Function

If the domain of the function is , then is equal to :
- A. 40
- B. 36
- C. 24
- D. 32
Let and . Then the function is
- A. neither one-one nor onto
- B. both one-one and onto
- C. one-one
- D. onto
If the function attains the maximum value at then :
- A. $\mathrm{e}^{\pi}<\pi^{\mathrm{e}}$
- B. $\mathrm{e}^{2 \pi}<(2 \pi)^{\mathrm{e}}$
- C. $(2 e)^{\pi}>\pi^{(2 e)}$
- D. $\mathrm{e}^{\pi}>\pi^{\mathrm{e}}$
Let be a function defined on . Then the range of the function is equal to :
- A. $\left[\frac{1}{8}, \frac{1}{5}\right]$
- B. $\left[\frac{1}{7}, \frac{1}{6}\right]$
- C. $\left[\frac{1}{7}, \frac{1}{5}\right]$
- D. $\left[\frac{1}{8}, \frac{1}{6}\right]$
The function is
- A. both one-one and onto
- B. onto but not one-one
- C. neither one-one nor onto
- D. one-one but not onto
Let be defined as :
and
Then the function is
- A. neither one-one nor onto
- B. one-one but not onto
- C. both one-one and onto
- D. onto but not one-one
If the domain of the function
is , then is equal to :
- A. 140
- B. 175
- C. 125
- D. 150
Let and be defined as
and
. Then, gof is:
- A. one-one but not onto
- B. neither one-one nor onto
- C. onto but not one-one
- D. both one-one and onto

If the range of the function $f(x) = \frac{5 - x}{x^2 - 3x + 2}$ , $x \neq 1, 2$ , is $(-\infty, \alpha] \cup [\beta, \infty)$ , then $\alpha^2 + \beta^2$ is equal to:
(1) 188
(2) 192
(3) 190
(4) 194
Let the domains of the functions $f(x) = \log_e \log_e \log_e (8 - \log_e (x^2 + 4x + 5))$ and $g(x) = \sin^{-1} \left( \frac{7x + 10}{x - 2} \right)$ be $(\alpha, \beta)$ and $[\gamma, \delta]$ respectively. Then $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is equal to:
(1) 15
(2) 13
(3) 16
(4) 14
Let $f, g : (1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x) = \frac{2x + 3}{5x - 2}$ and $g(x) = \frac{5x}{x - 1}$ . If the range of the function $f \circ g : [2, 4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$ , then $\frac{1}{\beta - \alpha}$ is equal to:
(1) 56
(2) 2
(3) 29
(4) 68
Let $f$ be a function such that $f(x) + 3f\left(\frac{4}{x}\right) = 4x$ , $x \neq 0$ . Then $f(3) + f(8)$ is equal to:
(1) 13
(2) 11
(3) 10
(4) 12
If the domain of the function $f(x) = \log_7 (1 - \log_4 (x^2 - 9x + 18))$ is $(\alpha, \beta) \cup (\gamma, \delta)$ , then $\alpha + \beta + \gamma + \delta$ is equal to:
(1) 17
(2) 15
(3) 16
(4) 18
If the domain of the function $\text{[math]}$ f(x) = \log_e \left( \frac{2x – 3}{5 + 4x} \right) + \sin^{-1} \left( \frac{4x – 3}{2 – x} \right) $\text{[math]}$ is $\text{[math]}$ [\alpha, \beta] $\text{[math]}$ , then $\text{[math]}$ \alpha^2 + 4\beta $\text{[math]}$ is equal to:
(1) 4
(2) 3
(3) 7
(4) 5
If the domain of the function $\text{[math]}$ f(x) = \frac{1}{\sqrt{10 + 3x – x^2}} + \frac{1}{\sqrt{x + |x|}} $\text{[math]}$ is $\text{[math]}$ [\alpha, \beta] $\text{[math]}$ , then $\text{[math]}$ (1 + \alpha)^2 + \beta^2 $\text{[math]}$ is equal to:
(1) 29
(2) 30
(3) 25
(4) 26
If the domain of the function $\text{[math]}$ \log_5 (18x – x^2 – 77) $\text{[math]}$ is $\text{[math]}$ (\alpha, \beta) $\text{[math]}$ and the domain of the function $\text{[math]}$ \log_e \left( \frac{2x^2 + 3x – 2}{(x – 1)(x^2 – 3x – 4)} \right) $\text{[math]}$ is $\text{[math]}$ (\gamma, \delta) $\text{[math]}$ , then $\text{[math]}$ \alpha^2 + \beta^2 + \gamma^2 + \delta^2 $\text{[math]}$ is equal to:
(1) 186
(2) 179
(3) 195
(4) 174
Let $\text{[math]}$ f : [0, 3] \rightarrow A $\text{[math]}$ be defined by $\text{[math]}$ f(x) = 2x^3 – 15x^2 + 36x + 7 $\text{[math]}$ and $\text{[math]}$ g : [0, \infty) \rightarrow B $\text{[math]}$ be defined by $\text{[math]}$ g(x) = \frac{x^{2025}}{x^{2025} + 1} $\text{[math]}$ . If both the functions are onto and $\text{[math]}$ S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} $\text{[math]}$ , then $\text{[math]}$ n(S) $\text{[math]}$ is equal to:
(1) 29
(2) 31
(3) 30
(4) 36
If $\text{[math]}$ f(x) = \frac{2x}{2 + \sqrt{x}} $\text{[math]}$ , $\text{[math]}$ x \in \mathbb{R} $\text{[math]}$ , then $\text{[math]}$ \sum_{k=1}^{81} f\left(\frac{k}{82}\right) $\text{[math]}$ is equal to:
(1) 82
(2) $\text{[math]}$ 81\sqrt{2} $\text{[math]}$
(3) 41
(4) $\text{[math]}$ \frac{41}{2} $\text{[math]}$
Let $\text{[math]}$ f : \mathbb{R} \rightarrow \mathbb{R} $\text{[math]}$ be a function defined by $\text{[math]}$ f(x) = (2 + 3a)x^2 + \left(\frac{2}{a}\right)x + b $\text{[math]}$ , $\text{[math]}$ a \neq 1 $\text{[math]}$ . If $\text{[math]}$ f(x + y) = f(x) + f(y) + 1 – \frac{2}{7}xy $\text{[math]}$ , then the value of $\text{[math]}$ 28 \sum_{i=1}^{5} |f(i)| $\text{[math]}$ is:
(1) 735
(2) 675
(3) 715
(4) 545
The function $ f: (-\infty, \infty) \to (-\infty, 1) $, defined by $ f(x) = \frac{2^x – 2^{-x}}{2^x + 2^{-x}} $ is:
\item[(A)] One-one but not onto
\item[(B)] Onto but not one-one
\item[(C)] Both one-one and onto
\item[(D)] Neither one-one nor onto
\end{itemize}
Let $ f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32} $. Then the value of $ 8 \left( f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \cdots + f\left(\frac{59}{15}\right) \right) $ is equal to
(1) 108
(2) 92
(3) 118
(4) 102
Let $ f(x) = \log_e x $ and $ g(x) = \frac{x^4 – 2x^3 + 3x^2 – 2x + 2}{2x^2 – 2x + 1} $. Then the domain of $ f \circ g $ is:
\item[(A)] $ (0, \infty) $
\item[(B)] $ [1, \infty) $
\item[(C)] $ \mathbb{R} $
\item[(D)] $ [0, \infty) $
\end{itemize}
Let $ A = \{1, 2, 3, 4\} $ and $ B = \{1, 4, 9, 16\} $. Then the number of many-one functions $ f: A \to B $ such that $ 1 \in f(A) $ is equal to:
\item[(A)] 151
\item[(B)] 139
\item[(C)] 163
\item[(D)] 127
\end{itemize}
Let the range of the function $\text{[math]}$ f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathbb{R} $\text{[math]}$ be $\text{[math]}$ [a, b] $\text{[math]}$ . If $\text{[math]}$ \alpha $\text{[math]}$ and $\text{[math]}$ \beta $\text{[math]}$ ar respectively the A.M. and the G.M. of $\text{[math]}$ a $\text{[math]}$ and $\text{[math]}$ b $\text{[math]}$ , then $\text{[math]}$ \frac{\alpha}{\beta} $\text{[math]}$ is equal to

A $\text{[math]}$ \pi $\text{[math]}$

B $\text{[math]}$ \sqrt{\pi} $\text{[math]}$

C $\text{[math]}$ \sqrt{2} $\text{[math]}$

D 2
If the do of the function $\text{[math]}$ f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right) $\text{[math]}$ is $\text{[math]}$ \mathbf{R}-(\alpha, \beta) $\text{[math]}$ , then $\text{[math]}$ 12 \alpha \beta $\text{[math]}$ is equal to :

A 40

(B) 36

(C) 24

D 32
Let $\text{[math]}$ f(x)=\left\{\begin{array}{cl}-\mathrm{a} & \text { if } \quad-\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } \quad 00 $\text{[math]}$ and $\text{[math]}$ \mathrm{g}(x)=(f(|x|)-|f(x)|) / 2 $\text{[math]}$ . Then the function $\text{[math]}$ g:[-a, a] \rightarrow[-a, a] $\text{[math]}$ is

(A) neither one-one nor onto

B both one-one and onto.

(C) one-one.

D onto
If the function $\text{[math]}$ f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0 $\text{[math]}$ attains the maximum value at $\text{[math]}$ x=\frac{1}{\mathrm{e}} $\text{[math]}$ then :

A $\text{[math]}$ \mathrm{e}^{\pi}<\pi^{\mathrm{e}} $\text{[math]}$

B $\text{[math]}$ \mathrm{e}^{2 \pi}\pi^{(2 e)} $\text{[math]}$

D $\text{[math]}$ \mathrm{e}^{\pi}>\pi^{\mathrm{e}} $\text{[math]}$
Let $\text{[math]}$ f(x)=\frac{1}{7-\sin 5 x} $\text{[math]}$ be a function defined on $\text{[math]}$ \mathbf{R} $\text{[math]}$ . Then the range of the function $\text{[math]}$ f(x) $\text{[math]}$ is equal to :

(A) $\text{[math]}$ \left[\frac{1}{8}, \frac{1}{5}\right] $\text{[math]}$

B $\text{[math]}$ \left[\frac{1}{7}, \frac{1}{6}\right] $\text{[math]}$

C $\text{[math]}$ \left[\frac{1}{7}, \frac{1}{5}\right] $\text{[math]}$

(D) $\text{[math]}$ \left[\frac{1}{8}, \frac{1}{6}\right] $\text{[math]}$
The function $\text{[math]}$ f(x)=\frac{x^{2}+2 x-15}{x^{2}-4 x+9}, x \in \mathbb{R} $\text{[math]}$ is

A both one-one and onto.

B onto but not one-one.

C neither one-one nor onto.

D one-one but not onto.
Let $\text{[math]}$ f, g: \mathbf{R} \rightarrow \mathbf{R} $\text{[math]}$ be defined as :

$\text{[math]}$ f(x)=|x-1| $\text{[math]}$ and $\text{[math]}$ g(x)= \begin{cases}\mathrm{e}^{x}, & x \geq 0 \\ x+1, & x \leq 0\end{cases} $\text{[math]}$

Then the function $\text{[math]}$ f(g(x)) $\text{[math]}$ is

A neither one-one nor onto.

B one-one but not onto.

C both one-one and onto.

