Let $${z_1}$$ and $${z_2}$$ be roots of the equation $${z^2} + pz + q = 0\,$$ , where the coefficients p and q may be complex numbers. Let A and B represent $${z_1}$$ and $${z_2}$$ in the complex plane. If $$\angle AOB = \alpha \ne 0\,$$ and OA = OB, where O is the origin, prove that $${p^2} = 4q\,{\cos ^2}\left( {{\alpha \over 2}} \right)$$.

Find all non-zero complex numbers Z satisfying $$\overline Z = i{Z^2}$$.

If $$i{z^3} + {z^2} - z + i = 0$$ , then show that $$\left| z \right| = 1$$.

If $$\left| {Z - W} \right| \le 1,\left| W \right| \le 1$$, show that $${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}$$