If $f: R \rightarrow R$ is defined as $f(x+y)=f(x)+f(y), \forall x, y \in R$ and $f(1)=10$, then, $\sum_{r=1}^n(f(r))^2=$

If $f(x)$ is a polynomial function satisfying $f(x) \cdot f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ and $f(4)=257$,then $f(3)=$

If $f(x + y) = f(x) + f(y)$ for all $x, y \in R$ and $f(1) = 1$,then find the value of $\lim_{x \to 0} \frac{2^{f(\tan x)} - 2^{f(\sin x)}}{f(\tan x) - f(\sin x)}$.

Let $R$ be the set of all real numbers. The number of continuous functions $f: R \rightarrow R$ such that for all real $x$,$f(x) + f(2x) = 0$ is

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 $\lim _{x \rightarrow 0} \frac{(f(x))^{\frac{1}{3}} - 1}{(f(x))^{\frac{1}{2}} - 1} =$

If $f(x + y, x - y) = xy$,then the arithmetic mean of $f(x, y)$ and $f(y, x)$ is