A function $f: R \rightarrow R$ defined by

$ f(x)=\left\{\begin{array}{c} 2 x+3, x \leq \frac{4}{3} \\ -3 x^2+8 x, x>\frac{4}{3} \end{array}\right. \text { is } $

If $2^{4 n+3}+3^{3 n+1}$ is divisible by $P$ for all natural numbers $n$, then $P$ is

Consider the following statements

Statement $\mathrm{I} \cosh ^{-1} x=\tanh ^{-1} x$ has no solution

Statement II $\cosh ^{-1} x=\operatorname{coth}^{-1} x$ has only one solution

The correct answer is

The domain of the real valued function $f(x)=\log _{\sqrt{2}}\left(\sqrt{x^2+x}+\sqrt{x^2-x}\right)$ is

If $\frac{x+1}{x^3(x-1)}=\frac{a}{x}+\frac{b}{x^2}+\frac{c}{x^3}+\frac{d}{x-1}$, then

Let $f: R \rightarrow R$ be defined by $f(x)=5^{-|x|}+\operatorname{sgn}\left(5^{-x}\right)$, where sgn $x$ denotes signum function of $x$. Then $f$ is

If the range of the real valued function $f(x)=\frac{x^2+x+k}{x^2-x+k}$ is $\left[\frac{1}{3}, 3\right]$, then $k=$

For a real number ' $a$ ', if a real valued function $f(x)=4 x^3+a x^2+3 x-2$ is monotonic in its domain, then the range of ' $a$ ' is

If $D \subseteq R$ and $f: D \rightarrow R$ defined by $f(x)=\frac{x^2+x+a}{x^2-x+a}$ is a surjection, then ' $a$ ' lies in the interval.

If the domain of the real valued function $f(x)=\frac{1}{\sqrt{\log _{\frac{1}{3}}\left(\frac{x-1}{2-x}\right)}}$ is $(a, b)$, then $2 b=$

A real valued function $f:[4, \infty) \rightarrow R$ is defined as $f(x)=\left(x^2+x+1\right)^{\left(x^2-3 x-4\right)}$, then $f$ is

If $f: R-\{0\} \rightarrow R$ is defined by $3 f(x)+4 f\left(\frac{1}{x}\right)=\frac{2-x}{x}$ then $f(3)=$

The inverse of the function $y=\frac{10^x-10^{-x}}{10^x+10^{-x}}+1$ is $x=$

If $f(x)=\tan \left(\frac{\pi}{\sqrt{x+1}+4}\right)$ is a real valued function, then the range of $f$ is

If $\frac{x^3+3}{(x-3)^3}=a+\frac{b}{x-3}+\frac{c}{(x-3)^2}+\frac{d}{(x-3)^3}$, then $(a+d)-(b+c)=$

$f(x)=a x^{2}+b x+c$ is an even function and

$g(x)=p x^{3}+q x^{2}+r x$ is an odd function.

If $h(x)=f(x)+g(x)$ and $h(-2)=0$, then $8 p+4 q+2 r=$

Let $f: R \rightarrow R$ be a function defined by

$ f(x)=\left\{\begin{array}{cc} x^2-4 x+3, & \text { if } x<2 \\ x-3, & \text { if } x \geq 2 \end{array}\right. $

Then, the number of real numbers $x$ for which $f(x)=8$ is

If $f(x)$ and $g(x)$ are two real valued functions such that $f(x)=3 x-2$ and $g(x)=x^2+2$, then $[(g \circ f)+(f \circ g)](x)=$

If $f(x)$ is a real valued function defined by $f(x)=\frac{a x^{10}+b x^8+c x^6+d x^4+e x^2+12 x+15}{x}(x \neq 0)$ and $f(4)=-4$, then $f(-4)=$

If ${ }^n C_r$ denotes the number of combinations of $n$ distinct things taken $r$ at a time, then the domain of the function $g(x)={ }^{(16-x)} C_{(2 x-1)}$ is

Let $X=\left\{\left.\left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \right\rvert\, a, b, c, d \in R\right\}$. If $f: X \rightarrow R$ is defined by $f(A)=\operatorname{det}(A) . \forall A \in X$, then $f$ is

The period of the function $f(x)=e^{\log (\sin x)}+(\tan x)^3-\operatorname{cosec}(3 x-5)$ is

Which one of the following functions is a bijection?

The domain of the real valued function $f(x)=\frac{\sqrt{|x|-x}}{\sqrt{x-[x]}}$ is

The range of the function defined by

$ f(x)=\left\{\begin{array}{lc} 2 x-3, & \text { if } x<-1 \\ 1-x^2, & \text { if }-1 \leq x \leq 1 \text { is } \\ 3 x^2+2, & \text { if } x>1 \end{array}\right. $

If $\sinh x=-\frac{4}{3}$, then $\sinh 2 x+\cosh 2 x=$

If the function $f: R \rightarrow R$ is defined by

$ f(x)= \begin{cases}2 x-3, & \text { if } x<-2 \\ x^2-1, & \text { if }-2 \leq x \leq 2 \\ 3 x+2, & \text { if } x>2\end{cases} $

then $f$ is

The domain of the real valued function

$ f(x)=\frac{\sqrt{\log _{10}\left(\frac{x}{x-2}\right)}}{\sqrt{[x]^2-5[x]+6}} \text { is } $

(Here, $[x]$ denotes the greatest integer function)

