# 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 NoQuestionsClass
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
330.
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
5Verify 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
7If ( 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
8If ( 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
10Evaluate ( 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
12State 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
13Differentiate: ( y=frac{x+2}{3 x+5} ) w.r.t ( x )12
14Differentiate the following w.r.t. ( x: )
( sin left(tan ^{-1} e^{-x}right) )
12
15Let ( 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
17Assertion ( (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
18If ( 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
19Examine 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
20Derivative 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
21If ( 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
22A 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
23If ( 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
24Differentiate: ( sin left(tan ^{-1} e^{x}right) )12
25Identify whether ( f(x)=frac{x^{2}-4}{x-2} ) is
continuous at ( x=2 ) or not
12
26Show 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
27Differentiate w.r.t ( x )
( boldsymbol{y}=boldsymbol{x}^{2} sin 2 boldsymbol{x} )
12
28Differentiate: ( mathbf{5}^{mathbf{5}^{5 x}}=boldsymbol{t} )12
29Let ( 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
30If ( sqrt{boldsymbol{x}}+sqrt{boldsymbol{y}}=4 ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) at ( boldsymbol{x}=mathbf{1} )12
31Find 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
32Find ( 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
34The 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
363.
The function f(x)=1+ sin xis (1986-2 Marks)
(a) continuous nowhere
(b) continuous everywhere
(d) not differentiable at x=0
(e) not differentiable at infinite number of points.
12
37Find ( 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
38Differentiate the following functions
with respect to ( boldsymbol{x} ) ( log sqrt{frac{boldsymbol{x}-mathbf{1}}{boldsymbol{x}+mathbf{1}}} )
12
39Find the derivatives of ( x cos x )12
40Find the derivative of ( x^{4}+4 )12
413.
(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
424.
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
439.
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
44If ( 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
46f ( boldsymbol{y}=tan (2 boldsymbol{x}+mathbf{3}) ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
47Assertion
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
4813. 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
49If ( 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
50Let ( 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
51Assertion
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
52If ( boldsymbol{x}=boldsymbol{y}(log boldsymbol{x} boldsymbol{y}) ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
53If ( 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
5428.
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=
mp4=
) p=5.q=
(a
12
55Differentiate the following functions
with respect to ( x ) :
( cos ^{-1}left{frac{x}{sqrt{x^{2}+a^{2}}}right} )
12
56The 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
57Find ( 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
58The value of ( f(2) ) is
( A cdot 2 )
B. 4
( c .6 )
D. 8
12
59Show that ( f(x)=|x-3| ) is continuous
but not differentiable at ( x=3 )
12
60If ( 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
61Suppose 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
62If ( y=frac{sin ^{2} x}{1+cot x}+frac{cos ^{2} x}{1+tan x} ) then ( y^{prime} ) is
equal to?
12
63Differentiate 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
65Find ( frac{d y}{d x} ) when ( x^{2}+y^{2}=c^{2} )12
66Find 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
67The 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
68f ( 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
69If ( x^{x}+x^{y}+y^{x}=a^{b}, ) then find ( frac{d y}{d x} )12
70If 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
71If 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
7219. Determine the values of x for which the following function
fails to be continuous or differentiable: (1997 – 5 Marks)
(1-x, xxl
[3-x,
x>2
12
73If ( 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
74Solve:
( frac{x cos ^{-1} x}{sqrt{1-x^{2}}} d x )
12
75Function ( 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
76Solve ( : frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(sin boldsymbol{3} boldsymbol{x})=? )12
77Find 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
78Find 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
79Differentiate the function with respect
to ( boldsymbol{x} )
( f(x)=x^{2 / 3}+7 e^{x}-frac{5}{x}+7 tan x )
12
80Assertion
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
81If ( 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
82Solve: ( frac{d}{d x} x sin ^{2} x )12
83Find the derivative by first principle
( cos 5 x )
12
84Differentiate ( 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
85The 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
86If 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
87If ( y= )
A ( cdot n^{2} y )
B. ( -n^{2} y )
( c cdot n y^{2} )
D cdot ( n^{2} y^{2} )
12
88If ( 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
89Given that, ( y=sin xleft(x^{2}right) e^{x} ), find ( y^{prime} ) at
( boldsymbol{x}=mathbf{0} )
12
90Derivative 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
91If ( 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
92If ( 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
93Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{y}=log left(boldsymbol{4} boldsymbol{x}-boldsymbol{x}^{5}right) )12
94ff ( y=x^{2} cos x ) then ( y_{8}(0) ) is
A . 72
B . 56
( c cdot 0 )
D. – 56
12
95Find the derivative of ( boldsymbol{y}=frac{1}{boldsymbol{x}}+frac{mathbf{1}}{boldsymbol{x}^{2}}+ )
( frac{mathbf{3}}{boldsymbol{x}^{mathbf{3}}} )
12
96A 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
97Find the derivative of ( sin x ) with respect
to ( x ) from first principles.
12
983.
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
99If ( 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
100Find the derivative of ( f(x) ) from the first
principles, where ( boldsymbol{f}(boldsymbol{x}) ) is
( sin x+cos x )
12
101The values of ( f^{prime}(1) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
12
102f ( sin theta+2 cos theta=1 ) then prove that
( 2 sin theta-cos theta=0 )
12
103Discuss 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
104Derivative of ( left(tan ^{-1} xright)^{2} ) wrt to ( x )12
105Assertion
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
107If ( 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
108Differentiate the following functions
w.r.t. ( boldsymbol{x} )
( e^{operatorname{cosec}^{2} x} )
12
109If ( 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
111Find 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
113Assertion
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
114Differentiate the following functions
with respect to ( boldsymbol{x} ) ( log left(frac{x^{2}+x+1}{x^{2}-x+1}right) )
12
115If ( f(x)=x sin x, ) then ( f^{prime}left(frac{pi}{2}right) ) is equal
to:
A.
B.
c. -1
D.
12
116If ( 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
117Differentiate with respect to ( x ) :
( boldsymbol{y}=boldsymbol{e}^{-mathbf{3} boldsymbol{x}}+sin mathbf{2} boldsymbol{x} )
12
118If ( 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
1191

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
120If ( 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
12211.
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
123f 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
1246. 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
125State 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
126Show that ( f(x)=e^{2 x} ) is increasing on ( R )12
127Differentiate
( 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
128If 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
129If ( 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
130Let ( 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
132If ( 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
133Let ( y=sin ^{-1}(cos x) ) then find ( frac{d y}{d x} )12
134If ( cos y=x cos (a+y), ) find ( frac{d y}{d x} )12
135The 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
136Consider 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
137ff ( 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
138If ( f(x)=frac{1+tan x}{1-tan x} ) then ( fleft(frac{pi}{6}right) )12
139Differentiate:
( sin ^{2} boldsymbol{y}+cos boldsymbol{x} boldsymbol{y}=boldsymbol{K} )
12
140ff ( 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
141Differentiation 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
142Discuss 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
143State 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
144Examine 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
145greatest 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
146If ( 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
147Find ( 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
149Differentiable ( log _{7}(log x) ) with respect
to ( boldsymbol{x} )
12
150If ( 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
151If ( 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
152Find 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
153Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{y}=boldsymbol{e}^{left(1+log _{e} boldsymbol{x}right)} )
12
154If ( 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
155Differentiate ( 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
157Assertion
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
158ff ( 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
159If ( 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
1607.
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
161Find the value ( : frac{d}{d x}left{cos x^{0}right}=? )12
162If ( 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
16316. Differentiation of x2 w.r.t. x is…12
164Discuss 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
165Find 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
1670
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
168Differentiate with respect to ( x ) :
( mathbf{3}^{e^{x}} )
12
169If ( 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
170Let ( 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
171Solution 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
172For 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
173Let ( 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
174Find 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
176If ( 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
179If ( 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
1806. If f(x) = mx + c,fo) =f(0) = 1 then f(2)=
(a) 1 (6) 2 (c) 3 (d) – 3
12
181The function ( y=f(x) ) is ?
A . odd
B. even
c. increasing
D. decreasing
12
182Identify 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
183Differentiate: ( log (log x), x>1 )12
184Diff: ( cos ^{-1}left(frac{2 x}{1+x^{2}}right) ) w.r.t. ( x )12
185Let ( 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
186Given, ( 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
187If ( 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
1893.
