Odd Function

Q. If $f\left(x\right)=\frac{\cos x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}},$ where $x$ is not an integral multiple of $\pi$ and $[.]$ denotes the greatest integer function, then

1. $f\left(x\right)$ is an even function
2. None of these
3. $f\left(x\right)$ is an odd function
4. $f\left(x\right)$ is neither even nor odd

Q. If $f:[-2,2]\to R$ defined by $f\left(x\right)=x^3+\tan x+\left[\frac{x^2+1}{p}\right]$ is an odd function, then the least value of $\left[p\right]$ is
($[.]$ represents the greatest integer function)

Q. If $f\left(x\right)={\sin }^{-1}\left[e^x\right]+{\sin }^{-1}\left[e^{-x}\right],$ where $[.]$ is the greatest integer function, then

1. domain of $f\left(x\right)=(-\ln 2,\ln 2)$
2. range of $f\left(x\right)=\left\{\pi \right\}$
3. $f\left(x\right)$ is discontinuous at $x=0$
4. $f\left(x\right)={\cos }^{-1}x$ has only one solution

Q.

If $I=\int \frac{8x-11}{\sqrt{5+2x-x^2}}dx=p\sqrt{5+2x-x^2}+q{\sin }^{-1}\left(\frac{x-1}{\sqrt{6}}\right)+C,$ then the value of $|p+q|$ ;$(p,q\in R)$ is
(where $C$ is integration constant)

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^2-3,& x\le -5\\ x+\lambda ,& -5<x<-1\\ (\mu -7)(|1-x|+|1+x|),& -1\le x\le 1\\ x+6,& 1<x<5\\ 3-x^2,& x\ge 5\end{array}$
If $f\left(x\right)$ is an odd function, then the value of $\lambda +\mu$ is

Q. Consider $f\left(x\right)={\sin }^{-1}(1-2\sqrt{x})+{\sec }^{-1}\left(\frac{1}{2\sqrt{\sqrt{x}-x}}\right)+{\tan }^{-1}\left(\frac{\sqrt{2}-1-\sqrt{x}}{1+\sqrt{2x}-\sqrt{x}}\right).$
Which of the following statements is (are) CORRECT?

1. $f\left(x\right)$ is a decreasing function
2. Minimum value of $f\left(x\right)$ is $\frac{-\pi }{8}$
3. $f^{\prime }\left(\frac{1^{-}}{4}\right)=\frac{-24}{5}$
4. $f^{\prime }\left(\frac{1^{+}}{4}\right)=\frac{-4}{5}$

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :
Let $[.]$ denote the greatest integer function. Let $f\left(x\right)=\left(\ln \right(a^2-3a-3\left)\right)|\sin x|+\left[\frac{a^2}{9}\right]\cos \pi x$ for all $x\in R$ and $a\in [-4,4]$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}f\left(x\right)\text{is periodic, then the number of integral values of}a\text{is}& \text{(P)}& 1\\ \text{(B)}& \text{If}f\left(x\right)\text{is periodic with period to be a rational number, then}& \text{(Q)}& 2\\ & \text{the number of integral values of}a\text{is}& & \\ \text{(C)}& \text{If}f\left(x\right)\text{is periodic with period to be an irrational number, then}& \text{(R)}& 3\\ & \text{the number of integral values of}a\text{is}& & \\ \text{(D)}& \text{If}f\left(x\right)\text{is non-periodic function, then the number of integral}& \text{(S)}& 4\\ & \text{values of}a\text{is}& & \end{array}$

Which of the following is a CORRECT combination?

1. $\left(A\right)\to \left(R\right),\left(B\right)\to \left(P\right)$
2. $\left(A\right)\to \left(R\right),\left(B\right)\to \left(Q\right)$
3. $\left(A\right)\to \left(P\right),\left(B\right)\to \left(Q\right)$
4. $\left(A\right)\to \left(S\right),\left(B\right)\to \left(P\right)$

Q. $f\left(x\right)=\frac{\cos x}{\left[\frac{2x}{\pi }\right]+\frac{1}{2}}$, where $x$ is not an integral multiple of $\pi$ and $[.]$ denotes the greatest integer function is

1. an odd function
2. an even function
3. neither odd nor even
4. none of the above

