If $a = \cos \alpha + i\sin \alpha$,$b = \cos \beta + i\sin \beta$,$c = \cos \gamma + i\sin \gamma$ and $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1$,then $\cos (\beta - \gamma) + \cos (\gamma - \alpha) + \cos (\alpha - \beta)$ is equal to

Among the statements:
$(S1) :$ The set $\{z \in \mathbb{C} - \{-i\} : |z|=1 \text{ and } \frac{z-i}{z+i} \text{ is purely real}\}$ contains exactly two elements,and
$(S2) :$ The set $\{z \in \mathbb{C} - \{-1\} : |z|=1 \text{ and } \frac{z-1}{z+1} \text{ is purely imaginary}\}$ contains infinitely many elements.

Let $z = 1 + ai$ be a complex number,$a > 0$,such that $z^3$ is a real number. Then the sum $1 + z + z^2 + .... + z^{11}$ is equal to

Solve: $i x^2 - 3 x - 2 i = 0$

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3$ is

The sum of the series $i - 2 - 3i + 4 + 5i - 6 - 7i + 8 + \dots$ up to $100$ terms,where $i = \sqrt{-1}$,is: