If the domain of the function $f(x)=\log _e$ $\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5 \beta-4 \alpha$ is 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 ${}^q{C_p}$.

Function ${\sin ^{ - 1}}\sqrt x $ is defined in the interval

The domain of $f(x) = \frac{1}{{\sqrt {{{\log }_{\frac{\pi }{4}}}({{\sin }^{ - 1}}x) - 1} }}$,is

Let $f ( x )=2 x ^{ n }+\lambda, \lambda \in R , n \in N$, and $f (4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $( f (3)- f (2))$ is

Let $f(\theta ) = \sin \theta (\sin \theta + \sin 3\theta )$, then $f(\theta )$