The number of solutions of the equation $2{e^{\left| x \right|}}{\tan ^{ - 1}}\left| x \right| = 1$ is

The function $f(x) = |px - q| + r|x|$,$x \in (-\infty, \infty)$,where $p > 0, q > 0, r > 0$ assumes its minimum value only at one point,if

If $f(x)=3[x]+\{x+1\}$,where $[x]$ is the greatest integer function of $x$ and $\{x\}$ is the fractional part function of $x$,then $f(-1.32)=$

There are three kinds of liquids $X, Y, Z$. Three jars $J_1, J_2, J_3$ contain $100 \, ml$ of liquids $X, Y, Z$ respectively. An operation consists of three steps in the following order:
- Stir the liquid in $J_1$ and transfer $10 \, ml$ from $J_1$ into $J_2$.
- Stir the liquid in $J_2$ and transfer $10 \, ml$ from $J_2$ into $J_3$.
- Stir the liquid in $J_3$ and transfer $10 \, ml$ from $J_3$ into $J_1$.
After performing the operation four times,let $x, y, z$ be the amounts of $X, Y, Z$ respectively in $J_1$. Then,

If $f(a) = \log \left| \frac{1-a}{1+a} \right|$ for $a \neq \{-1, 1\}$,then the set of values of all $a$,for which $f\left( \frac{2a}{1+a^2} \right) > 0$ is

Let $f, g: R \rightarrow R$ be functions defined by $f(x) = \begin{cases} [x] & x < 0 \\ |1-x| & x \geq 0 \end{cases}$ and $g(x) = \begin{cases} e^x - x & x < 0 \\ (x-1)^2 - 1 & x \geq 0 \end{cases}$ where $[x]$ denotes the greatest integer less than or equal to $x$. Then,the function $(f \circ g)(x)$ is discontinuous at exactly