Graph of [x]

Q.

The area(in sq. units) of the region $A=\left\{\left(x,y\right):\left|x\right|+\left|y\right|\le 1,2y^2=\left|x\right|\right\}$

1. $\frac{1}{6}$

2. $\frac{5}{6}$

3. $\frac{1}{3}$

4. $\frac{7}{6}$

Q. The function $f\left(x\right)$ is defined as $|\left[x\right]x|$ for $-1<x\le 2$.
For what values of $x$ is $f\left(x\right)$ discontinuous ?

1. $0$
2. $1$
3. $2$
4. $-1$

Q. Which of the following is/are periodic?

1. $f\left(x\right)=\{\begin{array}{cc}1,& x\text{is rational}\\ 0,& x\text{is irrational}\end{array}$
2. $f\left(x\right)=\{\begin{array}{cc}x-\left[x\right];& 2n\le x\le 2n+1\\ \frac{1}{2};& 2n+1\le x\le 2n+2\end{array},n\in Z$
3. $f\left(x\right)=(-1{)}^{\left[\frac{2x}{\pi }\right]}$
4. $f\left(x\right)=x-[x+3]+\tan \left(\frac{nx}{2}\right)$