Suppose $f$ is a function satisfying $f ( x + y )= f ( x )+ f ( y )$ for all $x , y \in N$ and $f (1)=\frac{1}{5}$. If $\sum \limits_{n=1}^m \frac{f(n)}{n(n+1)(n+2)}=\frac{1}{12}$, then $m$ is equal to $...............$.

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 $N$ be the set of positive integers. For all $n \in N$, let $f_n=(n+1)^{1 / 3}-n^{1 / 3} \text { and }$ $A=\left\{n \in N: f_{n+1}<\frac{1}{3(n+1)^{2 / 3}} < f_n\right\}$ Then,

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

If $f(x)$ is a polynomial function satisfying the condition $f(x) . f(1/x) = f(x) + f(1/x)$ and $f(2) = 9$ then :

If $f(x)=\frac{\left(\tan 1^{\circ}\right) x+\log _{\varepsilon}(123)}{x \log _{\varepsilon}(1234)-\left(\tan 1^{\circ}\right)}, x > 0$, then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is $...........$.