D onto but not one-one.
If the do of the function

$\text{[math]}$ f(x)=\frac{\sqrt{x^{2}-25}}{\left(4-x^{2}\right)}+\log _{10}\left(x^{2}+2 x-15\right) $\text{[math]}$ is $\text{[math]}$ (-\infty, \alpha) \cup[\beta, \infty) $\text{[math]}$ , then $\text{[math]}$ \alpha^{2}+\beta^{3} $\text{[math]}$ is equal to :

A 140

B 175

C 125

D 150
Let $\text{[math]}$ f: \mathbf{R} \rightarrow \mathbf{R} $\text{[math]}$ and $\text{[math]}$ g: \mathbf{R} \rightarrow \mathbf{R} $\text{[math]}$ be defined as

$\text{[math]}$ f(x)=\left\{\begin{array}{ll}\log _{\mathrm{e}} x, & x>0 \\ \mathrm{e}^{-x}, & x \leq 0\end{array}\right. $\text{[math]}$ and

$\text{[math]}$ g(x)=\left\{\begin{array}{ll}x, & x \geqslant 0 \\ \mathrm{e}^{x}, & x<0\end{array}\right. $\text{[math]}$ . Then, gof $\text{[math]}$ : \mathbf{R} \rightarrow \mathbf{R} $\text{[math]}$ is:

A one-one but not onto

B neither one-one nor onto

C onto but not one-one

D both one-one and onto
If $\text{[math]}$ f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3} $\text{[math]}$ and $\text{[math]}$ (f \circ f)(x)=g(x) $\text{[math]}$ , where $\text{[math]}$ g: \mathbb{R}-\left\{\frac{2}{3}\right\} \rightarrow \mathbb{R}-\left\{\frac{2}{3}\right\} $\text{[math]}$ , then $\text{[math]}$ ( $\text{[math]}$ gogog $\text{[math]}$ )(4) $\text{[math]}$ is equal to

A -4

B $\text{[math]}$ \frac{19}{20} $\text{[math]}$

C $\text{[math]}$ -\frac{19}{20} $\text{[math]}$

D 4
If the do of the function $\text{[math]}$ f(x)=\log _{e}\left(\frac{2 x+3}{4 x^{2}+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right) $\text{[math]}$ is $\text{[math]}$ (\alpha, \beta] $\text{[math]}$ , then the value of $\text{[math]}$ 5 \beta-4 \alpha $\text{[math]}$ is equal to

A 9

B 12

C 11

D 10
If the do of the function $\text{[math]}$ f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left\{\log _{e}(3-x)\right\}^{-1} $\text{[math]}$ is $\text{[math]}$ [-\alpha, \beta)-\{\gamma\} $\text{[math]}$ , then $\text{[math]}$ \alpha+\beta+\gamma $\text{[math]}$ is equal to :

(A) 11

B 12

C 9

D 8
If $\text{[math]}$ f(x)=\left\{\begin{array}{c}2+2 x, \quad-1 \leq x<0 \\ 1-\frac{x}{3}, \quad 0 \leq x \leq 3\end{array} ;g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0<x \leq 1\end{array}\right.\right. $\text{[math]}$ , then range of $\text{[math]}$ (f o g)(x) $\text{[math]}$ is

A $\text{[math]}$ [0,1) $\text{[math]}$

B $\text{[math]}$ [0,3) $\text{[math]}$

C $\text{[math]}$ (0,1] $\text{[math]}$

D $\text{[math]}$ [0,1] $\text{[math]}$
Let $\text{[math]}$ f: \mathbf{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathbf{R} $\text{[math]}$ and $\text{[math]}$ g: \mathbf{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathbf{R} $\text{[math]}$ be defined as $\text{[math]}$ f(x)=\frac{2 x+3}{2 x+1} $\text{[math]}$ and $\text{[math]}$ g(x)=\frac{|x|+1}{2 x+5} $\text{[math]}$ . Then, the do of the function fog is :

(A) $\text{[math]}$ \mathbf{R}-\left\{-\frac{7}{4}\right\} $\text{[math]}$

B $\text{[math]}$ \mathbf{R} $\text{[math]}$

C $\text{[math]}$ \mathbf{R}-\left\{-\frac{5}{2},-\frac{7}{4}\right\} $\text{[math]}$

D $\text{[math]}$ \mathbf{R}-\left\{-\frac{5}{2}\right\} $\text{[math]}$
The function $\text{[math]}$ f: \mathbf{N}-\{1\} \rightarrow \mathbf{N} $\text{[math]}$ ; defined by $\text{[math]}$ f(\mathrm{n})= $\text{[math]}$ the highest prime factor of $\text{[math]}$ \mathrm{n} $\text{[math]}$ , is :

A one-one only

B neither one-one nor onto

C onto only

D both one-one and onto
The range of $\text{[math]}$ f(x)=4 \sin ^{-1}\left(\frac{x^{2}}{x^{2}+1}\right) $\text{[math]}$ is

A $\text{[math]}$ [0,2 \pi] $\text{[math]}$

B $\text{[math]}$ [0,2 \pi) $\text{[math]}$

C $\text{[math]}$ [0, \pi) $\text{[math]}$

D $\text{[math]}$ [0, \pi] $\text{[math]}$
For $\text{[math]}$ x \in \mathbb{R} $\text{[math]}$ , two real valued functions $\text{[math]}$ f(x) $\text{[math]}$ and $\text{[math]}$ g(x) $\text{[math]}$ are such that, $\text{[math]}$ g(x)=\sqrt{x}+1 $\text{[math]}$ and $\text{[math]}$ f \circ g(x)=x+3-\sqrt{x} $\text{[math]}$ . Then $\text{[math]}$ f(0) $\text{[math]}$ is equal to

A 5

B 0

C -3

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

A 45

B 136

C 46

D nearly 135
The do of the function $\text{[math]}$ f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]-10}} $\text{[math]}$ is : ( where $\text{[math]}$ [\mathrm{x}] $\text{[math]}$ denotes the greatest integer less than or equal to $\text{[math]}$ x $\text{[math]}$ )

A $\text{[math]}$ (-\infty,-2) \cup[6, \infty) $\text{[math]}$

B $\text{[math]}$ (-\infty,-3] \cup[6, \infty) $\text{[math]}$

C $\text{[math]}$ (-\infty,-2) \cup(5, \infty) $\text{[math]}$

D $\text{[math]}$ (-\infty,-3] \cup(5, \infty) $\text{[math]}$
If $\text{[math]}$ f(x)=\frac{\left(\tan 1^{\circ}\right) x+\log _{e}(123)}{x \log _{e}(1234)-\left(\tan 1^{\circ}\right)}, x>0 $\text{[math]}$ , then the least value of $\text{[math]}$ f(f(x))+f\left(f\left(\frac{4}{x}\right)\right) $\text{[math]}$ is :

A 2

B 4

C 0

D 8
Let the sets $\text{[math]}$ \mathrm{A} $\text{[math]}$ and $\text{[math]}$ \mathrm{B} $\text{[math]}$ denote the do and range respectively of the function $\text{[math]}$ f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}} $\text{[math]}$ , where $\text{[math]}$ \lceil x\rceil $\text{[math]}$ denotes the smallest integer greater than or equal to $\text{[math]}$ x $\text{[math]}$ . Then among the statements

(S1) $\text{[math]}$ : A \cap B=(1, \infty)-\mathbb{N} $\text{[math]}$ and

$\text{[math]}$ (\mathrm{S} 2): A \cup B=(1, \infty) $\text{[math]}$

A only (S2) is true

B only (S1) is true

C neither (S1) nor (S2) is true

D both (S1) and (S2) are true
Let $\text{[math]}$ f: \mathbb{R}-0,1 \rightarrow \mathbb{R} $\text{[math]}$ be a function such that $\text{[math]}$ f(x)+f\left(\frac{1}{1-x}\right)=1+x $\text{[math]}$ . Then $\text{[math]}$ f(2) $\text{[math]}$ is equal to

(A) $\text{[math]}$ \frac{9}{4} $\text{[math]}$
B $\text{[math]}$ \frac{7}{4} $\text{[math]}$

C $\text{[math]}$ \frac{7}{3} $\text{[math]}$

D $\text{[math]}$ \frac{9}{2} $\text{[math]}$
Let $\text{[math]}$ f(x)=\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & \sin 2 x \\ \sin ^{2} x & 1+\cos ^{2} x & \sin 2 x \\ \sin ^{2} x & \cos ^{2} x & 1+\sin 2 x\end{array}\right|, x \in\left[\frac{\pi}{6}, \frac{\pi}{3}\right] $\text{[math]}$ . If $\text{[math]}$ \alpha $\text{[math]}$ and $\text{[math]}$ \beta $\text{[math]}$ respectively are the maximum and the minimum values of $\text{[math]}$ f $\text{[math]}$ , then

(A) $\text{[math]}$ \alpha^{2}-\beta^{2}=4 \sqrt{3} $\text{[math]}$

B $\text{[math]}$ \beta^{2}-2 \sqrt{\alpha}=\frac{19}{4} $\text{[math]}$

C $\text{[math]}$ \beta^{2}+2 \sqrt{\alpha}=\frac{19}{4} $\text{[math]}$

D $\text{[math]}$ \alpha^{2}+\beta^{2}=\frac{9}{2} $\text{[math]}$
Let $\text{[math]}$ f: \mathbb{R}-\{2,6\} \rightarrow \mathbb{R} $\text{[math]}$ be real valued function

defined as $\text{[math]}$ f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12} $\text{[math]}$ .

Then range of $\text{[math]}$ f $\text{[math]}$ is

A $\text{[math]}$ \left(-\infty,-\frac{21}{4}\right] \cup[1, \infty) $\text{[math]}$

B $\text{[math]}$ \left(-\infty,-\frac{21}{4}\right) \cup(0, \infty) $\text{[math]}$

C $\text{[math]}$ \left(-\infty,-\frac{21}{4}\right] \cup[0, \infty) $\text{[math]}$

D $\text{[math]}$ \left(-\infty,-\frac{21}{4}\right] \cup\left[\frac{21}{4}, \infty\right) $\text{[math]}$
The absolute minimum value, of the function

$\text{[math]}$ f(x)=\left|x^{2}-x+1\right|+\left[x^{2}-x+1\right] $\text{[math]}$ ,

where $\text{[math]}$ [t] $\text{[math]}$ denotes the greatest integer function, in the interval $\text{[math]}$ [-1,2] $\text{[math]}$ , is :

(A) $\text{[math]}$ \frac{3}{4} $\text{[math]}$

(B) $\text{[math]}$ \frac{3}{2} $\text{[math]}$

(c) $\text{[math]}$ \frac{1}{4} $\text{[math]}$

(D) $\text{[math]}$ \frac{5}{4} $\text{[math]}$
If the do of the function $\text{[math]}$ f(x)=\frac{[x]}{1+x^{2}} $\text{[math]}$ , where $\text{[math]}$ [x] $\text{[math]}$ is greatest integer $\text{[math]}$ \leq x $\text{[math]}$ , is $\text{[math]}$ [2,6) $\text{[math]}$ , then its range is

A $\text{[math]}$ \left(\frac{5}{37}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\} $\text{[math]}$

B $\text{[math]}$ \left(\frac{5}{37}, \frac{2}{5}\right] $\text{[math]}$

C $\text{[math]}$ \left(\frac{5}{26}, \frac{2}{5}\right] $\text{[math]}$

(D) $\text{[math]}$ \left(\frac{5}{26}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\} $\text{[math]}$
The range of the function $\text{[math]}$ f(x)=\sqrt{3-x}+\sqrt{2+x} $\text{[math]}$ is :

