf{c} cdot log a+log b )
D. none of these
12
878 Verify Rolle’s theorem for ( boldsymbol{f}(boldsymbol{x})= ) ( x sqrt{a^{2}-x^{2}} ) in ( [0, a] ) 12
879 Diffrentiate w.r.t ( x: ) ( boldsymbol{y}=cos ^{-1}left[frac{1-x}{1+x}right] ) 12
880 For ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1} ; boldsymbol{x} in[-1,1], ) the
constant of Meanvalue theorem is
( mathbf{A} cdot mathbf{0} )
B. ( frac{1}{2} )
( c cdot-frac{1}{2} )
D.
12
881 28. Let f:R →R and g:
R R be two non-constant
differentiable functions. If
f'(x)=(e ^)-(x)))g'(x) for all x eR,
and f(1) = g(2)=1, then which of the following
statement (s) is (are) TRUE?
(JEE Adv. 2018)
(a) f(2)1-loge 2
(c) g(1) >1-loge 2 (d) g(1) <1-loge 2
12
882 If ( boldsymbol{y}=sin ^{-1}left(3^{-x}right), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A ( cdot frac{-log 3}{sqrt{3^{2 x}-1}} )
в. ( frac{3^{x} log 3}{sqrt{3^{2 x}-1}} )
c. ( frac{-3^{-x} log 3}{sqrt{3^{2 x}-1}} )
D. ( frac{log 3}{3^{x} sqrt{3^{2 x}-1}} )
12
883 If ( f(x)=[x sin pi x], ) then which of the
following is incorrect?
A. ( f(x) ) is continuous at ( x=0 )
B. ( f(x) ) is continuous in (-1,0)
c. ( f(x) ) is differentiable at ( x=1 )
D. ( f(x) ) is differentiable in (-1,1)
12
884 Value of ( c ) of Rolles theorem for ( boldsymbol{f}(boldsymbol{x})= )
( sin x-sin 2 x ) on ( [0, pi] )
( ^{mathbf{A}} cdot cos ^{-1}left(frac{1+sqrt{33}}{8}right) )
B. ( cos ^{-1}left(frac{1+sqrt{35}}{8}right) )
( ^{mathbf{c}} cdot cos ^{-1}left(frac{1-sqrt{38}}{5}right) )
D. does not exist
12
885 If ( f(x)=left{begin{array}{l}frac{1-sqrt{2} sin x}{pi-4 x} x neq frac{pi}{4} frac{pi}{4} \ a, x=frac{pi}{4}end{array}right. )
is continuous at ( x=frac{pi}{4} ) then ( a= )
( mathbf{A} cdot mathbf{4} )
B. 2
c. 1
D.
12
886 If ( f ) is a real valued function defined by
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{4} boldsymbol{x}+mathbf{3}, ) then find ( boldsymbol{f}^{prime}(mathbf{1}) ) and
( boldsymbol{f}^{prime}(boldsymbol{3}) )
12
887 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( frac{e^{2 x}+e^{-2 x}}{e^{2 x}-e^{-2 x}} )
12
888 Solve:
( frac{d y}{d x}=x^{2}(x-2), ) given ( y=2 ) where
( boldsymbol{x}=mathbf{0} )
12
889 ff ( y=e^{4 x}+2 e^{-x} ) satisfies the equation
( boldsymbol{y}_{3}+boldsymbol{A} boldsymbol{y}_{1}+boldsymbol{B} boldsymbol{y}=mathbf{0} ) then the value of
( A B ) is
12
890 Evaluate ( lim _{x rightarrow 4} frac{3-sqrt{5+x}}{1-sqrt{5-x}} )
A ( cdot frac{1}{3} )
B. ( -frac{1}{3} )
( c cdot frac{2}{3} )
D. ( -frac{2}{3} )
12
891 If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{t}-sin t) ) and ( boldsymbol{y}=boldsymbol{a}(boldsymbol{1}+cos boldsymbol{t}) )
then the value of ( y_{2} ) at ( t=frac{pi}{2} ) is
( A )
B . a ( ^{2} )
( c cdot frac{1}{a} )
D. ( frac{1}{a^{2}} )
12
892 ( y=left(frac{2^{x+1}}{1+4^{x}}right) ) Find ( frac{d y}{d x} ) 12
893 ( f(x)=left{begin{array}{ll}frac{x^{2}-4}{x-2} & x neq 2 \ 4 & x=2end{array} ) discus right.
continuity at ( boldsymbol{f}(boldsymbol{2}) )
12
894 If ( boldsymbol{y}=boldsymbol{x}^{2}+mathbf{5} boldsymbol{x} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
895 State True or False, Differentiating the equation of the curve at a point gives the slope of the tangent to the curve at that point.
A . True
B. False
12
896 The value of ( boldsymbol{f}(mathbf{0}) ) so that ( boldsymbol{f}(boldsymbol{x})= )
( frac{sqrt{1+x}-sqrt[3]{1+x}}{x} ) is continuous is
A ( cdot frac{1}{6} )
B. ( frac{1}{4} )
c. ( frac{1}{3} )
D. –
12
897 if ( y^{2}=a x+b x+c, ) then ( y^{3} frac{d^{2} y}{d x^{2}} ) is
A . a constant
B. a function of ( x ) only
c. a function of ( y ) only
D. a function of ( x ) and ( y )
12
898 If a function ( f ) satisfy ( fleft(frac{x+y}{3}right)= ) ( frac{mathbf{2}+boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})}{mathbf{3}} ) for real ( mathbf{x} ) and ( mathbf{y} )
( boldsymbol{f}^{prime}(2)=3 ) then ( boldsymbol{f}(boldsymbol{x}) ) is equal to
A ( cdot-frac{1}{12} x^{3}+x^{2} )
B . ( 24 log (3 x+2) )
( mathbf{c} cdot(3 x+2) )
D. ( frac{3}{4} x^{2}+2 )
12
899 Find the value of the constant ( k ) so that
the function given below is continuous
at ( boldsymbol{x}=mathbf{0} )
( boldsymbol{f}(boldsymbol{x})= )
( left{frac{1-cos 2 x}{2 x^{2}}, x neq 0 quad k x=0right} )
12
900 14. A value of c for which conclusion of Mean Value Theorem
holds for the function f(x) = log, x on the interval [1, 3] is
[2007]
(a) logze
(b) log3
(C) 2 logze
(d) = log;e
12
901 Let ( g ) is the inverse function of ( f ) and ( boldsymbol{f}^{prime}(boldsymbol{x})=frac{boldsymbol{x}^{mathbf{1 0}}}{left(mathbf{1}+boldsymbol{x}^{2}right)} cdot ) If ( boldsymbol{g}(boldsymbol{2})=boldsymbol{a} ) then ( boldsymbol{g}^{prime}(boldsymbol{2}) )
is equal to
A ( cdot frac{5}{2^{10}} )
в. ( frac{1+a^{2}}{a^{10}} )
c. ( frac{a^{10}}{1+a^{2}} )
D. ( frac{1+a^{10}}{a^{2}} )
12
902 Find the derivatives of the following
functions.
( log _{2}left(2 x^{2}-3 x+1right) )
12
903 If ( f(x)=frac{sin 4 x}{5 x}+a, quad ) for ( x>0 )
( =boldsymbol{x}+mathbf{4}-boldsymbol{b} quad ) for ( boldsymbol{x}<mathbf{0} )
( =1 ) for ( x=0 ) is continuous
( a t x=0, ) find ( a ) and ( b )
12
904 For some constants ( a ) and ( b ) find the
derivative of
( frac{x-a}{x-b} )
12
905 ( f(x)=left{begin{array}{cl}m x+1, & x leq frac{pi}{2} \ sin x+n, & x>frac{pi}{2}end{array} ) is right.
continuous at ( x=frac{pi}{2}, ) then
A ( . m=1, n=0 )
B. ( m=frac{n pi}{2}+1 )
c. ( n=frac{m}{2} )
D. ( m=n=frac{n pi}{2} )
12
906 Find the derivative of tan ( x ) using first principle of derivatives 12
907 If ( e^{x}+e^{y}=e^{x+y}, ) show that ( frac{d y}{d x}= )
( -e^{y-x} )
12
908 By using ( L M V T ), prove that ( frac{beta-alpha}{1+beta^{2}}<tan ^{-1} beta-tan ^{-1} alpha< )
( frac{beta-alpha}{1+alpha^{2}}, beta-alpha<0 )
12
909 If ( (cos x)^{y}=(cos y)^{x} ) then find ( frac{d y}{d x} ) 12
910 Write an example of a function which is everywhere continuous but fails to be differentiable exactly at five points. 12
911 Let ( y=x^{3}-8 x+7 ) and ( x=f(t) . ) If
( frac{d y}{d t}=2 ) and ( x=3 ) at ( t=0, ) then ( frac{d x}{d t} ) at
( t=0 ) is given by
( mathbf{A} cdot mathbf{1} )
B. ( frac{19}{2} )
( c cdot frac{2}{19} )
D. None of these
12
912 If ( sqrt{1-x^{6}}+sqrt{1-y^{6}}=aleft(x^{3}-y^{3}right) )
and ( frac{d y}{d x}=f(x, y) sqrt{frac{1-y^{6}}{1-x^{6}}} ) then
( begin{array}{ll}text { A } cdot f(x, y)=frac{y}{x} & text { B. } f(x, y)=frac{x^{2}}{y^{2}} \ text { c. } f(x, y)=2 frac{y^{2}}{x^{2}} & \ text { D. } f(x, y)=frac{y^{2}}{x^{2}} & end{array} )
12
913 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{mathbf{3} boldsymbol{x}+boldsymbol{4}}{mathbf{2} boldsymbol{x}-mathbf{3}}right) )
A ( cdot frac{17}{(2 x-3)^{2}} )
B ( cdot frac{1}{(2 x-3)^{2}} )
c. ( frac{-1}{(2 x-3)^{2}} )
D. ( frac{-17}{(2 x-3)^{2}} )
12
914 If ( boldsymbol{y}=sin ^{-1}left[operatorname{atan}^{-1} sqrt{frac{1-x}{1+x}}right] ) then find
( frac{d boldsymbol{y}}{d x} )
12
915 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}=sin boldsymbol{y} ) 12
916 f ( y(n)=e^{x} e^{x^{2}} ldots e^{x^{n}}, 0<x<1 ) then
( lim _{n rightarrow infty} frac{boldsymbol{d} boldsymbol{y}(boldsymbol{n})}{boldsymbol{d} boldsymbol{x}} ) at ( frac{1}{2} ) is
( A )
B. ( 4 e )
( c cdot 2 e )
D. ( 3 e )
12
917 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( sin boldsymbol{x}-boldsymbol{3} boldsymbol{x}=mathbf{5} boldsymbol{y} )
12
918 If ( 2^{x}+2^{y}=2^{x+y}, ) then ( frac{d y}{d x} ) has the value
equal to
This question has multiple correct options
A ( cdot-frac{2^{y}}{2^{x}} )
B. ( frac{1}{1-2^{x}} )
( mathbf{C} cdot 1-2^{y} )
D ( cdot frac{2^{x}left(1-2^{y}right)}{2^{y}left(2^{x}-1right)} )
12
919 Show that ( f(x)=(x-1) e^{x}+1 ) is an
increasing function for all ( x>0 )
12
920 The value of ‘c’ in Rolle’s theorem for the
function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{c}boldsymbol{x}^{2} cos left(frac{1}{boldsymbol{x}}right), boldsymbol{x} neq mathbf{0} \ mathbf{0}, boldsymbol{x}=mathbf{0}end{array}right. )
the interval [-1,1] is?
A ( cdot frac{-1}{2} )
B. ( frac{1}{4} )
( c cdot 0 )
D. Non-existent in the interval
12
921 Number of points where ( boldsymbol{f}(boldsymbol{x})=(1- )
( x)left|x-x^{2}right|+x ) is not differentiable is
A .
B.
( c cdot 2 )
D.
12
922 If ( boldsymbol{f}(boldsymbol{x})=left(frac{boldsymbol{x}^{a}}{boldsymbol{x}^{b}}right)^{boldsymbol{a}+boldsymbol{b}} cdotleft(frac{boldsymbol{x}^{b}}{boldsymbol{x}^{c}}right)^{b+c} cdotleft(frac{boldsymbol{x}^{c}}{boldsymbol{x}^{a}}right)^{c+a} )
then ( f^{prime}(x) ) is equal to
( mathbf{A} cdot mathbf{1} )
B.
( mathbf{c} cdot x^{a+b+c} )
D. None of these
12
923 If ( boldsymbol{y}=mathbf{2}^{2^{x}}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A ( cdot y(log 2)^{2} times 2^{x} )
В . ( y times 2(log 2) 2^{x} )
C ( cdot y times 2(log 2)^{2} times 2^{x} )
D. ( -y(log 2) times 2^{x} )
12
924 If ( y=log [tan x], ) find ( frac{d y}{d x} ) 12
925 If ( y^{2}=a x^{2}+b x+c, ) where ( a, b, c ) are
constants, then ( y^{3} frac{d^{2} y}{d x^{2}} ) is equal to.
A ( cdot frac{-1}{4} )
в. ( frac{-1}{6} )
c. ( frac{-3}{4} )
D. ( frac{-1}{8} )
12
926 Let ( [x] ) denote the greatest integer less
than or equal to ( x ). If ( f(x)=[x sin pi x] )
( operatorname{then} f(x) ) is :
This question has multiple correct options
A. continuous at ( x=0 )
B. continuous in (-1,0)
c. differentiable at ( x=1 )
D. differentiable in (-1,1)
12
927 Let ( f(x) ) be a continuous function which satisfies ( boldsymbol{f}left(boldsymbol{x}^{2}+mathbf{1}right)=frac{mathbf{2}}{boldsymbol{f}left(mathbf{2}^{x}right)-mathbf{1}} boldsymbol{&} )
( boldsymbol{f}(boldsymbol{x})>mathbf{0} forall boldsymbol{x} varepsilon boldsymbol{R} ) Then ( lim _{boldsymbol{x} rightarrow mathbf{1}} boldsymbol{f}(boldsymbol{x}) ) is
( A cdot 4 )
B. 2
( c )
D. does not exist
12
928 Verify the Rolle’s theorem for the following functions:
( f(x)=x^{4}-1 ) on the interval [-1,1]
A. True
B. False
12
929 Let ( x y=x+y ) then prove that ( frac{d y}{d x}+ ) ( frac{1}{(x-1)^{2}}=0 ) 12
930 ( y=(sin x)^{cos x}+(cos x)^{sin x}, ) find ( frac{d y}{d x} ) 12
931 Examine the functions for continuity:
( f x=left{begin{array}{ll}frac{sin 2 x}{sin 3 x}, & text { when } x neq 0 \ 2, & text { when } x=0end{array} text { at } x=0right. )
12
932 If ( boldsymbol{y}=tan ^{-1}left(cot left(frac{pi}{2}-xright)right), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( A )
B. -1
c. 0
D.
12
933 Find: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[frac{sin (boldsymbol{x}+boldsymbol{a})}{cos boldsymbol{x}}right] ) 12
934 ( fleft(x^{2}+y^{2}right)^{2}=x y, ) then ( left(frac{d y}{d x}right) ) is 12
935 For a real number ( boldsymbol{y},[boldsymbol{y}] ) denotes the
greatest integer less than or equal to ( y ) then ( f(x)=frac{tan (pi[x-pi])}{1+[x]^{2}} ) is
A. discontinuous at some ( x )
B. continuous at all ( x ), but ( f^{prime}(x) ) does not exist for same ( x )
( mathbf{C} cdot f^{prime}(x) ) exists for all ( x ) but ( f^{prime}(x) ) does not exist
D. ( f^{prime}(x) ) exists for all ( x )
12
936 The equation of a curve is ( y=frac{e^{2 x}}{4 x+1} )
and the point ( P ) on the curve has ( y )
coordinate 10.

Find the gradient of the curve at ( P )

12
937 35.
If the function f: [0,4] → Ris differentiable then show that
(1) For a, b € (0,4), (S(4))2 – (0))2=8f'(a)fb)
(ii) [ f(t)dt = 2[af (a?)+BF (B2)]0<a,ß < 2
12
938 If ( boldsymbol{f}(boldsymbol{a})=boldsymbol{a}^{2}, boldsymbol{phi}(boldsymbol{a})=boldsymbol{b}^{2} ) and ( boldsymbol{f}^{prime}(boldsymbol{a})= )
( mathbf{3} phi^{prime}(boldsymbol{a}) ) then ( lim _{x rightarrow 0} frac{sqrt{boldsymbol{f}(boldsymbol{x})}-boldsymbol{a}}{sqrt{boldsymbol{phi}(boldsymbol{x})}-boldsymbol{b}} ) is
( mathbf{A} cdot b^{2} / a^{2} )
в. ( b / a )
c. ( 2 b / a )
D. None of these
12
939 Range of ( boldsymbol{f}(boldsymbol{x}) ) is ( ? )
A. ( R )
B . ( R-{0} )
( c cdot R^{+} )
D. ( (0, e) )
12
940 If ( y=frac{sin ^{-1} x}{sqrt{1-x^{2}}}, ) prove that
( left(1-x^{2}right) frac{d y}{d x}-x y=1 )
12
941 Show that ( f(x)=x^{9}+4 x^{7}+11 ) is an
increasing function for all ( boldsymbol{x} in boldsymbol{R} )
12
942 ff ( y=x-x^{2} ), then the derivative of
( boldsymbol{y}^{2} boldsymbol{w} cdot boldsymbol{r} cdot boldsymbol{t} cdot boldsymbol{x}^{2} ) is
A ( cdot 2 x^{2}+3 x-1 )
B . ( 2 x^{2}-3 x+1 )
c. ( 2 x^{2}+3 x+1 )
D. None of these
12
943 Derivate ( e^{sqrt{2 x+1}} ) where ( x=12 ) w.r.t. ( x ) 12
944 Compute the value of ( theta ) in the first mean
value theorem ( boldsymbol{f}(boldsymbol{x}+boldsymbol{h})=boldsymbol{f}(boldsymbol{x})+ )
( boldsymbol{h} boldsymbol{f}^{prime}(boldsymbol{x}+boldsymbol{theta h}) ) if ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} )
A ( cdot frac{1}{2} )
в. ( frac{1}{3} )
( c cdot frac{1}{4} )
D. ( frac{1}{5} )
12
945 an
D
>00
24. f’O)= lim wy(9) and $0) = 0. Using this find
diem (n + 1}{cos- ” () “).cs-4
ICOS
nOC
(2004 – 2 Marks)
12
946 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ccc}x & text { if } & x text { is rational } \ 2-x & text { if } & x text { is irrational }end{array} ) Then right.
fof ( (x) ) is continuous
A. everywhere
B. no where
c. at all irrational ( x )
D. at all rational ( x )
12
947 Let ( lim _{x rightarrow a} f(x) ) exists but it is not equal to ( f(a) ). Then ( f(x) ) is discontinuous
at ( x=a ) and a is called a removable
discontinuity. If ( lim _{x rightarrow a^{-}} f(x)= )
( l ) and ( lim _{x rightarrow a^{+}} f(x)=m ) exist but ( l neq )
( m . ) Then a is called a jump
discontinuity. If one of the limits (left hand limit or right hand limit ) does not exist, then a is called an infinite discontinuity.
( operatorname{Let} f(x)left{begin{array}{cc}2|x|, & x leq-1 \ 2 x, & -1 leq x leq 0 \ x+1, & 01end{array} ) Then right.
( f(x) ) at
This question has multiple correct options
A. ( x=-1 ) is a removable discontinuity
B. ( x=0 ) is a jump discontinuity
c. ( x=1 ) is a removable discontinuity
D. ( x=-1 ) is a jump discontinuity
12
948 Differentiate:
( boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=tan boldsymbol{x}+boldsymbol{y} ? )
12
949 ( f(x)=left{begin{array}{ll}frac{1-cos x}{x^{2}}, & text { when } x neq 0 \ 1, & text { when } x=0end{array}right. )
then show that ( f(x) ) is discontinuous at
( boldsymbol{x}=mathbf{0} )
12
950 Solve the different equation:-
( left(tan ^{-1} y-xright) d y=left(1+y^{2}right) d x )
12
951 If ( f(x)=frac{e^{1 / x}-1}{e^{1 / x}+1}, x neq 0 ) and ( f(0)=0 )
then ( f(x) ) is
A. Continuous at 0
B. Right continuous at 0
c. Discontinuous at 0
D. Left continuous at 0
12
952 Express a in terms of b if the function defined by ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{a} boldsymbol{x}+mathbf{1} & , boldsymbol{x} leq mathbf{3} \ boldsymbol{b} boldsymbol{x}+mathbf{3} & boldsymbol{x}>mathbf{3}end{array}right} )
is continuous at ( x=3 )
12
953 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{x}+boldsymbol{y}^{2}=log boldsymbol{y}+boldsymbol{x}^{2} )
12
954 Let ( boldsymbol{f}(boldsymbol{x}) ) be defined in the interval [-2,2] such that ( f(x)= )
( left{begin{array}{ll}-1, & -2 leq x leq 0 \ x-1, & 0<x leq 2end{array} text { and } g(x)=right. )
( boldsymbol{f}(|boldsymbol{x}|)+|boldsymbol{f}(boldsymbol{x})| ) Test the differentiablity
of ( g(x) ) in (-2,2)
A. not derivable at ( x=0 ) and ( x=1 )
B. derivable at all points
c. not derivable at ( x=0 )
D. not derivable at ( x=1 )
12
955 If ( e^{x y}=y+sin ^{2} x, ) then at ( x=0, d y / d x )
is equal to
12
956 Find derivative of ( tan ^{-1} frac{cos x}{1+sin x} )
A ( cdot frac{1}{2} )
B. ( -frac{1}{2} )
( c cdot frac{3}{2} )
D. ( -frac{3}{2} )
12
957 Let ( boldsymbol{f}(boldsymbol{x}) ) be a polynomial in ( mathbf{x} . ) The
second derivative of ( fleft(e^{x}right) ) at ( x=1 ) is
( mathbf{A} cdot e f^{prime prime}(e)+f^{prime}(e) )
В ( cdotleft(f^{prime prime}(e)+f^{prime}(e)right) e^{2} )
c. ( e^{2} f^{prime prime}(e) )
D. ( left(f^{prime prime}(e) e+f^{prime}(e)right) e )
12
958 ( f(x)=left{begin{array}{ll}frac{1-sin ^{3} x}{3 cos ^{2} x}, & text { if } quad xfrac{pi}{2}end{array}right. )
so that ( f(x) ) is continuous at ( x=frac{pi}{2} )
then
This question has multiple correct options
A ( a=frac{1}{2} )
в. ( b=4 )
( c cdot a=1 )
D. ( b=-4 )
12
959 Let ( f ) be a twice differentiable such that
( boldsymbol{f}^{prime prime}(boldsymbol{x})=-boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{f}^{prime}(boldsymbol{x})=boldsymbol{g}(boldsymbol{x}) . ) If
( boldsymbol{h}(boldsymbol{x})={boldsymbol{f}(boldsymbol{x})}^{2}+{boldsymbol{g}(boldsymbol{x})}^{2}, ) where
( h(5)=11 . ) Find ( h(10) )
( mathbf{A} cdot mathbf{1} )
B . 10
c. 11
D. 100
12
960 If ( f(x)=|x|^{3} . ) show that ( f^{prime prime}(x) ) exists for
all real ( x ) and find it.
12
961 4. If y = cos(sin x2), then
then at x = 12
(a) -2
(b) 2
12
962 f ( boldsymbol{y}=sin boldsymbol{x} ) and ( boldsymbol{x} ) changes from ( boldsymbol{pi} / mathbf{2} ) to
( 22 / 14, ) what is the approximate change
in ( boldsymbol{y} ? )
12
963 ff ( y=log _{sin x}(tan x), ) then ( left(frac{d y}{d x}right)_{pi / 4} ) is
equal to
A ( cdot frac{4}{log 2} )
B. ( -4 log 2 )
c. ( frac{-4}{log 2} )
D. None of these
12
964 State true or false:
The differential coefficient of ( boldsymbol{f}(log boldsymbol{x}) ) w.r.t. ( log x ) where ( f(x)=log x ) is ( frac{1}{log x} )
A. True
B. False
12
965 Find ( boldsymbol{f}^{prime}(mathbf{0}) ) for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}}+ )
( boldsymbol{a}^{2} log (boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}}) )
A . – 2a
B. 2a
( c cdot-a )
( D )
12
966 If ( boldsymbol{x}=boldsymbol{a} cos ^{3} boldsymbol{theta} ) and ( boldsymbol{y}=boldsymbol{a} sin ^{3} boldsymbol{theta}, ) then find
the value of ( frac{d^{2} y}{d x^{2}} ) at ( theta=frac{n}{6} )
12
967 Check the continuity of the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{2}|+boldsymbol{x} ) 12
968 Let ( f(x) ) be a function such that ( lim _{x rightarrow 0} frac{f(x)}{x}=1 . ) If
[
lim _{x rightarrow 0} frac{x(1+a cos x)-b sin x}{{f(x)}^{3}}=1
]
then ( |a+b|= )
12
969 If the function ( f(x)=2 x^{2}+3 x+5 )
satisfies LMVT at ( x=2 ) on the
closed interval ( [1, a] ) then the value of ‘ ( a^{prime} )
is equal to
A . 3
B. 4
( c cdot 6 )
D.
12
970 ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} ) and ( boldsymbol{f}^{prime}(boldsymbol{pi}) )
A . -1
B.
c. 1
D. None of these
12
971 ff ( y=frac{sin 4 x}{x^{2}+16}, ) then find ( frac{d y}{d x} ) 12
972 f ( boldsymbol{y}=cot ^{-1}left[frac{sqrt{1+x^{2}}+1}{x}right], ) then find
the value of ( frac{d y}{d x} )
12
973 ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=cos boldsymbol{x}+sin boldsymbol{x} ) 12
974 If ( 4 a+2 b+c=0 ) then the equation
( 3 a x^{2}+2 b x+c=0 ) has at least one
real root lying between
A. 0 and 1
B. 1 and 2
c. 0 and 2
D. none of these
12
975 If ( boldsymbol{y}=boldsymbol{e}^{sin ^{2} boldsymbol{x}+sin ^{4} boldsymbol{x}+sin ^{6} boldsymbol{x}+ldots infty}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A ( cdot e^{tan ^{2} x} )
B. ( e^{tan ^{2} x} sec ^{2} x )
C ( cdot 2 e^{tan ^{2} x} tan x cdot sec ^{2} x )
D. 1
12
976 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}int_{0}^{x}{1+|1-t|} d t & text { if } x>2 \ mathbf{5 x – 7} & text { if } x leq 2end{array} ) then right.
A. ( f ) is not continuous at ( x=2 )
B. ( f ) is continuous but not differentiable at ( x=2 )
c. ( f ) is differentiable everywhere
D ( cdot f^{prime}(2+) ) doesn’t exist
12
977 Match the columns 12
978 Differentiate:
( boldsymbol{y}=boldsymbol{c}^{2}+frac{boldsymbol{c}}{boldsymbol{x}} )
12
979 47.
f(x) is a differentiable function and g(x) is a double
function such that f(x) <1 and f'(x) = g(x). Iff-(0)73
Prove that there exists some CE-3, 3) such
on and g(x) is a double differentiable
1 and f '(x)=g(x). Iff2(0)+7(0)=9.
some ce(-3, 3) such that
gc.g"C) <0.
(2005 – 6 Marks)
12
980 Find the value of ( c ) of Rolle’s theorem for
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) in [-1,1]
( mathbf{A} cdot mathbf{0} )
B.
( c cdot-1 )
D. does not exist
12
981 ( x=frac{(n+1)^{n}}{(n+2)} )
( frac{d x}{d n}=? )
12
982 The value of ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} int_{2}^{boldsymbol{x}^{2}}(boldsymbol{t}-mathbf{1}) boldsymbol{d} boldsymbol{t} )
A ( cdotleft(x^{2}-1right) )
В. ( xleft(x^{2}-1right) )
c. ( 2 xleft(x^{2}-1right) )
D. none of these
12
983 Derivative of which function is ( boldsymbol{f}^{prime}(boldsymbol{x})= )
( x sin x ? )
This question has multiple correct options
A. ( x sin x+cos x )
B. ( x cos x+sin x )
c. ( x sin left(frac{pi}{2}-xright)+cos left(frac{pi}{2}-xright) )
D. ( x cos left(frac{pi}{2}-xright)+sin left(frac{pi}{2}-xright) )
12
984 Given ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{[{|boldsymbol{x}|}] boldsymbol{e}^{boldsymbol{x}^{2}}{[boldsymbol{x}+{boldsymbol{x}}]}}_{left(frac{1}{boldsymbol{e}^{x^{2}}-1}right)} boldsymbol{s} boldsymbol{g n}(sin boldsymbol{x}) \ boldsymbol{0} quad text { for } quad boldsymbol{x}=mathbf{0}end{array}right. )
Where ( {x} ) is the fractional part
function; ( [x] ) is the step up function and
( operatorname{sgn}(x) ) is the signum function of ( x ) then,
( boldsymbol{f}(boldsymbol{x}) )
A. Is continuous at ( x=0 )
B. Is discontinuous at ( x=0 )
C. Has a removable discontinuity at ( x=0 )
D. Has in irremovable disconitnuity at ( x=0 )
12
985 If ( boldsymbol{y}=mathbf{s e c}^{-1}left[frac{sqrt{boldsymbol{x}}+mathbf{1}}{sqrt{boldsymbol{x}}-mathbf{1}}right]+ )
( sin ^{-1}left[frac{sqrt{boldsymbol{x}}-1}{sqrt{boldsymbol{x}}+1}right] ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( A )
в.
c.
D.
12
986 Show that the Lagrange’s mean value theorem is not applicable to the function ( boldsymbol{f}(boldsymbol{x})=frac{1}{boldsymbol{x}} ) on [-1,1] 12
987 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} sec boldsymbol{x}= )
( mathbf{A} cdot sec x tan x )
( mathbf{B} cdot cos x tan x )
( c cdot sin x tan x )
( mathbf{D} cdot sec x cot x )
12
988 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
( boldsymbol{y}=-boldsymbol{x}^{mathbf{3}}+boldsymbol{x} )
12
989 ( operatorname{Let} g(x)= )
( left{begin{array}{ll}2(x+1), & -infty<x leq-1 \ sqrt{1-x^{2}}, & -1<x<1 \ |x+1|, & 1 leq x<inftyend{array} ) then right.
A ( cdot g(x) ) is discontinuous at exactly three points
B . ( g(x) ) is continuous in ( (-infty, 1] )
C ( cdot g(x) ) is continuous in ( [1, infty) )
D. ( g(x) ) has finite type of discontinuity at ( x=1, ) but continuous at ( x=-1 )
12
990 Which of the following given statements
is/are not correct?
This question has multiple correct options
A ( cdot frac{d}{d x}(operatorname{cosec} x)=operatorname{cosec} x cdot cot x )
B. ( frac{d}{d x}(sec x)=sec x . tan x )
C ( cdot frac{d}{d x}(3 cot x)=-3 operatorname{cosec}^{2} x )
D ( cdot frac{d}{d x}(2 tan x)=-2 sec ^{2} x )
12
991 Differentiate the following function
w.r.t.x.
( frac{1}{left(x^{2}+3^{2}right)} )
12
992 Illustration 2.26
Find the derivative of y=-
.