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
191If 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
193If ( 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
194Find 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
195If ( 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
196If ( 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
197Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
( boldsymbol{y}=sin boldsymbol{x} . cos boldsymbol{x} )
12
198Let ( 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
199Assertion
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
200If 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
201If ( 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
20221. If y = sin x, then
will be …
12
203Evaluate
( 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
205If 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
206Differentiate the following with respect
to ( boldsymbol{x} )
( cos ^{-1}[sqrt{frac{1+x}{2}}],-1<x<1 )
12
207If ( boldsymbol{y}=sqrt{boldsymbol{x}+boldsymbol{y}}, ) prove that ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}= )
( frac{1}{(2 y-1)} )
12
208Find ( frac{d y}{d x}, ) if ( y=sqrt{1+sin 2 x} )12
209Find the derivative of ( y=sqrt{x^{2}+1} )12
210Which 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
211If ( 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
212Find the second derivative of the
function ( log x )
12
213The 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
214If ( 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
216If ( 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
2191-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
220Solve ( : boldsymbol{y}=sin ^{-1}left(frac{1-boldsymbol{x}^{2}}{1+boldsymbol{x}^{2}}right), boldsymbol{0}<boldsymbol{x}<1 )12
221Find the derivative of ( cos ^{2} x, ) by using
first principle of derivatives.
12
222Find the derivative of
( x^{-4}left(3-4 x^{-5}right) )
12
223Find 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
224If ( 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
225differentiate :
( boldsymbol{y} log boldsymbol{x} )
12
226If ( 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
227Find the derivative of ( f(log x) ) with
respect to ( x ) where ( f(x)=log x )
12
228Identify 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
229Xis
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
230Say 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
232A 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
233If ( boldsymbol{y}=cos sqrt{boldsymbol{x}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
2342. 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
236Find 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
237Differentiate wirh respect to
( sqrt{x+frac{1}{x}} )
12
238Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
if ( boldsymbol{y}=(sin boldsymbol{x})^{boldsymbol{x}}+sin ^{-1} sqrt{boldsymbol{x}} )
12
239If ( 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
240If ( 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
241Differentiate ( frac{x^{2}+1}{x} ) w.r.t ( x )12
242Find ( 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
24325.
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
244Diffrentiate w.r.t ( x: )
( boldsymbol{y}=e^{2 x}(boldsymbol{a}+boldsymbol{b} boldsymbol{x}) )
12
245Let 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
246If ( 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
247Find the derivative of ( operatorname{cosec}^{2} x, ) by using
first principle of derivatives?
12
248The 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
249If 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
250Let ( 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
251Differentiate w.r.t. ( boldsymbol{x} )
( boldsymbol{y}=boldsymbol{e}^{cos (boldsymbol{6} boldsymbol{x}-mathbf{1})} )
12
252If ( 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
254Show 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
255Let ( 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
25628. 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
257Differentiate w.r.t ( boldsymbol{x}: boldsymbol{x}^{boldsymbol{y}}+boldsymbol{y}^{boldsymbol{x}}=mathbf{1} )12
258ff ( (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
259f ( 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
2605.
If x=ete
.
(20041
. x > 0, then
1+X
1-*
(b)
(d)
12
261If ( 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
262Find the differentiation of ( sec left(tan ^{-1} xright) )
w.r.t. ( boldsymbol{x} )
12
263Ify 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
265Find 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
266If ( 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
268ff ( 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
2697.
[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
271If ( 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
272Differentiate ( y=sin b x^{2} ) w.r.t ( x )12
273Differentiate the following functions
with respect to ( boldsymbol{x} ) ( frac{e^{x} sin x}{left(x^{2}+2right)^{3}} )
12
274Let ( 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
275Let ( 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
276The 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
277Find the derivative of ( f(x) ) from the first
principle. ( sin x div cos x )
12
278If ( (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
279Evaluate ( 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
280Prove 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
281Evaluate ( 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
283f ( 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
284Find 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
285Write 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
287If ( 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
288Discuss 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
289If ( 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
291Verify LMVT :
( boldsymbol{f}(boldsymbol{x})=frac{boldsymbol{x}+mathbf{1}}{boldsymbol{x}} ) for ( boldsymbol{x}=[mathbf{1}, boldsymbol{3}] )
12
292The 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
293If ( 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
294f ( 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
295If ( e^{x}+e^{y}=e^{x+y}, ) find ( frac{d y}{d x} )12
296Let ( 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
297Given 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
298If ( 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
299Find differentiation of ( sec ^{-1} tan x )12
300If ( 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
301If ( cos (x+y)=y sin x, ) then find ( frac{d y}{d x} )12
302If ( 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
303Range 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
304If ( y=e^{x}+sin x-4 x^{3}, ) find ( frac{d y}{d x} )12
305If 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
306If ( 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
307Let ( 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
308If ( 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
309Differentiate 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
311Differentiate w.r.t. ( x )
( boldsymbol{y}=(cos boldsymbol{x})left(1-sin ^{2} boldsymbol{x}right) )
12
312The 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
313If ( 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
314Let ( 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
315Find 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
316Differentiate the following function with
respect to ( x ) ( 1+3 x )
12
317If ( 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
318Differentiate the following functions
with respect to ( boldsymbol{x} ) ( sin ^{-1}left(2 x^{2}-1right), 0<x<1 )
12
3198. 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
321Differentiate the following w.r.t. ( x: ) ( e^{sin ^{-1} x} )12
322The 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
324Solve the following differential equation ( frac{d y}{d x}=3 x )12
325What 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
326If ( 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
327If ( 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
328If 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
330Solve: ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=cos (boldsymbol{x}+boldsymbol{y}) )12
331The 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
333If ( 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
334If ( y=sin left(log _{e} xright) ) prove that ( frac{d y}{d x}= )
( frac{sqrt{1-y^{2}}}{x} )
12
335If ( x^{2}+2 x y+y^{3}=42, ) find ( frac{d y}{d x} )12
336Differentiate ( 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
337Differentiate w.r.t. ( mathbf{x} )
( f(x)=sqrt{sin (cos x)} )
12
338Let ( 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
3392.
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
340From 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
342Derivative 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
344Differentiate the following w.r.t. ( x ) ( e^{x^{3}} )12
345Let ( 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
346If ( 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
347A 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
348The 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
349Find 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
351If ( 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
352If ( 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
354Differentiate 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
355The graph of ( f(x) ) is given below. Based
on this graph determine where the
function is discontinuous.
12
356Let ( 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
357If ( a x+b y^{2}=cos y, ) then find ( frac{d y}{d x} )12
358Domain 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
359Use 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
360Find 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
361if ( 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
362will 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
363Let ( 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
365If ( 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
366If 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
368ff ( 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
369Let ( 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
370If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
371If ( 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
373If ( 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
375f ( 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
377Let ( 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
378Find ( 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
379Illustration 2.35 If f(x) = x cos x, find f”(x).12
380If ( 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
381Differentiate the following expression
( boldsymbol{w} cdot boldsymbol{r} cdot boldsymbol{t} cdot boldsymbol{x} )
( boldsymbol{y}=csc ^{2}left(boldsymbol{x}^{2}right) )
12
382Differentiate 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
3846.
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
385If ( 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
386If ( 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
387If ( 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
388Find 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
390Examine 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
391The 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
392Consider 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
393In 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
394differentiate ( e^{-2 tan ^{-1} x^{2}} )12
395ff ( 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
396Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) of ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}=sin boldsymbol{y} )12
397Illustration 2.25 If y= 12++3]|2x* + 1], then find dy
Illustration 2.25
If y =
31|2x
+ 1), then find
12
398Differentiate with respect to ( x ) :
( log left(cos x^{2}right) )
12
399Find 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
400Let ( 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
40126. If y=x2 sin x, then
will be …
12
402If ( boldsymbol{y}=tan boldsymbol{x}+cot boldsymbol{x} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
403Differentiate the following
function with respect to ( x )
( tan h^{-1}(3 x+1) )
12
404Let ( 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
405Examine 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
406Let ( 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
407If ( 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
408If ( 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
409f ( 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
410If ( 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
411If ( 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
412If 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
413If ( 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
4147.
„Osx<1
Let f(x) =
(1983 – 2 Marks)
*+2,15×52
Discuss the continuity of f,f' and f" on [0,2].
12
415If ( f(x) ) is differentiable everywhere, then
( |boldsymbol{f}(boldsymbol{x})|^{2} ) is differentiable everywhere.
Enter ( 1 text { if true or } 0 text { otherwise }) )
12
416Dfferentiate w.r.t ( x ) :
( tan ^{2} 7 x )
12
417Find 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
418A 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
4194.
