The domain of the function $f(x) = \sqrt{x-1} + \sqrt{6-x}$ is

Let $f : R \rightarrow R$ be a function defined by $f(x) = \log_{\sqrt{m}}\{\sqrt{2}(\sin x - \cos x) + m - 2\}$,for some $m$,such that the range of $f$ is $[0, 2]$. Then the value of $m$ is $............$

If $[x]$ denotes the greatest integer function,then the domain of the function $f(x) = \sqrt{\frac{x-[x]}{\log(x^2-x)}}$ is

If the domain of the real valued function $f(x) = \frac{1}{\sqrt{\log_{\frac{1}{3}}\left(\frac{x-1}{2-x}\right)}}$ is $(a, b)$,then $2b =$

The range of the function $f(x) = \frac{x-3}{5-x}, x \neq 5$ is

Let $D$ be the domain of the function $f(x) = \sin^{-1} \left(\log_{3x} \left(\frac{6+2 \log_3 x}{-5x}\right)\right)$. If the range of the function $g: D \rightarrow R$ defined by $g(x) = x - [x]$ (where $[x]$ is the greatest integer function) is $(\alpha, \beta)$,then $\alpha^2 + \frac{5}{\beta}$ is equal to