+2 +1
12
993 Assertion
If ( boldsymbol{y}= )
( (1+x)left(1+x^{2}right)left(1+x^{4}right) ldotsleft(1+x^{2^{n}}right) )
then ( frac{d y}{d x} ) at ( x=0 ) is 1
Reason
( y=frac{1-x^{2^{n+1}}}{1-x} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
994 ff ( y=tan ^{-1}(3 x), ) then find ( frac{d^{2} y}{d x^{2}} ) 12
995 Left hand derivative and right hand
derivative of a function ( f(x) ) at a point ( x=a ) are defined as
( f^{prime}left(a^{-}right)=lim _{h rightarrow 0^{+}} frac{f(a)-f(a-h)}{h}= )
( lim _{h rightarrow 0^{-}} frac{f(a)-f(a-h)}{h}= )
( lim _{x rightarrow a^{+}} frac{f(a)-f(x)}{a-x} ) respectively
Let ( f ) be a twice differentiable function.
We also know that derivative of an even
function is odd function and derivative
of an odd function is even function.
If ( f ) is odd, which of the following is Lefthand derivative of ( f ) at ( x=a )
A ( cdot lim _{h rightarrow 0^{-}} frac{f(a-h)-f(a)}{-h} )
в. ( lim _{h rightarrow 0^{-}} frac{f(a-h)-f(a)}{h} )
c. ( lim _{h rightarrow 0^{+}} frac{f(a)+-f(a-h)}{-h} )
D. ( lim _{h rightarrow 0^{-}} frac{f(-a)-f(-a-h)}{-h} )
12
996 ( operatorname{Let} boldsymbol{F}(boldsymbol{x})=left|begin{array}{ccc}sin boldsymbol{x} & cos boldsymbol{x} & sin boldsymbol{x} \ cos boldsymbol{x} & -sin boldsymbol{x} & cos boldsymbol{x} \ boldsymbol{x} & boldsymbol{1} & boldsymbol{1}end{array}right| )
Which of the following statement hold true?
This question has multiple correct options
A ( cdot ) Range of ( F(x) ) is ( (-infty, infty) )
B ( cdot F^{prime}left(frac{pi}{2}right)= )
c. ( F(x) ) is bounded
D. ( F(x) ) is continuous and differentiable every where in its domain
12
997 If ( boldsymbol{y}=boldsymbol{x}+boldsymbol{e}^{boldsymbol{x}}, ) then what will be the
value of ( frac{d^{2} x}{d y^{2}} ? )
12
998 (x² + 2x)
Illustration 2.28
If y=
, then find
(3x – 4)
dx
12
999 18. Differentiation of 2×2 + 3x w.r.t. x is…
…fondra
12
1000 If ( boldsymbol{y}=sec ^{-1}left(frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}-mathbf{1}}right)+sin ^{-1}left(frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}right) )
then ( frac{d y}{d x} ) is equal to
A .
B. ( x+1 )
( c cdot 1 )
D. –
12
1001 Using the ahove approximation, the value ( sqrt{104} ) is
( mathbf{A} cdot 10.18 )
B. 10.49
c. 10.2
D. 10.28
12
1002 Assertion(A): Let ( boldsymbol{f}(boldsymbol{x}) ) be twice
differentiable function such that
( boldsymbol{f}^{prime prime}(boldsymbol{x})=-boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{f}^{prime}(boldsymbol{x})=boldsymbol{g}(boldsymbol{x}) . ) If
( boldsymbol{h}(boldsymbol{x})=[boldsymbol{f}(boldsymbol{x})]^{2}+[boldsymbol{g}(boldsymbol{x})]^{2} ) and ( boldsymbol{h}(mathbf{1})=mathbf{8} )
( operatorname{then} h(2)=8 )
Reason (R): Derivative of a constant
function is zero.
A. Both A and R are true R is correct reason of A
B. Both A and R are true R is not correct reason of A
c. A is true but R is false
D. A is false but R is true
12
1003 Let ( [t] ) denote the greatest integer ( leq t ) and ( lim _{x rightarrow 0} xleft[frac{4}{x}right]=A . ) Then the function,
( f(x)=left[x^{2}right] sin (pi x) ) is discontinuous,
when ( x ) is equal to:
( A cdot sqrt{A} )
B. ( sqrt{A+1} )
c. ( sqrt{A+5} )
D. ( sqrt{A+21} )
12
1004 f ( boldsymbol{y}=boldsymbol{f}left(boldsymbol{a}^{boldsymbol{x}}right) ) and ( boldsymbol{f}^{prime}(sin boldsymbol{x})=log _{e} boldsymbol{x}, ) then
find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if it exists, where ( frac{boldsymbol{pi}}{boldsymbol{2}}<boldsymbol{x}<boldsymbol{pi} )
12
1005 Differentiate the given function w.r.t. ( x ) ( frac{cos ^{-1} frac{x}{2}}{sqrt{2 x+7}},-2<x<2 ) 12
1006 ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}mathbf{3} boldsymbol{x}-mathbf{4}, & mathbf{0} leq boldsymbol{x} leq mathbf{2} \ mathbf{2} boldsymbol{x}+boldsymbol{lambda}, & mathbf{2}<boldsymbol{x} leq mathbf{3}end{array} . text { If } boldsymbol{f} ) is right.
continuous at ( x=2, ) then ( lambda ) is
( A )
B.
c. -2
( D )
12
1007 Ify=(x + V1+x? “, then (1+x) 2 x is 12002]
(b) – n’y
(d) 2xy
12
1008 5. f(x) = x2 – 3x, then the points at which f(x) = f(x) are
(a) 1,3
(b) 1, -3
(c)-1,3
(d) None of these
12
1009 Left hand derivative and right hand derivative of a function ( f(x) ) at a point ( boldsymbol{x}=boldsymbol{a} ) are defined as
( f^{prime}left(a^{-}right)=lim _{h rightarrow 0^{+}} frac{f(a)-f(a-h)}{h}= )
( lim _{h rightarrow 0^{-}} frac{f(a)-f(a-h)}{h}= )
( lim _{x rightarrow a^{+}} frac{f(a)-f(x)}{a-x} ) respectively
Let ( f ) be a twice differentiable function.
We also know that derivative of an even
function is odd function and derivative
of an odd function is even function
The statement
( lim _{h rightarrow 0} frac{f(-x)-f(-x-h)}{h}= )
( lim _{h rightarrow 0} frac{f(x)-f(x-h)}{-h} ) implies that for all
( mathbf{x} boldsymbol{epsilon} boldsymbol{R} )
( A . f ) is odd
B. ( f ) is even
c. ( f ) is neither odd nor ever
D. nothing can be concluded
12
1010 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(sqrt{boldsymbol{3}} sin left(boldsymbol{2} boldsymbol{x}+frac{boldsymbol{pi}}{boldsymbol{3}}right)+cos left(boldsymbol{2} boldsymbol{x}+frac{boldsymbol{pi}}{boldsymbol{3}}right)right) )
A. ( 4 cos 2 x )
B . ( -4 sin 2 x )
( c .4 sin 2 x )
D. ( -4 cos 2 x )
12
1011 If ( y=2 sin x-3 x^{4}+8, ) then ( frac{d y}{d x} ) is
B . ( 2 cos x-12 x^{3} )
( mathbf{c} cdot 2 cos x+12 x^{3} )
D. ( 2 sin x+12 x^{3} )
12
1012 ( boldsymbol{x}=sin boldsymbol{t} quad boldsymbol{y}=cos boldsymbol{m} boldsymbol{t} )
Prove ( :left(boldsymbol{1}-boldsymbol{x}^{2}right) boldsymbol{y}_{n+2}-(boldsymbol{2} boldsymbol{n}+boldsymbol{1}) boldsymbol{y}_{n+1}- )
( left(boldsymbol{n}^{2}-boldsymbol{m}^{2}right) boldsymbol{y}_{n}=boldsymbol{0} )
12
1013 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[frac{boldsymbol{operatorname} boldsymbol{operatorname { a n } boldsymbol { x }}-boldsymbol{operatorname { c o t } boldsymbol { x }}}{boldsymbol{operatorname { t a n } boldsymbol { x }}+boldsymbol{operatorname { c o t } boldsymbol { x }}}right]= )
( A cdot 2 sin 2 x )
B. – -2 ( sin 2 x )
( c cdot 2 cos 2 x )
D. ( -2 cos 2 x )
12
1014 If ( e^{y}+x y=e ) then at ( x=0, frac{d^{2} y}{d x^{2}}=e^{-lambda} )
then numerical quantity ( -lambda ) should be
equal to
A .2
B. 3
( c cdot 4 )
D. 5
12
1015 The differential coefficient of ( boldsymbol{f}left(log _{e} boldsymbol{x}right) )
with respect to ( x, ) where ( f(x)=log _{e} x )
is
A ( cdot frac{x}{log _{e} x} )
в. ( frac{1}{x} log _{e} x )
c. ( frac{1}{x log _{e} x} )
D. none of these
12
1016 Find ( boldsymbol{f}^{prime}(boldsymbol{3}) ) if ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+mathbf{5} boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}+mathbf{5} )
A . 28
B. 54
( c .32 )
D. None
12
1017 ( frac{boldsymbol{d}}{d x}left(e^{tan x}right) )
( mathbf{A} cdot e^{tan x} cdot sec ^{2} x )
B ( cdot e^{cot x} cdot sec ^{2} x )
( mathbf{C} cdot e^{cos x} cdot sec ^{2} x )
D ( cdot e^{sin x} cdot sec ^{2} x )
12
1018 Find the derivative of ( boldsymbol{y}= )
( n sqrt{frac{1-sin x}{1+sin x}} )
12
1019 If ( x=sin ^{-1} t ) and ( y=log left(1-t^{2}right), ) then
( frac{d^{2} y}{d x^{2}} ) at ( t=1 / 2 ) is
A ( cdot frac{-8}{3} )
в. ( frac{8}{3} )
( c cdot frac{3}{4} )
D. ( frac{-3}{4} )
12
1020 Find the derivative of the following functions (it is to be understood that ( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero
constants and ( m ) and ( n ) are integers) ( sec x-1 )
( sec x+1 )
12
1021 Find the derivative of ( x^{2} ) with respect to
( log boldsymbol{x} )
12
1022 Show that ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(tan boldsymbol{h}^{-1} boldsymbol{x}right)=frac{mathbf{1}}{mathbf{1 – x}^{mathbf{2}}} ) 12
1023 If ( y=ln sqrt{tan x} ) then the value of ( frac{d y}{d x} ) at ( x=frac{pi}{4} ) is
( A cdot infty )
B.
c.
D.
12
1024 Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a
point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem:
( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}-boldsymbol{x}^{2} ) on ( [mathbf{0}, mathbf{1}] )
12
1025 A metal sphere with radius of ( 10 mathrm{cm} ) is
to be covered with a ( 0.02 mathrm{cm} ) coating of
silver approximately silver required is
( left(operatorname{in} c m^{3}right) )
A ( .2 pi )
в. ( 10 pi )
( c .6 pi )
D. ( 8 pi )
12
1026 The number of real solutions of the
equation ( e^{x}=x ) is
( mathbf{A} cdot mathbf{1} )
B. 2
c. 0
D. none of these
12
1027 If ( S_{n} ) denotes the sum of ( n ) terms of ( g . p )
whose common ratio is ( r, ) then
( (r-1) frac{d S_{n}}{d r} ) is equal to
A ( cdot(n-1) S_{n}+n S_{n-1} )
B . ( (n-1) S_{n}-n S_{n-1} )
( mathbf{c} cdot(n-1) S_{n} )
D. None of these
12
1028 Differentiate the following w.r.t. ( x )
( left(2 x^{2}+9right)^{3} )
A ( cdot 4left(2 x^{2}+9right)^{2} x )
B. ( 12left(2 x^{2}+9right)^{2} x )
c. ( 12left(2 x^{2}+9right)^{3} x )
D. ( 6left(2 x^{2}+9right)^{2} x )
12
1029 If ( y=left(5 x^{3}-4 x^{2}-8 xright)^{9}, ) find ( frac{d y}{d x} ) 12
1030 ( boldsymbol{y}=cos ^{-1}left{frac{2 boldsymbol{x}-boldsymbol{3} sqrt{1-boldsymbol{x}^{2}}}{sqrt{mathbf{1 3}}}right}, ) find ( frac{boldsymbol{d} boldsymbol{2}}{boldsymbol{d}} ) 12
1031 Verify Rolle’s theorem for the function
( boldsymbol{y}=boldsymbol{x}^{2}+mathbf{2} . boldsymbol{x} boldsymbol{epsilon}|-mathbf{2}, mathbf{2}| )
12
1032 ( operatorname{Let} y=sqrt{x}+2 x^{frac{3}{4}}+3 x^{frac{5}{6}}(x>0) . ) Find
the derivative of ( y ) with respect to ( x )
12
1033 Solve ( : I_{n}=int_{0}^{frac{pi}{2}} e^{-x} sin ^{n} x d x ) 12
1034 Find the derivative ( x^{3} ) 12
1035 Find ( frac{d y}{d x}, ) when
( boldsymbol{x}=boldsymbol{a}(mathbf{1}-cos boldsymbol{theta}) ) and ( boldsymbol{y}=boldsymbol{a}(boldsymbol{theta}+sin boldsymbol{theta}) )
( operatorname{at} theta=frac{pi}{2} )
12
1036 If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{c}boldsymbol{a}^{2} cos ^{2} boldsymbol{x}+boldsymbol{b}^{2} sin ^{2} boldsymbol{x}, boldsymbol{x} leq mathbf{0} \ boldsymbol{e}^{boldsymbol{a} boldsymbol{x}+boldsymbol{b}}, boldsymbol{x}>mathbf{0}end{array}right. )
( f(x) ) is continuous at ( x=0 ) then
( mathbf{A} cdot 2 log |a|=b )
B. ( 2 log |b|=e )
c. ( log a=2 log mid b )
( mathbf{D} cdot a=b )
12
1037 For the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{6} boldsymbol{x}^{2}+ )
( boldsymbol{a} boldsymbol{x}+boldsymbol{b} . ) If Rolle’s theorem holds in ( [mathbf{1}, boldsymbol{3}] )
with ( c=2+frac{1}{sqrt{3}} ) then ( (a, b) )
A ( cdot(11,12) )
B . (11,11)
c. ( (11, text { any real value }) )
D. (any real value, 0 )
12
1038 Consider the function: ( f(-infty, infty) rightarrow )
( (-infty, infty) ) defined by ( boldsymbol{f}(boldsymbol{x})= )
( frac{x^{2}-a x+1}{x^{2}+a x+1}, 0<a<2 )
Which of the following is true?
A. ( f(x) ) is decreasing on (-1,1) and has a local minimum at ( x=1 )
B. ( f(x) ) is increasing on (-1,1) and has a local maximum at ( x=1 )
C. ( f(x) ) is increasing on (-1,1) and has neither a local maximum nor a local minimum at ( x=-1 )
D. ( f(x) ) is decreasing on (-1,1) and has neither a local maximum nor a local minimum at ( x=1 )
12
1039 The function ( f(x)=sin ^{-1}(tan x) ) is not
differentiable at-
A ( . x=0 )
В. ( x=-pi / 6 )
c. ( x=pi / 6 )
D. ( x=pi / 4 )
12
1040 Check continuity of the function ( x^{2}|x| ) at the origin. 12
1041 Find ( k, ) if ( f ) is continuous at ( x=0 ) wher
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}frac{16^{x}-2^{x}}{k^{x}-1} text { when } x neq 0 \ 3 text { when } x=0end{array}right. )
12
1042 Differentiate: ( sin ^{2} 3 x cdot tan ^{3} 2 x ) 12
1043 The ordered pair (a,b) such that ( f(x)= ) ( left{begin{array}{ll}frac{b e^{x}-cos x-x}{x} & , x>0 \ a & , x=0 \ frac{2 tan ^{-1}left(e^{x}right)-frac{pi}{4}}{x} & , x<0end{array}right. )
continuous at ( x-0 ) is
12
1044 If ( f(x)=left{begin{array}{ll}frac{1-cos x}{x^{2}}, & x<0 \ frac{1}{2} e^{x}, quad x geq 0end{array} ) then at right.
( boldsymbol{x}=mathbf{0}, boldsymbol{f} ) is
A. continuous
B. not continuous
( c . ) differentiable
D. none of these
12
1045 Find the derivative of the following functions from first principle ( frac{1}{x^{2}} ) 12
1046 If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin theta) ) and ( boldsymbol{y}=boldsymbol{a}(boldsymbol{1}+cos boldsymbol{theta}) )
then ( frac{d y}{d x} ) is
A . ( cot theta )
B. ( cot frac{theta}{2} )
( c cdot-cot frac{theta}{2} )
D. none of these
12
1047 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) when ( boldsymbol{y}=boldsymbol{x}^{boldsymbol{x} cos boldsymbol{x}}+left(frac{boldsymbol{x}^{2}+mathbf{1}}{boldsymbol{x}^{2}-mathbf{1}}right) ) 12
1048 [
f(x)=left{begin{array}{ll}
frac{1-sin ^{2} x}{3 cos ^{2} x}, & xfrac{pi}{2}
end{array}right.
]
then ( f(x) ) is continuous at ( x=frac{pi}{2} )
12
1049 ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}+mathbf{1}: & boldsymbol{x} leq mathbf{1} \ boldsymbol{3}-boldsymbol{a} boldsymbol{x}^{2}: & boldsymbol{x}>1end{array}right. )
Find the value of ( a ) if ( f ) is continuous at
( boldsymbol{x}=mathbf{1} )
12
1050 Consider the following statements in respect of the function ( boldsymbol{f}(boldsymbol{x})=sin left(frac{mathbf{1}}{boldsymbol{x}}right) )
for ( boldsymbol{x} neq mathbf{0} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} )
1. ( lim _{x rightarrow 0} f(x) ) exists
2. ( f(x) ) is continuous at ( x=0 )
Which of the above statements is/are
correct?
A. 1 only
B. 2 only
c. Both 1 and 2
D. Neither 1 nor 2
12
1051 Let ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y}), forall boldsymbol{x}, boldsymbol{y} in boldsymbol{R} )
suppose that ( boldsymbol{f}(mathbf{3})=mathbf{3}, boldsymbol{f}^{prime}(mathbf{0})=mathbf{1 1}, ) then
( f^{prime}(3) ) is given by
A .22
B. 44
c. 28
D. 33
12
1052 f ( mathbf{y}=boldsymbol{a} cos (log mathbf{x})+mathbf{b} sin (log mathbf{x}), ) then
( boldsymbol{x}^{2} boldsymbol{y}^{prime prime}+boldsymbol{x} boldsymbol{y}^{prime}= )
A. ( -y )
B.
( c cdot y )
D. ( y^{2} )
12
1053 If ( boldsymbol{y}= )
( (1+x)left(1+x^{2}right)left(1+x^{4}right) dotsleft(1+x^{2^{n}}right) )
then ( frac{d y}{d x} ) at ( x=0 ) is
( mathbf{A} cdot mathbf{1} )
B. – –
c. 0
D. none of these
12
1054 Let ( boldsymbol{f}=boldsymbol{R} rightarrow boldsymbol{R} ) be a continuous
function defined by ( boldsymbol{f}(boldsymbol{x})=frac{mathbf{1}}{boldsymbol{e}^{boldsymbol{x}}+mathbf{2} boldsymbol{e}^{-boldsymbol{x}}} )
Statement ( 1: f(c)=frac{1}{3}, ) for some ( c epsilon R ) Statement ( 2: 0<f(x) leq frac{1}{2 sqrt{2}}, ) for al
( boldsymbol{x} epsilon boldsymbol{R} )
12
1055 ( operatorname{Let} mathbf{f}(boldsymbol{x})=left{begin{array}{ll}frac{3 x+4 tan x}{x} & text { for } x neq 0 \ 7 & text { for } x=0end{array}right. )
( operatorname{then} f(x) ) is
A. continuous at ( x=0 )
B. not continuous at ( x=0 )
c. not determined at ( x=0 )
D. ( L t_{x rightarrow 0} f(x)=8 )
12
1056 If ( f(x) ) is continuous on ( [a, b] ) and ( boldsymbol{f}(boldsymbol{a}) neq boldsymbol{f}(boldsymbol{b}), ) then for
any value ( c ) belongs ( (f(a), f(b)), ) there is at least one number
( boldsymbol{x}_{o} ) in ( (boldsymbol{a}, boldsymbol{b}) ) for which ( boldsymbol{f}left(boldsymbol{x}_{boldsymbol{o}}right)=boldsymbol{c} )
A. ( fleft(x_{o}right)=b )
В. ( fleft(x_{o}right)=c )
c. ( fleft(x_{o}right)=f(c) )
D. ( fleft(x_{o}right)=0 )
12
1057 For ( boldsymbol{x} in boldsymbol{R}, boldsymbol{f}(boldsymbol{x})=|log 2-sin boldsymbol{x}| ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(boldsymbol{f}(boldsymbol{x})), ) then:
A. g is not differential at ( X=0 )
B ( cdot g^{prime}(0)=cos (log 2) )
C・ ( g^{prime}(0)=-cos (log 2) )
D. g is differentiable at ( x=0 ) and ( g^{prime}(0)=-sin (log 2) )
12
1058 Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be a continuous function such that ( boldsymbol{f}(boldsymbol{x})-mathbf{2} boldsymbol{f}left(frac{boldsymbol{x}}{mathbf{2}}right)+boldsymbol{f}left(frac{boldsymbol{x}}{mathbf{4}}right)=boldsymbol{x}^{mathbf{2}} )
The equation ( f(x)-x-f(0)=0 ) have
exactly:
A. no solution
B. one solution
c. two solutions
D. Infinite solutions
12
1059 Consider the functions, ( boldsymbol{f}(boldsymbol{x})=mid boldsymbol{x} )
( mathbf{2}|+| boldsymbol{x}-mathbf{5} mid, boldsymbol{x} in boldsymbol{R} )
Statement 1: ( boldsymbol{f}^{prime}(mathbf{4})=mathbf{0} )
Statement ( 2: f ) is continuous in [2,5] differentiable in (2,5) and ( f(2)=f(5) )
A. Statement 1 is false, Statement 2 is true
B. Statement 1 is true, Statement 2 is true; Statement 2 is correct explanation for Statement
c. statement 1 is true, statement 2 is true; Statement 2 is not a correct explanation for Statement 1
D. Statement 1 is true, Statement 2 is false
12
1060 The function ( boldsymbol{f}(boldsymbol{x})=frac{1+sin boldsymbol{x}-cos boldsymbol{x}}{1-sin boldsymbol{x}-cos boldsymbol{x}} )
is not defined at ( x=0 . ) The value of
( f(0) ) so that ( f(x) ) is continuous at ( x= )
( 0, ) is
( A )
B. –
( c cdot 0 )
D. none of these
12
1061 Find the derivative of ( frac{x}{(x-1)} ) 12
1062 Find the derivative of ( 99 x ) at ( x=100 ) 12
1063 31.
Consider the function, f(x)=bx-2+bx-51, XER.
Statement-1: f ‘(4)=0
Statement-2 :fis continuous in [2,5), differentiable in (2,5)
and f(2)=f(5).
[2012]
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, statement-2 is true; statement-2 is
a correct explanation for Statement-1.
Statement-1 is true, statement-2 is true; statement-2 is
not a correct explanation for Statement-1.
) Statement-1 is true, statement-2 is false.
12
1064 Illustration 2.31 Find the derivative of y=
– 21 Since
12
1065 ff ( boldsymbol{y}=boldsymbol{x}+frac{1}{boldsymbol{x}+frac{1}{boldsymbol{x}+frac{1}{boldsymbol{x}+cdots}}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A ( cdot frac{d y}{d x}=frac{x}{2 x-y} )
в. ( frac{d y}{d x}=frac{y}{2 y-x} )
c. ( frac{d y}{d x}=frac{2 y}{2 x-y} )
D. None of these
12
1066 0 000
Iffis a real valued differentiable function satisi
If(x)-) < (x – y)2, x,y e Randf(0) = 0, then f (1)
equals
[2005]
(a) -1 (6) 0 (0) 2 (d) 1.
12
1067 ( y=frac{2(x-sin x)^{frac{3}{2}}}{sqrt{x}}, ) find ( frac{d y}{d x} )
( mathbf{A} cdot frac{d y}{d x}=yleft{frac{3}{2} cdot frac{1-cos x}{x-sin x}-frac{1}{2 x}right} )
B. ( frac{d y}{d x}=yleft{frac{3}{4} cdot frac{1-sin x}{x-cos x}+frac{1}{2 x}right} )
( mathbf{c} cdot frac{d y}{d x}=yleft{frac{3}{4} cdot frac{1-cos x}{x-sin x}-frac{1}{2 x}right} )
D. ( frac{d y}{d x}=yleft{frac{3}{2} cdot frac{1-sin x}{x-cos x}-frac{1}{2 x}right} )
12
1068 f ( boldsymbol{y}=log sin boldsymbol{x}+boldsymbol{e}^{boldsymbol{x}} tan boldsymbol{x}, ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
1069 If ( y=sqrt{x}+frac{1}{sqrt{x}}, ) show that ( 2 x frac{d y}{d x}+ )
( boldsymbol{y}=mathbf{2} sqrt{boldsymbol{x}} )
12
1070 If ( y^{2}=4 a x, ) then ( frac{left(1+y_{1}^{2}right)^{3 / 2}}{y_{2}} ) at ( x=a )
is
A. ( -4 sqrt{2} a )
an ( a sqrt{2} ) an ( sqrt{2} )
B. ( 4 sqrt{2} a )
( c cdot frac{4 sqrt{2}}{a} )
D. ( -4 a )
12
1071 Show that the function ( f ) fiven below by:
( left{begin{array}{ll}frac{e^{1 / x}-1}{e^{1 / x}+1} & , text { if } x neq 0 \ -1 & , text { if } x=0end{array} ) is discontinuous at right.
( boldsymbol{x}=mathbf{0} )
12
1072 If the function ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}frac{1-cos 4 x}{x^{2}} & text { if } x0end{array}right. )
is continuous at ( x=0 ) then ( a= )
( mathbf{A} cdot mathbf{8} )
B.
( c .-8 )
( D )
12
1073 Find the value of ( ^{prime} a^{prime} ) for which the
function ( f ) defined by
( f(x)=left{begin{array}{l}a sin frac{pi}{2}(x+1), x leq 0 \ frac{tan x-sin x}{x^{3}}, x>0end{array}right. )
continuous at ( boldsymbol{x}=mathbf{0} )
12
1074 Consider the function for ( boldsymbol{x}=[-mathbf{2}, mathbf{3}] )
( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cc}boldsymbol{x}^{mathbf{3}}-mathbf{2} boldsymbol{x}^{mathbf{2}}-mathbf{5} boldsymbol{x}+mathbf{6} & boldsymbol{i} boldsymbol{f} boldsymbol{x} neq mathbf{1} \ boldsymbol{x}-mathbf{1} & boldsymbol{i} boldsymbol{f} boldsymbol{x}=mathbf{1}end{array}right. )
then
A. ( f ) is discontinuous at ( x=1 Rightarrow ) Rolles theorem is not
applicable in [-2,3]
B. ( f(-2) neq f(3) Rightarrow ) Rolles theorem is not applicable in [-2,3]
C. ( f ) is not derivable in (-2,3)( Rightarrow ) Rolles theorem is not applicable
D. Rolles theorem is applicable as ( f ) satisfies all the
conditions and ( c ) of Rolles theorem is
12
1075 Find the value of ( k ) if ( f(x)= ) ( left{begin{array}{ll}frac{1-cos 2 x}{1+cos 2 x}, & x neq 0 \ k, & x=0end{array} ) is continuous at right.
( boldsymbol{x}=mathbf{0} )
12
1076 Differentiate: ( e^{sin ^{-1} x} ) 12
1077 10. Let f(x) = 0, *<0
then for all x
, x20
(a) f' is differentiable (b) fis differentiable
(c) f'is continuous (d) fis continuous
12
1078 If ( boldsymbol{x}=cos ^{n} boldsymbol{theta}, boldsymbol{y}=sin ^{n} boldsymbol{theta} operatorname{then} frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}= )
A ( cdot frac{n}{n-1} cdot frac{cos ^{2 n-1} theta}{sin ^{n-3} theta} )
B. ( frac{n-2}{n} cdot frac{tan ^{n-3} theta}{cos ^{n+1} theta sin theta} )
c. ( n-1 . tan ^{n-2} theta cdot sec ^{2} theta )
D. ( quad n frac{sin ^{n-1} theta}{cos ^{n-2} theta} )
12
1079 If ( x sqrt{1+y}+y sqrt{1+x}=0, ) then ( frac{d y}{d x} ) is
equal to
( ^{A} cdot frac{1}{(1+x)^{2}} )
B. ( frac{-1}{(1+x)^{2}} )
c. ( frac{1}{(1-x)^{2}} )
D. None of these
12
1080 Differentiate the following from first principle. ( f(x)=cos left(x-frac{pi}{8}right) ) 12
1081 The value of ( c ) in lagrange’s theorem for
the function ( f(x)=log sin x ) in the interval ( left[frac{pi}{6}, frac{5 pi}{6}right] ) is
A ( cdot frac{pi}{4} )
в.
c. ( frac{2 pi}{3} )
D. None of these
12
1082 If, is continuous at ( x=2, ) find the value
of ( k )
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{boldsymbol{x}^{3}+boldsymbol{x}^{2}-mathbf{1 6} boldsymbol{x}+mathbf{2 0}}{(boldsymbol{x}-mathbf{2})^{2}}, & boldsymbol{x} neq mathbf{2} \ boldsymbol{k}, & boldsymbol{x}=mathbf{2}end{array}right. )
12
1083 If ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ccc}x^{alpha} cos left(frac{1}{x}right), & text { if } & x neq 0 \ 0 & , text { if } & x=0end{array}right. )
continuous at ( x=0 ) then
( mathbf{A} cdot alpha0 )
c. ( alpha=0 )
D. ( alpha geq 0 )
12
1084 If ( u ) and ( v ) are differentiable functions of
( x ) and if ( y=u+v ) then ( frac{d y}{d x}=frac{d u}{d x}+frac{d v}{d x} )
12
1085 Discuss the continuity of the following functions:
(a) ( f(x)=sin x+cos x )
(b) ( f(x)=sin x-cos x )
12
1086 Which of the following is(can be) continuous at each point of its
domain-
This question has multiple correct options
A ( . f(x) )
в. ( g(x) )
c. ( k(x) )
D. all three ( f, g, k )
12
1087 Let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}]^{2}+sqrt{{boldsymbol{x}}} ) where ( mathbb{D} boldsymbol{&} )
Orespectively denotes the greatest integer and fractional part functions, then which of the following is
correct?
