If $x+\frac{1}{x}=2,$ then find the value of $x^{4}+\frac{1}{x^{4}}.$

Resolve into factors: $81 x^{2} y^{2}+108 x y z+36 z^{2}$.

If $a+b+c=0,$ then the value of $\left(\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}\right)$ is:

If $x_{1} x_{2} x_{3} = 4(4 + x_{1} + x_{2} + x_{3})$,then what is the value of $\left[\frac{1}{(2 + x_{1})}\right] + \left[\frac{1}{(2 + x_{2})}\right] + \left[\frac{1}{(2 + x_{3})}\right]$?

When $(x^{3}-2 x^{2}+p x-q)$ is divided by $(x^{2}-2 x-3)$,the remainder is $(x-6)$. The values of $p$ and $q$ are:

If $\frac{x}{y} + \frac{y}{x} = 6,$ then find the value of $\frac{x^{3}}{y^{3}} + \frac{y^{3}}{x^{3}}.$