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

If $P(S)$ denotes the set of all subsets of a given set $S, $ then the number of one-to-one functions from the set $S = \{ 1, 2, 3\}$ to the set $P(S)$ is

Let $A=\{(x, y): 2 x+3 y=23, x, y \in N\}$ and $B=\{x:(x, y) \in A\}$. Then the number of one-one functions from $\mathrm{A}$ to $\mathrm{B}$ 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 $x$ )

Range of the function

$f(x) = \sqrt {\left| {{{\sin }^{ – 1}}\left| {\sin x} \right|} \right| – {{\cos }^{ – 1}}\left| {\cos x} \right|} $ is