A. ( f(x) ) is continuous at all integral points
B. ( f(x) ) is not differentiable ( forall x in I )
c. ( f(x) ) is discontinuous as ( x in I-{1} )
( f(x) ) is continuous ( & ) differentiable at ( x=0 ) 0
12
1088 37. Using Rolle’s theorem, prove that there is at least one root
in (451/100, 46) of the polynomial
P(x) = 51×101 – 2323(x)100 – 45x + 1035. (2004 – 2 Marks)
12
1089 Discuss the applicability of Rolle’s
theorem for the following function on the indicated interval:
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2 / 3} ) on [-1,1]
12
1090 In which of the following functions is Rolle’s theorem applicable
A ( , quad f(x)=left{begin{array}{l}x, 0 leq x<1 \ 0, x=1end{array} text { on }[0,1]right. )
в. ( quad f(x)=left{begin{array}{l}frac{sin x}{x},-pi leq x<0 \ 0, x=0end{array} text { on }[-pi, 0]right. )
c. ( f(x)=frac{x^{2}-x-6}{x-1} ) on [-2,3]
D ( quad f(x)=left{begin{array}{l}frac{x^{3}-2 x^{2}-5 x+6}{x-1}, text { if } x neq 1 \ -6, text { if } x=1end{array}right. )
12
1091 If ( y=a^{frac{1}{2} log _{a} cos x} . ) Find ( frac{d x}{d} ) 12
1092 Identify the graph of the polynomial function ( boldsymbol{f} )
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{boldsymbol{4}}+mathbf{1} )
begin{tabular}{|l|l|l|l|l|}
hline 1 & & & & & \
hline & & & & & \
hline & & & & & \
hline & & & & & \
hline & & & & ( mathrm{d} ) & & \
hline & & & & & & \
hline
end{tabular}
A. graph a
B. graph b
C. graph c
D. graph d
12
1093 Differentiate the following functions
with respect to ( x ) ( frac{sqrt{x^{2}+1}+sqrt{x^{2}-1}}{sqrt{x^{2}+1}-sqrt{x^{2}-1}} )
12
1094 The differential coefficient of
( a^{log _{10}left(operatorname{cosec}^{-1} xright)} ) is
A ( cdot frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathrm{x}} frac{1}{mathrm{x} sqrt{mathrm{x}^{2}-1}} log _{10} mathrm{a} )
( ^{mathrm{B}}-frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathrm{x}} frac{1}{|x| sqrt{mathrm{x}^{2}-1}} log _{10} mathrm{a} )
( ^{mathbf{C}} cdot frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathbf{x}} frac{1}{|x| sqrt{mathrm{x}^{2}-1}} log _{mathrm{a}} 10 )
D. ( frac{a^{log _{10}left(operatorname{cosec}^{-1} xright)}}{[operatorname{cosec}]^{-1} mathrm{x}} frac{1}{mathrm{x} sqrt{mathrm{x}^{2}-1}} log _{mathrm{a}} 10 )
12
1095 If ( boldsymbol{y}=boldsymbol{x}^{3} log left(frac{1}{x}right) . ) Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
1096 ( operatorname{Let} f(x)=left{begin{array}{cl}min left{x, x^{2}right}, & x geq 0 \ max left{2 x, x^{2}-1right}, & x<0end{array}right. )
Then which of the following is not true?
A. ( f(x) ) is continuous at ( x=0 )
B. ( f(x) ) is not differentiable at ( x=1 )
( mathrm{c} . f(x) ) is not differentiable at exactly three points
D. none of these
12
1097 Differentiate ( e^{sin x} )
( mathbf{A} cdot e^{sin x} cos x )
B. ( -e^{sin x} cos x )
( mathbf{c} cdot e^{-sin x} cos x )
( mathbf{D} cdot e^{cos x} sin x )
12
1098 Find ( lim _{x rightarrow 0} f(x), ) where ( f(x)= )
( left{begin{array}{ll}frac{x}{|x|}, & x neq 0 \ 0, & x=0end{array}right. )
12
1099 Let ( boldsymbol{f}(boldsymbol{x}) ) be a continuous function
whose range is ( [2,6,5] . ) If ( h(x)= ) ( left[frac{cos x+f(x)}{lambda}right], lambda in N ) be continuous
where [.] denotes the greatest integer
function, then the least value of ( lambda ) is
( A cdot 6 )
B. 7
c. 8
D. None of these
12
1100 For what triplets of real numbers ( (a, b, )
c) with ( a neq 0 ) the function ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cc}boldsymbol{x} & boldsymbol{x} leq mathbf{1} \ boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c} & text { otherwise }end{array}right. )
differentiable for all real x?
A ( cdot{(a, 1-2 a, a) / a in R, a neq 0} )
в. ( {(a, 1-2 a, c) / a, c in R, a neq 0} )
c. ( {(a, b, c) / a, b, c in R, a+b+c=1} )
D. ( {(a, 1=2 a, 0) / a in R, a neq 0} )
12
1101 Discuss the continuity of the following
function at ( x=0 . ) If the function has a
removable discontinuity, redefine the
function so as to remove the
discontinuity
( f(x)=left{begin{array}{ll}frac{4^{x}-e^{x}}{6^{x}}-1 & text { for } x neq 0 \ log left(frac{2}{3}right) & text { for } x=0end{array}right. )
12
1102 Examine the continuity of the function:
( f(x)=frac{log 100+log (0.01+x)}{3 x}, ) for
( boldsymbol{x} neq mathbf{0} )
( =frac{mathbf{1 0 0}}{mathbf{3}} quad ) for ( boldsymbol{x}= )
( mathbf{0} ; boldsymbol{a} boldsymbol{t} boldsymbol{x}=mathbf{0} )
12
1103 Let ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}x^{3}-x^{2}+10 x-5 quad, x leq 1 \ -2 x+log _{2}left(b^{2}-2right), x>1end{array} ) the set of right.
values of ( b ) for which ( f(x) ) has greatest
value at ( x=1 ) is given by:
A. ( 1 leq b leq 2 )
В . ( b={1,2} )
c. ( b in(-infty,-1) )
D ( cdot b in[-sqrt{130},-sqrt{2}] cup[sqrt{2}, sqrt{130}] )
12
1104 Which of the following functions is every where continuous-
A . ( x+|x| )
B . ( x-|x| )
c. ( x|x| )
D. All of the above
12
1105 A value of ( C ) for which the conclusion of
Mean Value Theorem holds for the
function ( f(x)=log _{e} x ) on the interval
[1,3] is
( mathbf{A} cdot 2 log _{3} e )
B. ( frac{1}{2} log _{3} e )
( mathbf{c} cdot log _{3} e )
D. ( 2 log _{e} 3 )
12
1106 If ( f(x)=operatorname{sgn}left(x^{5}right) ) then which of the
following is/are false (where sgn denotes signum function) This question has multiple correct options
( mathbf{A} cdot f^{prime}left(0^{+}right)=1 )
B. ( f^{prime}left(0^{-}right)=1 )
C. ( f ) is continuous but not differentiable at ( x=0 )
D. fis discontinuous at ( x=0 )
12
1107 Assertion
For the function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{3} boldsymbol{x}+boldsymbol{2} )
( boldsymbol{L} M boldsymbol{V} boldsymbol{T} ) is applicable in ( [mathbf{1}, boldsymbol{2}] ) and the
value of ( c ) is ( frac{3}{2} )
Reason
If ( L M V T ) is known to be applicable for
any quadratic polynomial in ( [a, b], ) then ( c ) of ( L M V T ) is ( frac{(a+b)}{2} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
1108 Let ( y=log (log (x)) ) then find ( frac{d y}{d x} ) 12
1109 ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b}, ) a,b being constants.
A ( cdot x frac{d^{2} y}{d x^{2}}-frac{d y}{d x}=0 )
в. ( x frac{d^{2} y}{d x^{2}}+frac{d y}{d x}=0 )
c. ( 2 x frac{d^{2} y}{d x^{2}}-frac{d y}{d x}=0 )
D. ( x frac{d^{2} y}{d x^{2}}-2 frac{d y}{d x}=0 )
12
1110 ff ( y=tan ^{-1}left[frac{sqrt{1+x^{2}}-sqrt{1-x^{2}}}{sqrt{1+x^{2}}+sqrt{1-x^{2}}}right] )
what would be ( frac{d y}{d x} )
A ( cdot frac{-x}{sqrt{1-x^{4}}} )
в. ( frac{1}{sqrt{1-x^{4}}} )
c. ( frac{x}{sqrt{1-x^{4}}} )
D. none of thes
12
1111 Find the value of ( frac{d y}{d x} quad ) a ( theta=frac{n}{4}, ) if ( x= )
( boldsymbol{a} e^{theta}(sin theta-cos theta) ) and ( y=a e^{theta}(sin theta- )
( cos theta) )
12
1112 Obtain the differential equation whose solution is
( boldsymbol{y}=boldsymbol{x} sin (boldsymbol{x}+boldsymbol{A}), mathbf{A} ) being constant
A ( cdotleft(x y_{1}-yright)^{2}+x^{2} y^{2}=x^{4} )
B. ( left(x y_{1}-yright)^{2}-x^{2} y^{2}=x^{4} )
c. ( left(x y_{1}-yright)^{2}+x^{2} y^{2}=x^{2} )
D. ( left(x y_{1}-yright)^{2}-x^{2} y^{2}=x^{2} )
12
1113 If ( boldsymbol{y}(boldsymbol{n})=boldsymbol{e}^{boldsymbol{x}} boldsymbol{e}^{boldsymbol{x}^{2}} ldots boldsymbol{e}^{boldsymbol{x}^{n}}, boldsymbol{0}<boldsymbol{x}<1 . ) Then
( lim _{n rightarrow infty} frac{boldsymbol{d} boldsymbol{y}(boldsymbol{n})}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=frac{mathbf{1}}{mathbf{2}} ) is
( A cdot e )
в. ( 4 € )
( c cdot 2 e )
D. ( 3 e )
12
1114 Solve:
( lim _{x rightarrow 3} frac{x^{2}-9}{x-3} )
12
1115 Let a function ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) satisfy the
equation ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y}) ) for
all ( x, y . ) If the function ( f(x) ) is
continuous at ( x=0, ) then
A. ( f(x)=0 ) continuous for all ( x )
B. ( f(x) ) is continuous for all positive real ( x )
c. ( f(x) ) is continuous for all ( x )
D. None of these
12
1116 Let ( mathbf{f}(mathbf{x}) ) be differentiable on the interval
( (0, infty) ) such that ( f(1)=1, ) and ( lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1 )
for each ( mathbf{x}>0 . ) Then ( mathbf{f}(mathbf{x}) ) is
A ( frac{1}{3 mathrm{x}}+frac{2 mathrm{x}^{2}}{3} )
B. ( -frac{1}{3 x}+frac{4 x^{2}}{3} )
c. ( -frac{1}{x}+frac{2}{x^{2}} )
( D cdot underline{1} )
12
1117 Differentiate ( log left(cos e^{x}right) ) w.r.t to ( x ) 12
1118 ( boldsymbol{f}(boldsymbol{x})=([boldsymbol{x}]-[-boldsymbol{x}]) boldsymbol{s i n}^{-1}|boldsymbol{x}-mathbf{1}| )
Which of the following statements is/are correct?
(Note : [.] denotes the greatest integer function)
This question has multiple correct options
A. ( f(x) ) is continuous at ( x=1 )
B. ( f(x) ) is differentiable at ( x=1 )
c. ( f(x) ) is not differentiable at ( x=1 )
D. ( f(x) ) is discontinuous at ( x=1 )
12
1119 Find ( frac{d y}{d x} ) of ( 2 x+3 y=sin x ) 12
1120 Let ( boldsymbol{f}(boldsymbol{x}) ) be a polynomial function of degree 2 and ( f(x)>0 ) for all ( x in R ) If ( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}^{prime}(boldsymbol{x})+boldsymbol{f}^{prime prime}(boldsymbol{x}), ) then for
any ( boldsymbol{x} )
A ( . g(x)0 )
( mathbf{c} cdot g(x)=0 )
D. ( g(x) geq 0 )
12
1121 Suppose that ( f(x)=x^{3}-3 x^{2}-4 x+12 )
( operatorname{and} h(x)=left{begin{array}{ll}frac{f(x)}{x-3} & x neq 3 \ K & x=3end{array}, ) then right.
find the value of ( mathrm{K} ) that makes
h’ continuous at ( x=3 )
12
1122 The set onto which the derivative of the
function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(log boldsymbol{x}-1) ) maps the
ray ( [1, infty) ) is ( ? )
A. ( [1, infty) )
(n)
в. ( (10, infty) )
( c cdot[0, infty) )
D. (0,0)
12
1123 Let ( f ) and ( g ) be differential functions
satisfying ( boldsymbol{g}^{prime}(boldsymbol{a})=mathbf{2}, boldsymbol{g}(boldsymbol{a}) boldsymbol{b} ) and ( boldsymbol{f} boldsymbol{o} boldsymbol{g}=boldsymbol{I} )
(identify function) then ( boldsymbol{f}^{prime}(boldsymbol{b})= )
A ( cdot 1 / 2 )
B . 2
( c cdot 2 / 3 )
D. None of these
12
1124 Find the derivative of the following functions (it is to be understood that
( a, b, c, d, p, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers) ( frac{boldsymbol{a}}{boldsymbol{x}^{4}}-frac{boldsymbol{b}}{boldsymbol{x}^{2}}+cos boldsymbol{x} )
12
1125 Prove that the difference of the infinite
continued fractions ( frac{1}{a+b+c} frac{1}{c+} dots, frac{1}{b+a+c+} dots ., ) is equal to
( frac{a-b}{1+a b} )
12
1126 If ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cc}frac{xleft(3 e^{1 / x}+4right)}{2-e^{1 / x}}, & x neq 0 text { then } f(x) \ 0 & , x=0end{array}right. )
A. ( f ) is not continuous
B. ( f ) is continuous but not differentiable at ( x=0 )
C ( cdot f^{prime prime}(0) ) exist
( mathbf{D} cdot f^{prime}(0+)=2 )
12
1127 If ( boldsymbol{f}(boldsymbol{x})= )
( frac{sin 3 x+A sin 2 x+B sin x}{x^{5}}, x neq 0 )
is continous at ( x=0 ) then
This question has multiple correct options
A. ( A=-4 )
B. ( B=5 )
c. ( f(0)=1 )
D. ( A=-3 )
12
1128 ( operatorname{Let} f(x)=left{begin{array}{ll}g(x) cdot cos frac{1}{x} & text { if } x neq 0 \ 0 & text { if } x=0end{array}right. )
where ( g(x) ) is an even function
differentiable at ( x=0, ) passing through
the origin. Then ( boldsymbol{f}^{prime}(mathbf{0}) )
A. is equal to 1
B. is equal to 0
c. is equal to 2
D. does not exist
12
1129 If ( y=frac{x^{2}}{2}+frac{1}{2} times sqrt{x^{2}+1}, ) then ( 2 y= )
( x y^{prime}, ) where ( y^{prime} ) denotes the derivative of ( y )
w.r.t. ( boldsymbol{x} )
A. True
B. False
12
1130 Verify Rolle’s Theorem for the function ( f(x)=e^{x}(sin x-cos x) ) on ( left[frac{pi}{4}, frac{5 pi}{4}right] ) 12
1131 Differentiate: ( left(sin ^{-1} x+frac{1}{2} log frac{1+x}{1-x}right) ) 12
1132 If ( y=tan ^{-1}left(frac{4 x}{1+5 x^{2}}right)+ )
( tan ^{-1}left(frac{2+3 x}{2-3 x}right), ) then ( frac{d y}{d x} ) is
( frac{1}{1+4 x^{2}} )
– s. ( frac{3}{1+4 x^{2}} )
( frac{5}{1+25 x^{2}} )
( frac{5}{left(1+25 x^{2}right)}-frac{1}{left(1+x^{2}right)}-frac{1.5}{left(1+2.25 x^{2}right)} )
12
1133 Suppose, ( A=frac{d y}{d x} ) of ( x^{2}+y^{2}=4 ) at ( (sqrt{2}, sqrt{2}), B=frac{d y}{d x} ) of ( sin y+sin x= )
( sin x cdot sin y operatorname{at}(pi, pi) ) and ( C=frac{d y}{d x} ) of
( 2 e^{x y}+e^{x} e^{y}-e^{x}=e^{x y+1} ) at ( (1,1), ) then
( (A-B-C) ) has the value equal to
A. ( frac{1}{2} )
в. ( frac{1}{3} )
c. 1
D. 2
12
1134 If ( y=log (log x) ) then ( frac{d^{2} y}{d x^{2}} ) is equal to
A ( cdot frac{-(1+log x)}{x^{2} log x} )
B. ( frac{(1+log x)}{x^{2} log x} )
c. ( frac{-(1+log x)}{(x log x)^{2}} )
D. ( frac{(1+log x)}{left(x^{2} log xright)^{2}} )
12
1135 ( lim _{x rightarrow 5} frac{2 x^{2}+9 x-5}{x+5} ) 12
1136 If the function ( f ) defined on ( left(frac{pi}{6}, frac{pi}{3}right) ) b ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{cl}frac{sqrt{mathbf{2}} cos boldsymbol{x}-mathbf{1}}{cot boldsymbol{x}-mathbf{1}}, & boldsymbol{x} neq frac{boldsymbol{pi}}{boldsymbol{4}} \ boldsymbol{k}, & boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{4}}end{array}right. )
continuous, then ( k ) is equal to?
( A cdot frac{1}{2} )
B.
( c cdot frac{1}{sqrt{2}} )
( D )
12
1137 ( mathbf{f} boldsymbol{y}=sin ^{-1}(mathbf{3} boldsymbol{x})+mathbf{s} mathbf{e} mathbf{c}^{-1}left(frac{mathbf{1}}{mathbf{3} boldsymbol{x}}right), ) find ( frac{boldsymbol{d} mathbf{2}}{boldsymbol{d}} ) 12
1138 Differentiate the following function with
respect to ( boldsymbol{x} )
( sin h^{-1}(sqrt{x}) )
12
1139 Find the values of ( k ) so that the function
( f ) is continuous at the indicated point:
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}
boldsymbol{k} boldsymbol{x}^{2}, & text { if } boldsymbol{x} leq boldsymbol{pi} \
cos boldsymbol{x}, & text { if } boldsymbol{x}>pi
end{array}right.
]
at ( boldsymbol{x}=boldsymbol{pi} )
12
1140 Find the second order derivative of the
following function:
( x^{3}+tan x )
12
1141 If ( boldsymbol{f}:[-boldsymbol{2}, boldsymbol{2}] rightarrow boldsymbol{R} ) is defined by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}frac{sqrt{1+e x}-sqrt{1-e x}}{x} & text { for }-2 leq x< \ frac{x+3}{x+1} & text { for } 0 leq x leqend{array}right. )
is continuous on ( [-2,2], ) then ( e= )
A ( cdot frac{2}{sqrt{3}} )
B. 3
( c cdot frac{3}{2} )
D. ( frac{3}{sqrt{2}} )
12
1142 The value of ( f(0) ) so that the function ( f(x)=frac{2 x-sin ^{-1} x}{2 x+tan ^{-1} x} ) is continuous at
each point in its domain, is equal to
A . 2
в. ( frac{1}{3} )
c. ( frac{2}{3} )
D. ( frac{-1}{3} )
12
1143 Consider the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cl}frac{tan k x}{x}, & x<0 \ 3 x+2 k^{2}, & x geq 0end{array} . ) What is the non- right.
zero value of k for which the function is
continuous at ( boldsymbol{x}=mathbf{0} ? )
A . ( 1 / 4 )
B. ( 1 / 2 )
c. 1
D.
12
1144 If ( y=left(x^{2}+1right) sin x ) then ( left(frac{pi}{2}right)^{2}- )
( y_{20}left(frac{pi}{2}right) ) is equal to
12
1145 Differentiate ( sin boldsymbol{h}^{-1}left(frac{mathbf{1}}{boldsymbol{x}}right) ) with respect
to ( boldsymbol{x}(boldsymbol{x}>mathbf{0}) )
12
1146 If ( f(x)=frac{sin 3 x}{sin x}, x neq 0 ) is continuous
[
=boldsymbol{K}, boldsymbol{x}=mathbf{0}
]
function, then ( boldsymbol{K}= )
A .
B. 3
( c cdot frac{1}{3} )
D.
12
1147 The constant ( c ) of Rolle’s theorem for the
function ( boldsymbol{f}(boldsymbol{x})=log frac{boldsymbol{x}^{2}+boldsymbol{a} boldsymbol{b}}{(boldsymbol{a}-boldsymbol{b}) boldsymbol{x}} ) in ( [boldsymbol{a}, boldsymbol{b}] )
where ( mathbf{0} notin[boldsymbol{a}, boldsymbol{b}] ) is
A. ( sqrt{a b} )
в. ( frac{a+b}{2} )
c. ( frac{a-b}{2} )
D. ( frac{b-a}{2} )
12
1148 Find ( mathrm{k} ) so that the function
[
begin{array}{cc}
boldsymbol{f}(boldsymbol{x})=left{1-cos 2 boldsymbol{x} / 2 boldsymbol{x}^{2}right. & boldsymbol{x} neq mathbf{0} \
{boldsymbol{k} & boldsymbol{x}=mathbf{0}
end{array}
]
is coutinous at ( x=0 )
12
1149 Evaluate:
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(boldsymbol{x}^{100}+boldsymbol{x}^{99}+boldsymbol{x}^{98}+ldots+boldsymbol{x}^{2}+boldsymbol{x}+right. )
( mathbf{1} )
12
1150 Among the following, the continuous function is ?
A. ( tan x )
B. ( sec x )
( c cdot sin 1 / x )
D. None of these
12
1151 ( f(x)=left{begin{array}{ll}cos x ; & x geq 0 \ x+k & ; x<0end{array} ) find the right.
value of ( k ) if ( f(x) ) is continuous at ( x=0 )
12
1152 Let ( f(x) ) be a continuous function
defined for ( 1 leq x leq 3 ). If ( f(x) ) takes
rational values for all ( x ) and ( f(2)=10 )
then the value of ( boldsymbol{f}(mathbf{1} . mathbf{5}) ) is
A . 7.5
B. 10
( c .5 )
D. none of these
12
1153 If ( f(x) ) is continuous for ( 0 leq x<infty )
then the most suitable values of ( a ) and ( b )
are
A ( . a=1, b=-1 )
B. ( a=-1, b=1+sqrt{2} )
c. ( a=-1, b=1 )
D. none of these
12
1154 Differentiate with respect to ( x ) :
( log (csc x-cot x) )
12
1155 f ( y=tan ^{-1}left(frac{1}{1+x+x^{2}}right)+ )
( tan ^{-1}left(frac{1}{x^{2}+3 x+3}right)+ )
( tan ^{-1}left(frac{1}{x^{2}+5 x+7}right)+—+ ) upto
n terms, then ( boldsymbol{y}^{prime}(mathbf{0})= )
A ( cdot frac{-1}{1+n^{2}} )
в. ( frac{-n^{2}}{1+n^{2}} )
c. ( frac{n^{2}}{1+n^{2}} )
D.
12
1156 Assertion
The function ( y=f(x), ) defined
parametrically as ( boldsymbol{y}=boldsymbol{t}^{2}+boldsymbol{t}|boldsymbol{t}|, boldsymbol{x}= )
( mathbf{2} t-|boldsymbol{t}|, boldsymbol{t} in boldsymbol{R}, ) is continuous for all rea
( boldsymbol{x} )
Reason
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}2 boldsymbol{x}^{2}, & boldsymbol{x} geq mathbf{0} \ mathbf{0}, & boldsymbol{x}<mathbf{0}end{array}right. )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
1157 The derivative of ( boldsymbol{y}=(1-boldsymbol{x})(2- )
( x) ldots(n-x) ) at ( x=1 ) is equal to
A . 0
в. (-1)( (n-1) ! )
c. ( n !-1 )
D ( cdot(-1)^{n-1}(n-1) ! )
E ( cdot(-1)^{n}(n-1) ! )
12
1158 Evaluate:
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{tan ^{-1} frac{boldsymbol{x}}{mathbf{1}+boldsymbol{x}^{2}}+tan ^{-1} frac{mathbf{1}+boldsymbol{x}^{2}}{boldsymbol{x}}right} )
( A )
в.
( c cdot frac{1}{2} )
( D )
12
1159 Suppose that ( f ) is differentiable
function with the property ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})= )
( f(x)+f(y)+x^{2} y^{2} ) and
( lim _{x rightarrow 0} frac{f(x)}{x}=10 ) then ( f^{prime}(0) ) is equal
to
12
1160 A point on the curve ( boldsymbol{f}(boldsymbol{x})=sqrt{boldsymbol{x}^{2}-boldsymbol{4}} )
defined in [2,4] where the tangent is
parallel to the chord joining two points
on the curve
A ( cdot(sqrt{2}, sqrt{6}) )
B . ( (sqrt{6}, sqrt{2}) )
C ( cdot(2,6) )
D ( cdot(6,2) )
12
1161 Let ( f(x)=left{begin{array}{cc}-2, & -3 leq x leq 0 \ x-2, & x<x leq 3end{array} ) and right.
( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(|boldsymbol{x}|)+mid boldsymbol{f}(boldsymbol{x}) )
Which of the following statements are correct?
1. ( g(x) ) is continuous at ( x=0 )
2. ( g(x) ) is continuous at ( x=2 )
3. ( g(x) ) is continuous at ( x=-1 )
Select the correct answer using the code given below
A. 1 and 2 only
B. 2 and 3 only
c. 1 and 3 only
D. 1,2 and 3
12
1162 If ( boldsymbol{y}=4 boldsymbol{x}^{4}+boldsymbol{2} boldsymbol{x}^{3}+frac{mathbf{5}}{boldsymbol{x}}+boldsymbol{9}, ) then find
( boldsymbol{d} boldsymbol{y} / boldsymbol{d} boldsymbol{x} )
12
1163 If ( f(x)=frac{log left(e^{x^{2}}+2 sqrt{x}right)}{tan sqrt{x}}, x neq 0, ) then
the value of ( f(0) ) so that ( f ) is
continuous at ( x=0 ) is
A ( cdot frac{1}{2} )
B. ( sqrt{2} )
( c cdot 2 )
D. ( frac{1}{sqrt{2}} )
12
1164 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} sqrt{cos boldsymbol{x}}= ) 12
1165 Differentiate :
( e^{e^{x}} )
12
1166 Differentiate ( boldsymbol{x}^{3} ) w.r.t ( boldsymbol{x} ) 12
1167 The function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{-|boldsymbol{x}|} ) is
A. continuous everywhere but not differentiable at ( x=0 )
B. continuous and differentiable everywhere
C. not continuous at ( x=0 )
D. None of the above
12
1168 For the discontinuous function given
below, find the value of ( boldsymbol{f}(-mathbf{3}) ) ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}^{2}+mathbf{1}, & boldsymbol{i f} quad boldsymbol{x}<mathbf{0} \ frac{boldsymbol{2} boldsymbol{x}}{mathbf{3}}-mathbf{1}, & boldsymbol{i f} quad boldsymbol{0}<boldsymbol{x}3end{array}right. )
( A cdot-3 )
B.
( c cdot 10 )
D. – 3, 7, and 10
12
1169 f ( boldsymbol{x}=sin t, boldsymbol{y}=sin boldsymbol{k} boldsymbol{t} ) satisfies
( left(1-x^{2}right) y_{2}-x y_{1}+A y=0 ) then ( A ) is
Equal to
( A cdot k )
B.
( mathbf{c} cdot k^{2} )
D. ( 1+k )
12
1170 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[(boldsymbol{x}+mathbf{1})left(boldsymbol{x}^{2}+mathbf{1}right)left(boldsymbol{x}^{4}+mathbf{1}right)left(boldsymbol{x}^{boldsymbol{8}}+right.right. )
1) ( ] )
( =frac{left(15 x^{p}-16 x^{q}+1right)}{(x-1)^{2}} Rightarrow(p, q)= )
A . (12,11)
B. (15, 14)
( c cdot(16,14) )
D. (16, 15)
12
1171 If ( y=A sin (omega t-k x), ) then the value of ( frac{d y}{d x} )
is
A. ( A cos (omega t-k x) )
B. ( -A omega cos (omega t-k x) )
c. ( A k cos (omega t-k x) )
D. ( -A k cos (omega t-k x) )
12
1172 ( fleft(x^{2}+y^{2}right)^{2}=x y, ) find ( frac{d y}{d x} ) 12
1173 If ( y=left[log log sin x^{circ}right]^{7}, ) find ( frac{d y}{d x} )
A ( cdot frac{6 pi}{180^{circ}}left[log left(log sin x^{circ}right)right]^{7} cdot frac{tan x^{circ}}{log sin x^{circ}} )
в. ( frac{7 pi}{180^{circ}}left[log left(log sin x^{circ}right)right]^{6} cdot frac{cot x^{circ}}{log sin x^{circ}} )
c. ( frac{pi}{30^{circ}}left[log left(log sin x^{circ}right)right]^{7} frac{cot x^{circ}}{log sin x^{circ}} )
D. none of these
12
1174 Differentiate ( sqrt{e^{sqrt{x}}}, x>0 ) 12
1175 ax
14. If y = a sin x + b cos x, then y2 +
(a) Function of x (b) Function of y
(c) Function of x and y (d) Constant
12
1176 The width of each of five continuous
classes in a frequency distribution is 5 and the lower class limit of the lowest
class is 10 The upper class limit of the highest class is
A . 25
B. 30
( c .35 )
D. 50
12
1177 If ( 27 a+9 b+3 c+d=0, ) then the
equation ( 4 a x^{3}+3 b x^{2}+2 c x+d=0 )
has atleast one real root lying between
A. 0 and 1
B. 1 and 3
c. 0 and 3
D. None
12
1178 If ( y=frac{5 x}{(1-x)^{2 / 3}}+cos ^{2}(2 x+1), ) find
( frac{d y}{d x} )
A. ( frac{5}{3(1-x)^{5 / 9}}(3-x)-2 sin (4 x+2) )
в. ( frac{-5}{3(1-x)^{5 / 9}}(3-x)-2 sin (4 x+2) )
c. ( frac{4}{3(1-x)^{5 / 9}}(3-x)-2 sin (4 x+2) )
D. ( frac{5}{3(1-x)^{5 / 9}}(3-x)-sin (4 x+2) )
12
1179 Find the derivative of the following functions (it is to be understood that
( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero constants and ( m ) and ( n ) are integers:
( (x+a) )
12
1180 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}[boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{g}(boldsymbol{x})]=boldsymbol{f}(boldsymbol{x}) frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{g}(boldsymbol{x})+ )
( g(x) frac{d}{d x} f(x) ) is known as ( _{–}—r u l e )
A. Product
B. Sum
c. Multiplication
D. None of these
12
1181 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{y}=log left(boldsymbol{e}^{boldsymbol{x}} sin ^{boldsymbol{5}} boldsymbol{x}right) ) 12
1182 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[sin ^{-1}left(frac{sqrt{mathbf{1}+boldsymbol{x}}+sqrt{mathbf{1}-boldsymbol{x}}}{mathbf{2}}right)right] mathbf{w} cdot mathbf{r} )
to ( x ) equals
A ( cdot frac{1}{2 sqrt{1-x^{2}}} )
в. ( frac{-2}{sqrt{1-x^{2}}} )
c. ( frac{-1}{2 sqrt{1-x^{2}}} )
D. None of these
12
1183 Let ( [x] ) denote the integral part of ( x in ) ( boldsymbol{R}, boldsymbol{g}(boldsymbol{x})=boldsymbol{x}-[boldsymbol{x}] . ) Let ( boldsymbol{f}(boldsymbol{x}) ) be any
continuous function with ( boldsymbol{f}(mathbf{0})=boldsymbol{f}(mathbf{1}) )
then the function ( h(x)=f(g(x)) )
A. has finitely many discontinuities
B. is discontinuous at some ( x=c, c in I )
c. is continuous on ( R )
D. is a constant function
12
1184 Differentiate the function with respect
to ( x )
( cos x^{3} cdot sin ^{2}left(x^{5}right) )
12
1185 Illustration 2.22 If y = 4x*+ 2x +
+ 9, then find dyldx.
