Let n denote the number of solutions of the equation z² + 3$$\overline z $$ = 0, where z is a complex number. Then the value of $$\sum\limits_{k = 0}^\infty {{1 \over {{n^k}}}} $$ is equal to :

If z and $$\omega$$ are two complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$\arg (z) - \arg (\omega ) = {{3\pi } \over 2}$$, then $$\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)$$ is :

(Here arg(z) denotes the principal argument of complex number z)

Let a complex number be w = 1 $$-$$ $${\sqrt 3 }$$i. Let another complex number z be such that |zw| = 1 and arg(z) $$-$$ arg(w) = $${\pi \over 2}$$. Then the area of the triangle with vertices origin, z and w is equal to :

If the equation $$a|z{|^2} + \overline {\overline \alpha z + \alpha \overline z } + d = 0$$ represents a circle where a, d are real constants then which of the following condition is correct?