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
420Verify 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
421Differentiate the following function with
respect to ( x ) ( left(1+x^{2}right) cos x )
12
422If ( 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
424Differentiate 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
425Suppose 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
426Let ( 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
427Illustration 2.30
Find the derivative of y = sin(x+ – 4).
a
2
1
1
12
428If ( 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
429Let ( 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
430Testify 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
431Verify 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
432Let ( 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
433The 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
434The 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
435Find the derivative of ( y=frac{2 x}{1-x^{2}} )12
436Find 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
437If ( 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
438Find ( 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
439Draw 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
440Evaluate ( : int(tan x-cot x)^{2} d x )12
441If ( 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
442The 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
443If ( 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
445The 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
446Differentiate 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
448If ( 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
449Differentiate ( boldsymbol{y}=sin ^{-1}left(frac{mathbf{2}^{x+1} mathbf{3}^{x}}{1+mathbf{3 6}^{x}}right) )12
450If ( 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
452If ( a x^{2}+2 x y+b y^{2}=0 ) then find ( frac{d y}{d x} )12
453Find 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
454Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{x} sin boldsymbol{y}+boldsymbol{y} sin boldsymbol{x}=mathbf{0} )12
455Sketch the graph ( y=|x+3| . ) Evaluate ( int_{-6}^{0}|x+3| d x . ) What does this integral represent on the graph?12
456Differentiate 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
457If ( 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
458Consider 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
459If ( 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
460By 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
461The 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
462Find the derivative of ( boldsymbol{y}=(boldsymbol{x}+ )
1) ( (x+2)^{2} )
12
463If ( 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
464Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: sin boldsymbol{x}+mathbf{7} boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{5} )12
465If ( 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
466Verify 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
467Assertion
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
468Let ( 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
469Differentiate with respect to ( x ) :
( frac{x^{2}+2}{sqrt{cos x}} )
12
470If ( 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
471If ( 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
472If ( 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
473Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{2} boldsymbol{y}^{2}+boldsymbol{6} boldsymbol{x}=mathbf{5} )
12
474Find the slope of tangent to the curve ( y=3 x^{2}-6 ) at the point on it whose ( x )
coordinate is 2
12
475If ( 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
476Differentiate ( boldsymbol{y}=log (log sqrt{boldsymbol{x}}) )12
477Explain Mean Value Theorem12
47815.
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
480If ( 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
48126. 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
482Differentiate with respect to ( x ) :
( frac{2^{x} cos x}{left(x^{2}+3right)^{2}} )
12
483A 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
485A 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
486Let ( 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
487Find ( frac{d y}{d x}, ) when ( y=x^{x}-2^{sin x} )12
488If ( 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
489Let ( 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
490If ( 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
491If 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
492If ( 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
493Differentiate: ( 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
494If ( 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
495Solve ( 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
496If ( 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
498Let ( 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
499Value 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
500Find 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
501If ( 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 to12
502If 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
503Find 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
504Let ( 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
505State Rolle’s theorem.12
506Let ( y=2^{x}+x^{2}+2 ) then find ( frac{d y}{d x} )12
507Differentiate the following w.r.t ( x: frac{e^{x}}{sin x} )12
508Say true or false.
Derivative of ( x^{n} ) is ( n x^{n-1} )
A. True
B. False
12
509Is 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
510Prove cos is continuous on R.12
511If ( 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
513f ( 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
515If ( 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
517Differentiate the functions with respect
to ( x )
( sec (tan (sqrt{x})) )
12
518Find ( frac{d y}{d x}=sin ^{-1} x )12
519Discuss the continuity of the function defined by ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{5}| )12
520Differentiate the following functions
with respect to ( x ) ( sin ^{-1}left{frac{x+sqrt{1-x^{2}}}{sqrt{2}}right},-1<x<1 )
12
521Answer 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
522If ( 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
523Find derivative of ( (boldsymbol{a} boldsymbol{x}+boldsymbol{b})^{n}(boldsymbol{c} boldsymbol{x}+boldsymbol{d})^{boldsymbol{n}} )12
524Differentiate the following with respect
to ( x )
( cos ^{-1} 2 x sqrt{1-x^{2}}, frac{1}{sqrt{2}}<x<1 )
12
525For every twice differentiable function f:R
2,2W
(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
526Let ( 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
527The 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
529If ( 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
53014. 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
531Differentiate
( frac{x^{4}}{4}-frac{x^{-3}}{3}-frac{2}{x}+C )
12
532The 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
533Let ( 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
534If ( 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
536If ( boldsymbol{y}=tan ^{2}left(log boldsymbol{x}^{3}right), ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
537Show 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
538Find 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
540If ( 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
541Illustration 2.20
If y= –
= (x)-1/2, then find dyldx.
12
542If ( 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
543Let ( (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
544If ( 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
545If ( 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
546If ( 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
547Show that ( boldsymbol{f}(boldsymbol{x})= )
( begin{array}{ll}frac{sin 3 x}{tan 2 x}, & text { if } x0end{array} )
12
548Find ( frac{d y}{d x}, x=aleft(cos t+log tan frac{t}{2}right), y= )
( a sin t )
12
549The 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
550Differentiate 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
551The 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
553Find:
( frac{d y}{d x}=sin (x+y)+cos (x+y) )
12
554If 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
555Let 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
556Differentiate w.r.t ( x, ) the following function:
( log sqrt{frac{1+cos x}{1-cos x}} )
12
557Consider 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
558Find ( 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
559Differentiate with respect to ( x ) :
( e^{tan ^{-1} sqrt{x}} )
12
560Differentiate the following functions
with respect to ( x: ) ( tan ^{-1}left(frac{a+b x}{b-a x}right) )
12
561If ( 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
562If ( 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
563Let ( 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
564Let ( 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
565COMO
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
566If ( 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
567What 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
568Differentiate with respect to ( x )
( boldsymbol{y}=log (1+sin boldsymbol{x}) )
12
569Find ‘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
570Differentiate ( sin ^{2} 3 x cdot tan ^{3} 2 x )12
571The 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
572What is the derivative of ( |x-1| ) at ( x= )
( mathbf{2} ? )
A . -1
B. 0
c. 1
D. Derivative does not exist
12
573If ( 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
574Differentiate
( 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
575Find 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
576If ( e^{y}(x+1)=1 ) show that ( frac{d y}{d x}=-e^{y} )12
577If ( 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
578If ( 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
579f ( 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
580If ( 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
581If ( 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
582A 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
583If ( 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
584If ( 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}= )
12
585Find 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
588Find ( frac{d y}{d x} ) of ( x y+y^{2}=tan x+y )12
589ff ( 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
590If ( 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
591Assertion 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
593If ( x^{y}-y^{x}=1, ) then the value of ( frac{d y}{d x} ) is :12
594If ( 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
595If ( 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
596ff ( x=a t^{4}, y=b t^{3}, ) then find ( frac{d y}{d x} )12
597If ( 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
600f ( f(x)=sqrt{1+cos ^{2}left(x^{2}right)}, ) then ( f^{prime}(x) ) is?12
601Find ( frac{d y}{d x} ) of ( a x+b y^{2}=cos y )12
602Differentiate 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
604If ( 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
606Solve the following differential equation ( frac{d y}{d x}=x^{2} )12
607If ( 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
608Find the derivation of ( sqrt{tan x} ) with respect to x using first principle.12
609Differentiate 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
612If ( 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
613Differentiate the following function with
respect to ( x )
( x^{3} sin x )
12
614Verify 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
615If ( 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
616Differentiate w.r.t ( boldsymbol{x} )
( e^{operatorname{cosec}^{2} x} )
12
61725. 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
618The 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
620Left 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
6211. 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
622Differentiate 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
623Find the derivative of the following functions: ( 3 cot x+5 cos e c x )12
624Find the derivative of the following
function.
( log (log x) )
12
625Let ( 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
626If ( 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
627Discuss 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} \
end{array}right.
]
( boldsymbol{x}=mathbf{0} )
12
628If ( 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
629Verify 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
630Let ( 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
631d?
dv2 equals
(2007-3 marks)
(d
12
632Find ( 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
633Let ( 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
6349.
The following functions are continuous on (0,7).
(1991 – 2 Marks)
(a) tan x
o
0<****
V
x sin x,
sin(+*), <x<*
12
635Solve:
( 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
636For instantaneous speed, the distance
traveled by the object and the time taken are both equal to zero.
A. True
B. False
12
637Find ( 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
639If ( 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
640Differentiate w.r.t ( mathbf{x} ) ( boldsymbol{y}=boldsymbol{x}^{2} ln (sqrt{frac{boldsymbol{x}^{2}+mathbf{9}}{boldsymbol{x}^{2}+boldsymbol{4}}} )12
641If ( 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
642Find ( 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
643If ( 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
644If ( 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
645Show that ( 3+frac{1}{1+} frac{1}{6+1+} frac{1}{6+} dots= )
( 3left(1+frac{1}{3+2+3+2+cdots}right) )
12
646Find ( 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
647Let ( 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
648Is every continuous function
differentiable?
