JEE Mathematics - Differential Calclus Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
If \(f(x)=cos[{ \Pi }^{ `2 }]x+cos[{ -\Pi }^{ `2 }]x,\)where [x] stands for the greatest integer function,then
- (a)
\(f(\frac { \Pi }{ 2 } )=-1\)
- (b)
\(f(\pi )=1\)
- (c)
\(f(-\pi )=1\)
- (d)
\(f(\frac { \Pi }{ 4 } )=2\)
Let {x} and [x] denote the fractional and integral parts of a real number x-respectively, and 4{x}=x+[x],then x=
- (a)
0 or \(\frac { 5 } { 3 }\)
- (b)
1 or \(\frac { 4 } { 3 }\)
- (c)
\(\frac { 3 } { 2}\)
- (d)
\(\frac { 5 } { 4}\)
If \(f(x)=log(\frac { 1+x }{ 1-x } )andg(x)=\frac { 3x+{ x }^{ 3 } }{ 1+3{ x }^{ 2 } } \)then f(g(x))is equal to
- (a)
-f(x)
- (b)
3f(x)
- (c)
\({ (f(x)) }^{ 3 }\)
- (d)
f(3x)
Let \(f(\theta )=\sin { \theta ( } \sin { \theta +\sin { 3\theta ), } } \)then \(f(\theta )\)
- (a)
\(\ge \)0 only when \(\theta\)\(\ge \)0
- (b)
\(\le \)0 for all\(\theta\)
- (c)
\(\ge \)0 for all real \(\theta\)
- (d)
\(\le \)0 only when \(\theta\)\(\le \)0
The value of the function \(f(x)={ cos }^{ 2 }x+{ cos }^{ 2 }(\frac { \pi }{ 3 } +x)-\cos { x\cos { (\frac { \pi }{ 3 } +x) } } \)is:
- (a)
0
- (b)
\(\frac { 3 } { 4 }\)
- (c)
1
- (d)
\(\frac { 4 } { 5}\)
Which of the following functions is an odd function
- (a)
\(f(x)=\sqrt { 1+x+{ x }^{ 2 } } -\sqrt { 1-x+{ x }^{ 2 } } \)
- (b)
f(x)=sinx+cosx
- (c)
f(x)=constant
- (d)
\(f(x)=1+x+{ 2x }^{ 3 }\)
Which of the following functions is periodic?
- (a)
\(f(x)=x+sinx\)
- (b)
\(f(x)=cos\sqrt { x } \)
- (c)
\(f(x)=cos{ x }^{ 2 }\)
- (d)
\(f(x)={ cos }^{ 2 }x\)
The period of the function \(\\ \\ \\ f(x)={ sin }^{ 4 }x+{ cos }^{ 4 }x\)is:
- (a)
\(\frac { \pi } { 4 }\)
- (b)
\(\frac { \pi } { 2 }\)
- (c)
\({ \pi }\)
- (d)
None of these
The period of the function \(f(x)=2\sin { (\frac { x-\pi }{ 3 } ) } \)is:
- (a)
\(\pi\)
- (b)
\(2\pi\)
- (c)
\(4\pi\)
- (d)
\(6\pi\)
The period of the function \(f(x)=3\sin { (4x+3)-4\cos { (8x-3)+\tan { (6x-2) } } } \)is:
- (a)
\(2\pi\)
- (b)
\(\pi\)
- (c)
\(\frac { \pi } { 2 }\)
- (d)
\(\frac { \pi } { 4 }\)
The real valued function \(f(x)=\frac { { a }^{ x }-1 }{ { x }^{ n }({ a }^{ x }+1) } \)is even; then n equals
- (a)
2
- (b)
\(\frac { 2 } { 3 }\)
- (c)
\(\frac { 1 } { 4 }\)
- (d)
\(-\frac { 1 } { 3 }\)
The value of the function \(f(x)=3\sin { (\sqrt { \frac { { \pi }^{ 2 } }{ 16 } -{ x }^{ 2 } } ) } \)lies in the interval
- (a)
\(-\frac { \pi }{ 4 } ,\frac { \pi }{ 4 } \)
- (b)
\((0,\frac { 3 }{ \sqrt { 2 } } )\)
- (c)
(-3,3)
- (d)
None of these
The domain of the function \(f(x)={ sin }^{ -1 }(\log _{ 2 }{ \frac { { x }^{ 2 } }{ 2 } } )\)is:
- (a)
[-2,2]
- (b)
[-2,-1]\(\cup \)[1,2]
- (c)
[-1,1]~{0}
- (d)
None of these
The domain of the function \(f(x)=\sin { [\log { (\frac { \sqrt { 4-{ x }^{ 2 } } }{ 1-x } ) } ] } \)is:
- (a)
(-2,1)
