The value of $\sum \limits_{n=0}^{1947} \frac{1}{2^n+\sqrt{2^{1994}}}$ is equal to

Let $S=\{1,2,3,4,5,6,7\} .$ Then the number of possible functions $f: S \rightarrow S$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in S$ and $m . n \in S$ is equal to $......$

The domain of the function $f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$ )

If $f:\left\{ {1,2,3,4} \right\} \to \left\{ {1,2,3,4} \right\}$ and $y=f(x)$ be a function such that $\left| {f\left( \alpha  \right) - \alpha } \right| \leqslant 1$,for $\alpha  \in \left\{ {1,2,3,4} \right\}$ then total number of such functions are

If $f(a) = a^2 + a+ 1$ , then number of solutions of equation $f(a^2) = 3f(a)$ is

The range of the function $f(x) = \frac{{\sqrt {1 - {x^2}} }}{{1 + \left| x \right|}}$ is