12
649If ( 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
650Examine 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
651Differentiate w.r.t. ( boldsymbol{x} )
( boldsymbol{y}=boldsymbol{e}^{boldsymbol{3} boldsymbol{x}} sin boldsymbol{4} boldsymbol{x} )
12
652If ( 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
653Find 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
655Form 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
656Check 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
657Is the function defined by ( f(x)=|x|, ) a
continuous function?
12
658If ( 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
659If ( 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
660Differentiate the following function with
respect to ( x )
( frac{boldsymbol{x}^{n}}{sin boldsymbol{x}} )
12
661Determine 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
662Differentiate the following w.r.t.x:
( 5^{x} cdot sec ^{-1} 2 x )
12
663The 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
664ffunction ( 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
665If ( 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
666Find 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
668Find 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
669If ( 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
670The function ( f(x)=[x], ) at ( x=5 ) is:
A. Ieft continuous
B. right continuous
c. continuous
D. cannot be determined
12
671Differentiate ( tan ^{-1}left(frac{sin x}{1+cos x}right) ) w.r.t.12
672If 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
673Derivative 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
674The 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
675Let ( 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
676Find the derivative of
( f(x)=left(x^{2}-5right)left(x^{3}-2 x+3right) )
12
677If ( y=a^{frac{1}{2} log _{a} cos x} . ) Find ( frac{d x}{d} )12
678If ( y=x^{2} tan x, ) find ( frac{d y}{d x} )12
679If ( 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
680Discuss 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
682Find ( frac{d y}{d x}, ) if ( y=log left(sqrt{x}-frac{1}{sqrt{x}}right) )12
683If ( y=log (log x)+2 sin x, ) find ( frac{d y}{d x} )12
684Show that ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{2}|+|boldsymbol{x}-mathbf{3}| ) is
not differentiable at ( x=2 )
12
68520. 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
687If ( 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
688Differentiate: ( y=sin (2 x+3) ) w.r.t ( x )12
689Suppose 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
690The 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
691Assertion
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
692Let ( 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
693In
[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
694If ( 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
695The 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
696If ( 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
697If ( 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
698If ( 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
699Examine 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
700Applying 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
701Suppose 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
702The 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
703If ( 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
704Discuss the applicability of Lagrange’s
mean value theorem for the function
( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}| ) on [-1,1]
12
705if ( 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
706Solve ( 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
708The 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
709The 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
710Let ( 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
711If ( 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
712Differentiate each of the functions with
respect to ( ^{prime} boldsymbol{x}^{prime} )
( frac{a x+b}{c x+d} )
12
function ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{-boldsymbol{x}} sin boldsymbol{x}, boldsymbol{x} boldsymbol{epsilon}[mathbf{0}, boldsymbol{pi}] )
12
714If ( 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
715Illustration 2.19 If y =

= x-10, then find dyldx.
12
716Answer 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
717Show that the function ( |x| ) is not
differentiable at ( x=0 )
12
718Identify 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
719If ( 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
720The 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
721Show 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
722If ( x=y sqrt{1-y^{2}}, ) then ( frac{d y}{d x} ) equals to?12
723If ( y=cos ^{3} x, ) then ( frac{d y}{d x}= )12
724Verify 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
725Find the derivative of
( frac{2}{x+1}-frac{x^{2}}{3 x-1} )
12
7269.
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
727The 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
728Let ( 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
729Given 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
730The 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
731The 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
732T 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
733Let ( 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
734Find 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
735If ( 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
736Solve-
( cos x^{3} cdot sin ^{2}left(x^{5}right) )
12
737Prove 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
738Let ( 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
740Show 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
741If ( 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
742f ( y=x^{x}+(sin x)^{cot x} ). find ( frac{d y}{d x} )12
743Differentiate the following functions
w.r.t. ( x )
( e^{log (log x)} )
12
744Find the derivative of the following functions from first principle:
( sin (x+1) )
12
745Differentiate 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
747Let ( 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
748Verify Rolle’s Theorem for the function
( f(x)=x(x-1)^{2} ) in the interval [0,1]
12
749Evaluate :
( 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
750Assertion
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
753If ( 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
754If ( 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
755f ( 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
757If ( 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
758If ( 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
759If ( x^{y}=e^{x-y}, ) then show that ( frac{d y}{d x}= ) ( frac{log boldsymbol{x}}{(1+log boldsymbol{x})^{2}} )12
760If ( 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
762Differentiate with respect to ( x ) :
( log _{7}(2 x-3) )
12
763Let ( 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
765If ( 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
766The 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
767Examine 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
768If 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
769If ( 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
770Find the slope of the tangent to the curve ( y=x^{3}-x ) at ( x=2 )12
771If ( y=tan ^{-1} frac{cos x+sin x}{cos x-sin x}, ) then find ( frac{d y}{d x} )12
772If ( y=frac{sin x+cos x}{sin x-cos x} ) find ( frac{d y}{d x} ) at ( x=frac{pi}{4} )12
773If ( x=a(cos theta+sin theta) ) and ( y= )
( a(sin theta-cos theta), ) then find ( frac{d^{2} y}{d x^{2}} )
12
774Differentiate 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
775If ( 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
776If ( 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
777If ( 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
778If ( 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
779Differentiate w.r.t. ( x )
( boldsymbol{y}=sin left(5 x^{3}+2 xright) )
12
780If ( 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
781The 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
782Find the derivative ( : cot x )12
783The 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
784f ( 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
785If ( 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
786Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for
( boldsymbol{y}=left(sin 20 x+a^{2 x}+10right) )
12
787Differentiate the following function with
respect to ( x ) ( frac{1}{sin x} )
12
788Differentiate with respect to ( x ) :
( sin ^{-1}left(frac{2^{x+1}}{1+4^{x}}right) )
12
789Let ( 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
790Examine 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} \
end{array}
]
[
boldsymbol{f}(boldsymbol{x})=mathbf{3}
]
12
791Find the diffrentiation of ( x sin x )12
7921.
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
(a) f(2)= 15
(6) h'(1)=666
(d) h(g(3))=36
(c)
h(0)=16
12
793Let ( 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
794If ( 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
796If ( tan (x+y)+tan (x-y)=1 ), then
find ( frac{d y}{d x} )
12
797Trace 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
799The 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
800If ( 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
801Statement 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
802If ( 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
803is 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
804ff ( 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
805If ( 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
806If ( 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
807If ( 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
808Find ( 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
809Check 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
810If ( y=ln left(x^{e^{x}}right) ) find ( frac{d y}{d x} )12
811Differentiate the given function w.r.t. ( x )
( boldsymbol{y}=log left(cos e^{x}right) )
12
812The 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
813If 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
815Draw 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
816Find the derivative of the following
function from first principle:
( -x )
12
817The 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
818For 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
819Find ( 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
820f ( y=log sqrt{frac{1+tan x}{1-tan x}}, ) prove that ( frac{d y}{d x}= )
( sec 2 x )
12
821If ( 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
822Find derivative of ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} cos boldsymbol{x} )12
823If ( 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
824Find 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
825If ( 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
826If ( 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
827Differentiate with respect to ( x ) :
( log left{cot left(frac{pi}{4}+frac{x}{2}right)right} )
12
828Differentiate ( 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
829For 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
830Find 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
831Differential 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
832If ( 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
833f ( boldsymbol{y}=log (log sin boldsymbol{x}), ) then evaluate ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
834If ( 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
835Verify 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
836If ( 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
837Examine the applicability of Mean Value Theorem for the following function. ( f(x)=x^{2}-1 ) for ( x epsilon[1,2] )12
838If ( 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
839Differentiate with respective to ( x ) ( log (sec x+tan x) )12
840If ( 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
841For ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{2} cdot sin boldsymbol{x}} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
842If 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
843If ( 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
844Solve: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}}(operatorname{cosec} boldsymbol{x})=? )12
845Let ( 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
847For some constants ( a ) and ( b ) find the
derivative of
( left(a x^{2}+bright)^{2} )
12
848Find 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
849If ( 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
850Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=boldsymbol{e}^{sin sqrt{tan boldsymbol{x}}} )12
851Differentiate w.r.t. ( boldsymbol{x} )
( frac{boldsymbol{x}}{boldsymbol{y}^{3}}=mathbf{1} )
12
852Find the second order derivatives of
( tan ^{-1} x )
12
853Find ( frac{d y}{d x} ) if ( 3 x+4 y=9 )12
854Find 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
855Show that the function ( boldsymbol{f}(boldsymbol{x})= )
( left[cos left(x^{10}+1right)right]^{1 / 3}, x in R ) is a continuous
function.