A $\text{[math]}$ [2 \sqrt{2}, \sqrt{11}] $\text{[math]}$

B $\text{[math]}$ [\sqrt{5}, \sqrt{13}] $\text{[math]}$

C $\text{[math]}$ [\sqrt{2}, \sqrt{7}] $\text{[math]}$

D $\text{[math]}$ [\sqrt{5}, \sqrt{10}] $\text{[math]}$
Consider a function $\text{[math]}$ f: \mathbb{N} \rightarrow \mathbb{R} $\text{[math]}$ , satisfying $\text{[math]}$ f(1)+2 f(2)+3 f(3)+\ldots+x f(x)=x(x+1) f(x) ; x \geq 2 $\text{[math]}$ with $\text{[math]}$ f(1)=1 $\text{[math]}$ . Then $\text{[math]}$ \frac{1}{f(2022)}+\frac{1}{f(2028)} $\text{[math]}$ is equal to

A 8000

B 8400

C 8100

D 8200
The do of $\text{[math]}$ f(x)=\frac{\log _{(x+1)}(x-2)}{e^{2 \log _{e} x}-(2 x+3)}, x \in \mathbb{R} $\text{[math]}$ is

A $\text{[math]}$ (-1, \infty)-\{3\} $\text{[math]}$

B $\text{[math]}$ \mathbb{R}-\{-1,3) $\text{[math]}$

C $\text{[math]}$ (2, \infty)-\{3\} $\text{[math]}$

D $\text{[math]}$ \mathbb{R}-\{3\} $\text{[math]}$
Let $\text{[math]}$ f: R \rightarrow R $\text{[math]}$ be a function such that $\text{[math]}$ f(x)=\frac{x^{2}+2 x+1}{x^{2}+1} $\text{[math]}$ . Then

A $\text{[math]}$ f(x) $\text{[math]}$ is many-one in $\text{[math]}$ (-\infty,-1) $\text{[math]}$

B $\text{[math]}$ f(x) $\text{[math]}$ is one-one in $\text{[math]}$ (-\infty, \infty) $\text{[math]}$

C $\text{[math]}$ f(x) $\text{[math]}$ is one-one in $\text{[math]}$ [1, \infty) $\text{[math]}$ but not in $\text{[math]}$ (-\infty, \infty) $\text{[math]}$

D $\text{[math]}$ f(x) $\text{[math]}$ is many-one in $\text{[math]}$ (1, \infty) $\text{[math]}$
The number of functions

$\text{[math]}$ f:\{1,2,3,4\} \rightarrow\{a \in Z|a| \leq 8\} $\text{[math]}$

satisfying $\text{[math]}$ f(n)+\frac{1}{n} f(n+1)=1, \forall n \in\{1,2,3\} $\text{[math]}$ is

A 2

B 3

C 1

D 4
Let $\text{[math]}$ f: \mathbb{R} \rightarrow \mathbb{R} $\text{[math]}$ be a function defined by $\text{[math]}$ f(x)=\log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\} $\text{[math]}$ , for some $\text{[math]}$ m $\text{[math]}$ , such that the range of $\text{[math]}$ f $\text{[math]}$ is $\text{[math]}$ [0 $\text{[math]}$ , 2]. Then the value of $\text{[math]}$ m $\text{[math]}$ is $\text{[math]}$ \qquad $\text{[math]}$ \\
A 4

B 3

C 5

D 2
Let $\text{[math]}$ f(x)=2 x^{n}+\lambda, \lambda \in R, n \in N $\text{[math]}$ , and $\text{[math]}$ f(4)=133, f(5)=255 $\text{[math]}$ . Then the sum of all the positive integer divisors of $\text{[math]}$ (f(3)-f(2)) $\text{[math]}$ is

A 60

B 58

C 61

D 59
Let $\text{[math]}$ f(x) $\text{[math]}$ be a function such that $\text{[math]}$ f(x+y)=f(x) $\text{[math]}$ . $\text{[math]}$ f(y) $\text{[math]}$ for all $\text{[math]}$ x, y \in \mathbb{N} $\text{[math]}$ . If $\text{[math]}$ f(1)=3 $\text{[math]}$ and $\text{[math]}$ \sum_{k=1}^{n} f(k)=3279 $\text{[math]}$ , then the value of $\text{[math]}$ \mathrm{n} $\text{[math]}$ is

A 9

B 7

C 6

D 8
If $\text{[math]}$ f(x)=\frac{2^{2 x}}{2^{2 x}+2}, x \in \mathbb{R} $\text{[math]}$ , then $\text{[math]}$ f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\ldots+f\left(\frac{2022}{2023}\right) $\text{[math]}$ is equal to

A 2011

B 2010

C 1010

D 1011
Let $\text{[math]}$ f(x)=a x^{2}+b x+c $\text{[math]}$ be such that $\text{[math]}$ f(1)=3, f(-2)=\lambda $\text{[math]}$ and $\text{[math]}$ f(3)=4 $\text{[math]}$ . If $\text{[math]}$ f(0)+f(1)+f(-2)+f(3)=14 $\text{[math]}$ , then $\text{[math]}$ \lambda $\text{[math]}$ is equal to :

A -4

B $\text{[math]}$ \frac{13}{2} $\text{[math]}$

C $\text{[math]}$ \frac{23}{2} $\text{[math]}$

D 4
Let $\text{[math]}$ \alpha, \beta $\text{[math]}$ and $\text{[math]}$ \gamma $\text{[math]}$ be three positive real numbers. Let $\text{[math]}$ f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R} $\text{[math]}$ and $\text{[math]}$ g: \mathbf{R} \rightarrow \mathbf{R} $\text{[math]}$ be such that $\text{[math]}$ g(f(x))=x $\text{[math]}$ for all $\text{[math]}$ x \in \mathbf{R} $\text{[math]}$ . If $\text{[math]}$ \mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{\mathrm{n}} $\text{[math]}$ be in arithmetic progression with mean zero, then the value of $\text{[math]}$ f\left(g\left(\frac{1}{\mathrm{n}} \sum_{i=1}^{\mathrm{n}} f\left(\mathrm{a}_{i}\right)\right)\right) $\text{[math]}$ is equal to :

(A) 0

B 3

(C) 9

D 27
Let $\text{[math]}$ f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N} $\text{[math]}$ be functions defined by $\text{[math]}$ f(a)=\alpha $\text{[math]}$ , where $\text{[math]}$ \alpha $\text{[math]}$ is the maximum of the powers of those primes $\text{[math]}$ p $\text{[math]}$ such that $\text{[math]}$ p^{\alpha} $\text{[math]}$ divides $\text{[math]}$ a $\text{[math]}$ , and $\text{[math]}$ g(a)=a+1 $\text{[math]}$ , for all $\text{[math]}$ a \in \mathbb{N}-\{1\} $\text{[math]}$ . Then, the function $\text{[math]}$ f+g $\text{[math]}$ is

A one-one but not onto

B onto but not one-one

C both one-one and onto

D neither one-one nor onto
The number of bijective functions $\text{[math]}$ f:\{1,3,5,7, \ldots, 99\} \rightarrow\{2,4,6,8, \ldots 100\} $\text{[math]}$ , such that $\text{[math]}$ f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots \ldots f(99) $\text{[math]}$ , is $\text{[math]}$ \qquad $\text{[math]}$ \\
A $\text{[math]}$ { }^{50} P_{17} $\text{[math]}$

B $\text{[math]}$ { }^{50} P_{33} $\text{[math]}$

C $\text{[math]}$ 33!\times 17 $\text{[math]}$ !

D $\text{[math]}$ \frac{50!}{2} $\text{[math]}$
The total number of functions,

$\text{[math]}$ f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\} $\text{[math]}$ such that $\text{[math]}$ f(1)+f(2)=f(3) $\text{[math]}$ , is equal to:

A 60

B 90

C 108

D 126
Let a function $\text{[math]}$ \mathrm{f}: \mathrm{N} \rightarrow \mathrm{N} $\text{[math]}$ be defined by

$\text{[math]}$ f(n)=\left[\begin{array}{c}2 n, \quad n=2,4,6,8, \ldots \ldots \\ n-1, \quad n=3,7,11,15, \ldots \ldots \\ \frac{n+1}{2}, \quad n=1,5,9,13, \ldots \ldots\end{array}\right. $\text{[math]}$

then, $\text{[math]}$ \mathrm{f} $\text{[math]}$ is

A one-one but not onto

B onto but not one-one

C neither one-one nor onto

D one-one and onto
Let $\text{[math]}$ \mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} $\text{[math]}$ be defined as $\text{[math]}$ \mathrm{f}(\mathrm{x})=\mathrm{x}-1 $\text{[math]}$ and $\text{[math]}$ \mathrm{g}: \mathrm{R}-\{1,-1\} \rightarrow \mathrm{R} $\text{[math]}$ be defined as $\text{[math]}$ g(x)=\frac{x^{2}}{x^{2}-1} $\text{[math]}$ .

Then the function fog is :

A one-one but not onto

B onto but not one-one

C both one-one and onto

D neither one-one nor onto
Let $\text{[math]}$ f(x)=\frac{x-1}{x+1}, x \in R-\{0,-1,1\} $\text{[math]}$ . If $\text{[math]}$ f^{n+1}(x)=f\left(f^{n}(x)\right) $\text{[math]}$ for all $\text{[math]}$ \mathrm{n} \in \mathrm{N} $\text{[math]}$ , then $\text{[math]}$ f^{6}(6)+f^{7}(7) $\text{[math]}$ is equal to :

(A) $\text{[math]}$ \frac{7}{6} $\text{[math]}$

B $\text{[math]}$ -\frac{3}{2} $\text{[math]}$

C $\text{[math]}$ \frac{7}{12} $\text{[math]}$

D $\text{[math]}$ -\frac{11}{12} $\text{[math]}$
Let $\text{[math]}$ \mathrm{f}: \mathrm{N} \rightarrow \mathrm{R} $\text{[math]}$ be a function such that $\text{[math]}$ f(x+y)=2 f(x) f(y) $\text{[math]}$ for natural numbers $\text{[math]}$ \mathrm{x} $\text{[math]}$ and $\text{[math]}$ \mathrm{y} $\text{[math]}$ . If $\text{[math]}$ \mathrm{f}(1)=2 $\text{[math]}$ , then the value of $\text{[math]}$ \alpha $\text{[math]}$ for which $\text{[math]}$ \sum_{k=1}^{10} f(\alpha+k)=\frac{512}{3}\left(2^{20}-1\right) $\text{[math]}$ holds, is :

A 2

B 3

C 4

D 6
Let $\text{[math]}$ f: R \rightarrow R $\text{[math]}$ and $\text{[math]}$ g: R \rightarrow R $\text{[math]}$ be two functions defined by $\text{[math]}$ f(x)=\log _{e}\left(x^{2}+1\right)-e^{-x}+1 $\text{[math]}$ and $\text{[math]}$ g(x)=\frac{1-2 e^{2 x}}{e^{x}} $\text{[math]}$ . Then, for which of the following range of $\text{[math]}$ \alpha $\text{[math]}$ , the inequality $\text{[math]}$ f\left(g\left(\frac{(\alpha-1)^{2}}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right) $\text{[math]}$ holds ?

