If $p$ and $q$ are positive real numbers such that $p^2 + q^2 = 1$,what is the maximum value of $(p + q)$?

If $x = \frac{1}{2} \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right)$, then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

$p$ and $q$ are two roots of the equation $x^2+7x+3=0$. If $\frac{3p}{1-2p}$ and $\frac{3q}{1-2q}$ are the roots of $lx^2+mx+n=0$ and the greatest common divisor of $l, m, n$ is $1$,then $l-m+n=$

Suppose $p, q, r$ are real numbers such that $q=p(4-p)$,$r=q(4-q)$,and $p=r(4-r)$. The maximum possible value of $p+q+r$ is

All the roots of the equation $x^5+15x^4+94x^3+305x^2+507x+353=0$ are increased by some real number $k$ in order to eliminate the $4^{th}$ degree term from the equation. Now,the coefficient of $x$ in the transformed equation is

Let $S = \{x \in [-6, 3] \setminus \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$ and $T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$. Then the number of elements in $S \cap T$ is $....$