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?

Let S₁, S₂ and S₃ be three sets defined as

S₁ = {z$$\in$$C : |z $$-$$ 1| $$ \le $$ $$\sqrt 2 $$}

S₂ = {z$$\in$$C : Re((1 $$-$$ i)z) $$ \ge $$ 1}

S₃ = {z$$\in$$C : Im(z) $$ \le $$ 1}

Then the set S₁ $$\cap$$ S₂ $$\cap$$ S₃ :