A $\text{[math]}$ (2,3) $\text{[math]}$

B $\text{[math]}$ (-2,-1) $\text{[math]}$

C $\text{[math]}$ (1,2) $\text{[math]}$

D $\text{[math]}$ (-1,1) $\text{[math]}$
The range of the function,

$\text{[math]}$ f(x)=\log _{\sqrt{5}}\left(3+\cos \left(\frac{3 \pi}{4}+x\right)+\cos \left(\frac{\pi}{4}+x\right)+\cos \left(\frac{\pi}{4}-x\right)-\cos \left(\frac{3 \pi}{4}-x\right)\right) $\text{[math]}$ is :

A $\text{[math]}$ (0, \sqrt{5}) $\text{[math]}$

B $\text{[math]}$ [-2,2] $\text{[math]}$

C $\text{[math]}$ \left[\frac{1}{\sqrt{5}}, \sqrt{5}\right] $\text{[math]}$

D $\text{[math]}$ [0,2] $\text{[math]}$
Let $\text{[math]}$ f: N \rightarrow N $\text{[math]}$ be a function such that $\text{[math]}$ f(m+n)=f(m)+f(n) $\text{[math]}$ for every $\text{[math]}$ m, n \in N $\text{[math]}$ . If $\text{[math]}$ f(6)=18 $\text{[math]}$ , then $\text{[math]}$ f(2) $\text{[math]}$ . $\text{[math]}$ f(3) $\text{[math]}$ is equal to :

A 6

B 54

C 18

D 36
Let $\text{[math]}$ \mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} $\text{[math]}$ be defined as $\text{[math]}$ f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 $\text{[math]}$ . Then, the value of $\text{[math]}$ \sum_{k=1}^{20} \frac{1}{\sin (k) \sin (k+f(k))} $\text{[math]}$ is equal to :

A $\text{[math]}$ \operatorname{cosec}^{2}(21) \cos (20) \cos (2) $\text{[math]}$

B $\text{[math]}$ \sec ^{2}(1) \sec (21) \cos (20) $\text{[math]}$

C $\text{[math]}$ \operatorname{cosec}^{2}(1) \operatorname{cosec}(21) \sin (20) $\text{[math]}$

D $\text{[math]}$ \sec ^{2}(21) \sin (20) \sin (2) $\text{[math]}$
Consider function $\text{[math]}$ \mathrm{f}: \mathrm{A} \rightarrow \mathrm{B} $\text{[math]}$ and $\text{[math]}$ \mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}(\mathrm{A}, \mathrm{B}, \mathrm{C} \subseteq \mathrm{R}) $\text{[math]}$ such that (gof) $\text{[math]}$ { }^{-1} $\text{[math]}$ exists, then :

A f and g both are one-one

B f and g both are onto

C $\text{[math]}$ f $\text{[math]}$ is one-one and $\text{[math]}$ g $\text{[math]}$ is onto

D $\text{[math]}$ f $\text{[math]}$ is onto and $\text{[math]}$ g $\text{[math]}$ is one-one
Let $\text{[math]}$ \mathrm{g}: \mathrm{N} \rightarrow \mathrm{N} $\text{[math]}$ be defined as

$\text{[math]}$ g(3 n+1)=3 n+2 $\text{[math]}$ ,

$\text{[math]}$ g(3 n+2)=3 n+3 $\text{[math]}$ ,

$\text{[math]}$ g(3 n+3)=3 n+1 $\text{[math]}$ , for all $\text{[math]}$ n \geq 0 $\text{[math]}$

Then which of the following statements is true?

A There exists an onto function $\text{[math]}$ \mathrm{f}: \mathrm{N} \rightarrow \mathrm{N} $\text{[math]}$ such that fog $\text{[math]}$ =\mathrm{f} $\text{[math]}$

B There exists a one-one function $\text{[math]}$ \mathrm{f}: \mathrm{N} \rightarrow \mathrm{N} $\text{[math]}$ such that fog $\text{[math]}$ =\mathrm{f} $\text{[math]}$

C gogog $\text{[math]}$ =g $\text{[math]}$

D There exists a function: $\text{[math]}$ \mathrm{f}: \mathrm{N} \rightarrow \mathrm{N} $\text{[math]}$ such that gof $\text{[math]}$ =\mathrm{f} $\text{[math]}$
Let $\text{[math]}$ f: R-\left\{\frac{\alpha}{6}\right\} \rightarrow R $\text{[math]}$ be defined by $\text{[math]}$ f(x)=\frac{5 x+3}{6 x-\alpha} $\text{[math]}$ . Then the value of $\text{[math]}$ \alpha $\text{[math]}$ for which (fof)(x)=x, for all $\text{[math]}$ x \in R-\left\{\frac{\alpha}{6}\right\} $\text{[math]}$ , is :

A No such $\text{[math]}$ \alpha $\text{[math]}$ exists

B 5

C 8

D 6
Let $\text{[math]}$ [\mathrm{x}] $\text{[math]}$ denote the greatest integer $\text{[math]}$ \leq \mathrm{x} $\text{[math]}$ , where $\text{[math]}$ \mathrm{x} \in \mathrm{R} $\text{[math]}$ . If the do of the real valued function $\text{[math]}$ f(x)=\sqrt{\frac{\| x] \mid-2}{\|x\| \mid-3}} $\text{[math]}$ is $\text{[math]}$ \left.(-\infty, \mathrm{a})\right] \cup[\mathrm{b}, \mathrm{c}) $\text{[math]}$ $\text{[math]}$ \cup[4, \infty), a<b<c $\text{[math]}$ , then the value of $\text{[math]}$ a+b+c $\text{[math]}$ is :

A 8

B 1

C -2

D -3
Let $\text{[math]}$ f: R-\{3\} \rightarrow R-\{1\} $\text{[math]}$ be defined by $\text{[math]}$ f(x)=\frac{x-2}{x-3} $\text{[math]}$ .

Let $\text{[math]}$ g: R \rightarrow R $\text{[math]}$ be given as $\text{[math]}$ g(x)=2 x-3 $\text{[math]}$ . Then, the sum of all the values of $\text{[math]}$ x $\text{[math]}$ for which $\text{[math]}$ f^{-1}(x)+g^{-1}(x)=\frac{13}{2} $\text{[math]}$ is equal to :

A 3

B 5

C 2

D 7
The real valued function

$\text{[math]}$ f(x)=\frac{\operatorname{cosec}^{-1} x}{\sqrt{x-[x]}} $\text{[math]}$ , where $\text{[math]}$ [\mathrm{x}] $\text{[math]}$ denotes the greatest integer less than or equal to $\text{[math]}$ \mathrm{x} $\text{[math]}$ , is defined for all $\text{[math]}$ \mathrm{x} $\text{[math]}$ belonging to :

A all real except integers

B all non-integers except the interval $\text{[math]}$ [-1,1] $\text{[math]}$

C all integers except $\text{[math]}$ 0,-1,1 $\text{[math]}$

D all real except the interval $\text{[math]}$ [-1,1] $\text{[math]}$
If the functions are defined as $\text{[math]}$ f(x)=\sqrt{x} $\text{[math]}$ and $\text{[math]}$ g(x)=\sqrt{1-x} $\text{[math]}$ , then what is the common do of the following functions :

$\text{[math]}$ \mathrm{f}+\mathrm{g}, \mathrm{f}-\mathrm{g}, \mathrm{f} / \mathrm{g}, \mathrm{g} / \mathrm{f}, \mathrm{g}-\mathrm{f} $\text{[math]}$ where $\text{[math]}$ (f \pm g)(x)=f(x) \pm g(x),(f / g) x=\frac{f(x)}{g(x)} $\text{[math]}$

A $\text{[math]}$ 0 \leq x \leq 1 $\text{[math]}$

B $\text{[math]}$ 0 \leq x<1 $\text{[math]}$

C $\text{[math]}$ 0<x<1 $\text{[math]}$

D $\text{[math]}$ 0<x \leq 1 $\text{[math]}$
The inverse of $\text{[math]}$ y=5^{\log x} $\text{[math]}$ is :

A $\text{[math]}$ x=5^{\log y} $\text{[math]}$

B $\text{[math]}$ x=y^{\frac{1}{\log 5}} $\text{[math]}$

C $\text{[math]}$ x=5^{\frac{1}{\log y}} $\text{[math]}$

D $\text{[math]}$ x=y^{\log 5} $\text{[math]}$
The range of $\text{[math]}$ a \in R $\text{[math]}$ for which the

function $\text{[math]}$ f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right), x \neq 2 n \pi, n \in N $\text{[math]}$ has critical points, is :

A $\text{[math]}$ [1, \infty) $\text{[math]}$

B $\text{[math]}$ (-3,1) $\text{[math]}$

C $\text{[math]}$ \left[-\frac{4}{3}, 2\right] $\text{[math]}$

D $\text{[math]}$ (-\infty,-1] $\text{[math]}$
Let $\text{[math]}$ A=\{1,2,3, \ldots, 10\} $\text{[math]}$ and $\text{[math]}$ f: A \rightarrow A $\text{[math]}$ be defined as

$\text{[math]}$ f(k)=\left\{\begin{array}{cc}k+1 & \text { if } k \text { is odd } \\ k & \text { if } k \text { is even }\end{array}\right. $\text{[math]}$

Then the number of possible functions $\text{[math]}$ g: A \rightarrow A $\text{[math]}$ such that $\text{[math]}$ g o f=f $\text{[math]}$ is :

A $\text{[math]}$ 5^{5} $\text{[math]}$

B $\text{[math]}$ 10^{5} $\text{[math]}$

C 5 !

D $\text{[math]}$ { }^{10} \mathrm{C}_{5} $\text{[math]}$
A function $\text{[math]}$ f(x) $\text{[math]}$ is given by $\text{[math]}$ f(x)=\frac{5^{x}}{5^{x}+5} $\text{[math]}$ , then the sum of the series $\text{[math]}$ f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right) $\text{[math]}$ is equal to :

A $\text{[math]}$ \frac{39}{2} $\text{[math]}$

B $\text{[math]}$ \frac{19}{2} $\text{[math]}$

C $\text{[math]}$ \frac{49}{2} $\text{[math]}$

D $\text{[math]}$ \frac{29}{2} $\text{[math]}$
Let $\text{[math]}$ \mathrm{x} $\text{[math]}$ denote the total number of one-one functions from a set $\text{[math]}$ \mathrm{A} $\text{[math]}$ with 3 elements to a set $\text{[math]}$ \mathrm{B} $\text{[math]}$ with 5 elements and $\text{[math]}$ \mathrm{y} $\text{[math]}$ denote the total number of one-one functions form the set $\text{[math]}$ \mathrm{A} $\text{[math]}$ to the set $\text{[math]}$ \mathrm{A} \times \mathrm{B} $\text{[math]}$ . Then :

A $\text{[math]}$ 2 y=273 x $\text{[math]}$

B $\text{[math]}$ y=91 x $\text{[math]}$

C $\text{[math]}$ 2 y=91 x $\text{[math]}$

D $\text{[math]}$ y=273 x $\text{[math]}$
Let $\text{[math]}$ f, g: N \rightarrow N $\text{[math]}$ such that $\text{[math]}$ f(n+1)=f(n)+f(1) \forall n \in N $\text{[math]}$ and $\text{[math]}$ g $\text{[math]}$ be any arbitrary function. Which of the following statements is NOT true?