12
856A 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
857If ( 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
858Differentiate the function with respect
to ( x )
( 2 sqrt{cot left(x^{2}right)} )
12
859If ( 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
860If ( 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
861f ( x sin (a+y)=sin y, ) then ( y^{prime}=? )12
862Solve:
( lim _{x rightarrow 0} frac{sin ^{3} x^{2}}{x^{6}} )
( mathbf{A} cdot mathbf{1} )
B.
( c cdot-1 )
D. ( infty )
12
863Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) for ( boldsymbol{y}=log left(frac{1+boldsymbol{x}}{1-boldsymbol{x}}right) )12
864Find ( 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
865If ( 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
866Assertion ( 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
867Differentiate ( 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
869If ( y=x sin y, ) then prove that ( frac{d y}{d x}= ) ( frac{boldsymbol{y}}{boldsymbol{x}(1-boldsymbol{x} cos boldsymbol{y})} )12
870If ( 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
871The 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
is (are) TRUE?
(6) f(x) <* _ xfor allx e (0,7)
© There exists a e (0, Tt) such that f'(a)=0
12
874Differentiation gives us the instantaneous rate of change of one variable with respect to another
A. True
B. False
12
875If ( 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
876The 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
877The 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
878Verify Rolle’s theorem for ( boldsymbol{f}(boldsymbol{x})= ) ( x sqrt{a^{2}-x^{2}} ) in ( [0, a] )12
879Diffrentiate w.r.t ( x: ) ( boldsymbol{y}=cos ^{-1}left[frac{1-x}{1+x}right] )12
880For ( 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
88128. 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?
(a) f(2)1-loge 2
(c) g(1) >1-loge 2 (d) g(1) <1-loge 2
12
882If ( 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
883If ( 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
884Value 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
885If ( 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
886If ( 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
887Differentiate the following functions
with respect to ( boldsymbol{x} ) ( frac{e^{2 x}+e^{-2 x}}{e^{2 x}-e^{-2 x}} )
12
888Solve:
( frac{d y}{d x}=x^{2}(x-2), ) given ( y=2 ) where
( boldsymbol{x}=mathbf{0} )
12
889ff ( 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
890Evaluate ( 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
891If ( 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
894If ( boldsymbol{y}=boldsymbol{x}^{2}+mathbf{5} boldsymbol{x} ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
895State 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
896The 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
897if ( 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
898If 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
899Find 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
90014. 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
901Let ( 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
902Find the derivatives of the following
functions.
( log _{2}left(2 x^{2}-3 x+1right) )
12
903If ( 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
904For 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
906Find the derivative of tan ( x ) using first principle of derivatives12
907If ( e^{x}+e^{y}=e^{x+y}, ) show that ( frac{d y}{d x}= )
( -e^{y-x} )
12
908By 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
909If ( (cos x)^{y}=(cos y)^{x} ) then find ( frac{d y}{d x} )12
910Write an example of a function which is everywhere continuous but fails to be differentiable exactly at five points.12
911Let ( 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
912If ( 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
914If ( boldsymbol{y}=sin ^{-1}left[operatorname{atan}^{-1} sqrt{frac{1-x}{1+x}}right] ) then find
( frac{d boldsymbol{y}}{d x} )
12
915Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}, ) if ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}=sin boldsymbol{y} )12
916f ( 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
917Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( sin boldsymbol{x}-boldsymbol{3} boldsymbol{x}=mathbf{5} boldsymbol{y} )
12
918If ( 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
919Show that ( f(x)=(x-1) e^{x}+1 ) is an
increasing function for all ( x>0 )
12
920The 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
921Number 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
922If ( 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
923If ( 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
924If ( y=log [tan x], ) find ( frac{d y}{d x} )12
925If ( 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
926Let ( [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
927Let ( 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
928Verify the Rolle’s theorem for the following functions:
( f(x)=x^{4}-1 ) on the interval [-1,1]
A. True
B. False
12
929Let ( 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
931Examine 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
932If ( 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
933Find: ( 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) ) is12
935For 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
936The 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
93735.
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
938If ( 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
939Range of ( boldsymbol{f}(boldsymbol{x}) ) is ( ? )
A. ( R )
B . ( R-{0} )
( c cdot R^{+} )
D. ( (0, e) )
12
940If ( 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
941Show that ( f(x)=x^{9}+4 x^{7}+11 ) is an
increasing function for all ( boldsymbol{x} in boldsymbol{R} )
12
942ff ( 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
943Derivate ( e^{sqrt{2 x+1}} ) where ( x=12 ) w.r.t. ( x )12
944Compute 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
945an
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
946Let ( 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
947Let ( 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
948Differentiate:
( 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
950Solve the different equation:-
( left(tan ^{-1} y-xright) d y=left(1+y^{2}right) d x )
12
951If ( 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
952Express 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
953Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{x}+boldsymbol{y}^{2}=log boldsymbol{y}+boldsymbol{x}^{2} )
12
954Let ( 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
955If ( e^{x y}=y+sin ^{2} x, ) then at ( x=0, d y / d x )
is equal to
12
956Find 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
957Let ( 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
959Let ( 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
960If ( f(x)=|x|^{3} . ) show that ( f^{prime prime}(x) ) exists for
all real ( x ) and find it.
12
9614. If y = cos(sin x2), then
then at x = 12
(a) -2
(b) 2
12
962f ( 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
963ff ( 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
964State 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
965Find ( 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
966If ( 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
967Check the continuity of the function ( boldsymbol{f}(boldsymbol{x})=|boldsymbol{x}-mathbf{2}|+boldsymbol{x} )12
968Let ( 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
969If 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
971ff ( y=frac{sin 4 x}{x^{2}+16}, ) then find ( frac{d y}{d x} )12
972f ( 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
974If ( 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
975If ( 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
976Let ( 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
977Match the columns12
978Differentiate:
( boldsymbol{y}=boldsymbol{c}^{2}+frac{boldsymbol{c}}{boldsymbol{x}} )
12
97947.
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
980Find 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
982The 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
983Derivative 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
984Given ( 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
985If ( 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
986Show 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
988Find ( 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
990Which 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
991Differentiate the following function
w.r.t.x.
( frac{1}{left(x^{2}+3^{2}right)} )
12
992Illustration 2.26
Find the derivative of y=-
.
+2 +1
12
993Assertion
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
994ff ( y=tan ^{-1}(3 x), ) then find ( frac{d^{2} y}{d x^{2}} )12
995Left 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
997If ( 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
99918. Differentiation of 2×2 + 3x w.r.t. x is…
…fondra
12
1000If ( 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
1001Using the ahove approximation, the value ( sqrt{104} ) is
( mathbf{A} cdot 10.18 )
B. 10.49
c. 10.2
D. 10.28
12
1002Assertion(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
1003Let ( [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
1004f ( 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
1005Differentiate 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
1007Ify=(x + V1+x? “, then (1+x) 2 x is 12002]
(b) – n’y
(d) 2xy
12
10085. 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
1009Left 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
1011If ( 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
1014If ( 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
1015The 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
1016Find ( 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
1018Find the derivative of ( boldsymbol{y}= )
( n sqrt{frac{1-sin x}{1+sin x}} )
12
1019If ( 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
1020Find 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
1021Find the derivative of ( x^{2} ) with respect to
( log boldsymbol{x} )
12
1022Show 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
1023If ( 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
1024Verify 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
1025A 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
1026The 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
1027If ( 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
1028Differentiate 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
1029If ( 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
1031Verify 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
1033Solve ( : I_{n}=int_{0}^{frac{pi}{2}} e^{-x} sin ^{n} x d x )12
1034Find the derivative ( x^{3} )12
1035Find ( 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
1036If ( 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
1037For 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
1038Consider 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
1039The 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
1040Check continuity of the function ( x^{2}|x| ) at the origin.12
1041Find ( 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
1042Differentiate: ( sin ^{2} 3 x cdot tan ^{3} 2 x )12
1043The 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
1044If ( 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
1045Find the derivative of the following functions from first principle ( frac{1}{x^{2}} )12
1046If ( 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
1047Find ( 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
1050Consider 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
1051Let ( 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
1052f ( 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
1053If ( 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
1054Let ( 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
1056If ( 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
1057For ( 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
1058Let ( 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
1059Consider 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
1060The 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
1061Find the derivative of ( frac{x}{(x-1)} )12
1062Find the derivative of ( 99 x ) at ( x=100 )12
106331.
