Let $f: \{1,2,3,4\} \to \{1,2,3,4\}$ and $g: \{1,2,3,4\} \to \{1,2,3,4\}$ be invertible functions such that $f \circ g = $ identity. Then
(A) $f = g^{-1}$
(B) $g = f^{-1}$
(C) $f \circ g \neq g \circ f$
(D) $f \circ g = g \circ f$

If the function $f(x)=x^{3}+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is

Let $f$ be a real-valued function defined on the interval $( - 1,1 )$ such that $e ^ { - x } f ( x ) = 2 + \int _ { 0 } ^ { x } \sqrt { t ^ { 4 } + 1 } d t$, for all $x \in ( - 1,1 )$, and let $f ^ { - 1 }$ be the inverse function of $f$. Then $\left( f ^ { - 1 } \right) ^ { \prime } ( 2 )$ is equal to
A) 1
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 1 } { e }$

If $$\lim _ { x \rightarrow 0 } \left[ 1 + x \ln \left( 1 + b ^ { 2 } \right) \right] ^ { \frac { 1 } { x } } = 2 b \sin ^ { 2 } \theta , b > 0 \text { and } \theta \in ( - \pi , \pi ]$$ then the value of $\theta$ is
(A) $\pm \frac { \pi } { 4 }$
(B) $\pm \frac { \pi } { 3 }$
(C) $\pm \frac { \pi } { 6 }$
(D) $\pm \frac { \pi } { 2 }$

Let $f ( x ) = x ^ { 2 }$ and $g ( x ) = \sin x$ for all $x \in \mathbb { R }$. Then the set of all $x$ satisfying $( f \circ g \circ g \circ f ) ( x ) = ( g \circ g \circ f ) ( x )$, where $( f \circ g ) ( x ) = f ( g ( x ) )$, is
(A) $\pm \sqrt { n \pi } , n \in \{ 0,1,2 , \ldots \}$
(B) $\pm \sqrt { n \pi } , n \in \{ 1,2 , \ldots \}$
(C) $\frac { \pi } { 2 } + 2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$
(D) $2 n \pi , n \in \{ \ldots , - 2 , - 1,0,1,2 , \ldots \}$

Let $f : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathbb{R}$ be given by $$f(x) = (\log(\sec x + \tan x))^3$$ Then
(A) $f(x)$ is an odd function
(B) $f(x)$ is a one-one function
(C) $f(x)$ is an onto function
(D) $f(x)$ is an even function

Let $f_1 : \mathbb{R} \rightarrow \mathbb{R}$, $f_2 : [0,\infty) \rightarrow \mathbb{R}$, $f_3 : \mathbb{R} \rightarrow \mathbb{R}$ and $f_4 : \mathbb{R} \rightarrow [0,\infty)$ be defined by
$$f_1(x) = \begin{cases} |x| & \text{if } x < 0 \\ e^x & \text{if } x \geq 0 \end{cases}$$
$$f_2(x) = x^2;$$
$$f_3(x) = \begin{cases} \sin x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$
and
$$f_4(x) = \begin{cases} f_2(f_1(x)) & \text{if } x < 0 \\ f_2(f_1(x)) - 1 & \text{if } x \geq 0 \end{cases}$$
List I (functions) P. $f_4$ is Q. $f_3$ is R. $f_2 \circ f_1$ is S. $f_2$ is
List II (properties)
1. onto but not one-one
2. neither continuous nor one-one
3. differentiable but not one-one
4. continuous and one-one
P Q R S
(A) 3142
(B) 1342
(C) 3124
(D) 1324

Let $g : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function with $g ( 0 ) = 0 , g ^ { \prime } ( 0 ) = 0$ and $g ^ { \prime } ( 1 ) \neq 0$. Let $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { | x | } g ( x ) , & x \neq 0 \\ 0 , & x = 0 \end{array} \right.$$ and $h ( x ) = e ^ { | x | }$ for all $x \in \mathbb { R }$. Let $( f \circ h ) ( x )$ denote $f ( h ( x ) )$ and $( h \circ f ) ( x )$ denote $h ( f ( x ) )$. Then which of the following is (are) true?
(A) $f$ is differentiable at $x = 0$
(B) $\quad h$ is differentiable at $x = 0$
(C) $f \circ h$ is differentiable at $x = 0$
(D) $h \circ f$ is differentiable at $x = 0$

