If $\omega$ is a complex cube root of unity,then $\cos \left(\sum_{k=1}^7(k-\omega)(k-\omega^2) \frac{\pi}{175}\right) =$

If $\alpha$ is a complex number such that $\alpha^{2}-\alpha+1=0$,then $\alpha^{2011}$ is equal to

If $n$ is an integer which leaves a remainder of $1$ when divided by $3$,then $(1+\sqrt{3}i)^n + (1-\sqrt{3}i)^n$ equals

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 $\omega$ is a non-real root of the equation $x^3 - 1 = 0$,then the value of $\sum_{r=1}^{5} (1 + \omega^r + \omega^{2r})$ is

If $z = \frac{\sqrt{3} + i}{2}$,then the value of $z^{69}$ is