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

Let $f(x) = \frac{x^2-6x+5}{x^2-5x+6}$. Match the conditions / expressions in Column $I$ with statements in Column $II$.

Column $I$	Column $II$
$(A)$ If $-1 < x < 1$,then $f(x)$ satisfies	$(p)$ $0 < f(x) < 1$
$(B)$ If $1 < x < 2$,then $f(x)$ satisfies	$(q)$ $f(x) < 0$
$(C)$ If $3 < x < 5$,then $f(x)$ satisfies	$(r)$ $f(x) > 0$
$(D)$ If $x > 5$,then $f(x)$ satisfies	$(s)$ $f(x) < 1$

If $f(x) = \tan \left(\frac{\pi}{\sqrt{x+1}+4}\right)$ is a real-valued function, then the range of $f$ is:

Let $f(x) = \frac{\tan^n x}{\sum_{r=0}^{2n} \tan^r x}$,$n \in N$,where $x \in [0, \frac{\pi}{2})$.

If the range of the function $f(x) = \frac{5-x}{x^2-3x+2}$,$x \neq 1, 2$,is $(-\infty, \alpha] \cup [\beta, \infty)$,then $\alpha^2 + \beta^2$ is equal to :

The range of the polynomial $P(x) = 4x^3 - 3x$ as $x$ varies over the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is