Number of solutions of the equation $$z^2+|z|^2=0$$ and $$z \neq 0$$ is

If $$z_1$$ and $$z_2$$ be nth root of unity which subtend a right angled at the origin. Then, $$n$$ must be of the form

If $$|w| = 2$$, then the set of points $$z = w - {1 \over w}$$ is contained in or equal to the set of points z satisfying

The smallest positive integral value of n such that $${\left[ {{{1 + \sin {\pi \over 8} + i\cos {\pi \over 8}} \over {1 + \sin {\pi \over 8} - i\cos {\pi \over 8}}}} \right]^n}$$ is purely imaginary, is equal to