The domain of the function

$f(x)=\frac{\cos ^{-1}\left(\frac{x^{2}-5 x+6}{x^{2}-9}\right)}{\log _{e}\left(x^{2}-3 x+2\right)} \text { is }$

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function which satisfies $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R} .$ If $\mathrm{f}(1)=2$ and $g(n)=\sum \limits_{k=1}^{(n-1)} f(k), n \in N$ then the value of $n,$ for which $\mathrm{g}(\mathrm{n})=20,$ is 

Let $A= \{1, 2, 3, 4\}$ and $R : A \to A$ be the relation defined by $R = \{ (1, 1), (2, 3), (3, 4), ( 4, 2) \}$. The correct statement is

If $f(x)$ satisfies $f(7 -x) = f(7 + x)\ \forall \,x\, \in \,R$ such that $f(x)$ has exactly $5$ real roots which are all distinct such that sum of the real roots is $S$ then $S/7$ is equal to

Let $[x]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $\mathrm{f}(\mathrm{x})=\sqrt{\frac{[\mathrm{x}] \mid-2}{\sqrt{[\mathrm{x}] \mid-3}}}$ is $(-\infty, \mathrm{a}) \cup[\mathrm{b}, \mathrm{c}) \cup[4, \infty), \mathrm{a}\,<\,\mathrm{b}\,<\,\mathrm{c}$, then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is:

Range of the function

$f(x) = \sqrt {\left| {{{\sin }^{ - 1}}\left| {\sin x} \right|} \right| - {{\cos }^{ - 1}}\left| {\cos x} \right|} $ is