The range of the real valued function $f(x)=\frac{1}{x-|x|}$ is

If $\frac{6 x^4+13 x^3+2 x^2-x+3}{2 x^2+3 x-2}=f(x)+\frac{A}{a x-1}+\frac{B}{x+b}$, then $f(\mathrm{l})+a \cdot B+b \cdot A=$

If $f: R \rightarrow R$ is defined by $f(x)=2 x+\sin x, x \in R$, then $f$ is

If $[x]$ represents the greatest integer function, then the set of all real values of $x$ for which $f(x)=\sqrt{\frac{[x]-x}{x-[x]}}$ is real is

If $[x]$ denotes the greatest integer $\leq x$, then the range of the real valued function $f(x)=\frac{1}{\sqrt{x-[x]}}$ is

Assertion (A) $\operatorname{coth} x=\frac{1-k}{1+k}(0 < k < 2)$.

Reason (R) The graph of $y=\tanh x$ always lies between the lines $y=-1$ and $y=1$

The correct option among the following is

The domain of the real valued function $f(x)=\sqrt{\frac{2 x^2-7 x+5}{3 x^2-5 x-2}}$ is

The range of the real valued function $f(x)=|x-2|+|x-3|$ is

Let $f: A \rightarrow B$ be defined as $f(x)=\frac{1}{2}-\tan \left(\frac{\pi x}{2}\right)$ and $g: B \rightarrow C$ be defined as $g(x)=\sqrt{3+4 x-4 x^2}$. If $A, B$ and $C$ are subsets of $R$ and $f$ is an onto function, then the range of the function $f(x)$ is

If $D$ is the domain and $G$ is the range of the real valued function $f(x)=\sqrt{\frac{1-x^2}{1+x^2}}$, then $D \cap G=$

The set of all real values of $x$ for which $f(x)=\log _2\left(2^x-2\right)+\sqrt{1-x}$ is also real is

Let $f(x)=1-x, g(x)=\frac{1}{1-x}, h(x)=\frac{1}{x}$ be three functions, for $x \neq(0,1)$. If a function $F(x)$ satisfies $f(F(h(x)))=g(x)$, then

If the minimum value of $\cos (\sinh (\log x)+\cosh (\log x))$ is $k$, then $\cosh (k+1)=$

Let $R$ be the set of all real number

Statement I The function $f:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$ defined by $f(x)=\sec x+\tan x$ is one-one function.

Statement II The function $f:[0, \infty) \rightarrow R$ defined by $f(x)=x^2$ is a one-one function

Which of the above statements is (are) true?

Let $R$ be the set of all real numbers. Let $f: R \rightarrow R$ be a function defined by

$ f(x)=\left\{\begin{array}{rcc} 2 x-5, & \text { if } & x<-3 \\ x+2, & \text { if } & -3 \leq x<5 \\ 3 x+1, & \text { if } & x \geq 5 \end{array}\right. $

Match the following

$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A } & f(-5)+f(0)+f(-1)= & \text { I } & 16 \\ \hline \text { B } & f(f(5)+10 f(-3))= & \text { II } & 40 \\ \hline \text { C } & f(|f(-4)|)= & \text { III } & -32 \\ \hline \text { D } & f(f(f(1)))= & \text { IV } & -12 \\ \hline & & \text { V } & 19 \\ \hline \end{array} $

The domain of the real valued function $f(x)=\frac{\sqrt{6 x^2+5 x-6}}{\sqrt{4-x}-\sqrt{x+4}}$ is

If $[x]$ represents the greatest integer $\leq x$, then the range of the real valued function $f(x)=\frac{1}{\sqrt{[x]^2+[x]-2}}$ is

The number of bijective functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ such that $f(x+y)=f(x)+f(y) \forall x, y \in \mathbf{Z}$, is

For each $n \in \mathbf{N}$, let $A_n=\{(n+1) k / k \in \mathbf{N}\}$ and $X=\bigcup_{n \in \mathbf{N}} A_n \cdot A$ mapping $f: X \rightarrow N$ defined by $f(x)=x$, $\forall x \in \mathbf{X}$, is

If $f: Z \rightarrow N$ is defined by

$ f(n)=\left\{\begin{array}{cll} 2 n, & \text { if } & n>0 \\ 1, & \text { if } & n=0, \text { then } f \text { is } \\ -2 n-1, & \text { if } & n<0 \end{array}\right. $

If $\frac{x^5-5}{x^3+x^2}=f(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$, then the larger value of $K$ for which $f(K)+A+B+C=1$, is

If $f(x)=x-\frac{1}{x}, x \neq 0$, then $3 f(x)=$

Let $[\cdot]$ denote greatest integer function. If $f(x)=[x]$ and $g(x)=3\left[\frac{x}{3}\right]$, then the set of all real $x$ such that $f(x)=g(x)$ is

A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is such that $f(\mathrm{l})=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed interms of $f(1), f(2)$ and $f(4)$ is

Let $f:[0,10] \rightarrow[1,20]$ be a function defined as

$ f(x)=\left\{\begin{array}{ll} \frac{60-5 x}{3}, & 0 \leq x \leq 6 \\ 10, & 6 \leq x \leq 7 \\ 31-3 x, & 7 \leq x \leq 10 \end{array} \text { then } f\right. \text { is } $

The domain of the function, $f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^2}{4}\right)}$ is