Let $f:\mathbb{R} \rightarrow \mathbb{R}, g:\mathbb{R} \rightarrow \mathbb{R}$ and $h:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $f(x) = x^3 + 3x + 2, g(f(x)) = x$ and $h(g(g(x))) = x$ for all $x \in \mathbb{R}$. Then
(A) $g'(2) = \frac{1}{15}$
(B) $h'(1) = 666$
(C) $h(0) = 16$
(D) $h(g(3)) = 36$

Let $[x]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f(x) = x\cos(\pi(x + [x]))$ is discontinuous?
[A] $x = -1$
[B] $x = 0$
[C] $x = 1$
[D] $x = 2$

If the function $f : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $f ( x ) = | x | ( x - \sin x )$, then which of the following statements is TRUE?
(A) $f$ is one-one, but NOT onto
(B) $f$ is onto, but NOT one-one
(C) $f$ is BOTH one-one and onto
(D) $f$ is NEITHER one-one NOR onto

Let the functions $f: (-1,1) \rightarrow \mathbb{R}$ and $g: (-1,1) \rightarrow (-1,1)$ be defined by $$f(x) = |2x - 1| + |2x + 1| \quad \text{and} \quad g(x) = x - [x],$$ where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ g: (-1,1) \rightarrow \mathbb{R}$ be the composite function defined by $(f \circ g)(x) = f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT differentiable. Then the value of $c + d$ is $\_\_\_\_$

Let $S = ( 0,1 ) \cup ( 1,2 ) \cup ( 3,4 )$ and $T = \{ 0,1,2,3 \}$. Then which of the following statements is(are) true?
(A) There are infinitely many functions from $S$ to $T$
(B) There are infinitely many strictly increasing functions from $S$ to $T$
(C) The number of continuous functions from $S$ to $T$ is at most 120
(D) Every continuous function from $S$ to $T$ is differentiable

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that $f ( x + y ) = f ( x ) + f ( y )$ for all $x , y \in \mathbb { R }$, and $g : \mathbb { R } \rightarrow ( 0 , \infty )$ be a function such that $g ( x + y ) = g ( x ) g ( y )$ for all $x , y \in \mathbb { R }$. If $f \left( \frac { - 3 } { 5 } \right) = 12$ and $g \left( \frac { - 1 } { 3 } \right) = 2$, then the value of $\left( f \left( \frac { 1 } { 4 } \right) + g ( - 2 ) - 8 \right) g ( 0 )$ is $\_\_\_\_$ .

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by
$$f ( x ) = \left\{ \begin{array} { l l } x | x | \sin \left( \frac { 1 } { x } \right) , & x \neq 0 , \\ 0 , & x = 0 , \end{array} \quad \text { and } \quad g ( x ) = \begin{cases} 1 - 2 x , & 0 \leq x \leq \frac { 1 } { 2 } \\ 0 , & \text { otherwise } \end{cases} \right.$$
Let $a , b , c , d \in \mathbb { R }$. Define the function $h : \mathbb { R } \rightarrow \mathbb { R }$ by
$$h ( x ) = a f ( x ) + b \left( g ( x ) + g \left( \frac { 1 } { 2 } - x \right) \right) + c ( x - g ( x ) ) + d g ( x ) , x \in \mathbb { R }$$
Match each entry in List-I to the correct entry in List-II.
List-I
(P) If $a = 0 , b = 1 , c = 0$, and $d = 0$, then
(Q) If $a = 1 , b = 0 , c = 0$, and $d = 0$, then
(R) If $a = 0 , b = 0 , c = 1$, and $d = 0$, then
(S) If $a = 0 , b = 0 , c = 0$, and $d = 1$, then
List-II
(1) $h$ is one-one.
(2) $h$ is onto.
(3) $h$ is differentiable on $\mathbb { R }$.
(4) the range of $h$ is $[ 0,1 ]$.
(5) the range of $h$ is $\{ 0,1 \}$.
The correct option is
(A) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (3)$
(C) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (2)$, $(\mathrm{S}) \rightarrow (4)$
(D) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (3)$