Q. Functions $P\left(x\right),Q\left(x\right),R\left(x\right)$ are differentiable on some open interval around $0$ and satisfy the below equations as well as the initial conditions.
$P^{\prime }\left(x\right)=2P^2\left(x\right)Q\left(x\right)R\left(x\right)+\frac{1}{Q\left(x\right)R\left(x\right)},$ $P\left(0\right)=1$
$Q^{\prime }\left(x\right)=P\left(x\right)Q^2\left(x\right)R\left(x\right)+\frac{4}{P\left(x\right)R\left(x\right)},$ $Q\left(0\right)=1$
$R^{\prime }\left(x\right)=3P\left(x\right)Q\left(x\right)R^2\left(x\right)+\frac{1}{P\left(x\right)Q\left(x\right)},$ $R\left(0\right)=1.$
Then $P\left(x\right)Q\left(x\right)R\left(x\right)=\tan (nx+\frac{\pi }{4}).$ The value of $n$ is

Q. $\text{List I}$ contains functions and $\text{List II}$ contains behaviours of functions in their respective domain. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& f\left(x\right)={\left(\right(\text{sgn}x{)}^{\text{sgn}x})}^n;x\ne 0& \text{(P)}& \text{odd function}\\ & \text{where}n\text{is an odd integer and}& & \\ & \text{sgn}x\text{denotes the signum function}& & \\ \text{(B)}& f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1& \text{(Q)}& \text{even function}\\ \text{(C)}& f\left(x\right)=\{\begin{array}{cc}0,& \text{if}x\text{is rational}\\ 1,& \text{if}x\text{is irrational}\end{array}& \text{(R)}& \text{neither odd nor}\\ & & & \text{even function}\\ \text{(D)}& f\left(x\right)=max\{\tan x,\cot x\}& \text{(S)}& \text{periodic}\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(B)}\to \text{(Q)}$
2. $\text{(B)}\to \text{(R)}$
3. $\text{(A)}\to \text{(Q)},\text{(S)}$
4. $\text{(A)}\to \text{(P)},\text{(S)}$

Q. If the function $f\left(x\right)=\{\begin{array}{ccc}\frac{kcosx}{\pi -2x},& when& x\ne \frac{\pi }{2}\\ 3,& when& x=\frac{\pi }{2}\end{array}$ be continuous at $x-\frac{\pi }{2}$, then k=

1. 3
2. 6
3. 12
4. None of these

Q. $\text{List I}$ contains functions and $\text{List II}$ contains behaviours of functions in their respective domain. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& f\left(x\right)={\left(\right(\text{sgn}x{)}^{\text{sgn}x})}^n;x\ne 0& \text{(P)}& \text{odd function}\\ & \text{where}n\text{is an odd integer and}& & \\ & \text{sgn}x\text{denotes the signum function}& & \\ \text{(B)}& f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1& \text{(Q)}& \text{even function}\\ \text{(C)}& f\left(x\right)=\{\begin{array}{cc}0,& \text{if}x\text{is rational}\\ 1,& \text{if}x\text{is irrational}\end{array}& \text{(R)}& \text{neither odd nor}\\ & & & \text{even function}\\ \text{(D)}& f\left(x\right)=max\{\tan x,\cot x\}& \text{(S)}& \text{periodic}\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(R)}$
2. $\text{(D)}\to \text{(P)},\text{(S)}$
3. $\text{(C)}\to \text{(Q)}$
4. $\text{(D)}\to \text{(R)},\text{(S)}$

Q. $\text{List I}$ contains functions and $\text{List II}$ contains behaviours of functions in their respective domain. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& f\left(x\right)={\left(\right(\text{sgn}x{)}^{\text{sgn}x})}^n;x\ne 0& \text{(P)}& \text{odd function}\\ & \text{where}n\text{is an odd integer and}& & \\ & \text{sgn}x\text{denotes the signum function}& & \\ \text{(B)}& f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1& \text{(Q)}& \text{even function}\\ \text{(C)}& f\left(x\right)=\{\begin{array}{cc}0,& \text{if}x\text{is rational}\\ 1,& \text{if}x\text{is irrational}\end{array}& \text{(R)}& \text{neither odd nor}\\ & & & \text{even function}\\ \text{(D)}& f\left(x\right)=max\{\tan x,\cot x\}& \text{(S)}& \text{periodic}\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(P)},\text{(S)}$
2. $\text{(B)}\to \text{(Q)}$
3. $\text{(A)}\to \text{(Q)},\text{(S)}$
4. $\text{(B)}\to \text{(R)}$

