If $x + \frac{1}{x} = 2\cos \alpha$,then $x^n + \frac{1}{x^n} = $

If $\alpha$ is a complex number satisfying the equation $\alpha^{2}+\alpha+1=0$,then $\alpha^{31}$ is equal to

Let $x = \alpha + \beta$,$y = \alpha \omega + \beta \omega^2$,and $z = \alpha \omega^2 + \beta \omega$,where $\omega$ is an imaginary cube root of unity. The product $xyz$ is equal to:

Let $z = \cos \theta + i \sin \theta$. Then,the value of $\sum_{m=1}^{15} \text{Im}(z^{2m-1})$ at $\theta = 2^{\circ}$ is

The value of $(1 - \omega + \omega^2)(1 - \omega^2 + \omega)^6$,where $\omega, \omega^2$ are the complex cube roots of unity.

If $\omega$ is a complex cube root of unity,then $225+(3 \omega+8 \omega^2)^2+(3 \omega^2+8 \omega)^2$ is equal to: