If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$ is $R-(\alpha, \beta)$ then $12 \alpha \beta$ is equal to :

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as

$f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 .$ Then, the value of $\sum_{\mathrm{k}=1}^{20} \frac{1}{\sin (\mathrm{k}) \sin (\mathrm{k}+\mathrm{f}(\mathrm{k}))}$ is equal to:

Let $f(x ) = x^3 - 2x + 2$. If real numbers $a$, $b$ and $c$ such that $\left| {f\left( a \right)} \right| + \left| {f\left( b \right)} \right| + \left| {f\left( c \right)} \right| = 0$ then the value of ${f^2}\left( {{a^2} + \frac{2}{a}} \right) + {f^2}\left( {{b^2} + \frac{2}{b}} \right) - {f^2}\left( {{c^2} + \frac{2}{c}} \right)$ equal to

Statement $1$ : If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^P$.
Statement $2$ : The total number of selections of $p$ different objects out of $q$ objects is ${}^q{C_p}$.

The minimum value of the function $f(x) = {x^{10}} + {x^2} + \frac{1}{{{x^{12}}}} + \frac{1}{{\left( {1\ +\ {{\sec }^{ - 1}}\ x} \right)}}$ is

Which of the following is true