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

Consider the following statements:
$(i)$ $A$ relation is a special case of a function.
$(ii)$ $A$ function is a special case of a relation.
$(iii)$ Both relation and function are the same.

Let $A = \{2, 3, 4, 5, 6\}$. Let $R$ be a relation on the set $A \times A$ defined by $(x, y) R (z, w)$ if and only if $x$ divides $z$ and $y \le w$. Then the number of elements in $R$ is . . . . . . .

Let $f(x) = \max(\sin x, \cos x)$ and $g(x) = \min(\cos x, \sin x)$. Define $h(y) = f(x)y^2 + ay + g(x)$. If the equation $h(y) = 0$ has real roots for all $x \in R$,find the complete set of values of $a$.

Let $f(x) = \begin{cases} x-1, & x \text{ is even} \\ 2x, & x \text{ is odd} \end{cases}$. If for some $a \in N, f(f(f(a))) = 21$,then $\lim_{x \rightarrow a^{-}} \left\{ \frac{|x|^3}{a} - \left[ \frac{x}{a} \right] \right\}$,where $[t]$ denotes the greatest integer less than or equal to $t$,is equal to:

Consider a function $f: R \to R$ such that $f(x + a) = \frac{1}{2} + \sqrt{f(x) - f^2(x)}$,where $a$ is a real constant. Then $f(x)$ must be