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

For real $x$, the greatest value of $\frac{x^{2}+2 x+4}{2 x^{2}+4 x+9}$ is

Let $x, y, z$ be positive integers such that $HCF(x, y, z)=1$ and $x^2+y^2=2z^2$. Which of the following statements are true?
$I$. $4$ divides $x$ or $4$ divides $y$.
$II$. $3$ divides $x+y$ or $3$ divides $x-y$.
$III$. $5$ divides $z(x^2-y^2)$.

If $x$ is real,the expression $\frac{x + 2}{2x^2 + 3x + 6}$ takes all values in the interval

The sum of the squares of the roots of $|x-2|^2+|x-2|-2=0$ and the squares of the roots of $x^2-2|x-3|-5=0$ is:

If $\left|\frac{x^2+kx+1}{x^2+x+1}\right| < 3$ for all real $x$,then $k$ is in the interval