A If $\text{[math]}$ g $\text{[math]}$ is onto, then fog is one-one

B f is one-one

C If $\text{[math]}$ f $\text{[math]}$ is onto, then $\text{[math]}$ f(n)=n \forall n \in N $\text{[math]}$

D If fog is one-one, then $\text{[math]}$ g $\text{[math]}$ is one-one\\
one-one but not onto

B onto but not one-one

C both one-one and onto

D neither one-one nor onto
For a suitably chosen real constant a, let a

function, $\text{[math]}$ f: R-\{-a\} \rightarrow R $\text{[math]}$ be defined by

$\text{[math]}$ f(x)=\frac{a-x}{a+x} $\text{[math]}$ . Further suppose that for any real number $\text{[math]}$ x \neq-a $\text{[math]}$ and $\text{[math]}$ f(x) \neq-a $\text{[math]}$ ,

$\text{[math]}$ (\mathrm{fof})(\mathrm{x})=\mathrm{x} $\text{[math]}$ . Then $\text{[math]}$ f\left(-\frac{1}{2}\right) $\text{[math]}$ is equal to :

(A) $\text{[math]}$ \frac{1}{3} $\text{[math]}$

B -3

(C) $\text{[math]}$ -\frac{1}{3} $\text{[math]}$

D 3
If $\text{[math]}$ \mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y}) $\text{[math]}$ and $\text{[math]}$ \sum_{x=1}^{\infty} f(x)=2, \mathrm{x}, \mathrm{y} \in \mathrm{N} $\text{[math]}$ , where $\text{[math]}$ \mathrm{N} $\text{[math]}$ is the set of all natural number, then the value of $\text{[math]}$ \frac{f(4)}{f(2)} $\text{[math]}$ is :

A $\text{[math]}$ \frac{2}{3} $\text{[math]}$

B $\text{[math]}$ \frac{1}{9} $\text{[math]}$

(c) $\text{[math]}$ \frac{1}{3} $\text{[math]}$

D $\text{[math]}$ \frac{4}{9} $\text{[math]}$
Let $\text{[math]}$ f: R \rightarrow R $\text{[math]}$ be a function which satisfies

$\text{[math]}$ f(x+y)=f(x)+f(y) \forall x, y \in R $\text{[math]}$ . If $\text{[math]}$ f(1)=2 $\text{[math]}$ and

$\text{[math]}$ \mathrm{g}(\mathrm{n})=\sum_{k=1}^{(n-1)} f(k), \mathrm{n} \in \mathrm{N} $\text{[math]}$ then the value of $\text{[math]}$ \mathrm{n} $\text{[math]}$ , for which $\text{[math]}$ \mathrm{g}(\mathrm{n})=20 $\text{[math]}$ , is :

A 20

B 9

C 5

D 4
Let $\text{[math]}$ a-2 b+c=1 $\text{[math]}$ .

If $\text{[math]}$ f(x)=\left|\begin{array}{lll}x+a & x+2 & x+1 \\ x+b & x+3 & x+2 \\ x+c & x+4 & x+3\end{array}\right| $\text{[math]}$ , then:

A $\text{[math]}$ f(50)=1 $\text{[math]}$

B $\text{[math]}$ f(-50)=-1 $\text{[math]}$

C $\text{[math]}$ f(50)=-501 $\text{[math]}$

D $\text{[math]}$ f(-50)=501 $\text{[math]}$
Let $\text{[math]}$ f:(1,3) \rightarrow R $\text{[math]}$ be a function defined by

$\text{[math]}$ f(x)=\frac{x[x]}{1+x^{2}} $\text{[math]}$ , where $\text{[math]}$ [\mathrm{x}] $\text{[math]}$ denotes the greatest integer $\text{[math]}$ \leq \mathrm{x} $\text{[math]}$ . Then the range of $\text{[math]}$ \mathrm{f} $\text{[math]}$ is

A $\text{[math]}$ \left(\frac{2}{5}, \frac{1}{2}\right) \cup\left(\frac{3}{4}, \frac{4}{5}\right] $\text{[math]}$

B $\text{[math]}$ \left(\frac{3}{5}, \frac{4}{5}\right) $\text{[math]}$

(c) $\text{[math]}$ \left(\frac{2}{5}, \frac{4}{5}\right] $\text{[math]}$

D $\text{[math]}$ \left(\frac{2}{5}, \frac{3}{5}\right] \cup\left(\frac{3}{4}, \frac{4}{5}\right) $\text{[math]}$
The inverse function of

$\text{[math]}$ f(x)=\frac{8^{2 x}-8^{-2 x}}{8^{2 x}+8^{-2 x}}, x \in(-1,1) $\text{[math]}$ , is :

A $\text{[math]}$ \frac{1}{4} \log _{e}\left(\frac{1-x}{1+x}\right) $\text{[math]}$

B $\text{[math]}$ \frac{1}{4}\left(\log _{8} e\right) \log _{e}\left(\frac{1-x}{1+x}\right) $\text{[math]}$

C $\text{[math]}$ \frac{1}{4}\left(\log _{8} e\right) \log _{e}\left(\frac{1+x}{1-x}\right) $\text{[math]}$

D $\text{[math]}$ \frac{1}{4} \log _{e}\left(\frac{1+x}{1-x}\right) $\text{[math]}$
If $\text{[math]}$ g(x)=x^{2}+x-1 $\text{[math]}$ and

$\text{[math]}$ (g \circ f)(x)=4 x^{2}-10 x+5 $\text{[math]}$ , then $\text{[math]}$ f\left(\frac{5}{4}\right) $\text{[math]}$ is equal to:

(A) $\text{[math]}$ \frac{1}{2} $\text{[math]}$

B $\text{[math]}$ \frac{3}{2} $\text{[math]}$

(c) $\text{[math]}$ -\frac{1}{2} $\text{[math]}$

(D) $\text{[math]}$ -\frac{3}{2} $\text{[math]}$
For $\text{[math]}$ \mathrm{x} \in(0,3 / 2) $\text{[math]}$ , let $\text{[math]}$ \mathrm{f}(\mathrm{x})=\sqrt{x}, \mathrm{~g}(\mathrm{x})=\tan \mathrm{x} $\text{[math]}$ and $\text{[math]}$ \mathrm{h}(\mathrm{x})=\frac{1-x^{2}}{1+x^{2}} $\text{[math]}$ . If $\text{[math]}$ \phi(\mathrm{x})=(( $\text{[math]}$ hof $\text{[math]}$ ) \circ \mathrm{g})(\mathrm{x}) $\text{[math]}$ , then $\text{[math]}$ \phi\left(\frac{\pi}{3}\right) $\text{[math]}$ is equal to :

A $\text{[math]}$ \tan \frac{7 \pi}{12} $\text{[math]}$

B $\text{[math]}$ \tan \frac{11 \pi}{12} $\text{[math]}$

C $\text{[math]}$ \tan \frac{\pi}{12} $\text{[math]}$

D $\text{[math]}$ \tan \frac{5 \pi}{12} $\text{[math]}$
Let $\text{[math]}$ f(x)=x^{2}, x \in R $\text{[math]}$ . For any $\text{[math]}$ A \subseteq R $\text{[math]}$ , define $\text{[math]}$ g(A)=\{x \in R: f(x) \in A\} $\text{[math]}$ . If $\text{[math]}$ S=[0,4] $\text{[math]}$ , then which one of the following statements is not true ?

A $\text{[math]}$ g(f(S)) \neq S $\text{[math]}$

B $\text{[math]}$ f(g(S))=S $\text{[math]}$

C $\text{[math]}$ \mathrm{f}(\mathrm{g}(\mathrm{S})) \neq \mathrm{f}(\mathrm{S}) $\text{[math]}$

D $\text{[math]}$ g(f(S))=g(S) $\text{[math]}$
Let $\text{[math]}$ f(x)=e^{x}-x $\text{[math]}$ and $\text{[math]}$ g(x)=x^{2}-x, \forall x \in R $\text{[math]}$ . Then the set of all $\text{[math]}$ x \in R $\text{[math]}$ , where the function $\text{[math]}$ h(x)=(f \circ g)(x) $\text{[math]}$ is increasing, is :

A $\text{[math]}$ [0, \infty) $\text{[math]}$

B $\text{[math]}$ \left[-1,-\frac{1}{2}\right] \cup\left[\frac{1}{2}, \infty\right) $\text{[math]}$

C $\text{[math]}$ \left[-\frac{1}{2}, 0\right] \cup[1, \infty) $\text{[math]}$

D $\text{[math]}$ \left[0, \frac{1}{2}\right] \cup[1, \infty) $\text{[math]}$
The do of the definition of the function

$\text{[math]}$ f(x)=\frac{1}{4-x^{2}}+\log _{10}\left(x^{3}-x\right) $\text{[math]}$ is

A $\text{[math]}$ (-1,0) \cup(1,2) \cup(2, \infty) $\text{[math]}$

B $\text{[math]}$ (-2,-1) \cup(-1,0) \cup(2, \infty) $\text{[math]}$

C $\text{[math]}$ (1,2) \cup(2, \infty) $\text{[math]}$

D $\text{[math]}$ (-1,0) \cup(1,2) \cup(3, \infty) $\text{[math]}$
Let $\text{[math]}$ \sum_{k=1}^{10} f(a+k)=16\left(2^{10}-1\right) $\text{[math]}$ where the function $\text{[math]}$ f $\text{[math]}$ satisfies

$\text{[math]}$ f(x+y)=f(x) f(y) $\text{[math]}$ for all natural numbers $\text{[math]}$ x $\text{[math]}$ , $\text{[math]}$ y $\text{[math]}$ and $\text{[math]}$ f(1)=2 $\text{[math]}$ . then the natural number ‘a’ is

A 2

B 16

C 4

D 3
If the function $\text{[math]}$ f: R-\{1,-1\} \rightarrow $\text{[math]}$ A defined by

$\text{[math]}$ f(x)=\frac{x^{2}}{1-x^{2}} $\text{[math]}$ , is surjective, then $\text{[math]}$ A $\text{[math]}$ is equal to

A $\text{[math]}$ R-(-1,0) $\text{[math]}$

B $\text{[math]}$ \mathrm{R}-\{-1\} $\text{[math]}$

C $\text{[math]}$ R-[-1,0) $\text{[math]}$

D $\text{[math]}$ [0, \infty) $\text{[math]}$
Let $\text{[math]}$ f(x)=a^{x}(a>0) $\text{[math]}$ be written as

$\text{[math]}$ f(x)=f_{1}(x)+f_{2}(x) $\text{[math]}$ , where $\text{[math]}$ f_{1}(x) $\text{[math]}$ is an even function of $\text{[math]}$ f_{2}(x) $\text{[math]}$ is an odd function.

Then $\text{[math]}$ f_{1}(x+y)+f_{1}(x-y) $\text{[math]}$ equals

A $\text{[math]}$ 2 f_{1}(x) f_{1}(y) $\text{[math]}$

B $\text{[math]}$ 2 f_{1}(x+y) f_{1}(x-y) $\text{[math]}$

C $\text{[math]}$ 2 f_{1}(x) f_{2}(y) $\text{[math]}$

D $\text{[math]}$ 2 f_{1}(x+y) f_{2}(x-y) $\text{[math]}$
If $\text{[math]}$ f(x)=\log _{e}\left(\frac{1-x}{1+x}\right),|x|<1 $\text{[math]}$ then $\text{[math]}$ f\left(\frac{2 x}{1+x^{2}}\right) $\text{[math]}$ is equal to

(A) $\text{[math]}$ 2 f\left(x^{2}\right) $\text{[math]}$

B $\text{[math]}$ 2 f(x) $\text{[math]}$

C $\text{[math]}$ (f(x))^{2} $\text{[math]}$

D $\text{[math]}$ -2 f(x) $\text{[math]}$
The number of functions $\text{[math]}$ f $\text{[math]}$ from $\text{[math]}$ \{1,2,3, \ldots ., 20\} $\text{[math]}$ onto $\text{[math]}$ \{1,2,3, \ldots ., 20\} $\text{[math]}$ such that $\text{[math]}$ f(k) $\text{[math]}$ is a multiple of 3 , whenever $\text{[math]}$ k $\text{[math]}$ is a multiple of 4 , is :

(A) $\text{[math]}$ 6^{5} \times(15) $\text{[math]}$ !

