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

If non-zero real numbers $b$ and $c$ are such that $min \,f\left( x \right) > \max \,g\left( x \right)$, where $f\left( x \right) = {x^2} + 2bx + 2{c^2}$  and $g\left( x \right) = {-x^2} - 2cx + {b^2}$$\left( {x \in R} \right)$; then $\left| {\frac{c}{b}} \right|$ lies in the interval

If $f(x)$ is a function satisfying $f(x + y) = f(x)f(y)$ for all $x,\;y \in N$ such that $f(1) = 3$ and $\sum\limits_{x = 1}^n {f(x) = 120} $. Then the value of $n$ is

Let $f(x) = {\cos ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right) + {\sin ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right)$ then the value of $f(1) + f(2)$, is -

The range of the function,

$\mathrm{f}(\mathrm{x})=\log _{\sqrt{5}}(3+\cos \left(\frac{3 \pi}{4}+\mathrm{x}\right)+\cos \left(\frac{\pi}{4}+\mathrm{x}\right)+\cos \left(\frac{\pi}{4}-\mathrm{x}\right)$

$-\cos \left(\frac{3 \pi}{4}-\mathrm{x}\right))$ is :

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