If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 = 1$,then the value of $\sum_{i=1}^4 z_i^3$ is

If $Z_{r} = \sin \frac{2 \pi r}{11} - i \cos \frac{2 \pi r}{11}$,then $\sum_{r=0}^{10} Z_{r}$ is equal to

If $z + z^{-1} = 1$,then $z^{100} + z^{-100}$ is equal to

If $n$ is a positive integer not a multiple of $3$,then $1 + \omega^n + \omega^{2n} = $

If $Z_r = \cos \left(\frac{\pi}{2^r}\right) + i \sin \left(\frac{\pi}{2^r}\right)$ for $r = 1, 2, 3, \ldots$,then the product $Z_1 Z_2 Z_3 \ldots \infty$ is equal to:

The smallest positive integral value of $n$ such that $\left[\frac{1+\sin \frac{\pi}{8}+i \cos \frac{\pi}{8}}{1+\sin \frac{\pi}{8}-i \cos \frac{\pi}{8}}\right]^{n} = 1$ is: