Let $f(x) = |x|$ and $g(x) = |x| + a$,where $a > 0$. For $0 \leq x \leq b$,the set $\{(x, y) \mid g(x) \leq y \leq f(x)\}$ represents all the points in the interior of:

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be a function such that $f(x) > 0$ for all $x \in R$,and $f(x+y)=f(x) f(y)$ for all $x, y \in R$. Let the real numbers $a_1, a_2, \ldots, a_{50}$ be in an arithmetic progression. If $f(a_{31})=64 f(a_{25})$,and $\sum_{i=1}^{50} f(a_i)=3(2^{25}+1)$,then the value of $\sum_{i=6}^{30} f(a_i)$ is:

Match the functions given in List-$I$ with their relevant characteristics from List-$II$.

List-$I$	List-$II$
$(A)$ $\sinh x$	$(I)$ Domain is $(-1, 1)$, even function
$(B)$ $\text{sech } x$	$(II)$ Domain is $[1, \infty)$, neither even nor odd function
$(C)$ $\tanh x$	$(III)$ Even function
$(D)$ $\text{cosech}^{-1} x$	$(IV)$ Range is $\mathbb{R}$, odd function
	$(V)$ Range is $(-1, 1)$, odd function

The correct answer is

The number of solutions of the equation $2^x = x^2$ is

Which one of the following is not bounded on the intervals as indicated?

Function $f(x) = \sin x + \tan x + \operatorname{sgn}(x^2 - 6x + 10)$ is (where $\operatorname{sgn}$ is the signum function):