If $\frac{1}{x} + x = 2\cos \theta,$ then ${x^n} + \frac{1}{{{x^n}}}$ is equal to

If $1, a, a^2, \ldots, a^{n-1}$ are the $n$th roots of unity,then $\sum_{i=1}^{n-1} \frac{1}{2-a^i}$ is equal to

$\frac{(-1 + i\sqrt{3})^{15}}{(1 - i)^{20}} + \frac{(-1 - i\sqrt{3})^{15}}{(1 + i)^{20}}$ is equal to

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

For a non-$0$ complex number $z$,let $\arg (z)$ denote the principal argument of $z$,with $-\pi < \arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg (\omega) < \pi$. Let $\alpha = \arg \left(\sum_{n=1}^{2025} (-\omega)^n\right)$. Then the value of $\frac{3 \alpha}{\pi}$ is $.....$ .

If $1, \omega$ and $\omega^2$ are the cube roots of unity,then $(a+b+c)(a+b \omega+c \omega^2)(a+b \omega^2+c \omega) = $