Q. If $f\left(x\right)=\{\begin{array}{c}{x}^2,x\ge 0\\ x,x<0\end{array}$ then

1. $f\left(f\right(x\left)\right)=$$$\{\begin{array}{c}{x}^2,x\ge 0\\ x,x<0\end{array}$$$
2. $f\left(f\right(x\left)\right)=$$$\{\begin{array}{c}{x}^4,x\ge 0\\ x^2,x<0\end{array}$$$
3. $f\left(f\right(x\left)\right)=$$$\{\begin{array}{c}{x}^4,x\ge 0\\ -x^2,x<0\end{array}$$$
4. $f\left(f\right(x\left)\right)=$$$\{\begin{array}{c}{x}^4,x\ge 0\\ x,x<0\end{array}$$$

Q. If $f\left(x\right)=\frac{\cos x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}},$ where $x$ is not an integral multiple of $\pi$ and $[.]$ denotes the greatest integer function, then

1. $f\left(x\right)$ is an even function
2. $f\left(x\right)$ is an odd function
3. $f\left(x\right)$ is neither even nor odd
4. None of these

Q. If the greatest and the least values of $f\left(x\right)={\sin }^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)-\ln x$ in $[\frac{1}{\sqrt{3}},\sqrt{3}]$ are $M$ and $m$ respectively, then

1. $M+m=\ln 3+\frac{\pi }{6}$
2. $M-m=\ln 3+\frac{\pi }{6}$
3. $M+m=\frac{\pi }{2}$
4. $M-m=\ln 3-\frac{\pi }{3}$

Q. The smallest and the largest values of
${\tan }^{-1}\left(\frac{1-x}{1+x}\right)$ , $0\le x\le 1$ are.

1. $0,\pi$
2. $-\frac{\pi }{4},\frac{\pi }{4}$
3. $0,\frac{\pi }{4}$
4. $\frac{\pi }{4},\frac{\pi }{2}$

Q. The least value of ${\csc }^{2}x+25{\sec }^{2}x$ is

1. $0$
2. $26$
3. $28$
4. $36$

Q. $f\left(x\right)=\{\begin{array}{cc}2x^2-5,& x\le -8\\ 3x+p,& -8<x<-3\\ (k-7)(|3-x|+|3+x|),& -3\le x\le 3\\ 3x+9,& 3<x<8\\ 5-2x^2,& x\ge 8\end{array}$

If $f\left(x\right)$ is an odd function then the value of $k-p$ is

Q. The solution set of $(2\cos x-1)(3+2\cos x)=0$ in the interval $0\le x\le 2\pi$ is-

1. $\left(\pi \right)/3,\left(5\pi \right)/3$
2. $\left(\pi \right)/3,\left(\pi \right)/2$
3. $\pi /3,{\cos }^{-1}(-3/2)$
4. $Noneofthese$

Q. Evaluate $\underset{-1/2}{\overset{1/2}{\int }}\cos x\ln \left(\frac{1+x}{1-x}\right)dx$

Q. If $f\left(x\right)=1+\left[\cos x\right]x$ in $0<x\le \frac{\pi }{2}$, where $[.]$ denotes the greatest integer function, then which of the following is correct regarding $f\left(x\right)$:

1. It is continuous and differentiable in $0<x<\frac{\pi }{2}$
2. $f\left(0\right)=0$
3. It is not differentiable in $0<x<\frac{\pi }{2}$
4. It is not differentiable at $x=1$

Q. Find the smallest and the largest values of ${\tan }^{-1}\left(\frac{1-x}{1+x}\right),0\le x\le 1$.

Q. $\text{List I}$ contains functions and $\text{List II}$ contains behaviours of functions in their respective domain. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& f\left(x\right)={\left(\right(\text{sgn}x{)}^{\text{sgn}x})}^n;x\ne 0& \text{(P)}& \text{odd function}\\ & \text{where}n\text{is an odd integer and}& & \\ & \text{sgn}x\text{denotes the signum function}& & \\ \text{(B)}& f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1& \text{(Q)}& \text{even function}\\ \text{(C)}& f\left(x\right)=\{\begin{array}{cc}0,& \text{if}x\text{is rational}\\ 1,& \text{if}x\text{is irrational}\end{array}& \text{(R)}& \text{neither odd nor}\\ & & & \text{even function}\\ \text{(D)}& f\left(x\right)=max\{\tan x,\cot x\}& \text{(S)}& \text{periodic}\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(R)}$
2. $\text{(D)}\to \text{(P)},\text{(S)}$
3. $\text{(C)}\to \text{(Q)}$
4. $\text{(D)}\to \text{(R)},\text{(S)}$