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

If $3 f(x) - f\left(\frac{1}{x}\right) = 8 \log_2 x^3$ for $x > 0$,then $f(2), f(4), f(8)$ are in

The function $f(x) = x - [x]$,where $[x]$ denotes the greatest integer function,is:

Let $f(x)$ be a polynomial of degree $3$ such that $f(k) = -\frac{2}{k}$ for $k = 2, 3, 4, 5$. Then the value of $52 - 10 f(10)$ is equal to:

Let $f: R^{+} \rightarrow R^{+}$ be a function satisfying $f(x) - x = \lambda$ (constant),$\forall x \in R^{+}$ and $f(x f(y)) = f(x y) + x, \forall x, y \in R^{+}$. Then $\lim _{x \rightarrow 0} \frac{(f(x))^{\frac{1}{3}} - 1}{(f(x))^{\frac{1}{2}} - 1} =$

Let $f$ be a function such that $f(x) + 3f\left(\frac{24}{x}\right) = 4x$,where $x \neq 0$. Then $f(3) + f(8)$ is equal to: