If $f(x) = \log_e \left( \frac{1-x}{1+x} \right)$,$|x| < 1$,then $f\left( \frac{2x}{1+x^2} \right)$ is equal to

The range of the function $f(x) = \left[ \frac{1}{\ln(x^2 + e)} \right] + \frac{1}{\sqrt{1 + x^2}}$ is,where $[*]$ denotes the greatest integer function and $e = \lim_{\alpha \to 0} (1 + \alpha)^{1/\alpha}$.

Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8 \left( f \left( \frac{1}{15} \right) + f \left( \frac{2}{15} \right) + \dots + f \left( \frac{59}{15} \right) \right)$ is equal to

Let $f:[0,2] \rightarrow R$ be the function defined by $f(x)=(3-\sin(2\pi x)) \sin(\pi x-\frac{\pi}{4})-\sin(3\pi x+\frac{\pi}{4})$. If $\alpha, \beta \in[0,2]$ are such that $\{x \in[0,2]: f(x) \geq 0\}=[\alpha, \beta]$,then the value of $\beta-\alpha$ is:

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 2x + |x|$,then $f(2x) + f(-x) - f(x) = $

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,