Let $f: N \rightarrow N$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m, n \in N$. If $f(6)=18$ then $f(2) \cdot f(3)$ is equal to :

Let $u+v+w=3$, where $u, v, w \in \mathbb{R}$ and $f(x)=u x^2+v x+w$ be such that $f(x+y)=f(x)+f(y)+x y$ for all $x, y \in \mathbb{R}$. Then $f(1)$ 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 $f : R - \{0, 1\} \rightarrow R$ be a function such that $f(x) + f\left(\frac{1}{1-x}\right) = 1 + x$. Then $f(2)$ is equal to:

If $f(x) + 2f(1/x) = 3x$ for $x \neq 0$ and $S = \{x \in R : f(x) = f(-x)\}$,then $S$:

Let $\sum\limits_{k = 1}^{10} {f(a + k)} = 16(2^{10} - 1),$ where the function $f$ satisfies $f(x + y) = f(x)f(y)$ for all natural numbers $x, y$ and $f(1) = 2.$ Then the natural number $a$ is