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
1064Illustration 2.31 Find the derivative of y=
– 21 Since
12
1065ff ( 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
10660 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
1068f ( 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
1069If ( y=sqrt{x}+frac{1}{sqrt{x}}, ) show that ( 2 x frac{d y}{d x}+ )
( boldsymbol{y}=mathbf{2} sqrt{boldsymbol{x}} )
12
1070If ( 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
1071Show 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
1072If 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
1073Find 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
1074Consider 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
1075Find 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
1076Differentiate: ( e^{sin ^{-1} x} )12
107710. 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
1078If ( 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
1079If ( 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
1080Differentiate the following from first principle. ( f(x)=cos left(x-frac{pi}{8}right) )12
1081The 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
1082If, 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
1083If ( 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
1084If ( 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
1085Discuss the continuity of the following functions:
(a) ( f(x)=sin x+cos x )
(b) ( f(x)=sin x-cos x )
12
1086Which 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
1087Let ( 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
108837. 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
1089Discuss 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
1090In 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
1091If ( y=a^{frac{1}{2} log _{a} cos x} . ) Find ( frac{d x}{d} )12
1092Identify 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
1093Differentiate 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
1094The 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
1095If ( 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
1097Differentiate ( 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
1098Find ( 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
1099Let ( 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
1100For 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
1101Discuss 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
1102Examine 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
1103Let ( 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
1104Which of the following functions is every where continuous-
A . ( x+|x| )
B . ( x-|x| )
c. ( x|x| )
D. All of the above
12
1105A 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
1106If ( 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
1107Assertion
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
1108Let ( 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
1110ff ( 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
1111Find 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
1112Obtain 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
1113If ( 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
1114Solve:
( lim _{x rightarrow 3} frac{x^{2}-9}{x-3} )
12
1115Let 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
1116Let ( 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
1117Differentiate ( 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
1119Find ( frac{d y}{d x} ) of ( 2 x+3 y=sin x )12
1120Let ( 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
1121Suppose 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
1122The 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
1123Let ( 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
1124Find 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
1125Prove 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
1126If ( 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
1127If ( 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
1129If ( 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
1130Verify 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
1131Differentiate: ( left(sin ^{-1} x+frac{1}{2} log frac{1+x}{1-x}right) )12
1132If ( 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
1133Suppose, ( 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
1134If ( 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
1136If 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
1138Differentiate the following function with
respect to ( boldsymbol{x} )
( sin h^{-1}(sqrt{x}) )
12
1139Find 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
1140Find the second order derivative of the
following function:
( x^{3}+tan x )
12
1141If ( 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
1142The 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
1143Consider 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
1144If ( 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
1145Differentiate ( sin boldsymbol{h}^{-1}left(frac{mathbf{1}}{boldsymbol{x}}right) ) with respect
to ( boldsymbol{x}(boldsymbol{x}>mathbf{0}) )
12
1146If ( 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
1147The 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
1148Find ( 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
1149Evaluate:
( 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
1150Among 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
1152Let ( 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
1153If ( 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
1154Differentiate with respect to ( x ) :
( log (csc x-cot x) )
12
1155f ( 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
1156Assertion
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
1157The 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
1158Evaluate:
( 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
1159Suppose 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
1160A 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
1161Let ( 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
1162If ( 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
1163If ( 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
1165Differentiate :
( e^{e^{x}} )
12
1166Differentiate ( boldsymbol{x}^{3} ) w.r.t ( boldsymbol{x} )12
1167The 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
1168For 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
1169f ( 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
1171If ( 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
1173If ( 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
1174Differentiate ( sqrt{e^{sqrt{x}}}, x>0 )12
1175ax
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
1176The 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
1177If ( 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
1178If ( 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
1179Find 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
1181Find ( 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
1183Let ( [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
1184Differentiate the function with respect
to ( x )
( cos x^{3} cdot sin ^{2}left(x^{5}right) )
12
1185Illustration 2.22 If y = 4x*+ 2x +
+ 9, then find dyldx.
12
1186Show that the function ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{cll}3-x, & text { if } & x1end{array}right. )
( boldsymbol{x}=mathbf{1} )
12
1187Differentiate the following
( (x+2)^{3} )
12
1188If ( 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
1190Find 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
1191Differentiate 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
1192The 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
1193Show 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
1195If ( 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
1196For 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
1197If ( 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
1198If ( 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
1199Let ( 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
1200Differentiate 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
1201Differentiate with respect to ( x ) :
( e^{sin ^{-1} 2 x} )
12
1202Verify Rolle’s theorem for the function ( f(x)=sin x+cos x-1 ) in the interval
( left[0, frac{pi}{2}right. )
12
1203If ( f(x)=sin 2 x-cos 2 x, ) find ( f^{prime}left(frac{pi}{6}right) )12
1204Consider 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
1205If ( 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
1206If ( 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
1207If ( 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
1208If ( 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
1209The 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
1210Let ( 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
1211Differentiate with respect to ( x ) :
( (log sin x)^{2} )
12
1212llustration 2.29 If y
Illustration 2.29
sin x
If y = –
, then find
x + cos x
sin x then finden
dx
12
1213Find 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
1214If ( 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
1215Differentiate
( -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
1217Find the inverse function of ( f(x)= )
( 2 x-3 )
12
1218Differentiate with respect to ( x ) :
( sin ^{2}(log (2 x+3)) )
12
1219A 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
1220Differentiate 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
1221Differentiate the following function w.r.t. ( x: cos ^{-1}left(1-2 sin ^{2} xright) )12
1222If 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
1224Find ( 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
1226Which 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
1227Find ( frac{d y}{d x} ) if ( y=frac{x^{2}+x}{2} )12
1228Using 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
1229Answer 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
1230If ( f(x)=frac{x-e^{x}+cos 2 x}{x^{2}}, x neq 0 ) is
continuous at ( x=0, ) then
B . ( [f(0)]=-2 )
c. ( f(0)=-0.5 )
D. ( [f(0)] . f(0)=-1.5 )
12
1231Find 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
1232ff ( 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
1233If ( 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
1234If ( 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
1235If ( 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
1236If ( 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
1237NI-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
1238The value of the derivative of ( |boldsymbol{x}-mathbf{1}|+ )
( |x-3| ) at ( x=2 ) is:
( A cdot 2 )
B.
( c .0 )
D. – 2
12
1239Verify Lagrange’s Mean Value Theorem for the function ( f(x)=x^{2}+x-1 ) in
the interval ( [mathbf{0}, mathbf{4}] )
12
1240The 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
1241The 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
1242If ( 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
1244ff ( 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
1245Differentiate: ( boldsymbol{y}=8 sin boldsymbol{x} cos boldsymbol{x} ) w.r.t ( boldsymbol{x} )12
1246Differentiate the following function with
respect to ( x ) ( boldsymbol{x}^{-4}left(boldsymbol{3}-boldsymbol{4} boldsymbol{x}^{-boldsymbol{5}}right) )
12
1247If ( 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
1248If ( 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
1249The 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
1250What is derivative of ( left[frac{1}{x}right]^{x} )12
1251If ( 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
1252Solve for ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{r}^{2} )12
1253The value of ( boldsymbol{f}^{prime}(boldsymbol{3}) ) is
( A cdot 8 )
B. 10
c. 12
D. 18
12
1254The 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
1255Differentiate the following function with
respect to ( x )
( boldsymbol{x}^{-3}(mathbf{5}+mathbf{3} boldsymbol{x}) )
12
1256Find 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
125710. 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
1259f ( 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
1260Find ( frac{d y}{d x} ) for ( y=log _{e}(x+sqrt{x^{2}-a^{2}}) )12
1261Let ( 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
1262If ( 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
1263The 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
1264Let
( 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
1265If ( 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
1266If ( 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
1268If 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
1269n 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
1270Let ( 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
1271Illustration 2.34 If x = ał”, y = bt”, then find ay
dx
12
1272If ( sin (x y)+frac{y}{x}=x^{2}-y^{2}, ) find ( frac{d y}{d x} )12
1273If ( 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
1274Differentiate the following functions
with respect to ( boldsymbol{x} )
( log {x+2+sqrt{x^{2}+4 x+1}} )
12
1275Find the derivative of ( csc ^{2} x, ) by using
first principle of derivatives.
12
1276Differentiate ( frac{x^{2} sin x}{1-x} ) w.r.t ( x )12
1277Let ( 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
1278Differentiate 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
1279Say 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
1281Solution 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
1282Assertion(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
1283Let ( 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
1284Differentiate 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
128516-1) 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
1286Discuss the continuity and differentiability of ( boldsymbol{f}(boldsymbol{x})=|log | boldsymbol{x} | )12
1287If ( 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
128841.
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
1289Differentiate 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
1290The 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
1291If ( 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
1294Verify 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
1295Differentiate the following function with
respect to ( x ) ( frac{4 x+5 sin x}{3 x+7 cos x} )
12
1296If ( 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
1299Identify 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
1300Differentiate ( -frac{4 x+5 sin x}{3 x+7 cos x} ) w.r.t ( x )12
1301Differentiate:
( frac{d}{d x}left(tan ^{-1} xright) )
12
1302The 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
1303If ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}^{3}}, ) find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
1304If 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
1305Given, ( 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
1306Differentiate sec ( x ) by first principle.12
1307TOPIL-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
1308Solve the differential equation ( cos left(frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}right)=boldsymbol{a},(boldsymbol{a} in boldsymbol{R}) )12
1309If ( |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
1310If ( 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
131136.