- (b)
(1,2)
- (c)
(2,1)
- (d)
None of these
Range of \(f(x)=\frac { { x }^{ 2 }+x+2 }{ { x }^{ 2 }+x+1 } \)is:
- (a)
\([1,\frac { 5 }{ 2 } ]\)
- (b)
\([1,\frac { 7 }{ 3 } ]\)
- (c)
\([1,\frac { 9 }{ 2 } ]\)
- (d)
\([1,\frac { 11 }{ 2 } ]\)
Which of the following functions is inverse of itself:
- (a)
\(f(x)=\frac { 1-x }{ 1+x } \)
- (b)
\(f(x)={ 5 }^{ \log { x } }\)
- (c)
\({ 2 }^{ x }(x-1)\)
- (d)
None of these
\(\underset { x->0 }{ lim } \frac { \sqrt { \frac { 1-cos2x }{ 2 } } }{ x } \)is:
- (a)
1
- (b)
-1
- (c)
0
- (d)
None of these
For \(x\epsilon R,\underset { x->\infty }{ lim } ({ \frac { x-3 }{ x+2 } ) }^{ x }=\)
- (a)
e
- (b)
\({ e } ^ { -1 }\)
- (c)
\({ e } ^ { -5 }\)
- (d)
\({ e } ^ { 5 }\)
If \(\underset { x->0 }{ lim } ({ \frac { \log { (3+x)-\log { (3-x) } } }{ x } ) }=k\) the value of k is
- (a)
0
- (b)
\(\frac { 2 } { 3 }\)
- (c)
\(-\frac { 2 } { 3 }\)
- (d)
\(-\frac { 1 } { 3 }\)
\(\underset { x->0 }{ lim } ({ \frac { \sin { (\pi { cos }^{ 2 }x) } }{ { x }^{ 2 } } ) }\)equals
- (a)
\(-\pi\)
- (b)
\(\pi\)
- (c)
\(\frac {\pi } { 2 }\)
- (d)
1
\(\underset { x->0 }{ lim } ({ \frac { { (1+x) }^{ 1/x }-e }{ { x } } ) }\)equals
- (a)
\(-\frac { e } { 2}\)
- (b)
e
- (c)
\(\frac { e } { 2 }\)
- (d)
None of these
Let \(f\left( x \right) =x\sin { \left( \frac { 1 }{ x } \right) } ;x\neq 0\) = 0; x = 0 Then \(f\left( x \right) \) is
- (a)
continuous at x = 0
- (b)
discontinuous at x = 0
- (c)
differential at x = 0
- (d)
NONE OF THESE
The derivative with respect to x of the function \(\tan { ^{ -1 }{ \left( \frac { \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } } \right) } } \) is
- (a)
\(\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } \)
- (b)
\(\frac { 1 }{ \sqrt { 1-{ x }^{ 4 } } } \)
- (c)
\(-\frac { 1 }{ \sqrt { 1-{ x }^{ 4 } } } \)
- (d)
NONE OF THESE
\(\underset { x->0 }{ lim } (\frac { x\cos { x-\sin { x } } }{ { x }^{ 2 }\sin { x } } )\)equals
- (a)
\(\frac { 1 } { 3 }\)
- (b)
\(\frac { 2 } { 3 }\)
- (c)
\(-\frac { 1 } { 3 }\)
- (d)
None of these
\(\underset { x->0 }{ lim } \frac { { e }^{ 1/x }-1 }{ { e }^{ 1/x }+1 } ,x\neq 0,\)equals
- (a)
0
- (b)
-1
- (c)
1
- (d)
does not exist
If \(y=f\left( \frac { 2x-1 }{ { x }^{ 2 }+1 } \right) \) and \(f^{ ' }\left( x \right) =\sin { { x }^{ 2 } } \) then \(\frac { dy }{ dx } \) at x = 0 equals
- (a)
\(\frac { 1 }{ 2 } \) sin 1
- (b)
sin 1
- (c)
2 sin 1
- (d)
NONE OF THESE
The differential coefficient of \(\tan ^{ -1 }{ \left( \frac { \sin { x } +\cos { x } }{ \cos { x } -\sin { x } } \right) } \) w.r.t. x is
- (a)
0
- (b)
\(\frac { 1 }{ 2 } \)
- (c)
1
- (d)
NONE OF THESE
If \(\sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ y }^{ 2 } } =a\left( x-y \right) \), then \(\frac { dy }{ dx } \) equals
- (a)
\(\sqrt { \left( 1+{ x }^{ 2 } \right) \left( 1-{ y }^{ 2 } \right) } \)