(B) $\text{[math]}$ 5^{6} \times 15 $\text{[math]}$

C (15)! $\text{[math]}$ \times 6 $\text{[math]}$ !

D $\text{[math]}$ 5!\times 6 $\text{[math]}$ !
Let a function $\text{[math]}$ f:(0, \infty) \rightarrow(0, \infty) $\text{[math]}$ be defined by $\text{[math]}$ f(x)=\left|1-\frac{1}{x}\right| $\text{[math]}$ . Then $\text{[math]}$ f $\text{[math]}$ is :

A not injective but it is surjective

B neiter injective nor surjective

C injective only

D both injective as well as surjective
Let $\text{[math]}$ \mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} $\text{[math]}$ be defined by $\text{[math]}$ \mathrm{f}(\mathrm{x})=\frac{x}{1+x^{2}}, x \in R $\text{[math]}$ . Then the range of $\text{[math]}$ \mathrm{f} $\text{[math]}$ is :

A $\text{[math]}$ \left[-\frac{1}{2}, \frac{1}{2}\right] $\text{[math]}$

B $\text{[math]}$ R-\left[-\frac{1}{2}, \frac{1}{2}\right] $\text{[math]}$

C $\text{[math]}$ (-1,1)-\{0\} $\text{[math]}$

D $\text{[math]}$ \mathrm{R}-[-1,1] $\text{[math]}$
Let $\text{[math]}$ \mathrm{f}_{\mathrm{k}}(\mathrm{x})=\frac{1}{k}\left(\sin ^{k} x+\cos ^{k} x\right) $\text{[math]}$ for $\text{[math]}$ \mathrm{k}=1,2,3, \ldots $\text{[math]}$ Then for all $\text{[math]}$ \mathrm{x} \in \mathrm{R} $\text{[math]}$ , the value of $\text{[math]}$ \mathrm{f}_{4}(\mathrm{x})-\mathrm{f}_{6}(\mathrm{x}) $\text{[math]}$ is equal to

(A) $\text{[math]}$ \frac{1}{4} $\text{[math]}$

(B) $\text{[math]}$ \frac{5}{12} $\text{[math]}$

(C) $\text{[math]}$ \frac{-1}{12} $\text{[math]}$

D $\text{[math]}$ \frac{1}{12} $\text{[math]}$
Let $\text{[math]}$ N $\text{[math]}$ be the set of natural numbers and two functions $\text{[math]}$ f $\text{[math]}$ and $\text{[math]}$ g $\text{[math]}$ be defined as $\text{[math]}$ f, g: N \rightarrow N $\text{[math]}$ such that

$\text{[math]}$ f(n)=\left\{\begin{array}{cc}\frac{n+1}{2} ; & \text { if } n \text { is odd } \\ \frac{n}{2} ; & \text { if } n \text { is even }\end{array}\right. $\text{[math]}$

and $\text{[math]}$ g(n)=n-(-1)^{n} $\text{[math]}$ .

Then fog is –
A
neither one-one nor onto

B
onto but not one-one

C
both one-one and onto

D one-one but not onto
Let $\text{[math]}$ N $\text{[math]}$ be the set of natural numbers and two functions $\text{[math]}$ f $\text{[math]}$ and $\text{[math]}$ g $\text{[math]}$ be defined as $\text{[math]}$ f, g: N \rightarrow N $\text{[math]}$ such that

$\text{[math]}$ \mathrm{f}(\mathrm{n})=\left\{\begin{array}{cc}\frac{n+1}{2} ; & \text { if } n \text { is odd } \\ \frac{n}{2} ; & \text { if } n \text { is even }\end{array}\right. $\text{[math]}$

and $\text{[math]}$ g(n)=n-(-1)^{n} $\text{[math]}$ .

Then fog is –

A neither one-one nor onto

B onto but not one-one

C both one-one and onto

D one-one but not onto
For $\text{[math]}$ x \in R-\{0,1\} $\text{[math]}$ , Let $\text{[math]}$ \mathrm{f}_{1}(\mathrm{x})=\frac{1}{x}, \mathrm{f}_{2}(\mathrm{x})=1-\mathrm{x} $\text{[math]}$

and $\text{[math]}$ f_{3}(x)=\frac{1}{1-x} $\text{[math]}$ be three given

functions. If a function, $\text{[math]}$ J(x) $\text{[math]}$ satisfies

$\text{[math]}$ \left(f_{2} \circ J \circ f_{1}\right)(x)=f_{3}(x) $\text{[math]}$ then $\text{[math]}$ J(x) $\text{[math]}$ is equal to:

A $\text{[math]}$ f_{1}(x) $\text{[math]}$

B $\text{[math]}$ \frac{1}{x} f_{3}(x) $\text{[math]}$

C $\text{[math]}$ f_{2}(x) $\text{[math]}$

D $\text{[math]}$ f_{3}(x) $\text{[math]}$
Let $\text{[math]}$ f: A \rightarrow B $\text{[math]}$ be a function defined as $\text{[math]}$ f(x)=\frac{x-1}{x-2} $\text{[math]}$ , Where $\text{[math]}$ A=\mathbf{R}-\{2\} $\text{[math]}$ and $\text{[math]}$ B=\mathbf{R}-\{1\} $\text{[math]}$ . Then $\text{[math]}$ f $\text{[math]}$ is :

A invertible and $\text{[math]}$ f^{-1}(y)=\frac{3 y-1}{y-1} $\text{[math]}$

B invertible and $\text{[math]}$ f^{-1}(y)=\frac{2 y-1}{y-1} $\text{[math]}$

C invertible and $\text{[math]}$ f^{-1}(y)=\frac{2 y+1}{y-1} $\text{[math]}$

D not invertible
The function $\text{[math]}$ \mathrm{f}: \mathbf{N} \rightarrow \mathbf{N} $\text{[math]}$ defined by $\text{[math]}$ \mathrm{f}(\mathrm{x})=\mathrm{x}-5\left[\frac{x}{5}\right] $\text{[math]}$ , Where $\text{[math]}$ \mathbf{N} $\text{[math]}$ is the set of natural numbers and $\text{[math]}$ [\mathrm{x}] $\text{[math]}$ denotes the greatest integer less than or equal to $\text{[math]}$ x $\text{[math]}$ , is :

A one-one and onto

B one-one but not onto.

C onto but not one-one.

D neither one-one nor onto.
Let $\text{[math]}$ f(x)=2^{10} \cdot x+1 $\text{[math]}$ and $\text{[math]}$ g(x)=3^{10} \cdot x-1 $\text{[math]}$ . If (fog) $\text{[math]}$ (x)=x $\text{[math]}$ , then $\text{[math]}$ x $\text{[math]}$ is equal to :

(A) $\text{[math]}$ \frac{3^{10}-1}{3^{10}-2^{-10}} $\text{[math]}$

B $\text{[math]}$ \frac{2^{10}-1}{2^{10}-3^{-10}} $\text{[math]}$

C $\text{[math]}$ \frac{1-3^{-10}}{2^{10}-3^{-10}} $\text{[math]}$

D $\text{[math]}$ \frac{1-2^{-10}}{3^{10}-2^{-10}} $\text{[math]}$
The function $\text{[math]}$ f: R \rightarrow\left[-\frac{1}{2}, \frac{1}{2}\right] $\text{[math]}$ defined as

$\text{[math]}$ f(x)=\frac{x}{1+x^{2}} $\text{[math]}$ , is

A invertible

B injective but not surjective.

C surjective but not injective

D neither injective nor surjective.
Let $\text{[math]}$ a, \mathrm{~b}, \mathrm{c} \in R $\text{[math]}$ . If $\text{[math]}$ f(\mathrm{x})=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c} $\text{[math]}$ is such that

$\text{[math]}$ a+\mathrm{b}+\mathrm{c}=3 $\text{[math]}$ and $\text{[math]}$ f(\mathrm{x}+\mathrm{y})=f(\mathrm{x})+f(\mathrm{y})+\mathrm{xy}, \forall x, y \in R $\text{[math]}$ ,

then $\text{[math]}$ \sum_{n=1}^{10} f(n) $\text{[math]}$ is equal to

A 165

B 190

C 255

D 330
For $\text{[math]}$ \mathrm{x} \in \mathbf{R}, \mathrm{x} \neq 0 $\text{[math]}$ , Let $\text{[math]}$ \mathrm{f}_{0}(\mathrm{x})=\frac{1}{1-x} $\text{[math]}$ and

$\text{[math]}$ f_{n+1}(x)=f_{0}\left(f_{n}(x)\right), n=0,1,2, \ldots $\text{[math]}$

Then the value of $\text{[math]}$ f_{100}(3)+f_{1}\left(\frac{2}{3}\right)+f_{2}\left(\frac{3}{2}\right) $\text{[math]}$ is equal to:

A $\text{[math]}$ \frac{8}{3} $\text{[math]}$

B $\text{[math]}$ \frac{5}{3} $\text{[math]}$

C $\text{[math]}$ \frac{4}{3} $\text{[math]}$

D $\text{[math]}$ \frac{1}{3} $\text{[math]}$
If $\text{[math]}$ f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0 $\text{[math]}$ , and $\text{[math]}$ \mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\} $\text{[math]}$ ; then $\text{[math]}$ \mathrm{S} $\text{[math]}$ :

A is an empty set.

B contains exactly one element.

C contains exactly two elements.

D contains more than two elements.
The do of the function $\text{[math]}$ f(x)=\frac{1}{\sqrt{|x|-x}} $\text{[math]}$ is

A $\text{[math]}$ (0, \infty) $\text{[math]}$

B $\text{[math]}$ (-\infty, 0) $\text{[math]}$

C $\text{[math]}$ (-\infty, \infty)-\{0\} $\text{[math]}$

D $\text{[math]}$ (-\infty, \infty) $\text{[math]}$
Let $\text{[math]}$ f(x)=(x+1)^{2}-1, x \geq-1 $\text{[math]}$

Statement – 1: The set $\text{[math]}$ \left\{x: f(x)=f^{-1}(x)\right\}=\{0,-1\} $\text{[math]}$ .

Statement – 2: $\text{[math]}$ f $\text{[math]}$ is a bijection.

A Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement – 1

B Statement – 1 is true, Statement – 2 is true; Statement – 2 is not a correct explanation for Statement – 1

C Statement – 1 is true, Statement -2 is false

D Statement -1 is false, Statement -2 is true
For real $\text{[math]}$ x $\text{[math]}$ , let $\text{[math]}$ f(x)=x^{3}+5 x+1 $\text{[math]}$ , then
A $\text{[math]}$ f $\text{[math]}$ is one-one but not onto $\text{[math]}$ R $\text{[math]}$

B $\text{[math]}$ f $\text{[math]}$ is onto $\text{[math]}$ R $\text{[math]}$ but not one-one

C $\text{[math]}$ f $\text{[math]}$ is one-one and onto $\text{[math]}$ R $\text{[math]}$

D f is neither one-one nor onto $\text{[math]}$ R $\text{[math]}$
Let $\text{[math]}$ f: N \rightarrow Y $\text{[math]}$ be a function defined as $\text{[math]}$ \mathrm{f}(\mathrm{x})=4 \mathrm{x}+3 $\text{[math]}$ where

$\text{[math]}$ Y=\{y \in N, y=4 x+3 $\text{[math]}$ for some $\text{[math]}$ x \in N\} $\text{[math]}$ .

