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) $(-\infty, 2]$

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 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) $R - \{ 0 \}$

The range of $f(x) = 4 \sin ^ { - 1 } \left( \frac { x ^ { 2 } } { x ^ { 2 } + 1 } \right)$ is
(1) $[ 0,2 \pi ]$
(2) $[ 0 , \pi ]$
(3) $[ 0,2 \pi )$
(4) $[ 0 , \pi )$

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

The domain of $f ( x ) = \frac { \log _ { ( x + 1 ) } ( x - 2 ) } { e ^ { 2 \log _ { e } x } - ( 2 x + 3 ) } , x \in R$ is
(1) $\mathbb { R } - \{ - 1,3 \}$
(2) $( 2 , \infty ) - \{ 3 \}$
(3) $( - 1 , \infty ) - \{ 3 \}$
(4) $\mathbb { R } - \{ 3 \}$

The range of the function $f(x) = \sqrt{3 - x} + \sqrt{2 + x}$ is
(1) $[\sqrt{5}, \sqrt{10}]$
(2) $[2\sqrt{2}, \sqrt{11}]$
(3) $[\sqrt{5}, \sqrt{13}]$
(4) $[\sqrt{2}, \sqrt{7}]$

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 $D$ be the domain of the function $f ( x ) = \sin ^ { - 1 } \left( \log _ { 3 x } \left( \frac { 6 + 2 \log _ { 3 } x } { - 5 x } \right) \right)$. If the range of the function $g : D \rightarrow \mathbb { R }$ defined by $g ( x ) = x - [ x ]$, ([x] is the greatest integer function), is $( \alpha , \beta )$, then $\alpha ^ { 2 } + \frac { 5 } { \beta }$ is equal to
(1) 135
(2) 45
(3) 46
(4) 136

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 $f ( x ) = \left\{ \begin{array} { l } 2 + 2 x , - 1 \leq x < 0 \\ 1 - \frac { x } { 3 } , 0 \leq x \leq 3 \end{array} ; g ( x ) = \left\{ \begin{array} { l } - x , - 3 \leq x \leq 0 \\ x , 0 < x \leq 1 \end{array} \right. \right.$, 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) $\mathrm { R } - - \frac { 5 } { 2 } , - \frac { 7 } { 4 }$

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

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 $[ x ]$ 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 $f : [ 0,3 ] \rightarrow \mathrm { A }$ 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

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

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

If the domain of the function $\frac { 1 } { \ln ( 10 - x ) } + \sin ^ { - 1 } \left( \frac { x + 2 } { 2 x + 3 } \right)$ is $( - \infty , - a ] \cup ( - 1 , b ) \cup ( b , c )$, then $( b + c - 3 a )$ is equal to
(A) $20 - \frac { 5 } { 3 }$
(B) $21$
(C) 23
(D) 24

The domain of $\operatorname { Sin } ^ { - 1 } \left( \frac { 1 } { x ^ { 2 } - 2 x - 1 } \right)$ is $( - \infty , \alpha ] \cup [ \beta , \gamma ] \cup [ \delta , \infty )$.
The value of $\alpha + \beta + \gamma + \delta$ is equal to:

Let $A , B$ be two sets, $B \backslash A \neq \emptyset$ and the Cartesian product set $( A \backslash B ) \times A$ has 14 elements.
Accordingly, what is the minimum number of elements in set B?
A) 1
B) 3
C) 4
D) 6
E) 8

A function $f$ on the set of real numbers is defined as $$f ( x ) = \frac { | x | } { 1 + | x | }$$ Accordingly, which of the following is the image set of the interval $[ - 2,1 )$ under the function $\mathbf{f}$?\ A) $[ 0,1 ]$\ B) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right]$\ C) $\left[ \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$\ D) $\left[ 0 , \frac { 1 } { 3 } \right]$\ E) $\left[ 0 , \frac { 2 } { 3 } \right]$