Functions - Calculus - Mathematics Previous Year Questions

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

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

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?

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

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

The range of the function defined by

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

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

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

Which one of the following functions is a bijection?

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

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

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 $f(\mathrm{l})+a \cdot B+b \cdot A=$

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)

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 $f(-4)=$

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 ${ }^n C_r$ 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

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