12
1186 Show that the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cll}3-x, & text { if } & x1end{array}right. )
( boldsymbol{x}=mathbf{1} )
12
1187 Differentiate the following
( (x+2)^{3} )
12
1188 If ( y=sin ^{-1} x operatorname{th} e nleft(1-x^{2}right) frac{d^{2} y}{d x^{2}}= )
A ( cdot-x frac{d y}{d x} )
B.
c. ( _{x} frac{d y}{d x} )
D. ( xleft(frac{d y}{d x}right)^{2} )
12
1189 ( lim _{x rightarrow 0} frac{sqrt{frac{1}{2}(1-cos x)}}{x}= )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. does not exist
12
1190 Find the value of ( f(0) ) so that the
function ( boldsymbol{f}(boldsymbol{x})= )
( frac{mathbf{9 6}left[log left(1+frac{boldsymbol{x}}{mathbf{1 2}}right)-log left(mathbf{1}-frac{boldsymbol{x}}{mathbf{8}}right)right]}{boldsymbol{x}}, boldsymbol{x} neq )
0 is continuous on [0,8]
12
1191 Differentiate the following w.r.t. ( x )
( sin ^{2} sqrt{x} )
A ( cdot frac{1}{2 sqrt{x}} sin (3 sqrt{x}) )
B. ( frac{1}{sqrt{x}} sin (2 sqrt{x}) )
c. ( frac{1}{2 sqrt{x}} sin (2 sqrt{x}) )
D. ( frac{1}{2 sqrt{x}} sin (4 sqrt{x}) )
12
1192 The length ( x ) of a rectangle is decreasing at a rate of ( 3 mathrm{cm} / mathrm{min} ) and
width ( y ) is increasing at a rate of ( 2 c m / m i n . ) When ( x=10 c m ) and ( y= )
( 6 c m, ) find the rates of change of (i) the
perimeter
(ii) the area of the rectangle.
12
1193 Show that the function is continuous at
[
begin{array}{l}
x=0, text { if } f(x)=frac{sin 3 x}{tan 2 x}, x0 \
=frac{3}{2}, x=0
end{array}
]
12
1194 ( operatorname{Let} f(x)=2 tan ^{-1} x+sin ^{-1}left(frac{2 x}{1+x^{2}}right) )
Then
A ( cdot f^{prime}(2)=f^{prime}(3) )
B . ( f^{prime}(2)=0 )
( mathbf{c} cdot f^{prime}(1 / 2)=16 / 5 )
D. All of these
12
1195 If ( f(x)= )
[
frac{sin 3 x+A sin 2 x+B sin x}{x^{5}} quad(x neq 0)
]
continuous at ( x=0, ) then find ( A+B )
12
1196 For the function ( f(x)=e^{cos x}, ) Rolle’s
theorem is
A ( cdot ) applicable, when ( frac{pi}{2} leq x leq frac{3 pi}{2} )
B. applicable, when ( 0 leq x leq frac{pi}{2} )
C . applicable, when ( 0 leq x leq pi )
D. applicable, when ( frac{pi}{4} leq x leq frac{pi}{2} )
12
1197 If ( frac{1}{2}left(e^{y}-e^{-y}right)=x, ) prove that
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{1}{sqrt{boldsymbol{x}^{2}+mathbf{1}}} )
12
1198 If ( f(x) ) and ( g(x) ) are both continuous at
( x=c ) then which of the following is/are
always continuous at ( x=c ? )
This question has multiple correct options
A. ( f(x)+g(x) )
B. ( (f(x)-g(x)) times f(x) )
c. ( g(x) times f(x) )
D. ( frac{f(x)-g(x)}{g(x)} )
12
1199 Let ( boldsymbol{f}_{boldsymbol{p}}(boldsymbol{a})=boldsymbol{e}^{frac{i a}{p^{2}}} cdot boldsymbol{e}^{frac{2 i a}{p^{2}}} cdot boldsymbol{e}^{frac{3 i a}{p^{2}}} ldots cdot boldsymbol{e}^{frac{i a}{p}} )
(Where ( i=sqrt{-1} ) and ( p in )
( N) ) then ( lim _{n rightarrow infty} f_{n}(pi) )
( mathbf{A} cdot mathbf{1} )
B.
c. -1
D. ( -i )
12
1200 Differentiate from first principle: ( y=x^{2} )
( mathbf{A} cdot 2 x )
B . ( (x-1)^{2} )
( mathbf{c} cdot x^{3} )
D. ( frac{1}{sqrt{x}} )
12
1201 Differentiate with respect to ( x ) :
( e^{sin ^{-1} 2 x} )
12
1202 Verify Rolle’s theorem for the function ( f(x)=sin x+cos x-1 ) in the interval
( left[0, frac{pi}{2}right. )
12
1203 If ( f(x)=sin 2 x-cos 2 x, ) find ( f^{prime}left(frac{pi}{6}right) ) 12
1204 Consider the function
( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}a x-2 & text { for }-2<x<-1 \ -1 & text { for }-1 leq x leq 1 \ a+2(x-1)^{2} & text { for } quad 1<x<2end{array}right. )
What is the value of a for which ( f(x) ) is
continuous at ( x=-1 ) and ( x=1 ? )
( A cdot-1 )
B.
( c cdot 0 )
( D )
12
1205 If ( boldsymbol{x}=cos t ) and ( boldsymbol{y}=ln t ; ) then at ( boldsymbol{t}=frac{boldsymbol{pi}}{2} )
( left(frac{d^{2} y}{d x^{2}}+left(frac{d y}{d x}right)^{2}right) ) is equal to
A . 0
B. – –
( c .1 )
( D )
12
1206 If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a}, boldsymbol{a} in boldsymbol{R}, ) then
( mathbf{A} cdot nabla f(x)=0 )
B ( cdot nabla f(x)=a )
( mathbf{c} cdot nabla f(x)=2 a )
D. ( nabla f(x)=a^{2} )
12
1207 If ( y=tan ^{-1}left(frac{2 x}{1-x^{2}}right)+ )
( tan ^{-1}left(frac{3 x-x^{3}}{1-3 x^{2}}right) )
( tan ^{1}left(frac{4 x-4 x^{3}}{1-6 x^{2}+x^{4}}right) ) then Show that
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{mathbf{1}}{mathbf{1}+boldsymbol{x}^{2}} )
12
1208 If ( y=frac{x}{a+frac{x}{b+y}}, ) then ( frac{d y}{d x} ) is
( mathbf{A} cdot frac{a}{a b+2 a y} )
B. ( frac{b}{a b+2 b y} )
c. ( frac{a}{a b+2 b y} )
D. ( frac{b}{a b+2 a y} )
12
1209 The number of points at which the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{0 . 5}|+|boldsymbol{x}-mathbf{1}|+ )
( tan x ) does not have a derivative in the
interval (0,2) is/are?
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D.
12
1210 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{g}^{prime}(1)+boldsymbol{g}^{prime prime}(2) ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}(1) cdot boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{f}^{prime}(boldsymbol{x})+boldsymbol{f}^{prime prime}(boldsymbol{x}) ) then
This question has multiple correct options
A ( cdot f^{prime}(1)+f^{prime}(2)=0 )
B . ( g^{prime}(2)=g^{prime}(1) )
c. ( g^{prime prime}(2)+f^{prime prime}(3)=6 )
D. none of these
12
1211 Differentiate with respect to ( x ) :
( (log sin x)^{2} )
12
1212 llustration 2.29 If y
Illustration 2.29
sin x
If y = –
, then find
x + cos x
sin x then finden
dx
12
1213 Find the value of ( k ) if ( f(x) ) is continuous
at ( boldsymbol{x}=boldsymbol{pi} / 2, ) where ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{k cos x}{pi-2 x}, x neq pi / 2 \ 3, quad x=pi / 2end{array}right. )
12
1214 If ( boldsymbol{y}=mathbf{1}+boldsymbol{x}+frac{boldsymbol{x}^{2}}{mathbf{2 !}}+frac{boldsymbol{x}^{mathbf{3}}}{mathbf{3 !}}+ldots+frac{boldsymbol{x}^{boldsymbol{n}}}{boldsymbol{n} !} )
then ( frac{d y}{d x} ) is equal to
( mathbf{A} cdot underline{y} )
B. ( y+frac{x^{n}}{n !} )
c. ( y-frac{x^{n}}{n !} )
D. ( y-1-frac{x^{n}}{n !} )
12
1215 Differentiate
( -3 x^{2} cdotleft(sin 2 x^{3}right)left{cos left[cos ^{2}left(x^{3}right)right]right} )
12
1216 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[csc ^{-1}left(frac{sqrt{boldsymbol{2}}}{boldsymbol{x}-sqrt{mathbf{1}-boldsymbol{x}^{2}}}right)right] ) 12
1217 Find the inverse function of ( f(x)= )
( 2 x-3 )
12
1218 Differentiate with respect to ( x ) :
( sin ^{2}(log (2 x+3)) )
12
1219 A particle moves along a straight line such that its displacement ( s ) at any
time ( t ) is given by ( s=t^{3}-6 t^{2}+3 t+ )
( 4 m, t ) being is seconds. Find the velocity
of the particle when the acceleration is
zero.
12
1220 Differentiate the following function with
respect to ( x )
( left(2 x^{2}-3right) sin x )
A ( cdot 4 x sin x+left(2 x^{2}+3right) cos x )
B. ( 4 x sin x+left(2 x^{2}-3right) sin x )
c. ( 4 x sin x+left(2 x^{2}-3right) cos x )
D. ( 4 x cos x+left(2 x^{2}-3right) cos x )
12
1221 Differentiate the following function w.r.t. ( x: cos ^{-1}left(1-2 sin ^{2} xright) ) 12
1222 If the transformation ( z=log tan left(frac{x}{2}right) )
reduces the differential equation ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+cos boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+mathbf{4} boldsymbol{y} operatorname{cosec}^{2} boldsymbol{x}=mathbf{0} ) int
( frac{d^{2} y}{d x^{2}}+A y=0 ) then the value of ( A ) is
12
1223 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[cos ^{2}left(tan ^{-1}left(sin left(cot ^{-1} xright)right)right)right]= )
A. ( frac{2}{left(x^{2}+2right)^{2}} )
B. ( frac{2 x}{left(x^{2}+2right)^{2}} )
c. ( frac{left(x^{2}+1right)}{left(x^{2}+2right)} )
D. ( frac{-2 x}{left(x^{2}-1right)^{2}} )
12
1224 Find ( frac{d y}{d x}, ) if ( x=a(theta-sin theta) ) and ( y= )
( boldsymbol{a}(mathbf{1}-cos boldsymbol{theta}) )
12
1225 ( frac{d y}{d x} ) for ( y=tan ^{-1}{sqrt{frac{1+cos x}{1-cos x}}}, ) where
( mathbf{0}<boldsymbol{x}<boldsymbol{pi}, ) is?
( A cdot frac{-1}{2} )
B.
( c cdot 1 )
D.
12
1226 Which one of the following is correct in
respect of the function ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}^{2}}{|boldsymbol{x}|} ) for
( boldsymbol{x} neq mathbf{0} ) and ( boldsymbol{f}(mathbf{0})=mathbf{0} ? )
A. ( f(x) ) is discontinuous everywhere
B. ( f(x) ) is continuous everywhere
c. ( f(x) ) is continues at ( x=0 ) only
D. ( f(x) ) is discontinuous at ( x=0 ) only
12
1227 Find ( frac{d y}{d x} ) if ( y=frac{x^{2}+x}{2} ) 12
1228 Using LMV Theorem, find a point on the curve ( y=(x-3)^{2} ), where the tangent is
parallel to the chord joining (3,0) and
( (mathbf{5}, mathbf{4}) )
12
1229 Answer the following question in one
word or one sentence or as per exact
requirement of the question. Write the value of ( lim _{x rightarrow a} frac{x f(a)-a f(x)}{x-a} )
12
1230 If ( f(x)=frac{x-e^{x}+cos 2 x}{x^{2}}, x neq 0 ) is
continuous at ( x=0, ) then
A ( quad f(0)=frac{-5}{2} )
B . ( [f(0)]=-2 )
c. ( f(0)=-0.5 )
D. ( [f(0)] . f(0)=-1.5 )
12
1231 Find the continuity of ( f(x) ), If ( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ccc}boldsymbol{x}+mathbf{1} & boldsymbol{i f} & boldsymbol{x} geq mathbf{1} \ boldsymbol{x}^{2}+mathbf{1} & boldsymbol{i f} & boldsymbol{x}<mathbf{1}end{array}right} ) 12
1232 ff ( y=4 x-5 ) is a tangent to the curve
( boldsymbol{y}^{2}=boldsymbol{p} boldsymbol{x}^{3}+boldsymbol{q} ) at ( (boldsymbol{2}, boldsymbol{3}), ) then ( (boldsymbol{p}+boldsymbol{q}) ) is
equal to
A . -5
B. 5
( c .-9 )
D.
E .
12
1233 If ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ccc}sqrt{1+k x}-sqrt{1-k x} & text { for } & -1 leq x<0 \ x & text { for } 0 leq x<1end{array}right. )
is continuous at ( boldsymbol{x}=mathbf{0} ) then ( boldsymbol{k}= )
12
1234 If ( y=cos left(m cos ^{-1} xright), ) show that
( left(1-x^{2}right) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}+m^{2} y=0 )
12
1235 If ( f(1)=3 ) and ( f^{prime}(1)=-frac{1}{3} ) then the
derivative of ( left(x^{11}+f(x)right)^{-2} ) at ( x=1 ) is
A. ( -frac{1}{2} )
B. – –
c. 1
D. ( f^{prime}(1) )
12
1236 If ( x e^{x y}+y e^{-x y}=sin ^{2} x, ) then ( frac{d y}{d x} ) at
( boldsymbol{x}=mathbf{0} ) is
A ( cdot 2 y^{2}-1 )
в. ( 2 y )
c. ( y^{2}-y )
D. ( y^{2}+1 )
E ( cdot y^{2}-1 )
12
1237 NI-86
9. Let g(x) = log f(x) where f(x) is twice differ
function on (0, 0) such that sex
N=1,2,3,
te() is twice differentible positive
hat (x+1)=xf(x). Then, for
(2008)
1
431+
1
1
(h)
+
9
+….. +
25′
(2N
4
1
+
-+-
+….. +
0
25
12
1238 The value of the derivative of ( |boldsymbol{x}-mathbf{1}|+ )
( |x-3| ) at ( x=2 ) is:
( A cdot 2 )
B.
( c .0 )
D. – 2
12
1239 Verify Lagrange’s Mean Value Theorem for the function ( f(x)=x^{2}+x-1 ) in
the interval ( [mathbf{0}, mathbf{4}] )
12
1240 The velocity ( v ) of a particle is given by the equation ( v=6 t^{2}-6 t^{3}, ) where ( v ) is in
the ( m s^{-1}, t ) is the instant of time in
seconds while 6 and 6 are suitable
dimensional constants. At what values
of ( t ) will the velocity be maximum and minimum? Determine these maximum
and minimum values of the velocity.
12
1241 The value of ( f(0) ) so that the function ( f(x)=frac{sqrt{1+x}-(1+x)^{1 / 3}}{x} ) becomes
continuous is equal to
( A cdot frac{1}{6} )
B.
c. 2
D.
12
1242 If ( boldsymbol{y}=cot ^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right) )
then ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) is equal to
A .
B. 1/2
c. ( frac{1}{1+sin x} )
D. ( frac{1}{sqrt{1+sin x}}+frac{1}{sqrt{1-sin x}} )
12
1243 ( f(x)=[x] ) is a greatest integer
function,then it is continuous at?
( A cdot R )
B. z
( c . )
D. R-z
12
1244 ff ( y=sec ^{-1}left(frac{sqrt{x}+1}{sqrt{x}-1}right)+sin ^{-1}left(frac{sqrt{x}-1}{sqrt{x}+1}right) )
then find ( frac{d y}{d x} )
12
1245 Differentiate: ( boldsymbol{y}=8 sin boldsymbol{x} cos boldsymbol{x} ) w.r.t ( boldsymbol{x} ) 12
1246 Differentiate the following function with
respect to ( x ) ( boldsymbol{x}^{-4}left(boldsymbol{3}-boldsymbol{4} boldsymbol{x}^{-boldsymbol{5}}right) )
12
1247 If ( f(x)=sqrt{2} x+frac{4}{sqrt{2 x^{prime}}} ) then ( f^{prime}(2) ) is
equal to
( mathbf{A} cdot mathbf{0} )
B. –
c. 1
D. 2
12
1248 If ( y=tan ^{-1}left(frac{a cos x-b sin x}{b cos x+a sin x}right) ) then
( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=? )
( A cdot frac{a}{b} )
B. ( frac{-b}{a} )
c.
D. –
12
1249 The derivative of ( sin ^{-1} x ) with respect to
( cos ^{-1} sqrt{1-x^{2}} ) is?
A ( cdot frac{1}{sqrt{1-x^{2}}} )
B. ( cos ^{-1} x )
c. 1
D. 0
12
1250 What is derivative of ( left[frac{1}{x}right]^{x} ) 12
1251 If ( boldsymbol{y}=|cos boldsymbol{x}|+|sin boldsymbol{x}|, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}= )
( frac{2 pi}{3} ) is
A ( cdot frac{1-sqrt{3}}{2} )
B.
c. ( frac{sqrt{3}-1}{2} )
D. None of these
12
1252 Solve for ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{r}^{2} ) 12
1253 The value of ( boldsymbol{f}^{prime}(boldsymbol{3}) ) is
( A cdot 8 )
B. 10
c. 12
D. 18
12
1254 The displacement of ( S ) of a particle at time ( t(O<t<pi) ) is given by ( S= )
( sin 2 t-6 cos t . ) Then the acceleration for
the value of ( t ) for which its velocity is
zero is
A .0 unit/ ( sec ^{2} )
B. 3 unit/ sec ( ^{2} )
c. 2 unit ( mid sec ^{2} )
D. 4 unit/ ( sec ^{2} )
12
1255 Differentiate the following function with
respect to ( x )
( boldsymbol{x}^{-3}(mathbf{5}+mathbf{3} boldsymbol{x}) )
12
1256 Find the value of ( f(0) ) so that the
function ( boldsymbol{f}(boldsymbol{x})= )
( 8left(frac{1-cos ^{2} x+sin ^{2} x}{x^{2}}right), x neq 0 ) is
continuous
12
1257 10. If x=
1-2
1+72 and y=-
1+
2, the
12
1258 ( operatorname{Let} mathbf{f}(mathbf{x})=frac{sin mathbf{4} boldsymbol{pi}[mathbf{x}]}{mathbf{1}+[mathbf{x}]^{2}}, mathbf{w h e r e}[x] ) is the
greatest integer less than or equal to ( x ) then
A ( cdot f(x) ) is not differentiable at some points
B. ( f(x) ) exists but is different from zero
( c cdot f(x)=0 ) for all ( x )
D. ( f^{prime}(x)=0 ) but ( f ) is not a constant function
12
1259 f ( boldsymbol{y}=mathbf{1}+boldsymbol{x} cdot boldsymbol{e}^{boldsymbol{y}}, ) show that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{e}^{boldsymbol{y}}}{boldsymbol{2}-boldsymbol{y}} ) 12
1260 Find ( frac{d y}{d x} ) for ( y=log _{e}(x+sqrt{x^{2}-a^{2}}) ) 12
1261 Let ( f ) be a continuous function on ( mathrm{R} )
satisfying ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x}) boldsymbol{f}(boldsymbol{y}) ) for all
( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) and ( boldsymbol{f}(1)=boldsymbol{4} ) then ( boldsymbol{f}(boldsymbol{3}) ) is
equal to
12
1262 If ( boldsymbol{y}=(mathbf{1}+boldsymbol{x})left(mathbf{1}+boldsymbol{x}^{2}right)left(mathbf{1}+boldsymbol{x}^{4}right) dots . .(mathbf{1}+ )
( left.x^{2^{n}}right), ) find ( frac{d y}{d x} ) at ( x=0 )
( mathbf{A} cdot 2^{n} )
B.
( c .1 )
D. ( 2 n )
12
1263 The set of points where ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{mathbf{1}+|boldsymbol{x}|} )
is differentiable, is
( mathbf{A} cdot(-infty, 0) cup(0, infty) )
B ( cdot(-infty,-1) cup(-1, infty) )
( c cdot(-infty, infty) )
D・ ( (0, infty) )
12
1264 Let
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}boldsymbol{x}^{2} & text { if } boldsymbol{x} leq boldsymbol{x}_{0} \ boldsymbol{a} boldsymbol{x}+boldsymbol{b} & text { if } boldsymbol{x}>boldsymbol{x}_{0}end{array}right. )
The values of the coefficients a and b for which the function is continuous and
has a derivative at ( x_{0} ). are
A ( cdot a=x_{0}, b=-x_{0} )
B . ( a=2 x_{0}, b=-x_{0}^{2} )
C ( . a=x_{0}^{2}, b=-x_{0} )
D. ( a=x_{0}, b=-x_{0}^{2} )
12
1265 If ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}}{sqrt{boldsymbol{x}+1}-sqrt{boldsymbol{x}}} ) be a real
values function
then
( A cdot f(x) ) is continuous, but ( f^{prime}(0) ) does not exist
B. ( f(x) ) is differentiable at ( x=0 )
( mathrm{c} cdot mathrm{f}(mathrm{x}) ) is not continuous at ( x=0 )
D. ( f(x) ) is not differentiable at ( x=0 )
12
1266 If ( boldsymbol{y}=tan (boldsymbol{2} boldsymbol{x}+boldsymbol{3}) cdot ) Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
1267 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{cos boldsymbol{x}^{0}right}=? ) 12
1268 If Rolles theorem holds for the function
( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{3}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{c} boldsymbol{x}, boldsymbol{x} in[-1,1] ) at
the point ( x=frac{1}{2} ) then ( 2 b+c ) equals
( A )
B.
( c cdot 2 )
( D cdot-3 )
12
1269 n figure a square ( O A B C ) is inscribed
in a quadrant ( O P B Q ) of a circle. If
( O A=21 c m . ) find the area of the shaded
region
12
1270 Let ( boldsymbol{f} cdot boldsymbol{R} rightarrow boldsymbol{R} ) be defined as ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cl}0, & x text { is rational } \ sin |x|, & x text { is rational }end{array}right. )
Then which of the following is true?
A. ( f ) is discontinuous for all ( x )
B. ( f ) is continuous for all ( x )
c. ( f ) is discontinuous at ( x=k pi ), where ( k ) is an integer
D. ( f ) is continuous at ( x=k pi, ) where ( k ) is an integer
12
1271 Illustration 2.34 If x = ał”, y = bt”, then find ay
dx
This is nolladimnlinit differentina:
12
1272 If ( sin (x y)+frac{y}{x}=x^{2}-y^{2}, ) find ( frac{d y}{d x} ) 12
1273 If ( f(x)=b e^{a x}+a e^{b x}, ) then ( f^{prime prime}(0)= )
( mathbf{A} cdot mathbf{0} )
в. ( 2 a )
( mathbf{c} cdot a b(a+b) )
( mathbf{D} cdot a b )
12
1274 Differentiate the following functions
with respect to ( boldsymbol{x} )
( log {x+2+sqrt{x^{2}+4 x+1}} )
12
1275 Find the derivative of ( csc ^{2} x, ) by using
first principle of derivatives.
12
1276 Differentiate ( frac{x^{2} sin x}{1-x} ) w.r.t ( x ) 12
1277 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{c} boldsymbol{x}+boldsymbol{d} ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x}-boldsymbol{2} )
If ( lim _{x rightarrow 1} frac{f(x)}{g(x)}=1 ) and ( lim _{x rightarrow 2} frac{f(x)}{g(x)}=4, ) then
find the value of ( frac{c^{2}+d^{2}}{a^{2}+b^{2}} )
12
1278 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( tan ^{-1}left{frac{5 x}{1-6 x^{2}}right},-frac{1}{sqrt{6}}<x<frac{1}{sqrt{6}} )
12
1279 Say true or false.
If ( y=2 sec x, ) then ( frac{d y}{d x} ) is ( 2 sec x tan x )
A. True
B. False
12
1280 ( sum_{n=0}^{infty}(-1)^{n} frac{x^{2 n+1}}{2 n+1} ) is equal to
( (-1<x<1) )
A ( cdot tan ^{-1} x-x+c )
B ( cdot log (1+x) )
c. ( frac{1}{1-x}+frac{1}{1+x} )
( mathbf{D} cdot sin ^{-1} x )
12
1281 Solution of ( boldsymbol{y}^{2} boldsymbol{x}+boldsymbol{y}-boldsymbol{x} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=mathbf{0} ) is
A ( cdot frac{y^{2}}{5}+frac{y^{2}}{4 x^{2}}=c )
в. ( frac{x^{2}}{2}+frac{x}{y}=C^{prime} )
c. ( frac{y^{2}}{4}+frac{y^{5}}{5 x^{4}}=c )
D. ( frac{x^{2}}{5}+frac{x^{4}}{4 y^{4}}=c )
12
1282 Assertion(A): ( f(x)= ) ( left{begin{array}{ll}x^{2} sin left(frac{1}{x}right), & x neq 0 \ 0, & x=0end{array} ) is continuous at right.
( boldsymbol{x}=mathbf{0} )
Reason(R): Both ( h(x)=x^{2}, g(x)= )
( left{begin{array}{ll}sin left(frac{1}{x}right), & x neq 0 \ 0, & x=0end{array} ) are continuous at right.
( boldsymbol{x}=mathbf{0} )
A. Both A and R are true and R is the correct explanation of
B. Both A and R are true and R is not the correct explanation of
c. ( A ) is true but ( R ) is false
D. R is true but A is false
12
1283 Let ( f(x) ) be differentiable on the interval
( (0, infty) ) such that ( f(1)=1, ) and ( lim _{t rightarrow x} frac{t^{2} f(x)-x^{2} f(t)}{t-x}=1 ) for ( operatorname{each} x>0 )
Then ( f(x) ) is
A ( frac{1}{3 x}+frac{2 x^{2}}{3} )
в. ( -frac{1}{3 x}+frac{4 x^{2}}{3} )
c. ( -frac{1}{x}+frac{2}{x^{2}} )
D.
12
1284 Differentiate the following w.r.t. ( x )
( left(3 x^{2}+2right)left(4 x-3 x^{3}right) )
( mathbf{A} cdot 45 x^{4}+18 x^{2}+8 )
B. ( -45 x^{4}+18 x^{2}+8 )
c. ( -45 x^{4}+15 x^{2}+8 )
D. ( -45 x^{4}+18 x^{2}+18 )
12
1285 16-1) sinifx+1
25. Let (8) – )(x-1)sinif x+1
[2008)
x-1
0 if x=1
Then which one of the following is true?
(a) fis neither differentiable at x=0 nor at x=1
(b) fis differentiable at x=0 and at x=1
(C) fis differentiable at x =0 but not atx=1
(d) fis differentiable at x = 1 but not at x=0
i function with
12
1286 Discuss the continuity and differentiability of ( boldsymbol{f}(boldsymbol{x})=|log | boldsymbol{x} | ) 12
1287 If ( y=sin (sin x), ) then prove that ( frac{d^{2} y}{d x^{2}}+ )
( tan x frac{d y}{d x}+y cos ^{2} x=0 )
12
1288 41.
For a twice differentiable function f (x), g(x) is defined as
g(x) = (f ‘(x)2 + f'(x)) f(x) on [a, e]. If for
a<b<c<d<e, f(a) = 0, f (b) = 2, f (C) = -1, f (d) = 2,
f(e)=0 then find the minimum number of zeros of g(x).
(2006 – 6M)
12
1289 Differentiate the following functions
with respect to ( boldsymbol{x} ) ( cos ^{-1}left{frac{cos x+sin x}{sqrt{2}}right},-frac{pi}{4}<x<frac{pi}{4} )
12
1290 The function ( mathbf{f}(mathbf{x})=frac{cos mathbf{3} boldsymbol{x}-cos mathbf{4} boldsymbol{x}}{boldsymbol{x} sin mathbf{2} boldsymbol{x}} ) for
( neq mathbf{0}, mathbf{f}(mathbf{0})=frac{mathbf{7}}{mathbf{4}} mathbf{a t} boldsymbol{x}=mathbf{0}, ) is
This question has multiple correct options
A. Continuous
B. discontinuous
c. left continuous
D. right continuous
12
1291 If ( y=tan ^{-1}left(frac{2^{x}}{1+2^{2 x+1}}right), ) then ( frac{d y}{d x} ) at
( boldsymbol{x}=mathbf{0} ) is?
A ( cdot frac{1}{10} log 2 )
B. ( frac{1}{5} log 2 )
c. ( -frac{1}{10} log 2 )
D. ( log 2 )
12
1292 ( operatorname{Let} f(x)=left{begin{array}{c}sin x, quad text { for } x geq 0 \ 1-cos x, & text { for } x<0end{array} ) and right.
( g(x)=e^{x} ). Then ( (g o f)^{prime}(0) ) is
12
1293 ( f(x)=left{begin{array}{cc}1-cos x & x neq 0 \ x & \ k & x=0end{array}right} )
continuous at ( x=0, ) then the value of ( k )
is:
A . 0
( B cdot frac{1}{2} )
( c cdot frac{1}{4} )
( D cdot-frac{1}{2} )
12
1294 Verify Lagrange’s Mean Value Theorem for the following function:
( boldsymbol{f}(boldsymbol{x})=2 sin boldsymbol{x}+sin 2 boldsymbol{x} ) on ( [mathbf{0}, boldsymbol{pi}] )
12
1295 Differentiate the following function with
respect to ( x ) ( frac{4 x+5 sin x}{3 x+7 cos x} )
12
1296 If ( boldsymbol{y}=cot ^{-1}left(frac{1-x}{1+x}right) ) then ( frac{d y}{d x}=? )
A ( cdot frac{-1}{left(1+x^{2}right)} )
B. ( frac{1}{left(1+x^{2}right)} )
c. ( frac{1}{left(1+x^{2}right)^{3 / 2}} )
D. none of these
12
1297 ( f(x) ) is defined as under: ( f(x)= ) ( left{begin{array}{cc}a x(x-1)+b, & x3end{array}right. )
( f^{prime}(x) ) is discontinuous at ( x=3 . ) Then ( boldsymbol{a} neq boldsymbol{k}, boldsymbol{b}=boldsymbol{m}, boldsymbol{c}=frac{mathbf{1}}{boldsymbol{h}}, boldsymbol{d}=-boldsymbol{p} . ) Find ( boldsymbol{k}+ )
( boldsymbol{m}+boldsymbol{h}+boldsymbol{p} ? )
12
1298 ( y=sin left(2 sin ^{-1} xright), frac{d y}{d x}= )
A ( cdot sqrt{left(frac{1-y^{2}}{1-x^{2}}right)} )
в. ( sqrt[2]{left(frac{1+y^{2}}{1-x^{2}}right)} )
( ^{mathrm{c}} cdot sqrt{left(frac{1-y^{2}}{1+x^{2}}right)} )
D. ( sqrt{left(frac{1+y^{2}}{1+x^{2}}right)} )
12
1299 Identify a possible graph for function ( boldsymbol{f} )
given by
( boldsymbol{f}(boldsymbol{x})=-sqrt{(boldsymbol{x}-mathbf{1})}-mathbf{1} )
A. graph a
B. graph b
c. graph c
D. graph d
12
1300 Differentiate ( -frac{4 x+5 sin x}{3 x+7 cos x} ) w.r.t ( x ) 12
1301 Differentiate:
( frac{d}{d x}left(tan ^{-1} xright) )
12
1302 The value of ( c ) in Lagrange’s theorem for
the function in the interval [-1,1] is
[
f(x)=left{begin{array}{cl}
x cos left(frac{1}{x}right), & x neq 0 \
0, & x=0
end{array}right.
]
( mathbf{A} cdot mathbf{0} )
B. ( frac{1}{2} )
( c )
[
-frac{1}{2}
]
D. Non existent in the interva
12
1303 If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
1304 If for a continuous function ( boldsymbol{f}, boldsymbol{f}(mathbf{0})= )
( boldsymbol{f}(1)=mathbf{0}, boldsymbol{f}^{prime}(1)=mathbf{2} ) and ( boldsymbol{g}(boldsymbol{x})= )
( fleft(e^{x}right) e^{f(x)}, ) then ( g^{prime}(0) ) is equal to
A . 1
B. 2
( c cdot 0 )
D. None of these
12
1305 Given, ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cl}tan 4 x times cos 3 x & x neq 0 \ x & x=0end{array} . text { If } f ) is right.
continuous at ( boldsymbol{x}=mathbf{0}, ) then ( boldsymbol{k}= )
( A cdot 0 )
B. 4
c. ( frac{4}{3} )
D.