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
1312Differentiate 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
1313Assertion
( 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
1314f ( e^{y}(x+1)=1, ) show that ( y_{2}=y_{1}^{2} )12
1315Let ( 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
1317Using 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
1318Find the derivative of the following functions from first principle:
( cos left(x-frac{pi}{8}right) )
12
1319If ( 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
1320Verify 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
1321Find 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
1322Verify 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
1323Consider 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
1324Differentiate 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
1325The 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
1326ff ( 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
1327Assertion
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
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
1329Differentiate from first principles.
(i) ( 3 x )
12
1330PICOU
23. If y=x then
is
12
1331Identify 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
1332Find the derivative of the following functions from the first principals w.r.t
to ( boldsymbol{x} )
( tan 2 x )
12
1333If ( 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
1334Is 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
1335Let ( 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
1336Derivative 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
1338Let ( 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
133917. 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
1340x20, 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
1341Differentiate the following functions
with respect to ( x ) :
( tan ^{-1}left{frac{x^{1 / 3}+a^{1 / 3}}{1-(a x)^{1 / 3}}right} )
12
1342Assertion
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
1343Check whether Lagrange’s mean value
theorem is applicable on ( f(x)=sin x+ ) ( cos x ) interval ( left[0, frac{pi}{2}right] )
12
1344The 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
1345Find 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
1346The 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
1347Find ( frac{d y}{d x} ) for ( 2 x^{2}+5 x y+3 y^{2}=1 )12
1348Differentiate ( boldsymbol{y}=mathbf{1 0}^{boldsymbol{x}}+mathbf{1 0}^{tan boldsymbol{x}} )12
1349Find from first principles the
differential coefficient of ( 2 x^{2}+3 x )
12
1350The 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
1351If ( 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
1352The range of the function ( Delta=f(|x|) ) is-
A ( cdot[0,1] )
в. [0,1)
c. (0,1]
D. None of these
12
1353If ( 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
1354Sketch 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
1355Differentiate ( 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
1358The function ( f(x)=sin ^{-1}(cos x) ) is :
A. discontinuous at ( =0 )
B. continuous at ( =0 )
c. differentiable ( =0 )
D. none of these
12
1359If ( 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
1361If ( 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
1362Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )
( boldsymbol{y}=frac{1}{boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}+boldsymbol{3}} )
12
1363If ( 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
1364Differentiate with respect to ( x ) :
( e^{sqrt{cot x}} )
12
1365Find 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
1366If 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
1367For a curve at which the tangent lines at
two distinct points coincide, then the curve cannot be
A. a cubic curve
c. a curve of 4th power
D. none of these
12
136811.
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
1369Sketch 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
1370Show 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
1371The 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
1372If ( y=frac{1}{sqrt{a^{2}-x^{2}}}, ) find ( frac{d y}{d x} )12
13732. 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
1374Which 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
137545.
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
1376Find 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
1377If ( 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
1378The 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
1380Prove that ( e^{x}-x>1, ) if ( x>0 )12
1381If ( x^{3} y^{5}=(x+y)^{8}, ) then show that
( frac{d y}{d x}=frac{y}{x} )
12
1382Differentiate ( frac{1}{3} tan ^{3} x-tan x+x ) w.r.t ( x )12
1383If ( left(x^{2}+y^{2}right)^{2}=x y, ) find ( frac{d y}{d x} )12
1384The 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
1385From 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
1386If ( 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
1387At 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
1388If ( 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
13893. If y = sin x & x = 3t then
will be
(a) 3 cos (x)
(c) 3 cos (x)
(b) cos x
(d) -cos x
12
1390The 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
1391Find 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
1393If ( 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
1394Find the derivative of ( y=log ^{3}left(x^{2}right) )12
1395Let ( 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
1396f ( 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
1397Differentiate ( e^{x}+e^{-x} ) with respect to ( x )12
1398ff ( 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
14003.
Given y= 5x
31(1 – x)2
+ cos2 (2x+1) ; Find
dy. (1980)
12
1401If ( 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
1402If ( 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
14033. 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
1404Differentiate w.r.t ( boldsymbol{x} ) ( sin ^{-1}left(frac{a cos x+b sin x}{sqrt{a^{2}+b^{2}}}right) )12
1405On 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
1406What 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
1407The 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
1408If ( 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
1409Let ( 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
1410If ( 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
1412The 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
1413Differentiate the following function from first principle:
( sin ^{-1}(2 x+3) )
12
1414If ( 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
1415Rolle’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
1416For 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
1417If 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
1419The 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
14202312
1421ff ( 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
1423Differentiate ( boldsymbol{x}=boldsymbol{y}+ )12
1424Using 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
1425If ( 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
1426Find ( 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
1427If ( 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
1430Differentiate 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
1431Find ( frac{d y}{d x} ) where ( boldsymbol{y}=sqrt{frac{boldsymbol{x}^{2}+mathbf{1}}{boldsymbol{x}^{2}+mathbf{2}}} )12
1432If 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
1433Differentiate ( 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
1435The 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
1436If ( 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
1437Let ( 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
1438The 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
1439Given ( 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
1440Discuss the continuity of the following function ( : f(x)=sin x . cos x )12
1441Differentiate ( 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
1442If ( 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
1443If ( 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
1444If ( 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
1445Differentiate: ( boldsymbol{y}=sin ^{-1}left(frac{1-boldsymbol{x}^{2}}{mathbf{1}+boldsymbol{x}^{2}}right), mathbf{0}< )
( boldsymbol{x}<mathbf{1} )
12
144615. 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
1447Test 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
1448If ( 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
1450Differentiate the following function w.r.t.
( boldsymbol{x} )
( sqrt[3]{left(2 x^{2}-7 x-4right)^{5}} )
12
1451Find ‘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} )
( ^{mathrm{c}} cdot_{C}=2-frac{sqrt{21}}{6} )
D. ( c=1+frac{sqrt{21}}{6} )
12
145217. Let \$(x) =
1-tan x
X
,X
E
f(x) is continuous
4x-tet
[2004]
(a) 165
(1) 1
12
1453If ( 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
1454If ( 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
1455Extend 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
1457The 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
1458If ( 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
1459Differentiate the following w.r.t ( x )
( sin left(x^{2}+5right) )
12
1460Find the second order derivatives of
( sin (log x) )
12
1461ff ( 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
1462If ( 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
1463Find the derivative of the following
functions:
( 5 sec x+4 cos x )
12
1464Given 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
1465Using 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
1466Differentiate the following function with
respect to ( x ) ( frac{x+cos x}{tan x} )
12
1467Solve ( lim _{x rightarrow-2} frac{left(frac{1}{x}+frac{1}{2}right)}{x+2} )12
1468Discuss 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
1469If ( 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
1470Find ( 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
1472If ( 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
147321. Let f : [a, b] → [1,00) be a continuous function and let
g: R → R be defined as
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
14742
( 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
1475The 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
1477Let ( 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
1478If 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
1479Solve: ( frac{boldsymbol{d}}{boldsymbol{d} boldsymbol{x}} mathbf{7} boldsymbol{x} )12
148014.
Let f(x)=x sin 7tx, x>0. Then for all natural numbers n, )
vanishes at
(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
1481Differentiate the following w.r.t. ( x ) ( e^{sin ^{-1} x} )12
1482If ( boldsymbol{y}+sin boldsymbol{y}=cos boldsymbol{x}, ) find ( frac{boldsymbol{d} boldsymbol{y} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} )12
1483Consider 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
1484If 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
1485Suppose ( 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
1486By 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
1487If ( 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
1489Find derivative of ( f(x) )
( f(x)=x sin x )
12
1490If ( 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
1491If ( 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
1492Discuss 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
1493Which 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
1494If ( 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
1495VJ (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
(d) all integers except 1
12
1496The 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
149824. If velocity of particle is given by v = 24, then it
acceleration (dv/dt) at any time t will be given by…
12
1499State 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
1500Letf: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
1501Differentiate the following w.r.t. ( x )
( log left(cos e^{x}right) )
12
1502If ( 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
1503If ( boldsymbol{y}=boldsymbol{x} sqrt{1-boldsymbol{x}^{2}}+sin ^{-1} boldsymbol{x}, ) then ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) is12
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
1505In the law of mean, the value of ( theta )
satisfies the condition
( mathbf{A} cdot theta>0 )
B . ( theta1 )
( 0.0<theta<1 )
12
1506Let [.] 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
1507If ( 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
1508If ( 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
1509Find 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
1511If ( 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
1512Differentiate ( boldsymbol{y}=cos (2 x-5) ) with
respect to ( x )
12
1513Obtain 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
15148.
