For some $a , b , c \in \mathbb { N }$, let $f ( x ) = a x - 3$ and $g ( x ) = x ^ { b } + c , x \in \mathbb { R }$. If $( f \circ g ) ^ { - 1 } ( x ) = \left( \frac { x - 7 } { 2 } \right) ^ { \frac { 1 } { 3 } }$, then $( f \circ g ) ( a c ) + ( g \circ f ) ( b )$ is equal to $\_\_\_\_$ .

Consider a function $\mathrm { f } : \mathbb { N } \rightarrow \mathbb { R }$, satisfying $f ( 1 ) + 2 f ( 2 ) + 3 f ( 3 ) + \ldots + x f ( x ) = x ( x + 1 ) f ( x ) ; x \geq 2$ with $f ( 1 ) = 1$. Then $\frac { 1 } { f ( 2022 ) } + \frac { 1 } { f ( 2028 ) }$ is equal to (1) 8200 (2) 8000 (3) 8400 (4) 8100

Let $f(x) = \begin{cases} x-1, & x \text{ is even,} \\ 2x, & x \text{ is odd,} \end{cases}$ $x \in \mathbb{N}$. If for some $a \in \mathbb{N}$, $f(f(f(a))) = 21$, then $\lim_{x \to a^-} \left(\frac{x^3}{a} - \left\lfloor\frac{x}{a}\right\rfloor\right)$, where $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$, is equal to:
(1) 121
(2) 144
(3) 169
(4) 225

If the domain of the function $f(x) = \log_e\frac{2x+3}{4x^2+x-3} + \cos^{-1}\frac{2x-1}{x+2}$ is $(\alpha, \beta]$, then the value of $5\beta - 4\alpha$ is equal to
(1) 10
(2) 12
(3) 11
(4) 9

If $f ( x ) = \left\{ \begin{array} { l } 2 + 2 x , - 1 \leq x < 0 \\ 1 - \frac { x } { 3 } , 0 \leq x \leq 3 \end{array} ; g ( x ) = \left\{ \begin{array} { l } - x , - 3 \leq x \leq 0 \\ x , 0 < x \leq 1 \end{array} \right. \right.$, then range of $( f \circ g ( x ) )$ is
(1) $( 0,1 ]$
(2) $[ 0,3 )$
(3) $[ 0,1 ]$
(4) $[ 0,1 )$

Let $\mathrm { f } : \mathrm { R } - \frac { - 1 } { 2 } \rightarrow \mathrm { R }$ and $\mathrm { g } : \mathrm { R } - \frac { - 5 } { 2 } \rightarrow \mathrm { R }$ be defined as $\mathrm { fx } = \frac { 2 \mathrm { x } + 3 } { 2 \mathrm { x } + 1 }$ and $\mathrm { gx } = \frac { | \mathrm { x } | + 1 } { 2 \mathrm { x } + 5 }$. Then the domain of the function fog is :
(1) $\mathrm { R } - - \frac { 5 } { 2 }$
(2) $R$
(3) $R - \frac { 1 } { 4 }$
(4) $\mathrm { R } - - \frac { 5 } { 2 } , - \frac { 7 } { 4 }$

Let $f , g : \mathbf { R } \rightarrow \mathbf { R }$ be defined as : $f ( x ) = | x - 1 |$ and $g ( x ) = \begin{cases} \mathrm { e } ^ { x } , & x \geq 0 \\ x + 1 , & x \leq 0 \end{cases}$ Then the function $f ( g ( x ) )$ is
(1) neither one-one nor onto.
(2) one-one but not onto.
(3) onto but not one-one.
(4) both one-one and onto.

Let $f ( x ) = \left\{ \begin{array} { c c c } - \mathrm { a } & \text { if } & - \mathrm { a } \leq x \leq 0 \\ x + \mathrm { a } & \text { if } & 0 < x \leq \mathrm { a } \end{array} \right.$ where $\mathrm { a } > 0$ and $\mathrm { g } ( x ) = ( f ( x \mid ) - | f ( x ) | ) / 2$. Then the function $g : [ - a , a ] \rightarrow [ - a , a ]$ is (1) neither one-one nor onto. (2) onto. (3) both one-one and onto. (4) one-one.

If the domain of the function $f(x) = \frac{\sqrt{x^2 - 25}}{4 - x^2} + \log_{10}(x^2 + 2x - 15)$ is $(-\infty, \alpha) \cup (\beta, \infty)$, then $\alpha^2 + \beta^3$ is equal to:
(1) 140
(2) 175
(3) 150
(4) 125

If the domain of the function $f ( x ) = \cos ^ { - 1 } \left( \frac { 2 - | x | } { 4 } \right) + \left( \log _ { e } ( 3 - x ) \right) ^ { - 1 }$ is $[ - \alpha , \beta ) - \{ \gamma \}$, then $\alpha + \beta + \gamma$ is equal to :
(1) 12
(2) 9
(3) 11
(4) 8

If $f(x) = \frac { 4 x + 3 } { 6 x - 4 } , \quad x \neq \frac { 2 } { 3 }$ and $(f \circ f)(x) = g(x)$, where $g : \mathbb{R} - \left\{\frac { 2 } { 3 }\right\} \rightarrow \mathbb{R} - \left\{\frac { 2 } { 3 }\right\}$, then $(g \circ g \circ g)(4)$ is equal to
(1) $- \frac { 19 } { 20 }$
(2) $\frac { 19 } { 20 }$
(3) $-4$
(4) 4

The function $f : \mathbb{R} \to \mathbb{R}$, $f ( x ) = \frac { x ^ { 2 } + 2 x - 15 } { x ^ { 2 } - 4 x + 9 } , x \in \mathbb { R }$ is
(1) one-one but not onto.
(2) both one-one and onto.
(3) onto but not one-one.
(4) neither one-one nor onto.

