Let C be the set of all complex numbers. Let

$${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} $$

$${S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\} $$ and

$${S_3} = \{ z \in C||z - \overline z | \ge 8\} $$.

Then the number of elements in $${S_1} \cap {S_2} \cap {S_3}$$ is equal to :

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 :