- (b)
\(\frac { \sqrt { 1-{ y }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } } \)
- (c)
\(\frac { \sqrt { 1+{ x }^{ 2 } } }{ \sqrt { 1-{ y }^{ 2 } } } \)
- (d)
NONE OF THESE
The integer n for which \(\underset { x->0 }{ lim } \frac { (cosx-1)(cosx-{ e }^{ x }) }{ { x }^{ n } } \), is a finite nm-zero number is,
- (a)
4
- (b)
3
- (c)
2
- (d)
1
The derivative of \(\tan ^{ -1 }{ \left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) } \) w.r.t. \(\sin ^{ -1 }{ \left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) } \) is
- (a)
0
- (b)
1
- (c)
2
- (d)
x
\(\underset { x->0 }{ lim } (\frac { { a }^{ x }+{ b }^{ x }+{ c }^{ x } }{ 3 } )\)equals
- (a)
\(( abc ) ^ { 3 }\)
- (b)
3(abc)
- (c)
a+b+c
- (d)
\({ (abc) } ^ { \frac { 1 } { 3 } }\)
\(\underset { h->0 }{ lim } \frac { f(2h+2+{ h }^{ 2 })-f(2) }{ f(h-{ h }^{ 2 }+1)-f(1) } \),given that \(f^{ ' }\left( 2 \right) =6\), and \(f^{ ' }\left( 1 \right) =4\)
- (a)
Does not exist
- (b)
is equal to \(-\frac { 3 } { 2 }\)
- (c)
is equal to \(\frac { 3 } { 2 }\)
- (d)
is equal to 3
If \(G\left( x \right) =-\sqrt { 25-{ x }^{ 2 } } \), then value of \(\underset { x\rightarrow 1 }{ lim } \left[ \frac { G\left( x \right) -G\left( 1 \right) }{ x-1 } \right] \) is
- (a)
1
- (b)
not defined
- (c)
\(\frac { 1 }{ 2\sqrt { 6 } } \)
- (d)
NONE OF THESE
The derivative of the function \(\sin ^{ -1 }{ \left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) } \) at x = 1 is
- (a)
1
- (b)
-1
- (c)
0
- (d)
NONE OF THESE
A function f(x) is defined as follows \(f(x)=\frac { \sin { (a+1)x+\sin { x } } }{ x } ;forx<0\quad f(x)=c;forx=0=\frac { \sqrt { x+bxz } }{ bx\sqrt { y } } \);for x>0 If f(x) is continuous at x=0, then
- (a)
\(a=\frac { -3 } { 2 },b=0,c=\frac { 1 } { 2 }\)
- (b)
\(a=\frac { -3 } { 2 },b=1,c=\frac { -1 } { 2 }\)
- (c)
\(a=\frac { -3 } { 2 },c=\frac { 1 } { 2 },b\)can have any value
- (d)
None of these
If y = (ax + b)m then yn equals (m > n)
- (a)
m! (ax + b)m-n an
- (b)
\(\frac { m!{ a }^{ x } }{ \left( m-n \right) ! } { \left( ax+b \right) }^{ m-n }\)
- (c)
n! (ax + b)m-n an
- (d)
NONE OF THESE
The equation of one of the tangents to the curve that is parallel to \(y=\cos { \left( x+y \right) } ,-2\pi \le x\le 2\pi \) the line x + 2y = 0, is
- (a)
x + 2y = 1
- (b)
x + 2y =\(\frac { \pi }{ 2 } \)
- (c)
x + 2y =\(\frac { \pi }{ 4 } \)
- (d)
NONE OF THESE
The normal to the curve \(x=a\left( t+\sin { t } \right) ,y=a\left( 1-\cos { t } \right) ,\left( 0
- (a)
2\(\pi \)
- (b)
\(\pi \)
- (c)
\(\frac { \pi }{ 2 } \)
- (d)
0
The angle of intersection of the curves x2 + y2 = 2a2 and x2 - y2 = a2 is
- (a)
30\(°\)
- (b)
45\(°\)
- (c)
60\(°\)
- (d)
90\(°\)
In a curve the product of the subtangent and subnormal at any point (x, y) equals
- (a)
x
- (b)
x2
- (c)
y
