If $\alpha$ is an imaginary cube root of unity,then for $n \in N$,the value of $\alpha^{3n + 1} + \alpha^{3n + 3} + \alpha^{3n + 5}$ is

Let $z_1, z_2, \ldots, z_7$ be the vertices of a regular heptagon inscribed in the unit circle with the center at the origin in the complex plane. If $w = \sum_{1 \leq i < j \leq 7} z_i z_j$,then $|w|$ is equal to:

If $\alpha$ and $\beta$ are the roots of $x^{2}+x+1=0$,then $\alpha^{16}+\beta^{16}$ is equal to

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

If $\alpha, \beta, \gamma, \delta$ are the roots of the equation $x^{4}+x^{3}+x^{2}+x+1=0$,then $\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}$ is equal to

If the cube roots of unity are $1, \omega, \omega^2$,then the roots of the equation $(x - 1)^3 + 8 = 0$ are