Consider function $f : A \rightarrow B$ and $g : B \rightarrow C ( A , B , C \subseteq R )$ such that $( g o f ) ^ { - 1 }$ exists, then:
(1) $f$ and $g$ both are one-one
(2) $f$ and $g$ both are onto
(3) $f$ is one-one and $g$ is onto
(4) $f$ is onto and $g$ is one-one

If the function $f ( x ) = \begin{cases} \frac { 1 } { x } \log _ { \mathrm { e } } \left( \frac { 1 + \frac { x } { a } } { 1 - \frac { x } { b } } \right) & , x < 0 \\ k & , x = 0 \\ \frac { \cos ^ { 2 } x - \sin ^ { 2 } x - 1 } { \sqrt { x ^ { 2 } + 1 } - 1 } & , x > 0 \end{cases}$ is continuous at $x = 0$, then $\frac { 1 } { a } + \frac { 1 } { b } + \frac { 4 } { k }$ is equal to:
(1) 4
(2) 5
(3) $- 4$
(4) $- 5$

The function $f : R \rightarrow R$ defined by $f ( x ) = \lim _ { n \rightarrow \infty } \frac { \cos ( 2 \pi x ) - x ^ { 2 n } \sin ( x - 1 ) } { 1 + x ^ { 2 n + 1 } - x ^ { 2 n } }$ is continuous for all $x$ in
(1) $R - \{ - 1 \}$
(2) $R - \{ - 1,1 \}$
(3) $R - \{ 1 \}$
(4) $R - \{ 0 \}$

Let a function $f : \mathbb{N} \rightarrow \mathbb{N}$ be defined by $$f(n) = \begin{cases} 2n, & n = 2,4,6,8,\ldots \\ n-1, & n = 3,7,11,15,\ldots \\ \frac{n+1}{2}, & n = 1,5,9,13,\ldots \end{cases}$$ then, $f$ is
(1) One-one and onto
(2) One-one but not onto
(3) Onto but not one-one
(4) Neither one-one nor onto

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as $f ( x ) = x - 1$ and $g : R \rightarrow \{ 1 , - 1 \} \rightarrow \mathbb { R }$ be defined as $g ( x ) = \frac { x ^ { 2 } } { x ^ { 2 } - 1 }$. Then the function $f o g$ is:
(1) One-one but not onto
(2) onto but not one-one
(3) Both one-one and onto
(4) Neither one-one nor onto

The domain of the function $f(x) = \sin^{-1}\left(\frac{x^2 - 3x + 2}{x^2 + 2x + 7}\right)$ is
(1) $[1, \infty)$
(2) $(-1, 2]$
(3) $[-1, \infty)$
(4) $(-\infty, 2]$

Let $f ( x ) = \frac { x - 1 } { x + 1 } , x \in R - \{ 0 , - 1 , 1 \}$. If $f ^ { n + 1 } ( x ) = f \left( f ^ { n } ( x ) \right)$ for all $n \in N$, then $f ^ { 6 } ( 6 ) + f ^ { 7 } ( 7 )$ is equal to
(1) $\frac { 7 } { 6 }$
(2) $- \frac { 3 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $- \frac { 11 } { 12 }$

If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { \log _ { e } \left( 1 - x + x ^ { 2 } \right) + \log _ { e } \left( 1 + x + x ^ { 2 } \right) } { \sec x - \cos x } , & x \in \left( \frac { - \pi } { 2 } , \frac { \pi } { 2 } \right) - \{ 0 \} \\ k & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then $k$ is equal to:
(1) 1
(2) $- 1$
(3) $e$
(4) 0

The domain of the function $f ( x ) = \sin ^ { - 1 } \left[ 2 x ^ { 2 } - 3 \right] + \log _ { 2 } \left( \log _ { \frac { 1 } { 2 } } \left( x ^ { 2 } - 5 x + 5 \right) \right)$, where $[ t ]$ is the greatest integer function, is
(1) $\left( - \sqrt { \frac { 5 } { 2 } } , \frac { 5 - \sqrt { 5 } } { 2 } \right)$
(2) $\left( \frac { 5 - \sqrt { 5 } } { 2 } , \frac { 5 + \sqrt { 5 } } { 2 } \right)$
(3) $\left( 1 , \frac { 5 - \sqrt { 5 } } { 2 } \right)$
(4) $\left[ 1 , \frac { 5 + \sqrt { } 5 } { 2 } \right)$

