Domain of the definition of function 

$f(x) = \sqrt {\frac{{4 - {x^2}}}{{\left[ x \right] + 2}}} $ is      $($ where $[.] \rightarrow G.I.F.)$

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,

Let $f\,:\,R \to R$ be a function such that $f\left( x \right) = {x^3} + {x^2}f'\left( 1 \right) + xf''\left( 2 \right) + f'''\left( 3 \right)$, $x \in R$. Then $f(2)$ equals

Let $R$ be the set of all real numbers and $f(x)=\sin ^{10} x\left(\cos ^8 x+\cos ^4 x+\cos ^2 x+1\right)$ $x \in R$. Let  $S=\{\lambda \in R$ there exists a point $c \in(0,2 \pi)$ with $\left.f^{\prime}(c)=\lambda f(c)\right\}$ Then,

If $f(x) = \sin \log x$, then the value of $f(xy) + f\left( {\frac{x}{y}} \right) - 2f(x).\cos \log y$ is equal to

If $f(x)=\frac{2^{2 x}}{2^{2 x}+2}, x \in R$ then $f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\ldots \ldots . .+f\left(\frac{2022}{2023}\right)$ is equal to