Consider the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { c l } \frac { \mathrm { a } \left( 7 \mathrm { x } - 12 - \mathrm { x } ^ { 2 } \right) } { \mathrm { b } \left| \mathrm { x } ^ { 2 } - 7 \mathrm { x } + 12 \right| } & , \mathrm { x } < 3 \\ 2 ^ { \frac { \sin ( \mathrm { x } - 3 ) } { \mathrm { x } - [ \mathrm { x } ] } } & , \mathrm { x } > 3 \\ \mathrm {~b} & , \mathrm { x } = 3 \end{array} \right.$, where $[ \mathrm { x } ]$ denotes the greatest integer less than or equal to x. If S denotes the set of all ordered pairs $( \mathrm { a } , \mathrm { b } )$ such that $\mathrm { f } ( \mathrm { x } )$ is continuous at $x = 3$, then the number of elements in S is:
(1) 2
(2) Infinitely many
(3) 4
(4) 1

Let $[ x ]$ denote the greatest integer less than or equal to $x$. Then the domain of $f ( x ) = \sec ^ { - 1 } ( 2 [ x ] + 1 )$ is :
(1) $( - \infty , - 1 ] \cup [ 0 , \infty )$
(2) $( - \infty , - 1 ] \cup [ 1 , \infty )$
(3) $( - \infty , \infty )$
(4) $( - \infty , \infty ) - \{ 0 \}$

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $\mathrm{m} + \mathrm{n}$ is equal to:
(1) 6
(2) 8
(3) 9
(4) 7

Let $f : [ 0,3 ] \rightarrow \mathrm { A }$ be defined by $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 7$ and $g : [ 0 , \infty ) \rightarrow B$ be defined by $\mathrm { g } ( x ) = \frac { x ^ { 2025 } } { x ^ { 2025 } + 1 }$. If both the functions are onto and $\mathrm { S } = \{ x \in \mathbf { Z } : x \in \mathrm {~A}$ or $x \in \mathrm {~B} \}$, then $\mathrm { n } ( \mathrm { S } )$ is equal to :
(1) 29
(2) 30
(3) 31
(4) 36

Let $f ( x ) = \log _ { \mathrm { e } } x$ and $g ( x ) = \frac { x ^ { 4 } - 2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 2 } { 2 x ^ { 2 } - 2 x + 1 }$. Then the domain of $f \circ g$ is
(1) $[ 0 , \infty )$
(2) $[ 1 , \infty )$
(3) $( 0 , \infty )$
(4) $\mathbb { R }$

Let $\mathrm { A } = \{ 1,2,3,4 \}$ and $\mathrm { B } = \{ 1,4,9,16 \}$. Then the number of many-one functions $f : \mathrm { A } \rightarrow \mathrm { B }$ such that $1 \in f ( \mathrm {~A} )$ is equal to :
(1) 151
(2) 139
(3) 163
(4) 127

Let $f : \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}$. If $\lim_{x \rightarrow 0}\left(\frac{1}{\alpha x} + f(x)\right) = \beta$; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to
(1) 5
(2) 3
(3) 4
(4) 6

The function $f : (-\infty, \infty) \rightarrow (-\infty, 1)$, defined by $f(x) = \frac{2^{x} - 2^{-x}}{2^{x} + 2^{-x}}$ is:
(1) Neither one-one nor onto
(2) Onto but not one-one
(3) Both one-one and onto
(4) One-one but not onto

Q67. Let $f ( x ) = x ^ { 2 } + 9 , g ( x ) = \frac { x } { x - 9 }$ and $\mathrm { a } = f \circ g ( 10 ) , \mathrm { b } = g \circ f ( 3 )$. If e and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$, then $8 \mathrm { e } ^ { 2 } + l ^ { 2 }$ is equal to.
(1) 8
(2) 16
(3) 6
(4) 12

Q70. If the domain of the function $f ( x ) = \sin ^ { - 1 } \left( \frac { x - 1 } { 2 x + 3 } \right)$ is $\mathbf { R } - ( \alpha , \beta )$, then $12 \alpha \beta$ is equal to :
(1) 32
(2) 40
(3) 24
(4) 36

Q71. If the domain of the function $\sin ^ { - 1 } \left( \frac { 3 x - 22 } { 2 x - 19 } \right) + \log _ { \mathrm { e } } \left( \frac { 3 x ^ { 2 } - 8 x + 5 } { x ^ { 2 } - 3 x - 10 } \right)$ is $( \alpha , \beta ]$, then $3 \alpha + 10 \beta$ is equal to:
(1) 100
(2) 95
(3) 97
(4) 98

Q71. Let $f ( x ) = x ^ { 5 } + 2 x ^ { 3 } + 3 x + 1 , x \in \mathbf { R }$, and $g ( x )$ be a function such that $g ( f ( x ) ) = x$ for all $x \in \mathbf { R }$. Then $\frac { g ( 7 ) } { g ^ { \prime } ( 7 ) }$ is equal to :
(1) 14
(2) 42
(3) 7
(4) 1

Q71. Let $f , g : \mathbf { R } \rightarrow \mathbf { R }$ be defined as : $f ( x ) = | x - 1 |$ and $g ( x ) = \begin{cases} \mathrm { e } ^ { x } , \text { MARA } & x \geq 0 \\ x + 1 , & x \leq 0 \end{cases}$ Then the function $f ( g ( x ) )$ is
(1) neither one-one nor onto.
(2) one-one but not onto.
(3) onto but not one-one.
(4) both one-one and onto.