Let $$\mathrm{D}$$ be the domain of the function $$f(x)=\sin ^{-1}\left(\log _{3 x}\left(\frac{6+2 \log _{3} x}{-5 x}\right)\right)$$. If the range of the function $$\mathrm{g}: \mathrm{D} \rightarrow \mathbb{R}$$ defined by $$\mathrm{g}(x)=x-[x],([x]$$ is the greatest integer function), is $$(\alpha, \beta)$$, then $$\alpha^{2}+\frac{5}{\beta}$$ is equal to

The domain of the function $$f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]-10}}$$ is : ( where $$[\mathrm{x}]$$ denotes the greatest integer less than or equal to $$x$$ )

If $$f(x) = {{(\tan 1^\circ )x + {{\log }_e}(123)} \over {x{{\log }_e}(1234) - (\tan 1^\circ )}},x > 0$$, then the least value of $$f(f(x)) + f\left( {f\left( {{4 \over x}} \right)} \right)$$ is :

Let the sets A and B denote the domain and range respectively of the function $$f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$$, where $$\lceil x\rceil$$ denotes the smallest integer greater than or equal to $$x$$. Then among the statements

(S1) : $$A \cap B=(1, \infty)-\mathbb{N}$$ and

(S2) : $$A \cup B=(1, \infty)$$