For $\theta \in [0, \pi]$,let $f(\theta) = \sin(\cos \theta)$ and $g(\theta) = \cos(\sin \theta)$. Let $a = \max_{0 \leq \theta \leq \pi} f(\theta)$,$b = \min_{0 \leq \theta \leq \pi} f(\theta)$,$c = \max_{0 \leq \theta \leq \pi} g(\theta)$,and $d = \min_{0 \leq \theta \leq \pi} g(\theta)$. The correct inequalities satisfied by $a, b, c, d$ are:

Let $S$ be a finite set. Then a non-identity function $f: S \rightarrow S$ can be

Let $f$ and $g$ be increasing and decreasing functions,respectively,from $[0, \infty)$ to $[0, \infty)$. Let $h(x) = f(g(x))$. If $h(0) = 0$,then $h(x) - h(1)$ is:

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

Let $f : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow \mathbb{R}$ be defined by $f(x) = (\log(\sec x + \tan x))^3$. Then:

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,