IISER Mathematics - Differential Calculus
Exam Duration: 45 Mins Total Questions : 30
A polynomial function f(x) satisfies the equation \(f(x)f(\frac { 1 }{ x } )=f(x)+f(\frac { 1 }{ x } )andf(3)=28\);then value of f(4) is:
- (a)
67
- (b)
65
- (c)
63
- (d)
29
Let g(x) be a function defined on [-1,1].If the area of the equilateral triangle with two of its vertices at (0,0) and [x,g(x)]is \(\frac { \sqrt { 3 } }{ 4 } \),then function g(x), is
- (a)
\(\pm \sqrt { 1-{ x }^{ 2 } } \)
- (b)
either \(\sqrt { 1-{ x }^{ 2 } } \)or -\(\sqrt { 1-{ x }^{ 2 } } \)
- (c)
\(\sqrt {1+{x} ^ {2 }}\)
- (d)
None of these
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 domain of the definition of the function \(f(x)=\sqrt { { sin }^{ -1 }(\log _{ 2 }{ x } ) } +\log _{ 2 }{ (\log _{ 3 }{ (\log _{ 4 }{ x)) } } } \)is:
- (a)
[1,2]
- (b)
\([4,\infty ]\)
- (c)
\([1,\infty ]\)
- (d)
None of these
The domain of the function y(x)given by the equation\({ 2 }^{ x }+{ 2 }^{ y }=2\)is:
- (a)
0<x<1
- (b)
\(0\le x\le 1\)
- (c)
\(-\infty <x<0\)
- (d)
\(-\infty <x<1\)
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 \(f(x)=\frac { 1 }{ x } +{ 2 }^{ { sin }_{ x }^{ -1 } }+\frac { 1 }{ \sqrt { x-2 } } \)is:
- (a)
(-1,1)
- (b)
(1,2)
- (c)
(-2,-1)
- (d)
None of these
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
\(\underset { x->0 }{ lim } ({ \frac { \sin { (\pi { cos }^{ 2 }x) } }{ { x }^{ 2 } } ) }\)equals
- (a)
\(-\pi\)
- (b)
\(\pi\)
- (c)
\(\frac {\pi } { 2 }\)
- (d)
1
\(\underset { x->0 }{ lim } ({ \frac { { (1+x) }^{ 1/x }-e+\frac { ex }{ 2 } }{ { { x }^{ 2 } } } ) }\)equals
- (a)
\(\frac { e } { 24 }\)
- (b)
\(\frac { 5e } { 24 }\)
- (c)
\(\frac { 11e } { 24 }\)
- (d)
\(\frac { 13e } { 24 }\)
The set of all points where the fuction \(f\left( x \right) =\)\(\frac { x }{ 1+\left| x \right| } \), is differentiable is
- (a)
\(\left( -\infty ,\infty \right) \)
- (b)
\(\left( 0,\infty \right) \sim \left\{ 0 \right\} \)
- (c)
\(\left( -\infty ,0 \right) \cup \left( 0,\infty \right) \)
- (d)
\(\left( 0,\infty \right) \)
\(\underset { x->1 }{ lim } f(x),\) where \(f(x)=4x-2\) when x<1 \(\underset { x->1 }{ lim } f(x),\)where\(f(x)=4x-3\) when x>1 equals
- (a)
2
- (b)
1
- (c)
0
- (d)
None of these
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
\(\underset { h->0 }{ lim } \frac { f(2h+2+{ h }^{ 2 })-f(2) }{ f(h-{ h }^{ 2 }+1)-f(1) } \),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 \(f(x)=\sqrt { { x }^{ 2 }+x } +\frac { \tan ^{ 2 }{ \alpha } }{ \sqrt { { x }^{ 2 }+x } } ;\alpha \epsilon (0,\frac { \pi }{ 2 } )\),is always greater than or equal to
- (a)
\(2\tan { \alpha } \)
- (b)
1
- (c)
2
- (d)
\(\sec ^{ 2 }{ \alpha } \)
In order that the function \(f(x)={ (x+1) }^{ \cot { x } }\)is continuous at x=0, f(0) must be defined as
- (a)
f(0)=0
- (b)
f(0)=e
- (c)
f(0)=\(\frac { 1} { e }\)
- (d)
None of these
