If domain of function $f(x) = \sqrt {\ln \left( {m\sin x + 4} \right)} $ is $R$ , then number of possible integral values of $m$ is

If $f( x + y )=f( x ) f( y )$ and $\sum \limits_{ x =1}^{\infty} f( x )=2, x , y \in N$ where $N$ is the set of all natural numbers, then the value of $\frac{f(4)}{f(2)}$ is

The number of functions $f :\{1,2,3,4\} \rightarrow\{ a \in Z :| a | \leq 8\}$ satisfying $f ( n )+$ $\frac{1}{ n } f ( n +1)=1, \forall n \in\{1,2,3\}$ is

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

If $y = 3[x] + 1 = 4[x -1] -10$, then $[x + 2y]$ is equal to (where $[.]$ is $G.I.F.$)

Range of ${\sin ^{ - 1\,}}\left( {\frac{{1 + {x^2}}}{{2 + {x^2}}}} \right)$ is