The domain of $f ( x ) = \frac { \cos ^ { - 1 } \left( \frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } - 9 } \right) } { \log \left( x ^ { 2 } - 3 x + 2 \right) }$ is
(1) $x \in \left[ \frac { - 1 } { 2 } , 1 \right) \cup ( 2 , \infty ) - \{ 3 \}$
(2) $x \in \left[ \frac { - 1 } { 2 } , 1 \right] \cup ( 2 , \infty ) - \{ 3 \}$
(3) $x \in \left( \frac { - 1 } { 2 } , 1 \right) \cup [ 2 , \infty ) - \{ 3 \}$
(4) $x \in \left[ \frac { - 1 } { 2 } , 1 \right) \cup [ 2 , \infty ) - \{ 3 \}$

Let $f : R \rightarrow R$ be defined as $f(x) = x ^ { 3 } + x - 5$. If $g(x)$ is a function such that $f( g(x) ) = x , \forall x \in R$, then $g ^ { \prime } (63)$ is equal to
(1) 49
(2) $\frac { 1 } { 49 }$
(3) $\frac { 43 } { 49 }$
(4) $\frac { 3 } { 49 }$

Let the function $f(x) = \begin{cases} \frac{\log_e(1 + 5x) - \log_e(1 + \alpha x)}{x} & \text{if } x \neq 0 \\ 10 & \text{if } x = 0 \end{cases}$ be continuous at $x = 0$. Then $\alpha$ is equal to
(1) 10
(2) $-10$
(3) 5
(4) $-5$

The value of $\cot \left( \sum _ { n = 1 } ^ { 50 } \tan ^ { - 1 } \left( \frac { 1 } { 1 + n + n ^ { 2 } } \right) \right)$ is
(1) $\frac { 25 } { 26 }$
(2) $\frac { 50 } { 51 }$
(3) $\frac { 26 } { 25 }$
(4) $\frac { 52 } { 51 }$

If $f ( x ) = \left\{ \begin{array} { l l } x + a , & x \leq 0 \\ | x - 4 | , & x > 0 \end{array} \right.$ and $g ( x ) = \left\{ \begin{array} { l l } x + 1 , & x < 0 \\ ( x - 4 ) ^ { 2 } + b , & x \geq 0 \end{array} \right.$ are continuous on $R$, then $( g \circ f ) ( 2 ) + ( f \circ g ) ( - 2 )$ is equal to:
(1) $- 10$
(2) 10
(3) 8
(4) $- 8$

If for $p \neq q \neq 0$, then function $f ( x ) = \frac { \sqrt [ 7 ] { p ( 729 + x ) } - 3 } { \sqrt [ 3 ] { 729 + q x } - 9 }$ is continuous at $x = 0$, then
(1) $7 p q f ( 0 ) - 1 = 0$
(2) $63 q f ( 0 ) - p ^ { 2 } = 0$
(3) $21 q f ( 0 ) - p ^ { 2 } = 0$
(4) $7 p q f ( 0 ) - 9 = 0$

The domain of the function $\cos ^ { - 1 } \left( \frac { 2 \sin ^ { - 1 } \left( \frac { 1 } { 4 x ^ { 2 } - 1 } \right) } { \pi } \right)$ is
(1) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right] \cup \left[ \frac { 1 } { \sqrt { 2 } } , \infty \right) \cup \{ 0 \}$
(2) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right] \cup \left[ \frac { 1 } { \sqrt { 2 } } , \infty \right)$
(3) $\left( - \infty , - \frac { 1 } { \sqrt { 2 } } \right) \cup \left( \frac { 1 } { 2 } , \infty \right) \cup \{ 0 \}$
(4) $R - \left\{ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right\}$