- (d)
y2
The length of the longest interval in which the function \(3\sin { x } -4\sin ^{ 3 }{ x } \), is increasing, is
- (a)
\(\frac { \pi }{ 3 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\frac { 3\pi }{ 3 } \)
- (d)
\( \pi \)
The function \(f\left( x \right) \) = x - [x], where [ ] denotes the greatest integer function, on the interval [0, 1), is
- (a)
strictly increasing
- (b)
strictly decreasing
- (c)
neither increasing nor decreasing
- (d)
monotonically increasing
The interval in which the function \(f\left( x \right) \) = 2x3 - 15x2 + 36x + 1, decreases, is
- (a)
[1, 2]
- (b)
[- 2, - 1]
- (c)
[2, 3]
- (d)
NONE OF THESE
Rolle's theorem is not applicable to the function \(f\left( x \right) \) = |sin x| in the interval because
- (a)
\(f\left( x \right) \) is not continuous in \(\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right] \)
- (b)
\(f\left( x \right) \) is not differentiable in \(\left[ -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right] \)
- (c)
\(f\left( -\frac { \pi }{ 2 } \right) \neq f\left( \frac { \pi }{ 2 } \right) \)
- (d)
NONE OF THESE
The value of C for Rolle's theorem for the function \(f\left( x \right) ={ \left( x-a \right) }^{ m }{ \left( x-b \right) }^{ m }\) in [a, b], is
- (a)
mb + na
- (b)
\(\frac { 1 }{ a+b } \)
- (c)
\(\frac { na+mb }{ m+n } \)
- (d)
\(\frac { a+b }{ 2 } \)
It is given that \(f\left( x \right) \)= x3 + bx2 + ax + 6 on the interal [1, 3] satisfies the Rolle's theorem for \(c=\frac { 2\sqrt { 3 } +1 }{ \sqrt { 3 } } \). The value of a and b are
- (a)
a = 11, b = 6
- (b)
a = - 11, b = 6
- (c)
a = 11, b = - 6
- (d)
NONE OF THESE
The function \(f\left( x \right) \) = x2 - 6x + 1, satifies the condition of L.M.V theorem. The coordinates of the point at which the tangent is parallel to the chord joining A(1, - 4) and B(3, - 8) are
- (a)
(2, 7)
- (b)
(2, - 7)
- (c)
(1, 7)
- (d)
(1, - 7)
The function \(f\left( x \right) ={ x }^{ 2 }\log { x } \), in the [1, e] has
- (a)
a point of maximum
- (b)
a point of minimum
- (c)
points of maximum as well as minimum
- (d)
neither a point of maximum nor minimum
If the function \(f\left( x \right) =2{ x }^{ 3 }-9a{ x }^{ 2 }+12{ a }^{ 2 }x+1\), where a > 0, attains its maximum and minimum at p and q respectively such that p2 = q, then a equals
- (a)
3
- (b)
1
- (c)
2
- (d)
\(\frac {1}{2}\)
Let \(f\left( x \right) ={ x }^{ 2 }-2bx+2{ c }^{ 2 }\) and \(g\left( x \right) ={ -x }^{ 2 }-2cx+{ b }^{ 2 }\); If the minimum value of \(f\left( x \right) \) is always greater than maximum value of \(g\left( x \right) \), then
- (a)
\(\left| c \right| >\sqrt { 3 } \left| b \right| \)
- (b)
\(\left| c \right| >\sqrt { 2 } \left| b \right| \)
- (c)
\(\left| c \right| <\sqrt { 3 } \left| b \right| \)
- (d)
\(\left| c \right| <\sqrt { 2b } \)