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
151612. 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
1517Answer 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
1519If ( 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
1520The 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
1521Let ( 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
1522Find 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
1523If 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
1524If ( 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
1526Compute 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
1528If ( y=3 cos x, ) then ( frac{d y}{d x} ) at ( x=frac{pi}{2} ) is
A. -3
B. 3
( c .0 )
D. –
12
1529Find drivative of ( boldsymbol{y}=(2-sin x)left(e^{x}+right. )
( left.x^{3}+2right) ) with respect to ( x )
12
1530The 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
1531Verify 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
1533Illustration 2.23
Find the derivatives of y=(x + 1) (x+3).
12
1534If ( 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
153535.
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
1536If ( 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
1537The 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
1538If 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
1539The 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
1540If 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
1541If ( x^{3}+y^{3}=3 a x y, ) find ( frac{d y}{d x} )12
1542et 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
1543Let ( 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
1544If 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
1545Differentiate w.r.t. ( boldsymbol{x} )
( boldsymbol{y}=e^{3 x-2} sin 3 x )
12
1546Let ( 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
154728. 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
1548Illustration 2.33
If y= cos x”, then find
12
1549If ( 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
155027. If y= tan x. cosx then
will be …
12
1551Let ( 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
1552Find the derivative of ( boldsymbol{y}= ) ( frac{1}{4} ln frac{x^{2}-1}{x^{2}+1} )12
1553For 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
1554If ( 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
1555f ( 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
1556If ( 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
1557If ( f(x)=tan x, ) find ( f^{prime}(x) ) and hence find ( boldsymbol{f}^{prime}left(frac{boldsymbol{pi}}{boldsymbol{4}}right) )12
1558Find the derivative of ( sin ^{2} x ) with respect to ( x ) using product rule12
1559State 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
1561If ( 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
1562Differentiate w.r.t ( boldsymbol{x} )
( e^{operatorname{cosec}^{2} x} )
12
1563Let ( 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
1564Find the derivative of the following functions from first principle ( (x-1)(x-2) )12
1565If ( 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
1566Solve : ( int frac{x^{2}+1}{(x+1)^{2}} d x )12
1567Differentiate 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
1569If ( 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
1570If ( 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
1571If ( 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
1572Solve :
( 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
1573If the derivatives of ( tan ^{-1}(a+b x) ) takes
the value 1 at ( x=0, ) prove that ( 1+ )
( a^{2}=b )
12
1574If ( 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
1575Find 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
1577Solve:
( 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
1578If 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
1579The 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
1580Consider ( 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
1581Illustration 2.21 If y= 3x + 2x, then find dyldx.12
1582Find ( 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
1584Differentiate 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
1585Differentiate 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
1587Find the derivative of
( left(5 x^{3}+3 x-1right)(x-1) )
12
1588If ( 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
1590Let 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
1591Applying 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
1592If ( 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
1593Let ( 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
1594At what point on the curve ( y=x(x-4) ) on [0,4] is the tangent parallel to ( X ) -axis.12
1595f ( 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
1596For 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
1597If 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
1598Function ( 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
1599Verify 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
1600If ( 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
1601If ( boldsymbol{y}=frac{1}{4}(boldsymbol{x} pm boldsymbol{A})^{2} )
Hence prove: ( boldsymbol{y}_{1}^{2}=boldsymbol{y} )
12
1602Let ( 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
1603ff ( 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
1604If ( 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
1605Find the second derivative of ( sin 3 x cos )
( 5 x )
12
1606Differentiate 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
1608Differentiate: ( frac{e^{x}}{sin x} )12
1609Differentiate :
( y^{x}=x^{y} )
12
1610is 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
1612Differentiate with respect to ‘t’ ( e^{-w t} )12
1613Verify LMVT for the function ( f(x)=x+ )
( frac{1}{x}, x in[1,3] )
12
1614Differentiate w.r.t. ( x ) in ( tan ^{-1}left(frac{5 x}{1-6 x^{2}}right) )12
161519. 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
1616Find ( 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
1618If ( 2^{x}+2^{y}=2^{x+y}, ) then find ( frac{d y}{d x} )12
1619If ( 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
16207.
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
1621If 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
1625Find 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
1626If ( 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
1627Examine 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
1628Assertion
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
1629Prove that ( f(x)=sin x+sqrt{3} cos x ) has
maximum value at ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{6}} )
12
1630Verify 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
1632The 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
1633The function given by ( y=| x|-1| ) is
differentiable for all real numbers
except the points.
B. ±1
( c cdot 1 )
D. –
12
1634if ( y=e^{x} cos x, ) prove that ( frac{d y}{d x}= )
( sqrt{2} e^{x} cos left(x+frac{pi}{4}right) )
12
1635Find ( frac{d y}{d x}, i f x^{y}=e^{x-y} )12
1636If ( 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
1637If ( 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
1638Differentiate the following w.r.t.x:
( sin ^{-1} x+cos ^{-1} x )
12
1639If ( 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
1640Find 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
1641Evaluate: ( 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
1643u
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
1644Find ( 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
1645Examine 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
1646If ( sqrt{1-x^{2}}+sqrt{1-y^{2}}=a ) find ( frac{d y}{d x} )12
164719. If square of x varies as cube of y and x = 3 when y = 4, the
value of y at
ill be…
12
1648Let 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
1650If ( 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
1651The 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
1652What 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
1653Differentiate ( log sqrt{frac{1+cos ^{2} x}{left(1-e^{2 x}right)}} ) w.r.t. ( x )12
1654The 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
1656Differentiate the following w.r.t. ( x: ) ( e^{x}+e^{x^{2}}+ldots+e^{x^{5}} )12
1657Differentiate: ( log left(cos e^{x}right) )12
1658If ( 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
1659If, ( 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
1660Differentiate ( (x)^{tan x}+(tan x)^{x} ) w.r.t ( x )12
1661Suppose ( 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
1662If ( 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
1663If ( frac{3}{2}+y^{3}=3 a x y, ) then find ( frac{d y}{d x} )12
1664Differentiate with respect to ( x ) :
( boldsymbol{y}=cos boldsymbol{x}+sin 2 boldsymbol{x} )
12
1665If ( 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
1667Differentiate with respect to ( x ) :
( e^{x} log sin 2 x )
12
1668Find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}: )
( boldsymbol{x}+boldsymbol{y}^{2}=log boldsymbol{y}+boldsymbol{x}^{2} )
12
1669Let ( [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
1670Find the derivatives of the following functions at the indicated points.
( boldsymbol{y}=ln (2-sqrt{2 x+1}), y^{prime}(0)=? )
12
1671Find ( 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
1672Find ( mathbf{k} ) if ( boldsymbol{f}(boldsymbol{x})= )
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
1674If ( 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
1676Discuss 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
1677Find ( 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
1678Find ( 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
1680For 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
1681Match the columns12
1682( y=e^{x}+e^{-x} ) prove that ( frac{d y}{d x}=sqrt{y^{2}-4} )12
1683Find ( 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
1684If ( 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
1685Find the second order derivatives of
( e^{x} sin 5 x )
12
1686If ( 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
1687Let ( 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
1688If ( 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
168941. 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
1691Differentiate with respect to ( x ) :
( left(sin ^{-1} x^{4}right)^{4} )
12
1692Differentiate ( 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
1693Let 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
1694If ( 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
1695If ( 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
16961.
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
1697The 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
1698Examine 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
1699Find 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
1701Find the derivate of ( e^{sqrt{2 x+1}} ) with respect
to ( x ) at ( x=12 )
12
1702If ( 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
1703Find the derivative of following functions using first principle with
respect to ( x )
( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} sin boldsymbol{x} )
12
1704If ( 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
1705A 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
1707If ( 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
1708If ( 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
1709Let ( 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
1711If ( 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
171230.
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
1713Let ( 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
1714Find ( 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
1715If ( 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
1716Let ( 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
1719Using 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
172044.
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
1721Assertion (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
1722Differentiate with respect to ( x ) :
( e^{-3 x} log (1+x) )
12
1723Write the derivative of ( f(x)=|x|^{3} ) at
( boldsymbol{x}=mathbf{0} )
12
1724Suppose 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
1725Illustration 2.18
If y=x”, then find dy/dx.
12
1726Solve:
( lim _{x rightarrow 2} frac{x^{2}-4}{sqrt{3 x-2}-sqrt{x+2}} )
12
1727If ( 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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