Show that $\text{[math]}$ f $\text{[math]}$ is invertible and its inverse is

A $\text{[math]}$ g(y)=\frac{3 y+4}{4} $\text{[math]}$

B $\text{[math]}$ g(y)=4+\frac{y+3}{4} $\text{[math]}$

C $\text{[math]}$ g(y)=\frac{y+3}{4} $\text{[math]}$

D $\text{[math]}$ g(y)=\frac{y-3}{4} $\text{[math]}$

\item

The largest interval lying in $\text{[math]}$ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $\text{[math]}$ for which the function

$\text{[math]}$ f(x)=4^{-x^{2}}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x) $\text{[math]}$ ,

is defined, is

A $\text{[math]}$ \left[-\frac{\pi}{4}, \frac{\pi}{2}\right) $\text{[math]}$

B $\text{[math]}$ \left[0, \frac{\pi}{2}\right) $\text{[math]}$

C $\text{[math]}$ [0, \pi] $\text{[math]}$

D $\text{[math]}$ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $\text{[math]}$
Let $\text{[math]}$ f:(-1,1) \rightarrow B $\text{[math]}$ , be a function defined by

$\text{[math]}$ f(x)=\tan ^{-1} \frac{2 x}{1-x^{2}} $\text{[math]}$ ,

then $\text{[math]}$ f $\text{[math]}$ is both one-one and onto when $\text{[math]}$ \mathrm{B} $\text{[math]}$ is the interval

A $\text{[math]}$ \left(0, \frac{\pi}{2}\right) $\text{[math]}$

B $\text{[math]}$ \left[0, \frac{\pi}{2}\right) $\text{[math]}$

C $\text{[math]}$ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $\text{[math]}$
D $\text{[math]}$ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $\text{[math]}$
A real valued function $\text{[math]}$ f(x) $\text{[math]}$ satisfies the functional equation

$\text{[math]}$ f(x-y)=f(x) f(y)-f(a-x) f(a+y) $\text{[math]}$

where $\text{[math]}$ a $\text{[math]}$ is given constant and $\text{[math]}$ f(0)=1, f(2 a-x) $\text{[math]}$ is equal to

A $\text{[math]}$ -f(x) $\text{[math]}$

B $\text{[math]}$ f(x) $\text{[math]}$

C $\text{[math]}$ f(a)+f(a-x) $\text{[math]}$

D $\text{[math]}$ f(-x) $\text{[math]}$
The range of the function $\text{[math]}$ \mathrm{f}(\mathrm{x})={ }^{7-x} P_{x-3} $\text{[math]}$ is

A $\text{[math]}$ \{1,2,3,4,5\} $\text{[math]}$

B $\text{[math]}$ \{1,2,3,4,5,6\} $\text{[math]}$

C $\text{[math]}$ \{1,2,3,4\} $\text{[math]}$

D $\text{[math]}$ \{1,2,3\} $\text{[math]}$
If $\text{[math]}$ f: R \rightarrow S $\text{[math]}$ , defined by

$\text{[math]}$ f(x)=\sin x-\sqrt{3} \cos x+1 $\text{[math]}$ ,

is onto, then the interval of $\text{[math]}$ S $\text{[math]}$ is

A $\text{[math]}$ [-1,3] $\text{[math]}$

B $\text{[math]}$ [-1,1] $\text{[math]}$

C $\text{[math]}$ [0,1] $\text{[math]}$

D $\text{[math]}$ [0,3] $\text{[math]}$
The graph of the function $\text{[math]}$ y=f(x) $\text{[math]}$ is symmetrical about the line $\text{[math]}$ x=2 $\text{[math]}$ , then

A $\text{[math]}$ f(x)=-f(-x) $\text{[math]}$

B $\text{[math]}$ f(2+x)=f(2-x) $\text{[math]}$

C $\text{[math]}$ f(x)=f(-x) $\text{[math]}$

D $\text{[math]}$ f(x+2)=f(x-2) $\text{[math]}$
The do of the function

$\text{[math]}$ f(x)=\frac{\sin ^{-1}(x-3)}{\sqrt{9-x^{2}}} $\text{[math]}$

A $\text{[math]}$ [1,2] $\text{[math]}$

B $\text{[math]}$ [2,3) $\text{[math]}$

C $\text{[math]}$ [1,2) $\text{[math]}$

D $\text{[math]}$ [2,3] $\text{[math]}$
The function $\text{[math]}$ f(x)=\log \left(x+\sqrt{x^{2}+1}\right) $\text{[math]}$ , is

A neither an even nor an odd function

B an even function

C an odd function

D a periodic function
A function $\text{[math]}$ f $\text{[math]}$ from the set of natural numbers to integers defined by

$\text{[math]}$ $\text{[math]}$
f(n)=\left\{\begin{array}{l}
\frac{n-1}{2}, \text { when } n \text { is odd } \\
-\frac{n}{2}, \text { when } n \text { is even }
\end{array}\right.
$\text{[math]}$ $\text{[math]}$

is

A neither one -one nor onto

B one-one but not onto

C onto but not one-one

D one-one and onto both
If $\text{[math]}$ f: R \rightarrow R $\text{[math]}$ satisfies $\text{[math]}$ f(\mathrm{x}+\mathrm{y})=f(\mathrm{x})+f(\mathrm{y}) $\text{[math]}$ , for all $\text{[math]}$ \mathrm{x}, \mathrm{y} \in \mathrm{R} $\text{[math]}$ and $\text{[math]}$ f(1)=7 $\text{[math]}$ , then $\text{[math]}$ \sum_{r=1}^{n} f(r) $\text{[math]}$ is

A $\text{[math]}$ \frac{7 n(n+1)}{2} $\text{[math]}$

B $\text{[math]}$ \frac{7 n}{2} $\text{[math]}$

C $\text{[math]}$ \frac{7(n+1)}{2} $\text{[math]}$

D $\text{[math]}$ 7 n+(n+1) $\text{[math]}$
Do of definition of the function $\text{[math]}$ \mathrm{f}(\mathrm{x})=\frac{3}{4-x^{2}}+\log _{10}\left(x^{3}-x\right) $\text{[math]}$ , is

A $\text{[math]}$ (-1,0) \cup(1,2) \cup(2, \infty) $\text{[math]}$

B $\text{[math]}$ (1,2) $\text{[math]}$

C $\text{[math]}$ (-1,0) \cup(1,2) $\text{[math]}$

D $\text{[math]}$ (1,2) \cup(2, \infty) $\text{[math]}$
The period of $\text{[math]}$ \sin ^{2} \theta $\text{[math]}$ is

(A) $\text{[math]}$ \pi^{2} $\text{[math]}$

(B) $\text{[math]}$ \pi $\text{[math]}$

C $\text{[math]}$ 2 \pi $\text{[math]}$

D $\text{[math]}$ \pi / 2 $\text{[math]}$

\item
Which one is not periodic?

A $\text{[math]}$ |\sin 3 x|+\sin ^{2} x $\text{[math]}$

B $\text{[math]}$ \cos \sqrt{x}+\cos ^{2} x $\text{[math]}$

C $\text{[math]}$ \cos 4 x+\tan ^{2} x $\text{[math]}$

D $\text{[math]}$ \cos 2 x+\sin x $\text{[math]}$

\item
The domain of $\text{[math]}$ \sin ^{-1}\left[\log _{3}\left(\frac{x}{3}\right)\right] $\text{[math]}$ is

A $\text{[math]}$ [1,9] $\text{[math]}$

B $\text{[math]}$ [-1,9] $\text{[math]}$

C $\text{[math]}$ [9,1] $\text{[math]}$

D $\text{[math]}$ [-9,-1] $\text{[math]}$
Let the domain of the function $\text{[math]}$ f(x) = \cos^{-1} \left(\frac{4x + 5}{3x – 7}\right) $\text{[math]}$ be $\text{[math]}$ [\alpha, \beta] $\text{[math]}$ and the domain of $\text{[math]}$ g(x) = \log_2 (2 – 6 \log_{27} (2x + 5)) $\text{[math]}$ be $\text{[math]}$ (\gamma, \delta) $\text{[math]}$ . Then $\text{[math]}$ [7(\alpha + \beta) + 4(\gamma + \delta)] $\text{[math]}$ is equal to
% Question 2
Let $\text{[math]}$ A = {(x, y) : 2x + 3y = 23, x, y \in \mathbb{N}} $\text{[math]}$ and $\text{[math]}$ B = {x : (x, y) \in A} $\text{[math]}$ . Then the number of one-one functions from $\text{[math]}$ A $\text{[math]}$ to $\text{[math]}$ B $\text{[math]}$ is equal to
% Question 3
If a function $\text{[math]}$ f $\text{[math]}$ satisfies $\text{[math]}$ f(m + n) = f(m) + f(n) $\text{[math]}$ for all $\text{[math]}$ m, n \in \mathbb{N} $\text{[math]}$ and $\text{[math]}$ f(1) = 1 $\text{[math]}$ , then the largest natural number $\text{[math]}$ \lambda $\text{[math]}$ such that $\text{[math]}$ \sum_{k=1}^{\lambda} f(\lambda + k) \leq (2022)^2 $\text{[math]}$ is equal to
% Question 4
If the range of $\text{[math]}$ f(\theta) = \frac{\sin^4 \theta + 3 \cos^2 \theta}{\sin^4 \theta + \cos^2 \theta} $\text{[math]}$ , $\text{[math]}$ \theta \in \mathbb{R} $\text{[math]}$ is $\text{[math]}$ [\alpha, \beta] $\text{[math]}$ , then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\text{[math]}$ \frac{\alpha}{\beta} $\text{[math]}$ , is equal to.
If $\text{[math]}$ S = { a \in \mathbb{R} : [2a – 1] = 3[a] + 2{a} } $\text{[math]}$ , where $\text{[math]}$ [t] $\text{[math]}$ denotes the greatest integer less than or equal to $\text{[math]}$ t $\text{[math]}$ and $\text{[math]}$ {t} $\text{[math]}$ represents the fractional part of $\text{[math]}$ t $\text{[math]}$ , then $\text{[math]}$ 72 \sum_{a \in S} a $\text{[math]}$ is equal to.
% Question 2
Consider the function $\text{[math]}$ f : \mathbb{R} \rightarrow \mathbb{R} $\text{[math]}$ defined by $\text{[math]}$ f(x) = \frac{2x}{\sqrt{1 + 9x^2}} $\text{[math]}$ . If the composition of
$\text{[math]}$ f, (f \circ f \circ f \circ \cdots \circ f)(x) (10-times)= \frac{2^{10}x}{\sqrt{1 + 9\alpha x^2}} $\text{[math]}$ , then the value of $\text{[math]}$ \sqrt{3\alpha} + 1 $\text{[math]}$ is equal to.