12
1306 Differentiate sec ( x ) by first principle. 12
1307 TOPIL-WIJL DU
12. Letf:(-1, 1) Rbe a differentiable function with f0=-1
and f” (0)=1. Let g(x)=[(2f(x) + 2)]2. Then g’O)=
120101
(a) 4
(a)
(6) o (c) 2 (d) 4
4
12
1308 Solve the differential equation ( cos left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)=boldsymbol{a},(boldsymbol{a} in boldsymbol{R}) ) 12
1309 If ( |y|=5 x-2 y, ) then which of the
following is incorrect?
A. ( y(x) ) is discontinuous at ( x=0 )
B. ( y(x) ) is continuous ( forall x in R )
c. ( y(x) ) is strictly increasing ( forall x epsilon R )
D. domain of ( y(x) ) is set of all real values
12
1310 If ( sqrt{boldsymbol{y}+boldsymbol{x}}+sqrt{boldsymbol{y}-boldsymbol{x}}=boldsymbol{c} text { (where } boldsymbol{c} neq mathbf{0}) )
then ( frac{d y}{d x} ) has the value equal to
This question has multiple correct options
A ( cdot frac{2 x}{c^{2}} )
в. ( frac{x}{y+sqrt{y^{2}-x^{2}}} )
c. ( frac{y-sqrt{y^{2}-x^{2}}}{x} )
D. ( frac{c^{2}}{2 y} )
12
1311 36.
Forx I R, f(x) = log2 – sinx and g(x)=f(f(x)), then:
JEEM 2016
(a) g'(0)=-cos(log2)
(b) gis differentiable at x=0 and g’O)=-sin(log2)
(c) g is not differentiable at x=0
(d) g’o= cos(log2)
12
1312 Differentiate with respect to ( times frac{(1+x)}{e^{x}} )
A ( .-x e^{-x} )
B . ( x e^{-x} )
( mathrm{c} cdot-x e^{-2 x} )
D. ( x^{2} e^{-x} )
12
1313 Assertion
( operatorname{Let} f(x)=left{begin{array}{ll}1 & text { if } 2 leq x leq 3 \ 3 & text { if } 3 leq x leq 5end{array}right. )
The mean value of ( boldsymbol{f} ) is attained
Reason
( f ) is a bounded function but not
continuous on ( [mathbf{2}, mathbf{5}] )
A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true
12
1314 f ( e^{y}(x+1)=1, ) show that ( y_{2}=y_{1}^{2} ) 12
1315 Let ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} ) be given by ( boldsymbol{f}(boldsymbol{x})=mathbf{5} boldsymbol{x}, ) if
( boldsymbol{x} in boldsymbol{Q} ) and ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{6} ) if ( boldsymbol{x} in boldsymbol{R}-boldsymbol{Q} )
then
A. f is continuous at ( x=1 ) and ( x=2 )
B. fis not continuous at ( x=1 ) and ( x=2 )
c. ( f ) is continuous at ( x=1 ) but not at ( x=2 )
D. fis continuous at ( x=2 ) but not at ( x=1 )
12
1316 ( f(x)=frac{1-cos (1-cos x)}{x^{4}} ) is continuous at
( x=0, ) then ( f(0)= )
A ( cdot frac{1}{2} )
B.
( c cdot frac{1}{6} )
D.
12
1317 Using the definition, show that the
function.
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sin (1 / boldsymbol{x}) ) if ( boldsymbol{x} neq mathbf{0}, boldsymbol{0} ) if ( boldsymbol{x}=mathbf{0} )
is continuous at the point ( boldsymbol{x}=mathbf{0} )
12
1318 Find the derivative of the following functions from first principle:
( cos left(x-frac{pi}{8}right) )
12
1319 If ( boldsymbol{y}=frac{boldsymbol{x}^{2}}{mathbf{2}}+frac{mathbf{1}}{mathbf{2}} boldsymbol{x} sqrt{boldsymbol{x}^{2}+mathbf{1}}+ )
( ln sqrt{x+sqrt{x^{2}+1}} ), then the value of
( boldsymbol{x} boldsymbol{y}^{prime}+log boldsymbol{y}^{prime} ) is
( mathbf{A} cdot underline{y} )
B. ( 2 y )
c. 0
D. ( -2 y )
12
1320 Verify L.M.V for the function? ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cc}2+x^{3} & x leq 1 \ 3 x & x>1end{array}right} o n[-1,2] ) 12
1321 Find the derivative of ( sin left(2 sin ^{-1} xright) )
A ( cdot frac{2 cos left(2 sin ^{-1} xright)}{sqrt{1-x^{2}}} )
B. ( frac{cos left(2 sin ^{-1} xright)}{sqrt{1-x^{2}}} )
c. ( frac{2 cos left(2 cos ^{-1} xright)}{sqrt{1-x^{2}}} )
D. ( -frac{cos left(2 cos ^{-1} xright)}{sqrt{1-x^{2}}} )
12
1322 Verify Rolle’s theorem for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}(boldsymbol{x}+ )
( mathbf{3}) e^{-boldsymbol{x} / 2} ) in ( (-mathbf{3}, mathbf{0}) )
A. Yes Rolle’s theorem is applicable and the stationary point is ( x=-2 )
B. Yes Rolle’s theorem is applicable and the stationary point is ( x=-1 )
C. No Rolle’s theorem is not applicable in the given interval
D. Both A and B
12
1323 Consider the function ( g(x)= ) ( left{begin{array}{ll}frac{1+a^{x}+x a^{x} ln a}{a^{x} x^{2}} & x0end{array}right. )
where ( a>0 )
Find the value of ( a & g(0) ) so that the function ( g(x) ) is continuous at ( x=0 )
( ^{A} cdot a=frac{1}{sqrt{2}}, g(0)=frac{(ln 2)^{2}}{8} )
B. ( a=-frac{1}{sqrt{2}}, g(0)=frac{(l n 2)^{2}}{8} )
c. ( a=-frac{1}{sqrt{2}}, g(0)=frac{-}{(l n 2)^{2}} 8 )
D. ( a=frac{1}{sqrt{2}}, g(0)=-frac{(l n 2)^{2}}{8} )
12
1324 Differentiate the following functions
with respect to ( boldsymbol{x} )
( tan ^{-1}left(frac{sqrt{boldsymbol{x}}+sqrt{boldsymbol{a}}}{1-sqrt{boldsymbol{x} boldsymbol{a}}}right) )
12
1325 The value of ( f(0), ) so that the function
( boldsymbol{f}(boldsymbol{x})= )
( frac{sqrt{a^{2}-a x+x^{2}}-sqrt{a^{2}+a x+x^{2}}}{sqrt{a+x}-sqrt{a-x}} )
becomes continuous for all ( x, ) is given
by
( mathbf{A} cdot a^{3 / 2} )
B. ( a^{1 / 2} )
c. ( -a^{1 / 2} )
D. ( -a^{3 / 2} )
12
1326 ff ( y=y(x) ) and it follows the relation ( mathbf{2} e^{x y^{2}}+boldsymbol{y} cos left(x^{2}right)=4, ) then ( left|boldsymbol{y}^{prime}(mathbf{0})right| ) is
equal to
12
1327 Assertion
If ( f(x)=0 ) has two distinct positive
real roots then number of non-
differentiable points of ( boldsymbol{y}=|boldsymbol{f}(-|boldsymbol{x}|)| ) is
( mathbf{1} )
Reason
Graph of ( boldsymbol{y}=boldsymbol{f}(|boldsymbol{x}|) ) is symmetrical
about y-axis
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
12
1328 ( x cos (a+y)=cos y )
then prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{cos ^{2}(boldsymbol{a}+boldsymbol{y})}{sin _{boldsymbol{a}}} )
12
1329 Differentiate from first principles.
(i) ( 3 x )
12
1330 PICOU
23. If y=x then
is
12
1331 Identify the graph of the polynomial
function ( boldsymbol{f} )
( f(x)=x^{4}+x^{3}-2 x^{2} )
begin{tabular}{|l|l|l|l|l|}
hline 1 & 1 & & & \
hline & & & & \
hline & & & & \
hline & & & & \
hline & & & & \
hline & & & & \
hline
end{tabular}
A. graph a
B. graph b
c. graph c
D. graph d
12
1332 Find the derivative of the following functions from the first principals w.r.t
to ( boldsymbol{x} )
( tan 2 x )
12
1333 If ( y=x^{-frac{1}{2}}+log _{5} x+frac{sin x}{cos x}+2^{x} ), then
find ( frac{d y}{d x} )
A. ( -frac{1}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+sec ^{2} x+2^{x} log 2 )
B. ( frac{1}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+sec ^{2} x+2^{x} log 2 )
c. ( -frac{3}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+sec ^{2} x+2^{x} log 2 )
D. ( -frac{1}{2} x^{-3 / 2}+frac{1}{x log _{e} 5}+cos ^{2} x+2^{x} log 2 )
12
1334 Is the function ( f ) defined by
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{x}, boldsymbol{i} boldsymbol{f} boldsymbol{x} leq 1 \ boldsymbol{5}, boldsymbol{i f} boldsymbol{x}>1end{array}right. )
continuous at ( boldsymbol{x}=mathbf{0} ? ) At ( boldsymbol{x}=mathbf{1} ) ? At ( boldsymbol{x}= )
( mathbf{2} ? )
12
1335 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{g}^{prime}(1)+boldsymbol{g}^{prime prime}(2) ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{x}^{2}+boldsymbol{x} boldsymbol{f}^{prime}(boldsymbol{2})+boldsymbol{f}^{prime prime}(boldsymbol{3}) . ) Then
A ( cdot f^{prime}(1)=4+f^{prime}(2) )
B ( cdot g^{prime}(2)=8+g^{prime}(1) )
C ( cdot g^{prime prime}(2)+f^{prime prime}(3)=4 )
D. all of these
12
1336 Derivative of ( 2 tan x-7 sec x ) with
respect to ( x ) is:
( mathbf{A} cdot 2 sec x+7 tan x )
B. ( sec x(2 sec x+tan x) )
c. ( 2 sec ^{2} x+sec x ). tan ( x )
D. ( sec x(2 sec x-7 tan x) )
12
1337 ( cos ^{-1} x=log (y)^{1 / m} )
Evaluate ( frac{d y}{d x} )
12
1338 Let ( f(x) ) be a real value function not identically zero satisifes the equation, ( fleft(x+y^{n}right)=f(x)+f(y)^{n} ) for all real
( x, y ) and ( f^{prime}(0) geq 0 ) where ( n(>1) ) is an odd natural number. ( boldsymbol{f}(mathbf{1 0})=mathbf{1 0 k . F i n d} )
( boldsymbol{k} ) value
12
1339 17. Let (9=
17. Let f(x) =
{1+ sin x |3a|sin xl <x<0
b
; x = 0
etan 2x/tan 3x ; 0<x<*
(1994 – 4 Marks)
Determine a and b such that f(x) is continuous at x=0
12
1340 x20, then show that f(x)=0 for all x 20. (2001 – 5 Marks)
21. Let a e R. Prove that a function f:
R R is differentiable at
a if and only if there is a function g:
R R which is
continuous at a and satisfies f(x)-f(a)=g(x) (x – a) for all
XER.
(2001 – 5 Marks)
12
1341 Differentiate the following functions
with respect to ( x ) :
( tan ^{-1}left{frac{x^{1 / 3}+a^{1 / 3}}{1-(a x)^{1 / 3}}right} )
12
1342 Assertion
If both functions ( f(t) ) and ( g(t) ) are
continuous on the closed interval ( [boldsymbol{a}, boldsymbol{b}] )
differentiable on the open interval ( (a, b) )
and ( g^{prime}(t) ) is not zero on that open
interval, then there exists some ( c ) in
( (a, b) ) such that ( frac{f^{prime}(c)}{g^{prime}(c)}=frac{f(b)-f(a)}{g(b)-g(a)} )
Reason
If ( f(t) ) and ( g(t) ) are continuous and
differntiable in ( [a, b], ) then there exists
some ( c ) in ( (a, b) ) such that ( f^{prime}(c)= ) ( frac{boldsymbol{f}(boldsymbol{b})-boldsymbol{f}(boldsymbol{a})}{boldsymbol{b}-boldsymbol{a}} ) and ( boldsymbol{g}^{prime}(boldsymbol{c})=frac{boldsymbol{g}(boldsymbol{b})-boldsymbol{g}(boldsymbol{a})}{boldsymbol{b}-boldsymbol{a}} )
from Lagrange’s mean value theorm.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
12
1343 Check whether Lagrange’s mean value
theorem is applicable on ( f(x)=sin x+ ) ( cos x ) interval ( left[0, frac{pi}{2}right] )
12
1344 The function ( f(x)=sin ^{-1}(cos x) ) is ( :- )
A. discontinuous at ( x=0 )
B. continuous at ( x=0 )
C. differentiable at ( x=0 )
D. none of these
12
1345 Find the derivative of the following (it is to be understood that ( a, b, c, d, p, q, r )
and ( s ) are fixed non-zero constants and
( m text { and } n text { are integers }): frac{sin x+cos x}{sin x-cos x} )
12
1346 The point(s) on the curve ( y^{3}+3 x^{2}= )
( 12 y ) where the tangent is vertical, is
( (operatorname{are}) )
A ( cdotleft(pm frac{4}{sqrt{3}},-2right) )
B ( cdotleft(pm frac{sqrt{11}}{3}, 1right) )
( mathbf{c} cdot(0,0) )
D. ( left(pm frac{4}{sqrt{3}}, 2right) )
12
1347 Find ( frac{d y}{d x} ) for ( 2 x^{2}+5 x y+3 y^{2}=1 ) 12
1348 Differentiate ( boldsymbol{y}=mathbf{1 0}^{boldsymbol{x}}+mathbf{1 0}^{tan boldsymbol{x}} ) 12
1349 Find from first principles the
differential coefficient of ( 2 x^{2}+3 x )
12
1350 The function ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cc}x^{2} / a, & 0 leq x<1 \ a, & 1 leq x<sqrt{2} \ frac{2 b^{2}-4 b}{x^{2}}, & sqrt{2} leq x<inftyend{array}right. )
continuous for ( 0 leq x<infty, ) then the
most suitable values of ( a ) and ( b ) are
B. ( a=-1, b=1+sqrt{2} )
c. ( a=-1, b=1 )
D. None of these
12
1351 If ( f ) and ( g ) are differentiable functions in
( [mathbf{0}, mathbf{1}] ) satisfying ( boldsymbol{f}(mathbf{0})=mathbf{2}= )
( g(1), g(0)=0 ) and ( f(1)=6, ) then for
some ( boldsymbol{c} in[mathbf{0}, mathbf{1}] )
A ( cdot 2 f^{prime}(c)=g^{prime}(c) )
B ( cdot 2 f^{prime}(c)=3 g^{prime}(c) )
c. ( f^{prime}(c)=g^{prime}(c) )
D. ( f^{prime}(c)=2 g^{prime}(c) )
12
1352 The range of the function ( Delta=f(|x|) ) is-
A ( cdot[0,1] )
в. [0,1)
c. (0,1]
D. None of these
12
1353 If ( y=e^{a sin ^{-1} x} ) then prove that
( left(1-x^{2}right) y_{2}-x y_{1}-a^{2} y=0, ) where ( y_{1} )
and ( y_{2} ) are first and second order
derivatives of ( y ) respectively.
12
1354 Sketch the graph ( y=|x-5| . ) Evaluate ( int_{0}^{1}|x-5| d x . ) What does this value of the integral represent on the graph? 12
1355 Differentiate ( log sqrt{frac{1+cos x}{1-cos x}} w . r . t . x ) 12
1356 ( boldsymbol{y}=sin left(boldsymbol{pi} / boldsymbol{6} e^{x y}right) ) putting ( boldsymbol{x}=mathbf{0} operatorname{than} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
1357 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(sec ^{2} boldsymbol{x}+operatorname{cosec}^{2} boldsymbol{x}right)= )
( mathbf{A} cdot-4 sec x cdot tan x cdot cos e c x cdot cot x )
B. ( 4 sec x cdot cos ) ec ( x )
c. ( 2 sec x cdot tan x-2 cos e c x cdot cot x )
D. ( 2 sec ^{2} cdot tan x-2 operatorname{cosec}^{2} x cdot cot x )
12
1358 The function ( f(x)=sin ^{-1}(cos x) ) is :
A. discontinuous at ( =0 )
B. continuous at ( =0 )
c. differentiable ( =0 )
D. none of these
12
1359 If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{a} boldsymbol{x} ) satisfies the
conditions of Rolles theorem on [1,3] with ( c=2+frac{1}{sqrt{3}} ) then ( (a, b)= )
( mathbf{A} cdot(11,6) )
В ( cdot(11,-6) )
c. (-6,11)
D. (6,11)
12
1360 ( operatorname{Let} f(x)=cos 2 x cdot cot left(frac{pi}{4}-xright) ) If ( f ) is
continuous at ( x=frac{pi}{4} ) then the value of ( fleft(frac{pi}{4}right) ) is equal to
A . 2
B. –
c. ( frac{-1}{2} )
D.
12
1361 If ( boldsymbol{f}(boldsymbol{x})= )
( left{x sin frac{1}{x}, x neq 0 quad k quad, x=0right} ) is
continuous at ( x=0, ) then the value of ( k )
is
( A cdot 1 )
B. – 1
c.
D. 2
12
1362 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
( boldsymbol{y}=frac{1}{boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}+boldsymbol{3}} )
12
1363 If ( x=phi(t), y=psi(t), ) then ( frac{d^{2} y}{d x^{2}} ) is
A ( cdot frac{phi^{prime} psi^{prime prime}-psi^{prime} phi^{prime prime}}{left(phi^{prime}right)^{2}} )
B. ( frac{phi^{prime} psi^{prime prime}-psi^{prime} phi^{prime prime}}{left(phi^{prime}right)^{3}} )
c. ( frac{phi^{prime prime}}{psi^{prime prime}} )
D. ( frac{psi^{prime prime}}{phi^{prime prime}} )
12
1364 Differentiate with respect to ( x ) :
( e^{sqrt{cot x}} )
12
1365 Find the minimum and maximum
values of the function ( y=x^{3}-3 x^{2}+6 )
Also find the values of ( x ) at which these
occur.
12
1366 If the function ( f(x)=x^{3}-6 x^{2}+a x+ )
( b ) satisfies Rolle’s theorem in the
interval [1,3] and ( f^{prime}left(frac{2 sqrt{3}+1}{sqrt{3}}right)=0, ) then
A ( . a=-11 )
B. ( b=-6 )
( mathbf{c} cdot a=6 )
D. ( a=11 )
12
1367 For a curve at which the tangent lines at
two distinct points coincide, then the curve cannot be
A. a cubic curve
B. a quadratic curve
c. a curve of 4th power
D. none of these
12
1368 11.
The function f(x)=[x]cos
2
The function f(x)=[x]cos 2x-1 , [.] denotes the greatest
integer function, is discontinuous at
(1995)
(a) Allx
(b) All integer points
(C) Nox
(d) x which is not an integer
12
1369 Sketch the graph ( y=|x-5| . ) Evaluate ( int_{0}^{1}|x-5| d x . ) What does this value of the integral represent on the graph? 12
1370 Show that ( left(frac{1}{a+} frac{1}{b+} frac{1}{c+} dotsright)left(c+frac{1}{b+} frac{1}{a+} frac{1}{c_{1}} dots .right)= )
( frac{1+b c}{1+a b} )
12
1371 The value of ( f(0) ) so that the function
( f(x) frac{log left(1+frac{x}{a}right)-log left(1-frac{x}{b}right)}{x},(x neq 0) )
is continuous at ( x=0 ) is
A ( cdot frac{a+b}{a b b b b b} )
B . ( frac{a-b}{a b b b b} )
c. ( frac{a b}{a+b b} )
D. ( frac{a b}{a-b} )
12
1372 If ( y=frac{1}{sqrt{a^{2}-x^{2}}}, ) find ( frac{d y}{d x} ) 12
1373 2. If y = 2 sin? 0 + tan 0 then dy will be
do
(a) 4 sin cos 0 + sec 0 tan O
(b) 2 sin 20+ seca e
(c) 4 sin + seca e
(d) 2 cos2 0 + sec2 e
dy
12
1374 Which of the following limits vanishes?
A ( cdot lim _{x rightarrow infty} x^{frac{1}{4}} sin frac{1}{sqrt{x}} )
B. ( lim _{x rightarrow pi^{2}}(1-sin x) cdot tan x )
C ( lim _{x rightarrow infty} frac{2 x^{2}+3}{x^{2}+x-5} cdot ) sgn
D. ( lim _{x rightarrow 3^{circ}} frac{[x]^{2}-9}{x^{2}-9} ) where [] denotes greatest integer function
12
1375 45.
Let f(x) = 15-x-10
(a)
©
{5, 10, 153
{5, 10, 15,20}
“(X)=15-x-101: X R. Then the set of all values of x, at
the function, g(x)=f(f(x)) is not differentiable, is:
JEEM 2019-9 April (M)
(b) {10, 15)
(d) {10}
12
1376 Find the derivative of the following functions(it is to be understood that
( a, b, c, d, p, q, r ) and ( s ) are fixed non-zero
constants and ( m text { and } n text { are integers }) ) ( frac{1}{a x^{2}+b x+c} )
12
1377 If ( f(x)=frac{log (1+a x)-log (1-b x)}{x} ) for
( boldsymbol{x} neq mathbf{0} ) and ( boldsymbol{f}(mathbf{0})=boldsymbol{k} ) and ( boldsymbol{f}(boldsymbol{x}) ) is
continuous at ( x=0, ) then ( k ) is equal to:
A ( cdot a+b )
в. ( a-b )
( c )
D.
12
1378 The number of points where the
function ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{cc}mathbf{1}+left[cos frac{pi x}{2}right], & mathbf{1}<boldsymbol{x} leq mathbf{2} \ mathbf{1}-{boldsymbol{x}}, & mathbf{0} leq boldsymbol{x}<mathbf{1} \ |boldsymbol{s i n} boldsymbol{pi} boldsymbol{x}|, & -mathbf{1} leq boldsymbol{x}<mathbf{0}end{array} ) and right.
( f(1)=0 ) is continuous but nondifferentiable is/are (where [.] and {
represent greatest integer and fractional part functions, respectively)
( A cdot O )
в.
( c cdot 2 )
D. none of these
12
1379 ( operatorname{Let} f(x)=left{begin{array}{cc}x^{2} / 2 & 0<x leq 1 \ 2 x^{2}-3 x+3, & 1<x<2end{array}right. )
then which is incorrect
This question has multiple correct options
A. ( f ) is continuous in (0,2)
B. ( f ) is not continuous at all points in (0,2)
c. ( f ) is differentiable in (0,2)
D. ( L t_{x rightarrow 1^{+}} f(x)=L t_{x rightarrow 1^{-}} f(x) )
12
1380 Prove that ( e^{x}-x>1, ) if ( x>0 ) 12
1381 If ( x^{3} y^{5}=(x+y)^{8}, ) then show that
( frac{d y}{d x}=frac{y}{x} )
12
1382 Differentiate ( frac{1}{3} tan ^{3} x-tan x+x ) w.r.t ( x ) 12
1383 If ( left(x^{2}+y^{2}right)^{2}=x y, ) find ( frac{d y}{d x} ) 12
1384 The value of ( c ) of mean value theorem
when ( f(x)=x^{3}-3 x-2 ) in [-2,3] is
A ( cdot sqrt{frac{7}{3}} )
B. ( sqrt{frac{3}{7}} )
c. ( frac{sqrt{7}}{3} )
D. ( frac{sqrt{3}}{7} )
12
1385 From mean value theorem, ( boldsymbol{f}(boldsymbol{b})- ) ( boldsymbol{f}(boldsymbol{a})=(boldsymbol{b}-boldsymbol{a}) boldsymbol{f}^{1}left(boldsymbol{x}_{1}right) ; boldsymbol{a}<boldsymbol{x}_{1}< )
bif ( f(x)=frac{1}{x} ) then ( x_{1}= )
( A cdot sqrt{a b} )
B. ( frac{a+b}{2} )
c. ( frac{a b}{a+b} )
D・ ( frac{a-b}{b-a-a-b} )
12
1386 If ( f(x) ) is a continuous function on [0,1] differentiable in (0,1) such that ( f(1)=0, ) then there exists some
( c epsilon(0,1) ) such that
A ( cdot c f^{prime}(c)-f(c)=0 )
B. ( f^{prime}(c)+c f(c)=0 )
c. ( f^{prime}(c)-c f(c)=0 )
D. ( c f^{prime}(c)+f(c)=0 )
12
1387 At the point ( x=1, ) the function ( f(x)= )
( left{begin{array}{ll}x^{3}-1, & 1<x<infty \ x-1, & -infty<x leq 1end{array}right. )
A. continuous and differentiable
B. Continuous and not differentiable
c. Discontinuous and differentiable
D. Discontinuous and not differentiable
12
1388 If ( boldsymbol{y}=cos ^{-1}left(frac{5 cos x-12 sin x}{13}right), x in )
( left(0, frac{pi}{2}right), ) then find the value of ( d y / d x )
12
1389 3. If y = sin x & x = 3t then
will be
(a) 3 cos (x)
(c) 3 cos (x)
(b) cos x
(d) -cos x
12
1390 The set of all points of differentiability of the function ( f(x)=frac{sqrt{x+1}-1}{sqrt{x}} ) for
( x neq 0 ) and ( f(0)=0 ) is
( mathbf{A} cdot(-infty, infty) )
B. ( [0, infty) )
( c cdot(0, infty) )
( mathbf{D} cdot(-infty, infty) sim{0} )
12
1391 Find the derivative of ( frac{x+cos x}{tan x} ) with
respect to ( x )
12
1392 ( boldsymbol{y}=frac{boldsymbol{x}}{2 sqrt{2}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
1393 If ( y=tan ^{-1}(cot x)+cot ^{-1}(tan x) )
then ( frac{d y}{d x} ) is equal to-
( A cdot 1 )
B. 0
c. -1
D. – 2
12
1394 Find the derivative of ( y=log ^{3}left(x^{2}right) ) 12
1395 Let ( f ) be differentiable for all ( x ). If
( boldsymbol{f}(1)=-2 ) and ( boldsymbol{f}^{prime}(boldsymbol{x}) geq 2 ) for ( boldsymbol{x} in[mathbf{1}, boldsymbol{6}] )
This question has multiple correct options
A. ( f(6)<8 )
B. ( f(6) geq 8 )
D. ( f(6) leq 5 )
12
1396 f ( p(x)=51 x^{101}-2323 x^{100}-45 x+ )
1035, then using Rolle’s Theorem. prove
that atleast one foot lies between
( left(45^{1 / 100}, 46right) )
12
1397 Differentiate ( e^{x}+e^{-x} ) with respect to ( x ) 12
1398 ff ( f(x)=left{begin{array}{lll}frac{e^{3 x}-1}{4 x} & text { for } & x neq 0 \ frac{k+x}{4} & text { for } & x=0end{array}right. )
continuous at ( x=0, ) then ( k= )
( A cdot )
B. 3
( c cdot 2 )
( D )
12
1399 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(boldsymbol{x}^{2} boldsymbol{e}^{boldsymbol{a} boldsymbol{x}}right) )
A ( cdot e^{a x}left(a x^{2}+2 xright) )
B ( cdot e^{a x}left(2 a x^{2}+2 xright) )
C ( cdot e^{a x}left(a x^{2}+2 a xright) )
D. ( e^{a x}left(a x^{2}-2 a xright) )
12
1400 3.
Given y= 5x
31(1 – x)2
+ cos2 (2x+1) ; Find
dy. (1980)
12
1401 If ( x+y=sin (x-y) ) then ( frac{d y}{d x} ) is equal
to
A ( cdot frac{1}{2} )
B. 0
c. -1
D.
12
1402 If ( boldsymbol{y}=cos ^{-1}(cos boldsymbol{x}), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, boldsymbol{a} boldsymbol{t} boldsymbol{x}= )
( frac{5 pi}{4} ) is equal to
( mathbf{A} cdot mathbf{1} )
B . –
c. ( frac{1}{sqrt{2}} )
D. ( -frac{1}{sqrt{2}} )
12
1403 3. If y = x sin x, then
(a) I dy 1
= – + cot x
= – + cotx
y dx x
(c) 1 dy-1 – cotx
y dx x
(d) None of these
12
1404 Differentiate w.r.t ( boldsymbol{x} ) ( sin ^{-1}left(frac{a cos x+b sin x}{sqrt{a^{2}+b^{2}}}right) ) 12
1405 On the interval ( boldsymbol{I}=[-2,2] ),for the
function ( left{begin{array}{ll}(x+1) e^{-left[frac{1}{[x]}+frac{1}{x}right]} & (x neq 0) \ 0 & (x=0)end{array}right. )
(where []( text { is } G I F) ) which one of the
following hold good?
This question has multiple correct options
A. is continuous for all values of ( x in I )
B. is continuous for all values of ( x in I-(0) )
c. assumes all intermediate values from ( f(-2) ) & ( f(2) )
D. has a maximum value equal to
12
1406 What are the value of ( c ) for which Rolle’s
theorem for the function ( f(x)=x^{3}- )
( 3 x^{2}+2 x ) in the interval [0,2] is
verified?
A ( . c=pm 1 )
B. ( _{c=1 pm frac{1}{sqrt{3}}} )
c. ( c=pm 2 )
D. None of these
12
1407 The function ( f(x)= ) ( frac{log (1+a x)-log (1-b x)}{x} ) is not
defined at ( x=0 . ) The value of which
should be assigned to ( f ) at ( x=0 ), is
( mathbf{A} cdot a-b )
B. ( a+b )
( mathbf{c} cdot log a+log b )
D. None of these
12
1408 If ( f(x)=left{begin{array}{ll}frac{x}{1+e^{1 / x}} & x neq 0 \ 0 & x=0end{array} ), then the right.
function ( f(x) ) is differentiable for:
( mathbf{A} cdot x in R^{+} )
B. ( x in R )
c. ( x in R-{0} )
D. ( x in R-{0,1} )
12
1409 Let ( f(x) ) be a continous and differentiable function on ( [0,1], ) such
that ( f(0) neq 0 ) and ( f(1)=0 . ) We can
conclude that there exists ( c in(0,1) )
such that
A ( cdot c . f^{prime}(c)-f(c)=0 )
B. ( f^{prime}(c)+c . f(c)=0 )
c. ( f^{prime}(c)-c . f(c)=0 )
D. ( c . f^{prime}(c)+f(c)=0 )
12
1410 If ( f(x)=frac{a cos x-cos b x}{x^{2}}, x neq 0 ) and
( boldsymbol{f}(mathbf{0})=mathbf{4} ) continuous at ( boldsymbol{x}=mathbf{0}, ) then the
ordered pair ( (a, b) ) is
( A cdot(neq 1,3) )
в. ( (1, neq 3) )
c. (-1,-3)
D. (1,±3)
12
1411 ( operatorname{Let} boldsymbol{f}left(frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2}}right)=frac{mathbf{1}}{mathbf{2}}[boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})] ) for
real ( x ) and ( y . ) If ( f^{prime}(0) ) exists and equals
-1 and ( f(0)=1 ) then the value of ( f(2) )
is
A . 1
B. –
c. ( 1 / 2 )
D. 2
12
1412 The value of ( frac{d^{2} y}{d x^{2}} ) at the point where ( t= ) 0 is
A . 1
B. 2
( c cdot-2 )
D. 3
12
1413 Differentiate the following function from first principle:
( sin ^{-1}(2 x+3) )
12
1414 If ( f(x) ) is a polynomial of degree ( n(>2) )
and ( f(x)=f(k-x), ) (where ( k ) is a
fixed real number), then degree of ( boldsymbol{f}^{prime}(boldsymbol{x}) )
is
( A )
B. ( n-1 )
( mathbf{c} cdot n-2 )
D. None of these
12
1415 Rolle’s theorem cannot be applicable
for:
A ( cdot f(x)=cos x-1 ) in ( (0,2 pi) )
B – ( f(x)=x(x-2)^{2} ) in (0,2)
c. ( f(x)=3+(x-1)^{frac{3}{5}} ) in (0,3)
D. ( f(x)=sin ^{2} x ) in ( (0, pi) )
12
1416 For what value of ( k ) the function ( f(x)= ) ( left{begin{array}{l}frac{sin 5 x}{3 x}, i f, x neq 0 \ k, text { if } x=0end{array} ) is continuous at right.