Let $S = \{ 1,2,3,4,5,6,7,8,9,10 \}$. Define $f : S \rightarrow S$ as $f ( n ) = \left\{ \begin{array} { c l } 2 n , & \text { if } n = 1,2,3,4,5 \\ 2 n - 11 & \text { if } n = 6,7,8,9,10 \end{array} \right.$ Let $g : S \rightarrow S$ be a function such that $f \circ g ( n ) = \left\{ \begin{array} { l l } n + 1 & , \text { if } n \text { is odd } \\ n - 1 & , \text { if } n \text { is even } \end{array} \right.$, then $g ( 10 ) ( g ( 1 ) + g ( 2 ) + g ( 3 ) + g ( 4 ) + g ( 5 ) )$ is equal to

The parabolas: $ax^{2} + 2bx + cy = 0$ and $dx^{2} + 2ex + fy = 0$ intersect on the line $y = 1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then
(1) $d, e, f$ are in A.P.
(2) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in G.P.
(3) $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
(4) $d, e, f$ are in G.P.

If $f(x) = \frac{\tan^{-1}x + \log_e 123}{x \log_e 1234 - \tan^{-1}x}$, $x > 0$, then the least value of $f(f(x)) + f\!\left(f\!\left(\frac{4}{x}\right)\right)$ is
(1) 0
(2) 8
(3) 2
(4) 4

Let $y = f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y = -\frac{1}{2}$. Then $S = \left\{x \in \mathbb{R} : \tan^{-1}\sqrt{f(x)} + \sin^{-1}\sqrt{f(x)+1} = \frac{\pi}{2}\right\}$:
(1) contains exactly two elements
(2) contains exactly one element
(3) is an infinite set
(4) is an empty set

If the domain of the function $f ( x ) = \log _ { e } \left( 4 x ^ { 2 } + 11 x + 6 \right) + \sin ^ { - 1 } ( 4 x + 3 ) + \cos ^ { - 1 } \left( \frac { 10 x + 6 } { 3 } \right)$ is $( \alpha , \beta ]$, then $36 | \alpha + \beta |$ is equal to
(1) 54
(2) 72
(3) 63
(4) 45

If the domain of the function $f(x) = \frac{x}{1+\lfloor x \rfloor^2}$, where $\lfloor x \rfloor$ is greatest integer $\leq x$, is $[2,6)$, then its range is
(1) $\left\{\frac{5}{26}, \frac{2}{5}\right\} \cup \left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
(2) $\left[\frac{5}{26}, \frac{2}{5}\right]$
(3) $\left\{\frac{5}{37}, \frac{2}{5}\right\} \cup \left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}$
(4) $\left[\frac{5}{37}, \frac{2}{5}\right]$

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \frac { 1 } { 1 - e ^ { - x } }$, and $g ( x ) = ( f ( - x ) - f ( x ) )$. Consider two statements (I) $g$ is an increasing function in $( 0,1 )$ (II) $g$ is one-one in $( 0,1 )$ Then,
(1) Only (I) is true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Both (I) and (II) are true

The number of functions $f : \{ 1,2,3,4 \} \rightarrow \{ \mathrm { a } \in \mathbb { Z } : | \mathrm { a } | \leq 8 \}$ satisfying $f ( \mathrm { n } ) + \frac { 1 } { \mathrm { n } } f ( \mathrm { n } + 1 ) = 1 , \forall \mathrm { n } \in \{ 1,2,3 \}$ is
(1) 3
(2) 4
(3) 1
(4) 2

Let $f : R \rightarrow R$ be a function such that $f ( x ) = \frac { x ^ { 2 } + 2 x + 1 } { x ^ { 2 } + 1 }$. Then
(1) $f ( x )$ is many-one in $( - \infty , - 1 )$
(2) $f ( x )$ is many-one in $( 1 , \infty )$
(3) $f ( x )$ is one-one in $[ 1 , \infty )$ but not in $( - \infty , \infty )$
(4) $f ( x )$ is one-one in $( - \infty , \infty )$