% Question 3
Let $\text{[math]}$ A = \{1, 2, 3, \ldots, 7\} $\text{[math]}$ and let $\text{[math]}$ P(A) $\text{[math]}$ denote the power set of $\text{[math]}$ A $\text{[math]}$ . If the number of functions $\text{[math]}$ f : A \rightarrow P(A) $\text{[math]}$ such that $\text{[math]}$ a \in f(a), \forall a \in A $\text{[math]}$ is $\text{[math]}$ m^n $\text{[math]}$ , $\text{[math]}$ m $\text{[math]}$ and $\text{[math]}$ n \in \mathbb{N} $\text{[math]}$ and $\text{[math]}$ m $\text{[math]}$ is least, then $\text{[math]}$ m + n $\text{[math]}$ is equal to.
% Question 4
Let $\text{[math]}$ A = \{1, 2, 3, 4, 5\} $\text{[math]}$ and $\text{[math]}$ B = \{1, 2, 3, 4, 5, 6\} $\text{[math]}$ . Then the number of functions $\text{[math]}$ f : A \rightarrow B $\text{[math]}$ satisfying $\text{[math]}$ f(1) + f(2) = f(4) – 1 $\text{[math]}$ is equal to.
Let $\text{[math]}$ R = \{a, b, c, d, e\} $\text{[math]}$ and $\text{[math]}$ S = \{1, 2, 3, 4\} $\text{[math]}$ . Total number of onto functions $\text{[math]}$ f : R \to S $\text{[math]}$ such that $\text{[math]}$ f(a) \neq 1 $\text{[math]}$ is equal to.
If domain of the function $\text{[math]}$ \log_e \left( \frac{6x^2 + 5x + 1}{2x – 1} \right) + \cos^{-1} \left( \frac{2x^2 – 3x + 4}{3x – 5} \right) $\text{[math]}$ is $\text{[math]}$ (\alpha, \beta] \cup (\gamma, \delta] $\text{[math]}$ , then $\text{[math]}$ 18 (\alpha^2 + \beta^2 + \gamma^2 + \delta^2) $\text{[math]}$ is equal to.
Let $\text{[math]}$ A = \{1, 2, 3, 5, 8, 9\} $\text{[math]}$ . Then the number of possible functions $\text{[math]}$ f : A \to A $\text{[math]}$ such that $\text{[math]}$ f(m \cdot n) = f(m) \cdot f(n) $\text{[math]}$ for every $\text{[math]}$ m, n \in A $\text{[math]}$ with $\text{[math]}$ m \cdot n \in A $\text{[math]}$ is equal to.
Let $\text{[math]}$ S = \{1, 2, 3, 4, 5, 6\} $\text{[math]}$ . Then the number of one-one functions $\text{[math]}$ f : S \to P(S) $\text{[math]}$ , where $\text{[math]}$ P(S) $\text{[math]}$ denote the power set of $\text{[math]}$ S $\text{[math]}$ , such that $\text{[math]}$ f(n) \subset f(m) $\text{[math]}$ where $\text{[math]}$ n < m $\text{[math]}$ is.
Suppose $\text{[math]}$ f $\text{[math]}$ is a function satisfying $\text{[math]}$ f(x + y) = f(x) + f(y) $\text{[math]}$ for all $\text{[math]}$ x, y \in \mathbb{N} $\text{[math]}$ and $\text{[math]}$ f(1) = \frac{1}{5} $\text{[math]}$ . If $\text{[math]}$ \sum_{n=1}^{m} \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12} $\text{[math]}$ , then $\text{[math]}$ m $\text{[math]}$ is equal to.
For some $\text{[math]}$ a, b, c \in \mathbb{N} $\text{[math]}$ , let $\text{[math]}$ f(x) = ax – 3 $\text{[math]}$ and $\text{[math]}$ g(x) = x^b + c, x \in \mathbb{R} $\text{[math]}$ . If $\text{[math]}$ (f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3} $\text{[math]}$ , then $\text{[math]}$ (f \circ g)(a) + (g \circ f)(b) $\text{[math]}$ is equal to.
For $\text{[math]}$ p, q \in \mathbb{R} $\text{[math]}$ , consider the real valued function $\text{[math]}$ f(x) = (x – p)^2 – q, x \in \mathbb{R} $\text{[math]}$ and $\text{[math]}$ q > 0 $\text{[math]}$ . Let $\text{[math]}$ a_1, a_2, a_3 $\text{[math]}$ and $\text{[math]}$ a_4 $\text{[math]}$ be in an arithmetic progression with mean $\text{[math]}$ p $\text{[math]}$ and positive common difference. If $\text{[math]}$ |f(a_i)| = 500 $\text{[math]}$ for all $\text{[math]}$ i = 1, 2, 3, 4 $\text{[math]}$ , then the absolute difference between the roots of $\text{[math]}$ f(x) = 0 $\text{[math]}$ is.
The number of functions $\text{[math]}$ f $\text{[math]}$ from the set $\text{[math]}$ A = \{ x \in \mathbb{N} : x^2 – 10x + 9 \leq 0 \} $\text{[math]}$ to the set $\text{[math]}$ B = \{ n^2 : n \in \mathbb{N} \} $\text{[math]}$ such that $\text{[math]}$ f(x) \leq (x – 3)^2 + 1 $\text{[math]}$ for every $\text{[math]}$ x \in A $\text{[math]}$ , is.
Let $\text{[math]}$ f(x) = 2x^2 – x – 1 $\text{[math]}$ and $\text{[math]}$ S = \{ n \in \mathbb{Z} : |f(n)| \leq 800 \} $\text{[math]}$ . Then, the value of $\text{[math]}$ \sum_{n \in S} f(n) $\text{[math]}$ is equal to.
Let $\text{[math]}$ f(x) $\text{[math]}$ be a quadratic polynomial with leading coefficient 1 such that $\text{[math]}$ f(0) = p, p \neq 0 $\text{[math]}$ , and $\text{[math]}$ f(1) = \frac{1}{3} $\text{[math]}$ . If the equations $\text{[math]}$ f(x) = 0 $\text{[math]}$ and $\text{[math]}$ f \circ f \circ f(x) = 0 $\text{[math]}$ have a common real root, then $\text{[math]}$ f(-3) $\text{[math]}$ is equal to.
Let $\text{[math]}$ f(x) $\text{[math]}$ and $\text{[math]}$ g(x) $\text{[math]}$ be two real polynomials of degree 2 and 1 respectively. If $\text{[math]}$ f(g(x)) = 8x^2 – 2x $\text{[math]}$ and $\text{[math]}$ g(f(x)) = 4x^2 + 6x + 1 $\text{[math]}$ , then the value of $\text{[math]}$ f(2) + g(2) $\text{[math]}$ is.
Let $\text{[math]}$ c, k \in \mathbb{R} $\text{[math]}$ . If $\text{[math]}$ f(x) = (c + 1)x^2 + (1 – c^2)x + 2k $\text{[math]}$ and $\text{[math]}$ f(x + y) = f(x) + f(y) – xy $\text{[math]}$ for all $\text{[math]}$ x, y \in \mathbb{R} $\text{[math]}$ , then the value of $\text{[math]}$ [2(f(1) + f(2) + f(3) + \cdots + f(20))] $\text{[math]}$ is equal to.
Let $\text{[math]}$ S = \{1, 2, 3, 4\} $\text{[math]}$ . Then the number of elements in the set $\text{[math]}$ \{ f : S \times S \to S : f $\text{[math]}$ is onto and $\text{[math]}$ f(a, b) = f(b, a) \geq a \vee (a, b) \in S \times S \} $\text{[math]}$ is.
Let $\text{[math]}$ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} $\text{[math]}$ . Define $\text{[math]}$ f : S \to S $\text{[math]}$ as
$\text{[math]}$ $\text{[math]}$ f(n) =
\begin{cases}
2n & \text{if } n = 1, 2, 3, 4, 5 \\
2n – 11 & \text{if } n = 6, 7, 8, 9, 10
\end{cases} $\text{[math]}$ $\text{[math]}$
Let $\text{[math]}$ g : S \to S $\text{[math]}$ be a function such that $\text{[math]}$ f \circ g(n) =
\begin{cases}
n + 1 & \text{if } n \text{ is odd} \\
n – 1 & \text{if } n \text{ is even}
\end{cases}
$\text{[math]}$ . Then $\text{[math]}$ g(10)g(1) + g(2) + g(3) + g(4) + g(5) $\text{[math]}$ is equal to.
Let $\text{[math]}$ f : \mathbb{R} \to \mathbb{R} $\text{[math]}$ be a function defined by $\text{[math]}$ f(x) = \frac{2e^{2x}}{e^{2x} + e^{-x}} $\text{[math]}$ . Then $\text{[math]}$ f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \cdots + f\left(\frac{99}{100}\right) $\text{[math]}$ is equal to.
Let $\text{[math]}$ f : \mathbb{R} \to \mathbb{R} $\text{[math]}$ be a function defined by $\text{[math]}$ f(x) = \left(2 \left(1 – \frac{x^{25}}{2}\right) (2 + x^{25})\right)^{\frac{1}{50}} $\text{[math]}$ . If the function $\text{[math]}$ g(x) = f(f(f(x))) + f(f(x)) $\text{[math]}$ , then the greatest integer less than or equal to $\text{[math]}$ g(1) $\text{[math]}$ is.
The number of one-one functions $\text{[math]}$ f : \{a, b, c, d\} \to \{0, 1, 2, \ldots, 10\} $\text{[math]}$ such that $\text{[math]}$ 2f(a) – f(b) + 3f(c) + f(d) = 0 $\text{[math]}$ is.
Let $\text{[math]}$ S = \{1, 2, 3, 4, 5, 6, 7\} $\text{[math]}$ . Then the number of possible functions $\text{[math]}$ f : S \to S $\text{[math]}$ such that $\text{[math]}$ f(m \cdot n) = f(m) \cdot f(n) $\text{[math]}$ for every $\text{[math]}$ m, n \in S $\text{[math]}$ and $\text{[math]}$ m \cdot n \in S $\text{[math]}$ is equal to.
Let $\text{[math]}$ A = \{0, 1, 2, 3, 4, 5, 6, 7\} $\text{[math]}$ . Then the number of bijective functions $\text{[math]}$ f : A \to A $\text{[math]}$ such that $\text{[math]}$ f(1) + f(2) = 3 – f(3) $\text{[math]}$ is equal to.
If $\text{[math]}$ f(x) $\text{[math]}$ and $\text{[math]}$ g(x) $\text{[math]}$ are two polynomials such that the polynomial $\text{[math]}$ P(x) = f(x^3) + g(x^3) $\text{[math]}$ is divisible by $\text{[math]}$ x^2 + x + 1 $\text{[math]}$ , then $\text{[math]}$ P(1) $\text{[math]}$ is equal to.
If $\text{[math]}$ a + \alpha = 1 $\text{[math]}$ , $\text{[math]}$ b + \beta = 2 $\text{[math]}$ and $\text{[math]}$ a f(x) + \alpha f\left(\frac{1}{x}\right) = bx + \frac{\beta}{x}, x \neq 0 $\text{[math]}$ , then the value of the expression $\text{[math]}$ \frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}} $\text{[math]}$ is.
Suppose that a function $\text{[math]}$ f : \mathbb{R} \to \mathbb{R} $\text{[math]}$ satisfies $\text{[math]}$ f(x + y) = f(x)f(y) $\text{[math]}$ for all $\text{[math]}$ x, y \in \mathbb{R} $\text{[math]}$ and $\text{[math]}$ f(1) = 3 $\text{[math]}$ . If $\text{[math]}$ \sum_{i=1}^{n} f(i) = 363 $\text{[math]}$ , then $\text{[math]}$ n $\text{[math]}$ is equal to.
Let $\text{[math]}$ A = \{a, b, c\} $\text{[math]}$ and $\text{[math]}$ B = \{1, 2, 3, 4\} $\text{[math]}$ . Then the number of elements in the set $\text{[math]}$ C = \{ f : A \to B \mid 2 \in f(A) \text{ and } f \text{ is not one-one} \} $\text{[math]}$ is.