( boldsymbol{x}=mathbf{0} )
12
1417 If the function
( left{begin{array}{l}boldsymbol{x}, quad text { if } quad boldsymbol{x} leq mathbf{1} \ boldsymbol{c x}+boldsymbol{k}, quad text { if } quad mathbf{1}<boldsymbol{x}<mathbf{4} \ -mathbf{2} boldsymbol{x}, quad text { if } quad boldsymbol{x} geq mathbf{4}end{array}right. )
is contionus everywhere, then the value
of ( c ) and ( k ) are respectively:
A. -3,-5
в. -3,5
c. -3,-4
D. -3,4
E . -3,3
12
1418 ( frac{d(tan x .)}{d x} )
( mathbf{A} cdot sec ^{2} x )
B. ( cot ^{2} x )
( mathbf{c} cdot cos ^{2} x )
( D cdot sin ^{2} x )
12
1419 The function ( f(x)=left{begin{array}{cc}frac{e^{1 / x}-1}{e^{1 / x}+1} & x neq 0 \ 0, & x=0end{array}right. )
is
A. continuous at ( x=0 )
B. discontinuous at ( x=0 )
c. discontinuous at ( x=0 ) but can be made continuous at
( x=0 )
D. None of these
12
1420 23 12
1421 ff ( y=operatorname{en}left{frac{x+sqrt{left(a^{2}+x^{2}right)}}{a}right}, ) then the
value of ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is
A ( cdot sqrt{a^{2}-x^{2}} )
B. ( a sqrt{a^{2}+x^{2}} )
c. ( frac{1}{sqrt{a^{2}+x^{2}}} )
D. ( x sqrt{a^{2}+x^{2}} )
12
1422 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left[(boldsymbol{x}+boldsymbol{a})left(boldsymbol{x}^{2}+boldsymbol{a}^{2}right)left(boldsymbol{x}^{4}+boldsymbol{a}^{4}right)right]=? )
( ^{mathbf{A} cdot frac{7 x^{8}+aleft(8 x^{7}-a^{7}right)}{(x-a)^{2}}} )
B. ( frac{7 x^{8}-aleft(8 x^{7}-a^{7}right)}{(x-a)^{2}} )
c. ( frac{7 x^{8}-aleft(8 x^{7}+a^{7}right)}{(x-a)^{2}} )
D. ( x^{4}+a^{4} )
12
1423 Differentiate ( boldsymbol{x}=boldsymbol{y}+ ) 12
1424 Using the fact that ( sin (boldsymbol{A}+boldsymbol{B})= )
( sin A cos B+cos A sin B ) and the
differentiation, obtain the sum formula
for cosines.
12
1425 If ( f(x)=frac{x-1}{x+2}, ) then ( frac{d f^{-1}(x)}{d x} ) is equal
to
A ( cdot frac{1}{(1-x)^{2}} )
B. ( frac{-3}{(1-x)^{2}} )
c. ( frac{3}{(1-x)^{2}} )
D. ( frac{-1}{(1-x)^{2}} )
12
1426 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of function ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}+frac{mathbf{1}}{mathbf{2}} log boldsymbol{x} )
A ( cdot 2 . e^{x^{3}} x^{2}+frac{1}{2 x} )
B. ( e^{x^{3}} x^{2}+frac{1}{2 x} )
c. ( 3 . e^{x^{3}} x^{2}+frac{1}{2 x} )
D. ( 3 . e^{x^{3}} x^{2}+frac{1}{x} )
12
1427 If ( boldsymbol{y}=boldsymbol{e}^{log left(1+boldsymbol{x}+boldsymbol{x}^{2}+boldsymbol{x}^{3}+cdotsright)}, ) where ( |boldsymbol{x}|<mathbf{1} )
then ( frac{d y}{d x} ) is equal to
A ( cdot frac{-1}{(1-x)^{2}} )
B. ( frac{1}{(1-x)^{2}} )
c. ( frac{1}{(1+x)^{2}} )
D. None of these
12
1428 [
begin{aligned}
boldsymbol{f}(boldsymbol{x})=& boldsymbol{x}, boldsymbol{i} boldsymbol{f} boldsymbol{x} leq 1 \
boldsymbol{5}, & boldsymbol{i} boldsymbol{f} boldsymbol{x} geq 1
end{aligned}
]
Check whether ( f(x) ) is continuous at
[
boldsymbol{x}=mathbf{0} ? boldsymbol{x}=mathbf{1} ? boldsymbol{x}=mathbf{2} ?
]
12
1429 ( f(x)=x^{2} ) in ( 2 leq x leq 3 ) Is Rolle’s
theorem applicable?
12
1430 Differentiate the following functions
with respect to ( x ) :
If ( y=tan ^{-1}left(frac{2 x}{1-x^{2}}right), x>0, ) prove
that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=frac{boldsymbol{4}}{boldsymbol{1}+boldsymbol{x}^{2}} )
12
1431 Find ( frac{d y}{d x} ) where ( boldsymbol{y}=sqrt{frac{boldsymbol{x}^{2}+mathbf{1}}{boldsymbol{x}^{2}+mathbf{2}}} ) 12
1432 If f(x) and g(x) are differentiable function for 0 < x <1 such
that f(0) = 2, g(0)=0, f(1) = 6; g(1)= 2, then show that there
exist c satisfying 0<c<1 and f'(c)=2g'(c).
12
1433 Differentiate ( f(x)=4 x^{2}-5 x )
( A cdot 8 x )
B. ( 8 x-5 )
( c .5 )
D. None of these
12
1434 ( y=sin x cos x ) find ( frac{d y}{d x} ) 12
1435 The displacement ( x ) of a particle along
the ( x ) -axis at time ( t ) is given by ( x= ) ( frac{a_{1}}{2} t+frac{a_{2}}{3} t^{2} . ) Find the acceleration of the
particle.
12
1436 If ( y=x^{2} sin x, ) then ( frac{d y}{d x} ) will be
A. ( x^{2} cos x+2 x sin x )
B. ( 2 x sin x )
C ( cdot x^{2} cos x )
D. ( 2 x cos x )
12
1437 Let ( f(x) ) be a real-valued differentiable
function not identically zero such that ( boldsymbol{f}left(boldsymbol{x}+boldsymbol{y}^{2 n+1}right)=boldsymbol{f}(boldsymbol{x})+ )
( {f(y)}^{2 n+1}, n epsilon N ) and ( x, y ) are any real
numbers and ( f^{prime}(0) geq 0 . ) Find the value
of ( f(5) )
A . 0
B. 1
c. 2
D. 5
12
1438 The function ( f(x)=x(x+3) e^{-(1 / 2) x} )
satisfies the condition of Rolle’s
theorem in ( [-3,0] . ) The value of ( c ) is
( mathbf{A} cdot mathbf{0} )
B. –
c. -2
D. – 3
12
1439 Given ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}3-left[cot ^{-1}left(frac{2 x^{3}-3}{x^{2}}right)right] & text { for } x>0 \ left{x^{2}right} cos left(e^{1 / x}right) & text { for } x<0end{array}right. )
where
{}( &[] ) denotes the fractional part and
the integral part functions respectively,
then which of the following statement
does not hold good –
This question has multiple correct options
( mathbf{A} cdot fleft(0^{-}right)=0 )
B . ( fleft(0^{+}right)=3 )
C ( . f(0)=0 Rightarrow ) continuity of ( f ) at ( x=0 )
D. irremovable discontinuity of ( f ) at ( x=0 )
12
1440 Discuss the continuity of the following function ( : f(x)=sin x . cos x ) 12
1441 Differentiate ( boldsymbol{f}(boldsymbol{x}) ) with respect to ( boldsymbol{g}(boldsymbol{x}) )
for the following.
( boldsymbol{f}(boldsymbol{x})=log _{e} boldsymbol{x}, boldsymbol{g}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} )
12
1442 If ( S_{n} ) denotes the sum of ( n ) terms of a
G.P. whose common ratio is ( r, ) then ( (r-1) frac{d S_{n}}{d r} ) is equal to
A ( cdot(n-1) S_{n}+n S_{n-1} )
B . ( (n-1) S_{n}-n S_{n-1} )
( mathbf{c} cdot(n-1) S_{n} )
D. None of these
12
1443 If ( boldsymbol{x}+boldsymbol{y}=boldsymbol{x}^{boldsymbol{y}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} boldsymbol{e q u a l s -} )
A ( frac{y x^{y-1}-1}{1-x^{y} log x} )
В. ( frac{y x^{y-1}-1}{x^{y} log x-1} )
c. ( frac{y x^{y-1}+1}{x^{y} log x+1} )
D. None of these
12
1444 If ( sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}}=10, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{y}=boldsymbol{4} )
( A cdot 4 )
в. -3
c. -4
D. 3
12
1445 Differentiate: ( boldsymbol{y}=sin ^{-1}left(frac{1-boldsymbol{x}^{2}}{mathbf{1}+boldsymbol{x}^{2}}right), mathbf{0}< )
( boldsymbol{x}<mathbf{1} )
12
1446 15. If f(x) = rex)
x+0 then f(x) is [2003]
„x=0
(a) discontinuous every where
(b) continuous as well as differentialble for all x
(c) continuous for all x but not differentiable at x=0
(d) neither differentiable nor continuous at x=0
12
1447 Test the continuity of the function ( f ) at ( mathbf{x}=mathbf{0}, ) where
[
begin{array}{l}
boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2} sin left(frac{1}{x}right) text { for } boldsymbol{x} neq mathbf{0} \
quad=mathbf{1} text { for } boldsymbol{x}=mathbf{0}
end{array}
]
12
1448 If ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-mathbf{5} boldsymbol{x}^{2}-boldsymbol{3} boldsymbol{x}, ) verify
conditions of the mean value theorem
satisfied for ( boldsymbol{a}=mathbf{1}, boldsymbol{b}=mathbf{3} . ) Find ( boldsymbol{c} boldsymbol{epsilon}(mathbf{1}, boldsymbol{3}) )
such that ( boldsymbol{f}^{prime}(boldsymbol{c})=frac{boldsymbol{f}(boldsymbol{3})-boldsymbol{f}(1)}{boldsymbol{3}-1} )
( A cdot 2 )
B.
( c .3 )
D.
12
1449 ( operatorname{Let} f(x)=frac{ln (1+x tan x)}{4 x}, x neq 0 ) is
continuous at ( boldsymbol{x}=mathbf{0}, ) then ( boldsymbol{f}(mathbf{0}) ) must be
equal to
A . 1
B. 0
( c .3 )
D.
12
1450 Differentiate the following function w.r.t.
( boldsymbol{x} )
( sqrt[3]{left(2 x^{2}-7 x-4right)^{5}} )
12
1451 Find ‘c’ of the mean value theorem,if
( f(x)=x(x-1)(x-2) )
( boldsymbol{a}=mathbf{0}, boldsymbol{b}=frac{mathbf{1}}{mathbf{2}} )
( ^{mathrm{A}} cdot_{C}=1-frac{sqrt{21}}{5} )
в. ( quad C=1-frac{sqrt{21}}{6} )
( ^{mathrm{c}} cdot_{C}=2-frac{sqrt{21}}{6} )
D. ( c=1+frac{sqrt{21}}{6} )
12
1452 17. Let $(x) =
1-tan x
X
,X
E
f(x) is continuous
4x-tet
[2004]
(a) 165
(1) 1
12
1453 If ( z=f ) of ( (x) ) where ( f(x)=x^{2}, ) then what
is ( frac{d z}{d x} ) equal to?
( mathbf{A} cdot x^{3} )
В. ( 2 x^{3} )
( c cdot 4 x^{3} )
D. ( 4 x^{2} )
12
1454 If ( cos ^{-1}left(frac{x^{2}-y^{2}}{x^{2}+y^{2}}right)=k ) (a constant)
then ( frac{d y}{d x}= )
A. ( frac{y}{x} )
в. ( frac{x}{y} )
c. ( frac{x^{2}}{y^{2}} )
D. ( frac{y^{2}}{x^{2}} )
12
1455 Extend the definition of the following by continuity. ( f(x)=frac{1-cos 7(x-pi)}{5(x-pi)^{2}} ) at the point
( boldsymbol{x}=boldsymbol{pi} )
12
1456 ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) in the interval [-1,1] Is Rolle’s
Theorem applicable?
12
1457 The Rolle’s theorem is applicable in the interval ( -1 leq x leq 1 ) for the function
( mathbf{A} cdot f(x)=x )
( mathbf{B} cdot f(x)=x^{2} )
( mathbf{c} cdot f(x)=2 x^{3}+3 )
( mathbf{D} cdot f(x)=|x| )
12
1458 If ( y=e^{m sin ^{-1} x},-1 leq x leq 1 ), show that
( left(1-x^{2}right) frac{d^{2} y}{d x^{2}}-x frac{d y}{d x}-a^{2} y=0 )
12
1459 Differentiate the following w.r.t ( x )
( sin left(x^{2}+5right) )
12
1460 Find the second order derivatives of
( sin (log x) )
12
1461 ff ( y=tan left(frac{5}{2} pi t+frac{pi}{6}right) ) then find the
value of ( frac{boldsymbol{a} boldsymbol{y}}{boldsymbol{d} boldsymbol{t}} ) at ( boldsymbol{t}=mathbf{0} )
12
1462 If ( y=cos ^{-1}left(frac{2 x}{1+x^{2}}right), ) then ( frac{d y}{d x} ) is equal
to
A ( cdot-frac{2}{1+x^{2}} ) of all ( |x|1 )
c. ( frac{2}{1+x^{2}} ) of all ( |x|<1 )
D. None of the above
12
1463 Find the derivative of the following
functions:
( 5 sec x+4 cos x )
12
1464 Given that ( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x} boldsymbol{g}(boldsymbol{x})}{|boldsymbol{x}|}, boldsymbol{g}(mathbf{0})= )
( boldsymbol{g}^{prime}(mathbf{0})=mathbf{0} ) and ( boldsymbol{f} ) is continuous at ( boldsymbol{x}=mathbf{0} )
the value of ( f^{prime}(0) ) is
12
1465 Using Rolle’s theorem, find points on the
curve ( boldsymbol{y}=mathbf{1 6}-boldsymbol{x}^{2}, boldsymbol{x} in[-mathbf{1}, mathbf{1}], ) where
tangent is parallel to ( x- ) axis.
12
1466 Differentiate the following function with
respect to ( x ) ( frac{x+cos x}{tan x} )
12
1467 Solve ( lim _{x rightarrow-2} frac{left(frac{1}{x}+frac{1}{2}right)}{x+2} ) 12
1468 Discuss the applicability of Rolle’s theorem for the following function on the indicated interval:
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}-mathbf{4} boldsymbol{x}+mathbf{5}, mathbf{0} leq boldsymbol{x} leq mathbf{1} \ mathbf{2} boldsymbol{x}-mathbf{3}, quad mathbf{1}<boldsymbol{x} leq mathbf{2}end{array}right. )
12
1469 If ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}], ) where ( [.] ) is the greatest
integer function, and ( boldsymbol{g}(boldsymbol{x})= )
( xleft(1-x^{2}right)left(4-x^{2}right), ) then ( g[f(x)] ) is
A. discontinuous at all integer
B. continuous at all integer
c. continuous at ( x=0,pm 1,pm 2 )
D. discontinuous at ( x=0,pm 1,pm 2 )
12
1470 Find ( frac{d y}{d x}, ) when ( y=frac{x cos ^{-1} x}{sqrt{1-x^{2}}} ) 12
1471 ( frac{mathbf{d}^{2} mathbf{x}}{mathbf{d} mathbf{y}^{2}} ) equals
A ( cdotleft(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)^{-1} )
B ( cdot quad-left(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)^{-1}left(frac{mathrm{d} mathbf{y}}{mathrm{d} mathbf{x}}right)^{-3} )
C ( cdotleft(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)left(frac{mathrm{d} mathbf{y}}{mathrm{d} mathbf{x}}right)^{-2} )
D ( cdot )
( -left(frac{mathrm{d}^{2} mathbf{y}}{mathrm{d} mathbf{x}^{2}}right)left(frac{mathrm{d} mathbf{y}}{mathrm{d} mathbf{x}}right)^{-3} )
12
1472 If ( sqrt{boldsymbol{y}+boldsymbol{x}}+sqrt{boldsymbol{y}-boldsymbol{x}}=boldsymbol{c}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is
equal to
This question has multiple correct options
A ( cdot frac{2 x}{c^{2}} )
B. ( frac{x}{y+sqrt{y^{2}-x^{2}}} )
c. ( frac{y-sqrt{y^{2}-x^{2}}}{x} )
D. ( frac{c^{2}}{2 y} )
12
1473 21. Let f : [a, b] → [1,00) be a continuous function and let
g: R → R be defined as
(JEE Adv. 2014)
if x <a,
t, if a < x b.
(a) g(x) is continuous but not differentiable at a
(b) g(x) is differentiable on R
c) g(x) is continuous but not differentiable at b
(d) g(x) is continuous and differentiable at either (a) or (b)
but not both
For everunoir ofation….
12
1474 2
( 10 полапс спасен
24.
The function f :R/{0} → R given by
[2007
f(x) = 1
2
r e2x – 1
can be made continuous at x =0 by defining f (0) as
(a) o (6) 1 (c) 2 (d) -1
w
12
1475 The graph of the function ( f(x)=x^{3}+1 )
after translation 4 units to the right and 2 units up, resulted in a new graph ( l(x) ) What is the value of ( l(3.7) ? )
A. 0.973
B. 1.784
c. 1.973
D. 2.027
E . 2.973
12
1476 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(boldsymbol{3} cos left(frac{boldsymbol{pi}}{boldsymbol{6}}+boldsymbol{x}^{0}right)-boldsymbol{4} cos ^{3}left(frac{boldsymbol{pi}}{boldsymbol{6}}+boldsymbol{x}^{0}right)right. )
A ( cdot cos left(3 x^{0}right) )
B. ( frac{pi}{60} sin left(3 x^{0}right) )
c. ( frac{pi}{60} cos left(3 x^{0}right) )
D. ( -frac{pi}{60} sin left(3 x^{0}right) )
12
1477 Let ( y=left(1+x^{2}right) tan ^{-1}(x-x) ) and
( f(x)=frac{1}{2 x} frac{d y}{d x}, ) then ( f(x)+cot ^{-1} x ) is
equal to
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
c. ( -frac{pi}{2} )
D.
12
1478 If the function ( f(x)=x^{3}+e^{x} ) and
( boldsymbol{g}(boldsymbol{x})=boldsymbol{f}^{-1}(boldsymbol{x}), ) then the value of ( boldsymbol{g}^{prime}(mathbf{1}) )
is
12
1479 Solve: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} mathbf{7} boldsymbol{x} ) 12
1480 14.
Let f(x)=x sin 7tx, x>0. Then for all natural numbers n, )
vanishes at
(JEE Adv. 2013)
(a)
A unique point in the interval | n9n+
+
1
(b) A unique point in the interval
2
(c) A unique point in the interval (n, n+1)
(d) Two points in the interval (n, n+1)
12
1481 Differentiate the following w.r.t. ( x ) ( e^{sin ^{-1} x} ) 12
1482 If ( boldsymbol{y}+sin boldsymbol{y}=cos boldsymbol{x}, ) find ( frac{boldsymbol{d} boldsymbol{y} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) 12
1483 Consider the functions defined
implicitly by the equation ( y^{3}-3 y+ )
( mathbf{x}=mathbf{0} ) on various intervals in the real
line. If ( x in(-infty,-2) cup(2, infty), ) the equation implicitly defines a unique real valued differentiable function ( mathbf{y}= ) ( f(x) ). If ( x in(-2,2) ), the equation implicitly defines a unique real valued differentiable function ( mathbf{y}=mathbf{g}(mathbf{x}) )
satisfying ( mathbf{g}(mathbf{0})=mathbf{0} )
f ( mathrm{f}(-10 sqrt{2})=2 sqrt{2}, ) then ( mathrm{f}^{prime prime}(-10 sqrt{2})= )
A ( cdot frac{4 sqrt{2}}{7^{3} 3^{2}} )
в. ( -frac{4 sqrt{2}}{7^{3} 3^{2}} )
c. ( frac{4 sqrt{2}}{7^{3} 3} )
D. ( -frac{4 sqrt{2}}{7^{3} 3} )
12
1484 If Rolle’s theorem holds for the function
( boldsymbol{f}(boldsymbol{x})=mathbf{2} boldsymbol{x}^{mathbf{3}}+boldsymbol{b} boldsymbol{x}^{2}+boldsymbol{c} boldsymbol{x}, boldsymbol{x} in[-mathbf{1}, mathbf{1}], ) at
the point ( x=frac{1}{2}, ) then ( 2 b+c ) equals:
A . -1
B.
c. -3
D.
12
1485 Suppose ( f ) is differentiable on ( R ) and
( boldsymbol{a} leq boldsymbol{f}^{prime}(boldsymbol{x}) leq boldsymbol{b} ) where ( boldsymbol{x} in boldsymbol{R} ) where
( boldsymbol{a}, boldsymbol{b}>mathbf{0} . ) If ( boldsymbol{f}(mathbf{0})=mathbf{0}, ) then
A ( cdot f(x) leq min (a x, b x) )
B . ( f(x) geq min (a x, b x) )
c. ( a leq f(x) leq b )
D. ( a x leq f(x) leq b x )
12
1486 By Rolles theorem for ( f(x)=(x- )
( a)^{m}(x-b)^{n} ) on ( [a, b] ; m, n ) being
positive integer. Find the value of ( c ) which lies between ( a )
& b.
A ( cdot c=frac{m b+n a}{m+n} )
в. ( c=frac{m b-n a}{m+n} )
c. ( _{c}=frac{n b+m a}{m+n} )
D. ( c=frac{n b-m a}{m+n} )
12
1487 If ( f(x)=x+log x ) find ( f^{prime}(x) ) 12
1488 ( f(x)=x^{2}left(1-cos left(frac{2}{x}right)right) ) for ( x neq 0 )
and ( f(0)=k . ) If ( f(x) ) is continuos at
( x=0 ) then find ( k )
12
1489 Find derivative of ( f(x) )
( f(x)=x sin x )
12
1490 If ( f ) is a continuous function on the real
line. Given that ( x^{2}+(f(x)-2) x- ) ( sqrt{3} cdot f(x)+2 sqrt{3}-3=0 . ) Then the value
of ( f(sqrt{3}) )
A. can not be determined
B. ( 2(1-sqrt{3} )
c. zero
D. ( frac{2(sqrt{3}-2)}{sqrt{3}} )
12
1491 If ( x^{2}+y^{2}=a^{2} ) and ( k=1 / a, ) then ( k ) is
equal to?
A ( cdot frac{y prime}{sqrt{1+y^{prime}}} )
B. ( frac{|y prime prime|}{sqrt{left(1+y^{prime 2}right)^{3}}} )
c. ( frac{2 y prime}{sqrt{1+y prime}} )
D. ( frac{y prime}{2 sqrt{left(1+y^{prime 2}right)^{3}}} )
12
1492 Discuss the applicability of Rolle’s theorem to ( f(x)=log left[frac{x^{2}+a b}{(a+b) x}right], ) in
the interval ( [a, b] )
A. Yes Rolle’s theorem is applicable and the stationary point is ( x=sqrt{a b} )
B. No Rolle’s theorem is not applicable due to the discontinuity in the given interval
C. Yes Rolle’s theorem is applicable and the stationary point is ( x=a b )
D. none of these
12
1493 Which of the following functions is differentiable at x=0?
(a) cos(xl) + bx
(b) cos(xl) – bx (20015)
c) sin (xl) + 1x
(d) sin(xD) – x
12
1494 If ( x^{2}+y^{2}=R^{2} ) and ( K=frac{1}{R} ) then ( K= )
A. ( frac{y_{1}}{x sqrt{1+y_{1}^{2}}} )
в. ( frac{left|y_{2}right|}{sqrt{left(1+y_{1}^{2}right)^{3}}} )
c. ( frac{2left|y_{2}right|}{sqrt{1+y_{1}^{2}}} )
D. ( frac{3left|y_{2}right|}{sqrt{left(1+y_{1}^{3}right)^{3}}} )
12
1495 VJ (0)
13. The function f(x)=
he function f(x)=x2-x21(where [y] is the greatest integer
ss than or equal to y), is discontinuous at (1999 – 2 Marks)
(a) all integers
(6) all integers except 0 and 1
© all integers except 0
(d) all integers except 1
12
1496 The function ( y=sin ^{-1}(cos x) ) is not
differentiable at
This question has multiple correct options
( mathbf{A} cdot x=pi )
В. ( x=-2 pi )
c. ( x=2 pi )
D. None of these
12
1497 ( operatorname{Let} boldsymbol{f}(boldsymbol{x})= )
( frac{boldsymbol{x}(mathbf{1}+boldsymbol{a} cos boldsymbol{x})-boldsymbol{b} sin boldsymbol{x}}{boldsymbol{x}^{3}}, boldsymbol{x} neq mathbf{0} ) and
( f(0)=1 . ) The value of ( a ) and ( b ) so that ( f )
is a continuous function are-
A ( cdot frac{5}{2}, frac{3}{2} )
В. ( frac{5}{2},-frac{3}{2} )
c. ( -frac{5}{2},-frac{3}{2} )
D. None of these
12
1498 24. If velocity of particle is given by v = 24, then it
acceleration (dv/dt) at any time t will be given by…
12
1499 State whether the given statement is True or False.

Derivative of ( y=2 x^{5} ) with respect to ( x )
is ( 10 x^{4} )
A. True
B. False

12
1500 Letf:R R be a function such that f (x + y) = f(x) + f(y),
X, y E R. If f(x) is differentiable at x=0, then (2011)
(a) f(x) is differentiable only in a finite interval containing
zero
(b) f(x) is continuous x eR
(c) f'(x) is constant x ER
(d) f(x) is differentiable except at finitely many points.
12
1501 Differentiate the following w.r.t. ( x )
( log left(cos e^{x}right) )
12
1502 If ( y=f(x) ) is continuous on [0,6] differentiable on ( (0,6), f(0)=-2 ) and
( f(6)=16, ) then at some point between
( boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}=mathbf{6}, mathbf{f}^{prime}(mathbf{x}) ) must be equal to?
A . -18
B. -3
( c .3 )
D. 14
12
1503 If ( boldsymbol{y}=boldsymbol{x} sqrt{1-boldsymbol{x}^{2}}+sin ^{-1} boldsymbol{x}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is 12
1504 ( boldsymbol{y}=boldsymbol{A} cos boldsymbol{n} boldsymbol{x}+boldsymbol{B} sin boldsymbol{n} boldsymbol{x} )
Prove that ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}+boldsymbol{n}^{2} boldsymbol{y}=mathbf{0} )
12
1505 In the law of mean, the value of ( theta )
satisfies the condition
( mathbf{A} cdot theta>0 )
B . ( theta1 )
( 0.0<theta<1 )
12
1506 Let [.] denote the greatest integer
function and ( f(x)=left[tan ^{2} xright] . ) Then
( mathbf{A} cdot lim _{x rightarrow 0} f(x) ) does not exis
B. ( f(x) ) is continuous at ( x=0 )
c. ( f(x) ) is not differentiable at ( x=0 )
12
1507 If ( y=left(tan ^{-1} xright)^{2} ) and ( left(x^{2}+1right)^{2} frac{d^{2} y}{d x^{2}}+ )
( 2 xleft(x^{2}+1right) frac{d y}{d x}=k, ) then the value of ( k )
is
A . 3
B. 2
c. 1
D.
12
1508 If ( y=x sqrt{a^{2}+x^{2}}+ )
( a^{2} log (x+sqrt{a^{2}+x^{2}}) ) then ( frac{d y}{d x}= )
( 2 sqrt{a^{2}+x^{2}} )
12
1509 Find the derivative of ( f(x)= ) ( frac{x+cos x}{tan x} w . r . t . x ) 12
1510 ( f frac{a_{0}}{n+1}+frac{a_{1}}{n}+frac{a_{2}}{n-1}+dots+frac{a_{n-1}}{2}+ )
( a_{n}=0, ) then the equation ( a_{0} x^{n}+ )
( a_{1} x^{n-1}+cdots+a_{n-1} x+a_{n}=0 ) has, in
the interval ( (mathbf{0}, mathbf{1}) )
A. Exactly one root
B. Atleast one root
c. Atmost one root
D. No root
12
1511 If ( boldsymbol{x}=boldsymbol{a} boldsymbol{t}^{2}, quad boldsymbol{y}= )
2at then ( boldsymbol{d}^{2} boldsymbol{y} / boldsymbol{d} boldsymbol{x}^{2}= )
A ( cdot frac{-1}{t^{2}} )
в. ( frac{1}{t^{2}} )
c. 0
D. ( frac{1}{2 a^{3}} )
12
1512 Differentiate ( boldsymbol{y}=cos (2 x-5) ) with
respect to ( x )
12
1513 Obtain the differential equation of the
family of circles ( x^{2}+y^{2}+2 g x+ )
( 2 f y+c=0 ; ) where ( g, f ) and ( c ) are
arbitrary constants.
( ^{A} cdotleft[1+left(y^{prime}right)^{2}right] y^{prime prime}-3 y^{prime}left(y^{prime prime}right)^{2}=0 )
( ^{mathrm{B}}left[1+left(y^{prime prime}right)^{3}right] y-2 y^{prime}left(y^{prime prime}right)^{2}=0 )
( ^{mathbf{C}}left[1+left(y^{prime prime}right)^{2}right] y^{prime prime}-3 y^{prime}left(y^{prime prime}right)^{2}=0 )
D. None of these
12
1514 8.
f(x) and g(x) are two differentiable functions on [0,2] such
that f”(x)-g”(x) = 0, f'(1) = 2g'(1) = 4 f(2)=3g(2)=9
then f(x)-g(x) at x =3/2 is
[2002]
(a) o (6) 2 (c) 10 (d) 5
12
1515 ( lim _{n rightarrow infty}left(frac{(n+1)(n+2) dots 3 n}{n^{2 n}}right)^{frac{1}{n}} )
equal to :-
12
1516 12. x/1+ y + y/1+ x = 0, then dy
dar
(a) 1+ x
(b) (1+x)?
