The domain of the real valued function $f(x) = \frac{\sqrt{\log _{0.5}(x-3)}}{\sqrt{x-1}}$ is

If the domain of the function $\log _5(18 x-x^2-77)$ is $(\alpha, \beta)$ and the domain of the function $\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)$ is $(\gamma, \delta)$,then $\alpha^2+\beta^2+\gamma^2$ is equal to :

Let $f$ be a function such that $f(x) = \sum_{r=1}^n [r + \cos(\frac{x}{r})]$,where $[.]$ denotes the greatest integer function and $x \in [0, \pi]$. Then the range of $f(x)$ is:

Consider the following lists.

$A$. $f(x)=\frac{|x+2|}{x+2}, x \neq-2$	$1$. $[\frac{1}{3}, 1]$
$B$. $g(x)=|[x]|, x \in R$	$2$. $Z$
$C$. $h(x)=|x-[x]|, x \in R$	$3$. $W$
$D$. $f(x)=\frac{1}{2-\sin 3x}, x \in R$	$4$. $[0, 1)$
	$5$. $\{-1, 1\}$

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

The domain of the function $f(x) = \sin^{-1}\left[\log_4\left(\frac{x}{4}\right)\right] + \sqrt{17x - x^2 - 16}$ is