If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$ is $R-(\alpha, \beta)$ then $12 \alpha \beta$ is equal to :

The domain of $f(x) = [\sin x] \cos \left( {\frac{\pi }{{[x - 1]}}} \right)$ is (where $[.]$ denotes $G.I.F.$)

The domain of definition of the function $y(x)$ given by ${2^x} + {2^y} = 2$ is

If $x = {\log _2}\left( {\sqrt {56 + \sqrt {56 + \sqrt {56 +  .... + \infty } } } } \right)$ then 

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

Let $f(\theta ) = \sin \theta (\sin \theta + \sin 3\theta )$, then $f(\theta )$