(c) -(1 + x)-1
(d) -(1 + x)-2
I
do
12
1517 Answer the following question in one word or one sentence or as per exact requirement of the question.
If ( boldsymbol{x}<mathbf{2}, ) then write the value of ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(sqrt{boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+boldsymbol{4}}) )
12
1518 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} tan ^{-1}left[frac{boldsymbol{x}^{mathbf{1} / 3}+boldsymbol{a}^{mathbf{1} / 3}}{mathbf{1}-boldsymbol{x}^{mathbf{1} / 3} boldsymbol{a}^{mathbf{1} / 3}}right] ) 12
1519 If ( boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{2} ) and ( boldsymbol{y}_{2}=boldsymbol{A} boldsymbol{y}^{-3} ) then ( mathbf{A}= )
A . -2
B. –
( c cdot 0 )
( D )
12
1520 The value of ( k ) for which the function
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{ll}frac{1-cos 4 x}{8 x^{2}} & , x neq 0 \ k & , x=0end{array}right. )
continuous at ( boldsymbol{x}=mathbf{0}, ) is
( mathbf{A} cdot k=0 )
B. ( k=1 )
c. ( k=-1 )
D. None of the above
12
1521 Let ( f ) be differentiable for all ( x ). If ( f(1)= )
-2 and ( f^{prime}(x) geq 2 ) for ( x in[1,6], ) then
( mathbf{A} cdot mathbf{f}(6) geq 8 )
B. ( f(6)<8 )
( mathrm{c} cdot mathrm{f}(6)<5 )
D. ( f(6)=5 )
12
1522 Find derivative of ( sin ^{-1}left(x^{2}right) ) using first
principle.
A ( cdot frac{2 x}{sqrt{1-x^{2}}} )
в. ( frac{2 x}{sqrt{1-x}} )
c. ( frac{2 x}{sqrt{1-x^{4}}} )
D. ( frac{x}{sqrt{1-x^{4}}} )
12
1523 If the derivative of the functions ( f(x)= ) ( left{begin{array}{cc}b x^{2}+a x+4 ; & x geq-1 \ a x^{2}+b ; & x<-1end{array}right} ) is
everywhere continuous then
A ( . a=2, b=3 )
В. ( a=3, b=2 )
C ( . a=-2, b=-3 )
D. ( a=-3, b=-2 )
12
1524 If ( boldsymbol{f}(boldsymbol{x})=mathbf{1} ) for ( boldsymbol{x}<mathbf{0}=mathbf{1}+sin boldsymbol{x} ) for
( 0 leq x<pi / 2, ) then at ( x=0, ) then show
that the derivative ( f^{prime}(x) ) does not exist.
12
1525 ( operatorname{Let} F(x)=f(x) g(x) h(x) ) for all real ( x )
where ( boldsymbol{f}(boldsymbol{x}), boldsymbol{g}(boldsymbol{x}), boldsymbol{h}(boldsymbol{x}) ) are
differentiable functions. At some point ( boldsymbol{x}_{0}, ) if ( boldsymbol{F}^{prime}left(boldsymbol{x}_{0}right)=mathbf{2 1} boldsymbol{F}left(boldsymbol{x}_{0}right), boldsymbol{f}^{prime}left(boldsymbol{x}_{0}right)= )
( 4 fleft(x_{0}right), g^{prime}left(x_{0}right)=-7 gleft(x_{0}right) ) and
( h^{prime}left(x_{0}right)=lambda hleft(x_{0}right), ) then ( lambda= )
A . 12
B. -12
( c cdot 24 )
D. -24
12
1526 Compute the derivative of ( 6 x^{100}- )
( boldsymbol{x}^{boldsymbol{5} boldsymbol{5}}+boldsymbol{x} )
12
1527 ( boldsymbol{x}=boldsymbol{e}^{boldsymbol{theta}}(sin boldsymbol{theta}+cos boldsymbol{theta}), boldsymbol{y}=boldsymbol{e}^{boldsymbol{theta}}(sin boldsymbol{theta}-boldsymbol{1}) )
( cos boldsymbol{theta}) )
Fine ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
12
1528 If ( y=3 cos x, ) then ( frac{d y}{d x} ) at ( x=frac{pi}{2} ) is
A. -3
B. 3
( c .0 )
D. –
12
1529 Find drivative of ( boldsymbol{y}=(2-sin x)left(e^{x}+right. )
( left.x^{3}+2right) ) with respect to ( x )
12
1530 The degree and order of differential
equation ( left(frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}right)^{2}=left(boldsymbol{y}+frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)^{frac{1}{2}} )
which of the following?
12
1531 Verify Rolle’s theorem for the following
function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{2}-mathbf{5} boldsymbol{x}+mathbf{9}, boldsymbol{x} varepsilon[mathbf{1}, boldsymbol{4}] )
12
1532 ( operatorname{Let} boldsymbol{f}(boldsymbol{x})=boldsymbol{a}_{5} boldsymbol{x}^{5}+boldsymbol{a}_{4} boldsymbol{x}^{4}+boldsymbol{a}_{3} boldsymbol{x}^{3}+ )
( a_{2} x^{2}+a_{1} x, ) where ( a_{i}^{prime} s ) are real and
( f(x)=0 ) has a positive root ( alpha_{0} . ) Then
This question has multiple correct options
A ( cdot f^{prime}(x)=0 ) has a root ( alpha_{1} ) such that ( 0<alpha_{1}<alpha_{0} )
B . ( f^{prime}(x)=0 ) has at least one real root
C ( cdot f^{prime prime}(x)=0 ) has at least one real root
D. All of the above
12
1533 Illustration 2.23
Find the derivatives of y=(x + 1) (x+3).
12
1534 If ( boldsymbol{y}=log _{e}left(frac{boldsymbol{x}^{2}}{e^{2}}right), ) then ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} ) equals
( A cdot-frac{1}{x} )
B. ( -frac{1}{x^{2}} )
c. ( frac{2}{x^{2}} )
D. ( -frac{2}{x^{2}} )
12
1535 35.
If the function.
g(x)=
JkVx+1, 0553
[mx+2, 3 **ss is differentiable, then the value
ofk+ mis :
JEE M 20151
(2) 10
(6) 4
(0) 2
(2) 16
12
1536 If ( f(x)=x^{2}-6 x+8 ) and there exists
a point ( c ) in the interval [2,4] such that
( boldsymbol{f}^{prime}(boldsymbol{c})=mathbf{0}, ) then what is the value of ( boldsymbol{c} ? )
A . 2.5
B. 2.8
( c cdot 3 )
D. 3.5
12
1537 The value of ( f(0) ) so that the function ( f(x)=frac{2 x-sin ^{-1} x}{2 x+tan ^{-1} x} ) is continuous at
each point in its domain, is equal to
A . 2
в. ( frac{1}{3} )
c. ( frac{2}{3} )
D. ( frac{-1}{3} )
12
1538 If the function ( g(x) ) is defined by ( boldsymbol{g}(boldsymbol{x})=frac{boldsymbol{x}^{200}}{200}+frac{boldsymbol{x}^{199}}{199}+frac{boldsymbol{x}^{198}}{198}+ldots .+ )
( frac{x^{2}}{2}+x+5, ) then ( g^{prime}(0)= )
( A )
в. 200
( c .100 )
D. 5
12
1539 The interval on which ( f(x)=sqrt{1-x^{2}} ) is continuous is:
A. ( (0, infty) )
(i)
В. ( (1, infty) )
c. [-1,1]
D. ( (-infty,-1) )
12
1540 If for all ( x, y ) the function ( f ) is defined by
( boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})+boldsymbol{f}(boldsymbol{x}) cdot boldsymbol{f}(boldsymbol{y})=1 ) and
( boldsymbol{f}(boldsymbol{x})>mathbf{0}, ) then
A ( cdot f^{prime}(x) ) does not exist
B ( cdot f^{prime}(x)=0 ) for all ( x )
c. ( f^{prime}(0)<f^{prime}(1) )
D. None of these
12
1541 If ( x^{3}+y^{3}=3 a x y, ) find ( frac{d y}{d x} ) 12
1542 et f: R
→ R be a continuous function defined by
28
ex + 2ex
[2010]
Statement-1:f
some c ER
Statement -2:0<f(x) s, for all x ER
(a) Statement -1 is true, Statement -2 is true ; Statement-2
is not a correct explanation for Statement -1.
(b) Statement -1 is true, Statement -2 is false.
(C) Statement-1 is false, Statement -2 is true .
(d) Statement – 1 is true, Statement 2 is true; Statement -2
is a correct explanation for Statement -1.
12
1543 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{x}^{2}+boldsymbol{x}+mathbf{1} ) and
( boldsymbol{g}(boldsymbol{x})= )
( left{begin{array}{l}max {f(t)}, quad 0 leq t leq x quad 0 leq x leq 1 \ 3-x, quad 1<x leq 2end{array}right. )
Then in the interval ( [0,2], g(x) ) is
This question has multiple correct options
A. Continuous for all ( x )
B. Differentiable for all ( x )
c. Discontinuous at ( x=1 )
D. Not differentiable at ( x=1 )
12
1544 If the functions ( f(x) ) and ( g(x) ) are continuous on ( [a, b] ) and differentiable on ( (a, b), ) then in the interval ( (a, b), ) the
equation ( left|begin{array}{ll}boldsymbol{f}^{prime}(boldsymbol{x}) & boldsymbol{f}(boldsymbol{a}) \ boldsymbol{g}^{prime}(boldsymbol{x}) & boldsymbol{g}(boldsymbol{a})end{array}right|=frac{1}{boldsymbol{a}-boldsymbol{b}}left|begin{array}{ll}boldsymbol{f}(boldsymbol{a}) & boldsymbol{f}(boldsymbol{b}) \ boldsymbol{g}(boldsymbol{a}) & boldsymbol{g}(boldsymbol{b})end{array}right| )
A. has at least one root
B. has exactly one root
c. has at most one root
D. no root
12
1545 Differentiate w.r.t. ( boldsymbol{x} )
( boldsymbol{y}=e^{3 x-2} sin 3 x )
12
1546 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{x}^{2}+boldsymbol{x}+1 ) and
( boldsymbol{g}(boldsymbol{x})= )
( left{begin{array}{lll}max (boldsymbol{f}(boldsymbol{t})) & text { for } & mathbf{0} leq boldsymbol{t} leq boldsymbol{x} \ boldsymbol{3}-boldsymbol{x}+boldsymbol{x}^{2} & text { for } & mathbf{1}<boldsymbol{x} leq mathbf{2}end{array}right. )
then
A. ( g(x) ) is continuous and derivable at ( x=1 )
B. ( g(x) ) is continuous but not derivable at ( x=1 )
c. ( g(x) ) is neither continuous nor derivable at ( x=1 )
D. ( g(x) ) is derivable but not continuous at ( x=1 )
12
1547 28. If f (x) is continuous and differentiable function and
f(1/n)=0 n land nel, then
(2005)
(a) f(x)=0, x € (0, 1]
(b) 10=0, 0) = 0
(C) FO)=0= f'(O), X € (0,1]
(d) f0 = 0 and f’o need not to be zero
12
1548 Illustration 2.33
If y= cos x”, then find
12
1549 If ( y=frac{x sin ^{-1} x}{sqrt{1-x^{2}}}, ) prove that ( (1- )
( left.x^{2}right) frac{d y}{d x}=x+frac{y}{x} )
12
1550 27. If y= tan x. cosx then
will be …
12
1551 Let ( f ) and ( g ) be functions satisfying
( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} boldsymbol{g}(boldsymbol{x}), boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+ )
( boldsymbol{f}(boldsymbol{y}), boldsymbol{g}(mathbf{0})=mathbf{0}, boldsymbol{g}^{prime}(mathbf{0})=mathbf{4}, boldsymbol{g} ) and ( boldsymbol{g}^{prime} ) are
continuous at 0
Then
A. ( f(x)=0 ) for all ( x )
B. ( f(x)=x ) for all ( x )
c. ( f(x)=x+4 ) for all ( x )
D. ( f(x)=4 x ) for all ( x )
12
1552 Find the derivative of ( boldsymbol{y}= ) ( frac{1}{4} ln frac{x^{2}-1}{x^{2}+1} ) 12
1553 For the function ( boldsymbol{f}(boldsymbol{x})= )
( (x-1)(x-2)(x-3) ) in [0,4] value of
c’ in Lagrange’s mean value theorem is
A ( cdot 2 pm frac{2}{sqrt{3}} )
B. ( _{1-frac{sqrt{21}}{6}} )
c. ( 1+frac{sqrt{21}}{6} )
D. ( 4-2 sqrt{3} )
12
1554 If ( boldsymbol{x}=boldsymbol{a}left(boldsymbol{1}-cos ^{3} boldsymbol{theta}right), boldsymbol{y}=boldsymbol{a} sin ^{3} boldsymbol{theta}, ) prove
that ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}=frac{boldsymbol{3} mathbf{2}}{mathbf{2 7} boldsymbol{a}} ) at ( boldsymbol{theta}=frac{boldsymbol{pi}}{boldsymbol{6}} )
12
1555 f ( boldsymbol{y}=boldsymbol{a} cos (sin 2 boldsymbol{x})+boldsymbol{b} sin (sin 2 boldsymbol{x}) )
then ( boldsymbol{y}^{prime prime}+(2 tan 2 boldsymbol{x}) boldsymbol{y}^{prime}= )
( A )
B ( cdot 4left(cos ^{2} 2 xright) y )
c. ( -4left(cos ^{2} 2 xright) y )
( D cdot-left(cos ^{2} 2 xright) y )
12
1556 If ( sin x=frac{2 t}{1+t^{2}}, ) tany ( =frac{2 t}{1-t^{2}}, ) then
( d y ) is equal to ( A_{n} )
A . –
B. 2
( c cdot 0 )
D.
12
1557 If ( f(x)=tan x, ) find ( f^{prime}(x) ) and hence find ( boldsymbol{f}^{prime}left(frac{boldsymbol{pi}}{boldsymbol{4}}right) ) 12
1558 Find the derivative of ( sin ^{2} x ) with respect to ( x ) using product rule 12
1559 State whether the following statement is true or false.
Enter 1 for true and 0 for false
( f(x) ) is differentiable at a point ( P, ) if
there exists a unique tangent at point ( boldsymbol{P} )
12
1560 ( boldsymbol{y}=sqrt{frac{boldsymbol{1}-boldsymbol{x}}{mathbf{1}+boldsymbol{x}}} )
Prove that ( left(mathbf{1}-boldsymbol{x}^{2}right) frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}+boldsymbol{y}=mathbf{0} )
12
1561 If ( f(x)=frac{x^{2}-9}{x^{2}-2 x-3}, x neq 3 ) is
continuous at ( x=3, ) then which one of
the following is correct?
A. ( f(3)=0 )
B. ( f(3)=1.5 )
c. ( f(3)=3 )
D. ( f(3)=-1.5 )
12
1562 Differentiate w.r.t ( boldsymbol{x} )
( e^{operatorname{cosec}^{2} x} )
12
1563 Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-boldsymbol{x}+boldsymbol{p}(boldsymbol{0} geq boldsymbol{x} geq mathbf{2}) )
where ( p ) is a constant. The value ( c ) of
mean value theorem is :
A ( cdot frac{sqrt{3}}{2} )
B. ( frac{sqrt{6}}{2} )
c. ( frac{sqrt{3}}{3} )
D. ( frac{sqrt{2}}{3} )
E ( cdot frac{2 sqrt{3}}{3} )
12
1564 Find the derivative of the following functions from first principle ( (x-1)(x-2) ) 12
1565 If ( boldsymbol{f}(boldsymbol{x})= )
[
left{begin{array}{cc}
frac{x^{2}}{2}, & text { if } 0 leq x leq 1 \
2 x^{2}-3 x+frac{3}{2}, & text { if } 1<x leq 2
end{array},text { Show }right.
]
that ( f ) is continuous at ( x=1 )
12
1566 Solve : ( int frac{x^{2}+1}{(x+1)^{2}} d x ) 12
1567 Differentiate the given function w.r.t. ( x ) ( boldsymbol{y}=sqrt{e^{sqrt{x}}}, boldsymbol{x}>0 ) 12
1568 ( mathbf{f}_{boldsymbol{f}(boldsymbol{x})}=left{begin{array}{ll}frac{boldsymbol{x}^{2}-boldsymbol{9}}{boldsymbol{x}-mathbf{3}}+boldsymbol{alpha} & , text { for } boldsymbol{x}>mathbf{3} \ mathbf{5} & , text { for } boldsymbol{x}=mathbf{3} \ mathbf{2} boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}+boldsymbol{beta} & , text { for } boldsymbol{x}<mathbf{3}end{array}right. )
is continuous at ( x=3, ) find ( alpha ) and ( beta )
12
1569 If ( boldsymbol{y}=sec left(tan ^{-1} xright), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{1} ) is
equal to.
A ( cdot frac{1}{sqrt{2}} )
B. ( frac{1}{2} )
( c cdot 1 )
D. ( sqrt{2} )
12
1570 If ( y=sin left(m sin ^{-1} xright) ) then
( left(1-x^{2}right) y^{prime prime}-x y^{prime} ) is equal to
A ( cdot m^{2} y )
в. ( m y )
c. ( -m^{2} y )
D. None of these
12
1571 If ( y ) is expressed in terms of a variable ( x ) as ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), ) then ( boldsymbol{y} ) is called
A. Explicit function
B. Implicit function
c. Linear function
D. Identity function
12
1572 Solve :
( int frac{1+x cos x}{xleft(1-x^{2} e^{2 sin x}right)} d x= )
( k ell n sqrt{frac{x^{2} e^{2 sin x}}{1-x^{2} e^{2 sin x}}}+C ) then ( k ) is equal
to
12
1573 If the derivatives of ( tan ^{-1}(a+b x) ) takes
the value 1 at ( x=0, ) prove that ( 1+ )
( a^{2}=b )
12
1574 If ( f(x)= ) ( left{begin{array}{c}frac{1-sin x}{(pi-2 x)^{2}} cdot frac{log sin x}{log left(1+pi^{2}-4 pi x+x^{2}right)} \ kend{array}right. )
is continuous at ( x=frac{pi}{2}, ) then ( k ) is equal
to.
A ( cdot-frac{1}{16} )
B. ( -frac{1}{32} )
( c cdot-frac{1}{64} )
D. ( -frac{1}{28} )
12
1575 Find all points of discontinuity of ( boldsymbol{f} )
where ( f ) is defined by ( f(x)= )
( left{begin{array}{ll}2 x+3, & x leq 2 \ 2 x-3, & x>2end{array}right. )
12
1576 ( f(x)=x^{4}-3 x^{2}+4 ) in the interval [-4
4]. Is Rolle’s theorem applicable?
12
1577 Solve:
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(sin ^{-1}left{frac{sqrt{mathbf{1}+boldsymbol{x}}+sqrt{mathbf{1}-boldsymbol{x}}}{mathbf{2}}right}right) )
A ( cdot frac{-1}{2 sqrt{1-x^{2}}} )
в. ( frac{1}{2 sqrt{1-x^{2}}} )
c. ( frac{1}{sqrt{1-x^{2}}} )
D. ( frac{-1}{sqrt{1-x^{2}}} )
12
1578 If the function ( mathbf{f}(boldsymbol{x})= ) ( frac{log (1+boldsymbol{a} boldsymbol{x})-log (1-boldsymbol{b} boldsymbol{x})}{boldsymbol{x}} ) for ( boldsymbol{x} neq mathbf{0} ) is
continuous at ( x=0 ) then ( f(0)= )
A ( . a-b )
B. ( a+b )
( mathbf{c} cdot log a+log b )
D. ( log a-log b )
12
1579 The set of all points of continuity of ( f o ) fo( f, ) where ( f(x)=operatorname{sgn} x ) is
A ( . R sim{0} )
в. ( R sim{1,0,1} )
c. ( R sim{-1,1} )
D. none of these
12
1580 Consider ( mathbf{f}(mathbf{x})=left{begin{array}{l}frac{x^{2}}{|x|}, mathbf{x} neq 0 \ mathbf{0}, mathbf{x}=0end{array}right. )
Then find the continuity of the function
( f(x) )
( mathbf{A} cdot f(mathrm{x}) ) is discontinuous every where
B. ( f(x) ) is continuous only at ( x=0 )
C ( . f(x) ) is discontinuous everywhere except at ( x=0 )
D. ( f(x) ) is continuous everywhere
12
1581 Illustration 2.21 If y= 3x + 2x, then find dyldx. 12
1582 Find ( frac{d y}{d x} ) while:
( boldsymbol{x}^{boldsymbol{y}}+boldsymbol{y}^{boldsymbol{x}}=boldsymbol{a}^{boldsymbol{b}} )
12
1583 ( operatorname{If}left(1+3 x+3 x^{2}right)^{20}=a_{0}+a_{1} x+ )
( boldsymbol{a}_{2} boldsymbol{x}^{2}+boldsymbol{a}_{3} boldsymbol{x}^{3}+boldsymbol{a}_{4} boldsymbol{x}^{4}+boldsymbol{a}_{5} boldsymbol{x}^{5}+ldots ldots+ )
( a_{40} x^{40}, ) then find the value of ( 2 a_{2}- )
( 6 a_{3}+12 a_{4}-20 a_{5} dots dots+1560 a_{40} )
A . 3450
B. 3350
( c .3540 )
D. 2150
12
1584 Differentiate with respect to ( x: e^{(5 x+2)} )
( A cdot 5 e^{5 x+2} )
B. ( 10 e^{5 x+2} )
( mathbf{c} cdot 25 e^{5 x+2} )
D. ( e^{5 x+2} )
12
1585 Differentiate the following ( x^{2}(3 x-2)^{4} cos x ) 12
1586 ( frac{d}{d x}left{cos ^{-1} x+sin ^{-1} sqrt{1-x^{2}}right}= )
A .
B.
c. ( frac{2}{sqrt{1-x^{2}}} )
D. ( frac{-2}{sqrt{1-x^{2}}} )
12
1587 Find the derivative of
( left(5 x^{3}+3 x-1right)(x-1) )
12
1588 If ( f(x)=cos ^{2} x+cos ^{2}left(frac{pi}{3}+xright) )
( cos x cos left(frac{pi}{3}+xright) ) then ( 4 fleft(frac{pi}{8}right) ) is equal
to
12
1589 ( boldsymbol{f}(boldsymbol{x})=(sin boldsymbol{x}+cos boldsymbol{x}) ) Find ( boldsymbol{f}^{prime}(boldsymbol{x}) ) 12
1590 Let then ( boldsymbol{f}(boldsymbol{x})= )
( (x-4)(x-5)(x-6)(x-7) ) then
A ( cdot f^{prime}(x)=0 ) has four real roots
( mathbf{B} cdot ) three roots of ( f^{prime}(x)=0 ) lie in (4,5)( cup(5,6) cup(6,7) )
C ( cdot ) the euation ( f^{prime}(x)= ) has only two roots ( >d ). three roots
of ( f^{prime}(x)=0(3,4) cup(4,5) cup(5,6) )
D. three roots of ( f^{prime}(x)=0(3,4) cup(4,5) cup(5,6) )
12
1591 Applying mean value theorem on ( boldsymbol{f}(boldsymbol{x})=log boldsymbol{x} ; boldsymbol{x} in[mathbf{1}, boldsymbol{e}] ) the value of ( boldsymbol{c}= )
( mathbf{A} cdot log (e-1) )
в. ( e-1 )
( mathbf{c} cdot 1-e )
( D )
12
1592 If ( boldsymbol{y}=cot ^{-1}left[frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right] )
where ( 0<x<frac{pi}{2}, ) then ( frac{d y}{d x} ) is equal to
A. ( -frac{1}{2} )
B. 2
( c cdot sin x+cos x )
( mathbf{D} cdot sin x-cos x )
12
1593 Let ( boldsymbol{f}: mathbb{R} rightarrow(0,1) ) be a continuous
function. Then, which of the following function(s) has (have) the value zero at
some point in the interval (0,1)( ? ) This question has multiple correct options
( mathbf{A} cdot f(x)+int_{0}^{frac{pi}{2}} f(t) sin t d t )
B . ( e^{x}-int_{0}^{x} f(t) sin t d t )
C ( cdot x-int_{0}^{frac{pi}{2}-x} f(t) cos t d t )
D. ( x^{9}-f(x) )
12
1594 At what point on the curve ( y=x(x-4) ) on [0,4] is the tangent parallel to ( X ) -axis. 12
1595 f ( f: R rightarrow R ) is defined by ( f(x)= ) ( left{begin{array}{ccc}frac{boldsymbol{x}+mathbf{2}}{boldsymbol{x}^{2}+mathbf{3} boldsymbol{x}+mathbf{2}} & boldsymbol{i f} & boldsymbol{x} in boldsymbol{R}-{-mathbf{1},-mathbf{2}} \ -mathbf{1} & boldsymbol{i f} & boldsymbol{x}=-mathbf{2} \ mathbf{0} & boldsymbol{i f} & boldsymbol{x}=-mathbf{1}end{array}right. )
then ( f ) is continuous on the set.
( A )
в. ( R-{-2} )
c. ( R-{-1} )
D. ( R-{-1,-2} )
12
1596 For the function ( boldsymbol{f}(boldsymbol{x})=left|boldsymbol{x}^{2}-mathbf{5} boldsymbol{x}+boldsymbol{6}right| )
the right hand derivative ( f^{prime}(2+) ) is
equal to.
12
1597 If it is possible to make ( f(x) ) continuous ( operatorname{at} x=2 ) then ( f(2) ) is equal to
A. 0
B. 2
( c cdot 3 )
( D )
12
1598 Function ( f(x)=left{begin{array}{ll}5 x-4 & text { for } 0<x leq 1 \ 4 x^{2}-3 x & text { for } 1<x<2 \ 3 x+4 & text { for } x geq 2end{array}right. )
A. continuous at ( x=1 ) and ( x=2 )
B. continuous at ( x=1 ) but not derivable at ( x=z )
c. continuous at ( x=2 ) but not derivable at ( x=1 )
D. none of these
12
1599 Verify Lagrange’s mean value theorem for the following function on the indicated interval. In each case find a
point ( ^{prime} c^{prime} ) in the indicated interval as stated by the Lagrange’s mean value theorem:
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{3}-mathbf{5} boldsymbol{x}-boldsymbol{3} ) on ( [mathbf{1}, boldsymbol{3}] )
12
1600 If ( f(x)=sin x, ) find ( frac{d y}{d x} )
A ( cdot cos x )
B. ( -cos x )
c. ( cot x )
D. ( -c o t^{2} x )
12
1601 If ( boldsymbol{y}=frac{1}{4}(boldsymbol{x} pm boldsymbol{A})^{2} )
Hence prove: ( boldsymbol{y}_{1}^{2}=boldsymbol{y} )
12
1602 Let ( f(x) ) be a function satisfying ( f(x+ ) ( boldsymbol{y})=mathbf{f}(mathbf{x}) mathbf{f}(mathbf{y}) ) for all ( boldsymbol{x}, boldsymbol{y} in mathbf{R} ) and
( mathbf{f}(mathbf{x})=mathbf{1}+mathbf{x} mathbf{g}(mathbf{x}), ) where ( lim _{x rightarrow 0} mathbf{g}(mathbf{x})=mathbf{1} )
then ( f^{prime}(x) ) is equal to
A. ( x g(x) )
в. ( mathrm{g}^{prime}(mathrm{x}) )
c. ( f(x) )
D.
12
1603 ff ( boldsymbol{y}=tan ^{-1} frac{1}{1+x+x^{2}}+ )
( tan ^{-1} frac{1}{x^{2}+3 x+3}+ )
( tan ^{-1} frac{1}{x^{2}+5 x+7}+ldots+ ) upto ( n ) terms
then ( y^{prime}(0) ) is equal to
A ( cdot-frac{1}{1+n^{2}} )
B. ( -frac{n^{2}}{1+n^{2}} )
c. ( frac{n}{1+n^{2}} )
D. none of these
12
1604 If ( y=frac{x^{4}-x^{2}+1}{x^{2}+sqrt{3} x+1} ) and ( frac{d y}{d x}=a x+b )
then the value of ( a+b ) is equal to
A ( cdot cot frac{5 pi}{8} )
B. ( cot frac{5 pi}{12} )
c. ( tan frac{5 pi}{12} )
D. ( tan frac{5 pi}{8} )
12
1605 Find the second derivative of ( sin 3 x cos )
( 5 x )
12
1606 Differentiate the following function w.r.t
( x )
( sqrt{x+frac{1}{x}} )
12
1607 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{boldsymbol{x}}{log boldsymbol{x}^{2}+mathbf{1}}right) ) 12
1608 Differentiate: ( frac{e^{x}}{sin x} ) 12
1609 Differentiate :
( y^{x}=x^{y} )
12
1610 is continuous but not derivable at ( x=0 )
A. ( m in[0,3] )
B . ( m in(0,2) )
c. ( m in(0,1] )
D. ( m=0,1 )
12
1611 ( boldsymbol{i} boldsymbol{f} boldsymbol{x} sqrt{boldsymbol{1}+boldsymbol{y}}+boldsymbol{y} sqrt{boldsymbol{1}+boldsymbol{x}}= )
( 0, ) then ( frac{d y}{d x} ) is equal to
A ( cdot frac{1}{(1+x)^{2}} )
B. ( -frac{1}{(1+x)^{2}} )
c. ( frac{1}{left(1+x^{2}right)} )
D. ( frac{1}{(1+x)} )
12
1612 Differentiate with respect to ‘t’ ( e^{-w t} ) 12
1613 Verify LMVT for the function ( f(x)=x+ )
( frac{1}{x}, x in[1,3] )
12
1614 Differentiate w.r.t. ( x ) in ( tan ^{-1}left(frac{5 x}{1-6 x^{2}}right) ) 12
1615 19. Letf:
R R be a function defined by f(x)=max {x,x}. The
set of all points where f() is NOT differentiable is (20015)
(a) {-1,1} (b) -1,0; (c) {0,1; (d){-1,0,1)
20 VL1
12
1616 Find ( frac{d y}{d x}, ) if ( y=sqrt{cos (3 x+1)} ) 12
1617 ( operatorname{Let} f(x)=(x+1) 2^{-left(frac{1}{[x]}+frac{1}{x}right)} ) and
( boldsymbol{f}(mathbf{0})=mathbf{0} )
A. ( f ) is continuous at ( x=0 )
B. ( lim _{x rightarrow 0^{+}} f(x) ) exists
C. ( lim _{x rightarrow 0^{+}} f(x) ) does not exist
D. ( lim _{x rightarrow 0} f(x) neq lim _{x rightarrow 0^{-}} f(x) )
12
1618 If ( 2^{x}+2^{y}=2^{x+y}, ) then find ( frac{d y}{d x} ) 12
1619 If ( y=sec ^{-1}left(frac{1}{2 x^{2}-1}right) ) then ( frac{d y}{d x}=? )
A ( cdot frac{-2}{left(1+x^{2}right)} )
B. ( frac{-2}{left(1-x^{2}right)} )
c. ( frac{-2}{sqrt{1-x^{2}}} )
D. none of these
12
1620 7.
