For complex numbers z and w, prove that $${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$ if and only if $$ z = w\,or\,z\overline {\,w} = 1$$.

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 $$\left| z \right| = 1$$.