Let $\mathbb { N }$ denote the set of all natural numbers, and $\mathbb { Z }$ denote the set of all integers. Consider the functions $f : \mathbb { N } \rightarrow \mathbb { Z }$ and $g : \mathbb { Z } \rightarrow \mathbb { N }$ defined by
$$f ( n ) = \begin{cases} ( n + 1 ) / 2 & \text { if } n \text { is odd } \\ ( 4 - n ) / 2 & \text { if } n \text { is even } \end{cases}$$
and
$$g ( n ) = \begin{cases} 3 + 2 n & \text { if } n \geq 0 \\ - 2 n & \text { if } n < 0 \end{cases}$$
Define $( g \circ f ) ( n ) = g ( f ( n ) )$ for all $n \in \mathbb { N }$, and $( f \circ g ) ( n ) = f ( g ( n ) )$ for all $n \in \mathbb { Z }$.
Then which of the following statements is (are) TRUE?

(A)	$g \circ f$ is NOT one-one and $g \circ f$ is NOT onto
(B)	$f \circ g$ is NOT one-one but $f \circ g$ is onto
(C)	$g$ is one-one and $g$ is onto
(D)	$f$ is NOT one-one but $f$ is onto

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow ( 0,4 )$ be functions defined by
$$f ( x ) = \log _ { e } \left( x ^ { 2 } + 2 x + 4 \right) , \text { and } g ( x ) = \frac { 4 } { 1 + e ^ { - 2 x } }$$
Define the composite function $f \circ g ^ { - 1 }$ by $\left( f \circ g ^ { - 1 } \right) ( x ) = f \left( g ^ { - 1 } ( x ) \right)$, where $g ^ { - 1 }$ is the inverse of the function $g$.
Then the value of the derivative of the composite function $f \circ g ^ { - 1 }$ at $x = 2$ is $\_\_\_\_$.

The function $f : R \sim \{ 0 \} \rightarrow R$ 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
(1) 2
(2) - 1
(3) 0
(4) 1

The largest interval lying in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ for which the function $\left[ f ( x ) = 4 ^ { - x ^ { 2 } } + \cos ^ { - 1 } \left( \frac { x } { 2 } - 1 \right) + \log ( \cos x ) \right]$ is defined, is
(1) $[ 0 , \pi ]$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
(3) $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(4) $\left[ 0 , \frac { \pi } { 2 } \right)$

If $g(x) = x^{2} + x - 2$ and $\frac{1}{2}\,g\circ f(x) = 2x^{2} - 5x + 2$, then $f(x)$ is equal to
(1) $2x-3$
(2) $2x+3$
(3) $2x^{2}+3x+1$
(4) $2x^{2}-3x-1$

Let $A$ and $B$ be non empty sets in $\mathbb{R}$ and $f : A \rightarrow B$ is a bijective function. Statement 1: $f$ is an onto function. Statement 2: There exists a function $g : B \rightarrow A$ such that $f \circ g = I _ { B }$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is false, Statement 2 is true.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.

The range of the function $f ( x ) = \frac { x } { 1 + | x | } , x \in R$, is
(1) $R$
(2) $( - 1,1 )$
(3) $R - \{ 0 \}$
(4) $[ - 1,1 ]$

Let $f : R \rightarrow R$ be defined by $f ( x ) = \frac { | x | - 1 } { | x | + 1 }$, then $f$ is
(1) one-one but not onto
(2) neither one-one nor onto
(3) both one-one and onto
(4) onto but not one-one

If $g$ is the inverse of a function $f$ and $f ^ { \prime } ( x ) = \frac { 1 } { 1 + x ^ { 5 } }$, then $g ^ { \prime } ( x )$ is equal to
(1) $\frac { 1 } { 1 + \{ g ( x ) \} ^ { 5 } }$
(2) $1 + \{ g ( x ) \} ^ { 5 }$
(3) $1 + x ^ { 5 }$
(4) $5 x ^ { 4 }$

Let $f(x) = x^2$, $g(x) = \sin x$ for all $x \in \mathbb{R}$ and $h(x) = (gof)(x) = g(f(x))$. Statement I: $h$ is not differentiable at $x = 0$. Statement II: $(hog)(x) = \sin^2(\sin x)$. Which of the following is correct?
(1) Statement I is false, Statement II is true
(2) Statement I is true, Statement II is false
(3) Both Statement I and Statement II are true
(4) Both Statement I and Statement II are false