If $\alpha, \beta \in \mathbb{C}$ are distinct roots of the equation $x^2 - x + 1 = 0$,then $\alpha^{101} + \beta^{107}$ is equal to:

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 $z = \frac{\sqrt{3}}{2} + \frac{i}{2}$, where $i = \sqrt{-1}$, then $(z^{201} - i)^{8}$ is equal to:

If $\omega$ is a complex cube root of unity and $a, b, c$ are distinct real numbers,then $\frac{a+b \omega+c \omega^2}{c+a \omega+b \omega^2}+\frac{a+b \omega+c \omega^2}{b+c \omega+a \omega^2} = $

If $\alpha, \beta$ are the roots of the equation $x^2-2x+4=0$ and for any $n \in N, \alpha^n+\beta^n=k \cos \frac{n \pi}{3}$, then $k=$

The value of $\sum_{k=1}^{6}\left(\sin \frac{2 k \pi}{7}-i \cos \frac{2 k \pi}{7}\right)$ is