Iff(x) is a twice differentiable function and given that
(1) = 1;/2) = 4,/3)=9, then
(2005)
(a) F”(x) = 2 for xe (1,3)
(b) f(x)=f()=5 for some x = (2,3)
c) S”(x)=3 for xe (2,3)
(d) {“(x)=2 for some x = (1,3)
12
1621 If the function ( f ) defined on ( left(-frac{1}{3}, frac{1}{3}right) ) by ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}frac{1}{x} log _{e}left(frac{1+3 x}{1-2 x}right), & text { when } x neq 0 \ k, & text { when } x=0end{array}right. )
is continuous, then k is equal to
12
1622 [
begin{array}{rlr}
text { If } boldsymbol{f}(boldsymbol{x}) & =frac{sin 4 boldsymbol{x}}{mathbf{5} boldsymbol{x}}+boldsymbol{a} & text { for } boldsymbol{x}>mathbf{0} \
& =boldsymbol{x}+mathbf{4}-boldsymbol{b} & text { for } boldsymbol{x}<mathbf{0} \
& =mathbf{1} & text { for } boldsymbol{x}=mathbf{0}
end{array}
]
is continuous at ( x=0 . ) Find ( a & b )
12
1623 ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}a tan ^{-1}left(frac{1}{x-4}right) text { if } 0 leq x<4 \ b tan ^{-1}left(frac{2}{x-4}right) text { if } 4<x<6 \ sin ^{-1}(7-x)+a frac{pi}{4} quad text { if } sin ^{-1}(7-2)end{array}right. )
and ( f(4)=pi / 2 ) is continuous on (0,8)
then
A . (1,1)
B. (1,-1)
c. (-1,1)
D. (-1,-1)
12
1624 ( int_{0}^{1} frac{e^{x}}{1+e^{2 x}} d x )
( mathbf{A} cdot tan ^{-1} e-frac{pi}{4} )
B ( cdot tan ^{-1} e+frac{pi}{4} )
( mathrm{C} cdot tan e-frac{pi}{4} )
D. None of these
12
1625 Find the value of the constant ( k ) so that
the given function is continuous at the
indicated point:
( boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}boldsymbol{k} boldsymbol{x}+mathbf{1}, text { if } boldsymbol{x} leq mathbf{5} \ boldsymbol{3} boldsymbol{x}-mathbf{5}, text { if } boldsymbol{x}>mathbf{5}end{array} text { at } boldsymbol{x}=mathbf{5}right. )
12
1626 If ( x^{y}+y^{x}=a^{b} ) then show that ( frac{d y}{d x}= ) ( -left[frac{boldsymbol{y} boldsymbol{x}^{boldsymbol{y}-mathbf{1}}+boldsymbol{y}^{boldsymbol{x}} log boldsymbol{y}}{boldsymbol{x}^{boldsymbol{y}} log boldsymbol{x}+boldsymbol{x} boldsymbol{y}^{boldsymbol{x}-1}}right] ) 12
1627 Examine the Rolles theorem is
applicable to the followng function. Find the number of points the following function is not continous?
( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] ) for ( boldsymbol{x} boldsymbol{epsilon}[boldsymbol{2}, boldsymbol{2}] )
12
1628 Assertion
Statement -1: If ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{c}boldsymbol{x} cos boldsymbol{x} cdot sin left(frac{1}{boldsymbol{x} cos boldsymbol{x}}right), quad text { whenever de } \ mathbf{0}end{array}right. )
then ( f(x) ) is continuous
Reason
Statement – ( 2: lim _{x rightarrow infty} frac{sin x}{x}=0 )
A. Statement -1 is True, Statement -2 is True ; Statement
-2 is a correct explanation for Statement –
B. Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-
c. Statement- -1 is True, Statement-2 is False
D. Statement- -1 is False, Statement-2 is True
12
1629 Prove that ( f(x)=sin x+sqrt{3} cos x ) has
maximum value at ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{6}} )
12
1630 Verify Rolle’s theorem the function ( f(x)=x^{3}-4 x ) on ( [-2,2] . ) If you think
it is applicable in the given interval then find the stationary point?
A. Yes Rolle’s theorem is applicable and stationary point is ( x=pm frac{2}{sqrt{3}} )
B. No Rolle’s theorem is not applicable
c. yes Rolle’s theorem is applicable and ( x=2 ) or -2
D. none of these
12
1631 ( frac{boldsymbol{d}^{20}(2 cos boldsymbol{x} cos mathbf{3} boldsymbol{x})}{boldsymbol{d} boldsymbol{x}^{20}}= )
A ( cdot 2^{20}left(cos 2 x-2^{20} cos 4 xright) )
B . ( -2^{20}left(cos 2 x+2^{20} cos 4 xright) )
C ( cdot 2^{20}left(sin 2 x+2^{20} sin 4 xright) )
D. ( 2^{20}left(sin 2 x-2^{20} sin 4 xright) )
12
1632 The values of ( p ) and ( q ) for which the
function ( mathbf{f}(mathbf{x})= )
( left{begin{array}{cl}frac{sin (mathbf{p}+1) mathbf{x}+sin mathbf{x}}{mathbf{x}} & , mathbf{x}0end{array}right. )
is continuous for all ( mathbf{x} ) in ( mathbf{R} ), are
A ( cdot p=frac{1}{2}, q=-frac{3}{2} )
B. ( _{mathrm{p}}=frac{5}{2}, mathrm{q}=frac{1}{2} )
( ^{mathbf{C}} cdot_{mathrm{p}}=-frac{3}{2}, mathrm{q}=frac{1}{2} )
D. ( _{mathrm{p}}=frac{1}{2}, mathrm{q}=frac{3}{2} )
12
1633 The function given by ( y=| x|-1| ) is
differentiable for all real numbers
except the points.
B. ±1
( c cdot 1 )
D. –
12
1634 if ( y=e^{x} cos x, ) prove that ( frac{d y}{d x}= )
( sqrt{2} e^{x} cos left(x+frac{pi}{4}right) )
12
1635 Find ( frac{d y}{d x}, i f x^{y}=e^{x-y} ) 12
1636 If ( g ) is the inverse function of ( f ) and ( f^{prime}(x)=frac{1}{1+x^{n}}, ) then ( g^{prime}(x) ) is equal to
A ( cdot 1+[g(x)]^{n} )
в. ( 1-g(x) )
c. ( 1+g(x) )
D ( cdot-g(x)^{n} )
12
1637 If ( boldsymbol{y}= )
( sqrt{sin x+sqrt{sin x+sqrt{sin x+cdots cdot t o infty}}} )
then ( frac{d y}{d x} ) is
A ( cdot frac{cos x}{1+2 y} )
B. ( -frac{sin x}{1-2 y} )
c. ( frac{cos x}{1-2 y} )
D. ( frac{cos x}{2 y-1} )
12
1638 Differentiate the following w.r.t.x:
( sin ^{-1} x+cos ^{-1} x )
12
1639 If ( boldsymbol{y}=boldsymbol{x}^{4} boldsymbol{e}^{2 x} ) then ( boldsymbol{y}_{10}(boldsymbol{0}) ) is equal to
( A cdot 2^{10} )
B . ( 315 times 2^{10} )
c. ( 195 times 2^{10} )
D. ( 315 times 2^{8} )
12
1640 Find the ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) by implicit differentiation
( boldsymbol{x}^{2}-mathbf{8} boldsymbol{x} boldsymbol{y}+boldsymbol{y}^{2}=boldsymbol{8} )
12
1641 Evaluate: ( 1.2+2.3+3.4+ldots+ )
( n(n+1)=frac{n}{3}(n+1)(n+2) )
12
1642 ( boldsymbol{f}(boldsymbol{x})=mathbf{1} /left(mathbf{1}-boldsymbol{e}^{-mathbf{1} / boldsymbol{x}}right), boldsymbol{x} neq mathbf{0} ) If ( mathbf{f} ) is
continuous at ( x=0 ) then, Find ( f(0) )
12
1643 u
2
27.
The function given by y=||x-1| is differentiable for all real
numbers except the points
(2005)
(a) {0, 1,-1} (b) +1 (c) 1
(d) -1
12
1644 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of ( boldsymbol{x}^{3}+boldsymbol{x}^{2} boldsymbol{y}+boldsymbol{x} boldsymbol{y}^{2}+boldsymbol{y}^{3}=boldsymbol{8} mathbf{1} ) 12
1645 Examine the following functions for continuity.
(i) ( f(x)=x-5 )
(ii) ( f(x)=frac{1}{x-5} )
( boldsymbol{x} neq mathbf{5} )
(iii) ( f(x)=frac{x^{2}-25}{x+5}, x neq-5 )
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{5}| )
12
1646 If ( sqrt{1-x^{2}}+sqrt{1-y^{2}}=a ) find ( frac{d y}{d x} ) 12
1647 19. If square of x varies as cube of y and x = 3 when y = 4, the
value of y at
ill be…
12
1648 Let f (x + y)=f(x) + f(y) for all x an
is continuous at x = 0, then show that
allx.
) + f() for all x and y. If the function f(x)
-0, then show that f(x) is continuous at
(1981 – 2 Marks)
12
1649 ( f(x)=left{begin{array}{cl}frac{left(1-sin ^{3} xright)}{3 cos ^{2} x}, & xfrac{pi}{2}end{array}right. )
continuous at ( x=frac{pi}{2}, ) then the value of
( left(frac{b}{a}right)^{5 / 3} ) is
( mathbf{A} )
B.
( c cdot 32 )
D. 54
12
1650 If ( 2 a+3 b+6 c=0, ) then at least one
root of the equation ( boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c}=mathbf{0} )
lies in the interval
( mathbf{A} cdot(0,1) )
B ( cdot(1,2) )
( mathbf{c} cdot(2,3) )
D ( cdot(-1,0) )
12
1651 The graph of any Quadratic polynomial is such that the chord joining the points ( x=a ) and ( x=b ) is parallel to the
tangent line at ( boldsymbol{x}=? )
A. A.M. of ( a ) and ( b )
B. G.M. of ( a ) and ( b )
C . H.M. of ( a ) and ( b )
D. AGP
12
1652 What is the nature of the graph: ( y= )
( -4 x^{2}+6 )
A. parabola not passing through origin
B. Hyperbola not passing through origin
c. Ellipse not passing through origin
D. it is not a conic
12
1653 Differentiate ( log sqrt{frac{1+cos ^{2} x}{left(1-e^{2 x}right)}} ) w.r.t. ( x ) 12
1654 The value of ( c ) in Lagranges mean value
theorem for ( boldsymbol{f}(boldsymbol{x})=boldsymbol{l} boldsymbol{x}^{2}+boldsymbol{m} boldsymbol{x}+ )
( boldsymbol{n},(boldsymbol{l} neq mathbf{0}) ) on ( [boldsymbol{a}, boldsymbol{b}] ) is
A ( cdot frac{a}{2} )
B. ( frac{b}{2} )
c. ( frac{(a-b)}{2} )
D. ( frac{(a+b)}{2} )
12
1655 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} int_{boldsymbol{f}(boldsymbol{x})}^{boldsymbol{g}(boldsymbol{x})} boldsymbol{h}(boldsymbol{t}) boldsymbol{d} boldsymbol{t}= )
A ( cdot g^{prime}(x) h(g(x)) )
B. ( h(g(x))-h(f(x)) )
c. ( h(g(x)) . g^{prime}(x)-h(f(x)) . f^{prime}(x) )
D. none of these
12
1656 Differentiate the following w.r.t. ( x: ) ( e^{x}+e^{x^{2}}+ldots+e^{x^{5}} ) 12
1657 Differentiate: ( log left(cos e^{x}right) ) 12
1658 If ( boldsymbol{y}=log _{e}left(boldsymbol{x}+log _{e}(boldsymbol{x}+ldots .)right), ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
at ( left(x=e^{2}-2, y=sqrt{2}right) ) is
A ( cdot frac{1}{e^{sqrt{2}}-1} )
B. ( frac{log 2}{2 sqrt{2}left(e^{2}-1right)} )
( ^{mathbf{c}} cdot frac{sqrt{2} log frac{e}{2}}{left(e^{2}-1right)} )
D. None of these
12
1659 If, ( boldsymbol{f}(boldsymbol{x})= )
( left[begin{array}{cc}boldsymbol{x} tan ^{-1} boldsymbol{x}+sec ^{-1} frac{1}{x} & , boldsymbol{x} boldsymbol{epsilon}(-1,1)-mathbf{0} \ frac{boldsymbol{pi}}{2} & boldsymbol{i} boldsymbol{f} boldsymbol{x}=mathbf{0}end{array}right] )
then ( boldsymbol{f}^{prime}(mathbf{0}) ) is
A. equal to –
B. equal to 0
c. equal to 1
D. non existent
12
1660 Differentiate ( (x)^{tan x}+(tan x)^{x} ) w.r.t ( x ) 12
1661 Suppose ( f ) is differentiable at ( x=1 ) and ( lim _{h rightarrow 0} frac{1}{h} f(1+h)=5, ) then
( mathbf{A} cdot f^{prime}(1)=4 )
B ( cdot f^{prime}(1)=3 )
( mathbf{c} cdot f^{prime}(1)=6 )
D. None of these
12
1662 If ( y=log (sec x+tan x), ) then ( frac{d y}{d x}= )
( mathbf{A} cdot sec x )
в. ( frac{1}{sec x+tan x} )
C. ( log left(cos x+sec ^{2} xright) )
D. none of these
12
1663 If ( frac{3}{2}+y^{3}=3 a x y, ) then find ( frac{d y}{d x} ) 12
1664 Differentiate with respect to ( x ) :
( boldsymbol{y}=cos boldsymbol{x}+sin 2 boldsymbol{x} )
12
1665 If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{2}}, ) then what is ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=boldsymbol{pi} )
equal to?
( mathbf{A} cdot(1+pi) e^{pi^{2}} )
В . ( 2 pi e^{pi^{2}} )
( mathbf{c} cdot 2 e^{pi^{2}}^{2} )
D cdot ( e^{pi^{2}} )
12
1666 ( operatorname{Let} f(x)=left{begin{array}{ll}frac{1-cos 2 x}{2 x^{2}} & : x neq 0 \ k & : x=0end{array}right. )
Then the value of ( k ) for which, ( f(x) ) will
be continuous at ( x=0 ) is
A . 0
B.
( c cdot 2 )
D. none of these
12
1667 Differentiate with respect to ( x ) :
( e^{x} log sin 2 x )
12
1668 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{x}+boldsymbol{y}^{2}=log boldsymbol{y}+boldsymbol{x}^{2} )
12
1669 Let ( [x] ) be the greatest integer function
( f(x)=frac{sin frac{1}{4} pi[x]}{[x]} ) is-
This question has multiple correct options
A. not continuous at any point
B. continuous at ( frac{3}{2} )
c. discontinuous at 2
D. differentiable at ( frac{4}{3} )
12
1670 Find the derivatives of the following functions at the indicated points.
( boldsymbol{y}=ln (2-sqrt{2 x+1}), y^{prime}(0)=? )
12
1671 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ; ) if ( boldsymbol{y}=tan ^{-1}left(frac{sin boldsymbol{x}}{mathbf{1}+cos boldsymbol{x}}right) ) 12
1672 Find ( mathbf{k} ) if ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{l}frac{2^{x+2}-16}{4^{x}-16}, quad text { if } quad x neq 2 \ k, quad text { if } quad x=2end{array}right. )
continuous at ( quad x=2 )
12
1673 ( boldsymbol{f}(boldsymbol{x})=sec boldsymbol{x}-cos boldsymbol{x}, boldsymbol{x} boldsymbol{epsilon}(mathbf{0}, boldsymbol{pi} / mathbf{2}) )
find ( mathrm{f}^{prime}(mathbf{x}) )
12
1674 If ( x=a t^{2}, y=2 a t, ) then ( frac{d^{2} y}{d x^{2}}= )
A. ( -frac{1}{t^{2}} )
в. ( frac{1}{2 a t^{3}} )
c. ( -frac{1}{t^{3}} )
D. ( -frac{1}{2 a t^{3}} )
12
1675 [
f(x)=left{begin{array}{ll}
frac{sin a x}{sin b x}, & x neq 0 \
frac{a}{b}, & x=0
end{array}right.
]
Test the continuity of function at ( x=0 )
12
1676 Discuss the applicability of Rolle’s theorem for the following function on the indicated interval:
( boldsymbol{f}(boldsymbol{x})=mathbf{3}[boldsymbol{x}] ) for ( -mathbf{1} leq boldsymbol{x} leq mathbf{1}, ) where ( [boldsymbol{x}] )
denotes the greatest integer not
exceeding ( boldsymbol{x} )
12
1677 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) when ( boldsymbol{y}=(boldsymbol{1}+boldsymbol{x})left(boldsymbol{1}+boldsymbol{x}^{2}right)(boldsymbol{1}+ )
( left.boldsymbol{x}^{4}right)left(boldsymbol{1}+boldsymbol{x}^{6}right) )
12
1678 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{e}^{sqrt{boldsymbol{x}}} ) 12
1679 ( operatorname{Let} f(x)=frac{sqrt{operatorname{sgn}left(alpha x^{2}+alpha x+1right)}}{cot ^{-1}left(x^{2}-alpharight)} )
( f(x) ) is continuous for all ( x in R, ) then
number of integer in the range of ( alpha ), is [Note : sgn k denotes signum function of k.
( mathbf{A} cdot mathbf{0} )
B. 4
c. 5
D. 6
12
1680 For a differentiable function ( phi(x) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{e}^{sin phi(boldsymbol{x})} ) 12
1681 Match the columns 12
1682 ( y=e^{x}+e^{-x} ) prove that ( frac{d y}{d x}=sqrt{y^{2}-4} ) 12
1683 Find ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}} quad ) If ( boldsymbol{x}=boldsymbol{a}(boldsymbol{theta}-sin boldsymbol{theta}), boldsymbol{y}= )
( boldsymbol{a}(1+cos boldsymbol{theta}) )
12
1684 If ( sqrt{frac{boldsymbol{v}}{boldsymbol{mu}}}+sqrt{frac{boldsymbol{mu}}{boldsymbol{v}}}=boldsymbol{6}, ) then ( frac{boldsymbol{d} boldsymbol{v}}{boldsymbol{d} boldsymbol{mu}}= )
A. ( frac{17 mu-v}{mu-17 v} )
в. ( frac{mu-17 v}{17 mu-v} )
c. ( frac{17 mu+v}{mu-17 v} )
D. ( frac{mu+17 v}{17 mu-v} )
12
1685 Find the second order derivatives of
( e^{x} sin 5 x )
12
1686 If ( y=frac{(a-x) sqrt{a-x}-(b-x) sqrt{x-b}}{sqrt{a-x}+sqrt{x-b}} )
then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) wherever defined
A ( frac{2 x-(a+b)}{2 sqrt{(a-x)(x-b)}} )
в. ( frac{x}{2 sqrt{(a-x)(x-b)}} )
c. ( frac{2 x-(a+b)}{4 sqrt{(a-x)(x-b)}} )
D. ( frac{2 x+(a+b)}{2 sqrt{(a-x)(x-b)}} )
12
1687 Let ( boldsymbol{f}:(-mathbf{1}, mathbf{1}) rightarrow boldsymbol{R} ) be a differentiable
function satisfying
[
begin{array}{c}
left(f^{prime}(x)right)^{4}=16(f(x))^{2} text { for all } x in \
(-1,1) \
f(0)=0
end{array}
]
The number of such functions is
( A cdot 2 )
B. 3
( c cdot 4 )
D. more than 4
12
1688 If ( boldsymbol{y}=(boldsymbol{x}+sqrt{boldsymbol{x}^{2}+boldsymbol{a}^{2}})^{n} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( A cdot y )
в. ( n y )
c. ( frac{n y}{sqrt{x^{2}+a^{2}}} )
D. ( frac{y}{sqrt{x^{2}+a^{2}}} )
12
1689 41. Let S = {TER:f(x) = x-Tem
differentiable at t. Then the set Sis equal to : JED
(a) {0}
(b) {}
(c) {0,7}
(d) • (an empty set)
ER:f(x) = x-T ex – 1)sin x is not
ven the set S is equal to : JEE M 2018||
12
1690 ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{boldsymbol{x}} sin boldsymbol{x} ) in the interval ( [mathbf{0}, boldsymbol{pi}] )
Is Rolle’s theorem applicable?
12
1691 Differentiate with respect to ( x ) :
( left(sin ^{-1} x^{4}right)^{4} )
12
1692 Differentiate ( frac{tan ^{-1} x}{1+tan ^{-1} x} ) w.r.t. ( tan ^{-1} x . )
A. ( frac{1}{left(1+tan ^{-1} xright)^{2}} )
s. ( frac{1}{left(1-tan ^{-1} xright)^{4}} )
c. ( frac{1}{left(1+tan ^{-1} xright)^{4}} )
D. ( frac{1}{left(1-tan ^{-1} xright)^{2}} )
12
1693 Let a function be defined as ( boldsymbol{f}(boldsymbol{x})= ) ( frac{boldsymbol{x}-|boldsymbol{x}|}{boldsymbol{x}} . ) Then ( boldsymbol{f}(boldsymbol{x}) ) is
A. continuous nowhere
B. continuous everywhere
c. continuous for all ( x ) except ( x=1 )
D. continuous for all ( x ) except ( x=0 )
12
1694 If ( f(x) ) is continuous function such that
( int_{0}^{x} f(t) d t rightarrow infty ) as ( x rightarrow infty, ) show that
every line ( y=m x ) intersect the curve
( boldsymbol{y}^{2}+int_{0}^{x} boldsymbol{f}(boldsymbol{t}) boldsymbol{d} boldsymbol{t}=boldsymbol{a} ) where ( boldsymbol{a} in boldsymbol{R}^{+} )
12
1695 If ( boldsymbol{y}=boldsymbol{e}^{-boldsymbol{x}} cos boldsymbol{x}, ) show that ( frac{boldsymbol{d}^{2} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}^{2}}= )
( 2 e^{-x} sin x )
12
1696 1.
If x+ y = 2y, then y as a function of x is (1984-3 Marks)
(a) defined for all real x
(b) continuous at x = 0
(c) differentiable for all x
dy 1
(d) such that = for x<0
dx
3
12
1697 The derivative of ( ln (x+sin x) ) with
respect to ( (x+cos x) ) is
A ( cdot frac{1+cos x}{(x+sin x)(1-sin x)} )
B. ( frac{1-cos x}{(x+sin x)(1+sin x)} )
c. ( frac{1-cos x}{(x-sin x)(1+cos x)} )
D. ( frac{1+cos x}{(x-sin x)(1-cos x)} )
12
1698 Examine the following curve for continuity and differentiability: ( boldsymbol{y}=boldsymbol{x}^{2} )
for ( boldsymbol{x} leq mathbf{0} ; boldsymbol{y}=mathbf{1} ) for ( mathbf{0} boldsymbol{x} leq mathbf{1} ) and ( boldsymbol{y}=mathbf{1} / boldsymbol{x} )
for ( x>1 . ) Also draw the graph of the
function.
12
1699 Find the differential coefficient of ( sin x )
by first principle.
12
1700 ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left(frac{sin boldsymbol{x}}{boldsymbol{x}}right) )
A. ( frac{x cos x-sin x}{x^{2}} )
B. ( frac{x cos x+sin x}{x^{2}} )
c. ( frac{x cos x+sin x}{x^{3}} )
D. ( frac{x cos x-sin x}{x^{3}} )
12
1701 Find the derivate of ( e^{sqrt{2 x+1}} ) with respect
to ( x ) at ( x=12 )
12
1702 If ( 2 f(sin x)+f(cos x)=x, ) then
( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} boldsymbol{f}(boldsymbol{x}) ) is
( mathbf{A} cdot sin x+cos x )
B. 2
c. ( frac{1}{sqrt{1-x^{2}}} )
D. none of these
12
1703 Find the derivative of following functions using first principle with
respect to ( x )
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sin boldsymbol{x} )
12
1704 If ( boldsymbol{x}^{boldsymbol{m}} cdot boldsymbol{y}^{boldsymbol{n}}=(boldsymbol{x}+boldsymbol{y})^{boldsymbol{m}+boldsymbol{n}} ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
A. ( frac{y}{x} )
B. ( -frac{y}{x} frac{y}{x} )
c. ( frac{m y}{x} )
D. ( frac{n y}{x} )
12
1705 A Funtion ( f ) is defined as ( f(x)= ) ( frac{x^{2}-4 x+3}{x^{2}-1} ) for ( x neq 1,=2 ) for ( x=1 ) Is
the function continuous at ( x=1 . ? )
A . True
B. False
12
1706 ( x sqrt{1+y}+y sqrt{1+x}=0, ) then ( frac{d y}{d x} ) equal
to
( mathbf{A} cdot 1+x )
В. ( (1-x)^{-2} )
C. ( -(1+x)^{-1} )
D. ( -(1+x)^{-2} )
12
1707 If ( left(x^{2}+x y+3 y^{2}right)=1, ) what will be the
value of ( (x+6 y)^{3} cdot frac{d^{2} y}{d x^{2}} ? )
12
1708 If ( y=frac{sqrt{x}(2 x+3)^{2}}{sqrt{x+1}}, ) then ( frac{d y}{d x} ) is equal to
A ( cdot yleft[frac{1}{2 x}+frac{4}{2 x+3}-frac{1}{2(x+1)}right] )
в. ( yleft[frac{1}{3 x}+frac{4}{2 x+3}+frac{1}{2(x+1)}right] )
c. ( yleft[frac{1}{3 x}+frac{4}{2 x+3}+frac{1}{x+1}right. )
D. None of these
12
1709 Let ( f(x) ) be defined as follows:
[
boldsymbol{f}(boldsymbol{x})=left{begin{array}{l}
boldsymbol{x}^{boldsymbol{6}}, boldsymbol{x}^{2}>mathbf{1} \
boldsymbol{x}^{3}, boldsymbol{x}^{2} leq mathbf{1}
end{array}right.
]
Then ( boldsymbol{f}(boldsymbol{x}) ) is?
This question has multiple correct options
A. continuous everywhere
B. differentiable everywhere
c. discontinuous at ( x=-1 )
D. not differentiable at ( x=1 )
12
1710 ( frac{d}{d x} csc ^{-1}left(frac{1+x^{2}}{2 x}right) ) is equal to
A ( cdot frac{-2}{left(1+x^{2}right)}, x neq 0 )
в. ( frac{2}{left(1+x^{2}right)}, x neq 0 )
c. ( frac{2left(1-x^{2}right)}{left(1+x^{2}right)left|1-x^{2}right|}, x neqpm 1,0 )
D. None of the above
12
1711 If ( boldsymbol{f}(boldsymbol{x})= )
( left{begin{array}{ll}(mathbf{1}+|sin boldsymbol{x}|)^{frac{a}{|sin |}}, & -boldsymbol{pi} / boldsymbol{6}<boldsymbol{x}<mathbf{0} \ boldsymbol{b}, & boldsymbol{x}=mathbf{0} \ boldsymbol{e}^{frac{tan 2 x}{tan x}}, & boldsymbol{0}<boldsymbol{x}<boldsymbol{pi} / boldsymbol{6}end{array}right. )
continuous at ( x=0, ) find the values of
( a ) and ( b )
A ( cdot frac{3}{2}, e^{3 / 2} )
B. ( frac{-2}{3}, e^{-3 / 2} )
c. ( frac{2}{3}, e^{2 / 3} )
D. None of these
12
1712 30.
Let f (x) be differentiable on the interval (0,0) such that
and lim f(x) – xf(t) – 1 for
-=1 for each x > 0. Then
– t-x
(2007 – 3 marks)
1-
X
f(x) is
3 x
3
12
1713 Let ( a, b ) be two distinct roots of a polynomial ( f(x) ). Then there exists at least one root lying between a and b of the polynomial
A ( . f(x) )
B. ( f^{prime}(x) )
c. ( f^{prime prime}(x) )
D. ( f^{prime prime prime}(x) )
12
1714 Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) if ( boldsymbol{y}=sin ^{-1}(boldsymbol{2} boldsymbol{x} sqrt{mathbf{1}-boldsymbol{x}^{2}}) )
( frac{mathbf{- 1}}{sqrt{mathbf{2}}}<boldsymbol{x}<frac{mathbf{1}}{sqrt{mathbf{2}}} )
12
1715 If ( y=sin ^{2} x, ) then ( frac{d y}{d x}= )
( mathbf{A} cdot cos ^{2} x )
B. ( 2 sin x )
( mathbf{c} cdot sin x cos x )
D. ( sin 2 x )
12
1716 Let ( boldsymbol{f}(boldsymbol{x})= )
( lim _{n rightarrow infty} frac{left(x^{2}+2 x+3+sin pi xright)^{n}-1}{left(x^{2}+2 x+3+sin pi xright)^{n}+1} . ) Then
A. ( f(x) ) is continuous and differentiable for all ( x in R )
B. ( f(x) ) is continuous but not differentiable for all ( x in R )
( mathrm{c} cdot f(x) ) is discontinuous at infinite number of points
D. ( f(x) ) is discontinuous at finite number of points
12
1717 ( operatorname{Let} f(x)=left{begin{array}{cc}-1, & -2 leq x<0 \ x^{2}-1, & 0<x leq 2end{array} ) and right.
( boldsymbol{g}(boldsymbol{x})=|boldsymbol{f}(boldsymbol{x})|+boldsymbol{f}|boldsymbol{x}| ) then the number
of points which ( g(x) ) is non
differentiable, is
A. at most one point
B. 2
c. exactly one point
D. infinite
12
1718 ( frac{d}{d x}[log sqrt{frac{1-cos x}{1+cos x}}]= )
A . sec
B. ( csc x )
c. ( operatorname{cosec} frac{x}{2} )
D. ( sec frac{x}{2} )
12
1719 Using Rolle’s theorem, the equation ( boldsymbol{a}_{0} boldsymbol{x}^{boldsymbol{n}}+boldsymbol{a}_{1} boldsymbol{x}^{boldsymbol{n}-1}+ldots+boldsymbol{a}_{boldsymbol{n}}=mathbf{0} ) has
atleast one root between 0 and ( 1, ) if
A ( cdot frac{a_{0}}{n}+frac{a_{1}}{n-1}+ldots .+a_{n-1}=0 )
в. ( frac{a_{0}}{n-1}+frac{a_{1}}{n-2}+ldots+a_{n-2}=0 )
c. ( n a_{0}+(n-1) a_{1}+ldots .+a_{n-1}=0 )
D. ( frac{a_{0}}{n+1}+frac{a_{1}}{n}+ldots .+a_{n}=0 )
12
1720 44.
If the function f defined on
V2cosx-1
f(x)=
cotx-1
**
k,
x = 1
is continuous, then k is equal to:
JEEM 2019-9 April (M)
(2) 2
(6) 1
(1) 1
(2) J
12
1721 Assertion (A):The derivative of ( (log x)^{x} )
w.r.t ( boldsymbol{x} ) is ( (log boldsymbol{x})^{x-1}[mathbf{1}+log boldsymbol{x} log (log boldsymbol{x})] )
Reason
( (mathrm{R}): frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}left{boldsymbol{f}(boldsymbol{x})^{g(boldsymbol{x})}right}=boldsymbol{f}(boldsymbol{x})^{g(boldsymbol{x})} )
( left(g(x) frac{f^{prime}(x)}{f(x)}+g^{prime}(x) log (f(x))right) )
A. Both A and R are true R is correct reason of A
B. Both A and R are true R is not correct reason of
c. A is true but R is false
D. A is false but R is true
12
1722 Differentiate with respect to ( x ) :
( e^{-3 x} log (1+x) )
12
1723 Write the derivative of ( f(x)=|x|^{3} ) at
( boldsymbol{x}=mathbf{0} )
12
1724 Suppose that on the interval [-2,4] the function ( f ) is differentiable, ( f(2)=1 )
and ( |boldsymbol{f}(boldsymbol{x})| leq mathbf{5} . ) Find the bounding
function of ( boldsymbol{f} ) on ( [-mathbf{2}, mathbf{4}], ) using LMVT.
A. ( y=-5 x-9 ) and ( y=5 x+11 )
B. ( y=-5 x+9 ) and ( y=5 x+11 )
c. ( y=5 x-9 ) and ( y=5 x-11 )
D. ( y=5 x+9 ) and ( y=5 x-11 )
12
1725 Illustration 2.18
If y=x”, then find dy/dx.
12
1726 Solve:
( lim _{x rightarrow 2} frac{x^{2}-4}{sqrt{3 x-2}-sqrt{x+2}} )
12
1727 If ( f(x)=frac{1-cos a x}{1-cos b x} ) for ( x neq 0, ) is
continuous at ( boldsymbol{x}=mathbf{0} ) then ( boldsymbol{f}(mathbf{0})= )
A ( cdot frac{a^{2}}{2} )
B. ( frac{a}{b^{2}} )
c. ( frac{a}{b} )
D. ( frac{a^{2}}{b^{2}} )
12

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