If domain of function $f(x) = \sqrt {\ln \left( {m\sin x + 4} \right)} $ is $R$ , then number of possible integral values of $m$ is

Let $[x]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $\mathrm{f}(\mathrm{x})=\sqrt{\frac{[\mathrm{x}] \mid-2}{\sqrt{[\mathrm{x}] \mid-3}}}$ is $(-\infty, \mathrm{a}) \cup[\mathrm{b}, \mathrm{c}) \cup[4, \infty), \mathrm{a}\,<\,\mathrm{b}\,<\,\mathrm{c}$, then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is:

Let $P(x)$ be a polynomial with real coefficients such that $P\left(\sin ^2 x\right)=P\left(\cos ^2 x\right)$ for all $x \in[0, \pi / 2)$. Consider the following statements:

$I.$ $P(x)$ is an even function.

$II.$ $P(x)$ can be expressed as a polynomial in $(2 x-1)^2$

$III.$ $P(x)$ is a polynomial of even degree.

Then,

The graph of $y = f(x)$ is shown then number of solutions of the equation $f(f(x)) =2$ is

Let $R$ be the set of all real numbers and let $f$ be a function from $R$ to $R$ such that $f(x)+\left(x+\frac{1}{2}\right) f(1-x)=1$, for all $x \in R$. Then $2 f(0)+3 f(1)$ is equal to

Domain of $f (x)$ = $\sqrt {{{\log }_2}\left( {\frac{{10x - 4}}{{4 - {x^2}}}} \right) - 1} $ , is