The sum of the $162^{\text{th}}$ power of the roots of the equation $x^{3}-2x^{2}+2x-1=0$ is

The number of common roots among the $12^{\text{th}}$ and $30^{\text{th}}$ roots of unity is

If $\omega$ is a complex cube root of unity,then the value of $\left[\frac{51+73 \omega+87 \omega^2}{73+87 \omega+51 \omega^2}+\frac{51+73 \omega+87 \omega^2}{87+51 \omega+73 \omega^2}\right]^{15}$ is:

If $\alpha \neq 1$ is any $n^{th}$ root of unity,then $S = 1 + 3\alpha + 5\alpha^2 + \dots$ up to $n$ terms,is equal to

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 27 = 0$,find the quadratic equation whose roots are $\left( \frac{\gamma}{\alpha} \right)^2$ and $\left( \frac{\beta}{\alpha} \right)^2$.

If $\omega$ is an imaginary cube root of unity,$(1 + \omega - \omega^2)^7$ equals