If \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } =a{ e }^{ \tan ^{ -1 }{ \left( y/x \right) } }\); a > 0, y (0) > 0, then y''(0) equals
- (a)
\(\frac { a }{ 2 } { e }^{ \pi /2 }\)
- (b)
\(a{ e }^{ \pi /2 }\)
- (c)
\(-\frac { 2 }{ a } { e }^{ -\pi /2 }\)
- (d)
\(-\frac { a }{ 2 } { e }^{ -\pi /2 }\)
Let \(f\) be twice differentiable function such that \(f^{ '' }\left( x \right) =-f\left( x \right) \) and \(f^{ ' }\left( x \right) =g\left( x \right) \) and \(h\left( x \right) ={ \left\{ f\left( x \right) \right\} }^{ 2 }+{ \left\{ g\left( x \right) \right\} }^{ 2 }\), h(5) = 11, then h(10) equals
- (a)
0
- (b)
5
- (c)
10
- (d)
11
In the curve \(y=\frac { a }{ 2 } \left( { e }^{ x/a }+{ e }^{ -x/a } \right) \), the length of the portion of the normal intercepted between the curve and x-axis is
- (a)
y
- (b)
\(\frac{y}{a}\)
- (c)
\(\frac { { y }^{ 2 } }{ { a }^{ 2 } } \)
- (d)
\(\frac { { y }^{ 2 } }{ a } \)
If \(f\left( x \right) =x{ e }^{ x\left( 1-x \right) }\), then \(f\left( x \right) \) is
- (a)
increasing on \(\left[ -\frac { 1 }{ 2 } ,1 \right] \)
- (b)
decreasing on R
- (c)
increasing on R
- (d)
decreasing on \(\left[ -\frac { 1 }{ 2 } ,1 \right] \)
The function \(f\left( x \right) =\frac { \log { \left( \pi +x \right) } }{ \log { \left( e+x \right) } } \) in the interval \(\left( 0,\infty \right) \), is always
- (a)
decreasing
- (b)
increasing
- (c)
neither increasing nor decreasing
- (d)
a constant
If \(f\left( x \right) =\frac { x }{ \sin { x } } ,g\left( x \right) =\frac { x }{ \tan { x } } \), when \(0\le x\le 1\), then in this interval
- (a)
both \(f\left( x \right) \) and \(g\left( x \right) \) are increasing functions
- (b)
both \(f\left( x \right) \) and \(g\left( x \right) \) are decreasing functions
- (c)
\(f\left( x \right) \) is an increasing function
- (d)
\(g\left( x \right) \) is an increasing function
The least value of a so that the function \(f\left( x \right) \) = x2 + ax + 1 is increasing on the interval [1, 2], is
- (a)
2
- (b)
- 2
- (c)
1
- (d)
- 1
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
If 2a + 3b + 6c = 0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval
- (a)
(0, 1)
- (b)
(1, 2)
- (c)
(2, 3)
- (d)
NONE OF THESE
If \(f\left( x \right) \) and \(g\left( x \right)\) are differentiable functions for \(0\le x\le 1\) such that \(f\left( 0 \right) =2,g\left( 0 \right) =0,f\left( 1 \right) =6,g\left( 1 \right) =2\) then for 0 < c < 1, the value of \(2g^{ ' }\left( c \right) \) is
- (a)
\(f\left( c \right) \)
- (b)
\(f^{ ' }\left( c \right) \)
- (c)
\(f^{ '' }\left( c \right) \)
- (d)
NONE OF THESE
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 difference between the greatest and least values of the function \(f\left( x \right) =\cos { x } +\frac { 1 }{ 2 } \cos { 2x } -\frac { 1 }{ 3 } \cos { 3x } \), is
- (a)
\(\frac{2}{3}\)
- (b)
\(\frac{8}{7}\)
- (c)
\(\frac{9}{4}\)
- (d)
\(\frac{3}{8}\)