The domain of the function $f(x) = \frac{{{{\sin }^{ - 1}}(x - 3)}}{{\sqrt {9 - {x^2}} }}$ is

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

Let $f$ be a function defined on the set of all positive integers such that $f(x y)=f(x)+f(y)$ for all positive integers $x, y$. If $f(12)=24$ and $f(8)=15$. The value of $f(48)$ is

Let $S=\{1,2,3,4,5,6\}$. Then the number of oneone functions $f: S \rightarrow P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n) \subset f(m)$ where $n < m$ is $..................$

Let $f(x) = \frac{{x\,\, - \,\,1}}{{2\,{x^2}\,\, - \,\,7x\,\, + \,\,5}}$ . Then :

The number of one-one function $f :\{ a , b , c , d \} \rightarrow$ $\{0,1,2, \ldots ., 10\}$ such that $2 f(a)-f(b)+3 f(c)+$ $f ( d )=0$ is