Let $$z$$ and $$\omega $$ be two non zero complex numbers such that
$$\left| z \right| = \left| \omega \right|$$ and $${\rm A}rg\,z + {\rm A}rg\,\omega = \pi ,$$ then $$z$$ equals

If $$\omega \,\left( { \ne 1} \right)$$ is a cube root of unity and $${\left( {1 + \omega } \right)^7} = A + B\,\omega $$ then $$A$$ and $$B$$ are respectively

Let $$z$$ and $$\omega $$ be two complex numbers such that
$$\left| z \right| \le 1,$$ $$\left| \omega \right| \le 1$$ and $$\left| {z + i\omega } \right| = \left| {z - i\overline \omega } \right| = 2$$ then $$z$$ equals

$${\rm{z }} \ne {\rm{0}}$$ is a complex number

Column I

(A) Re z = 0
(B) Arg $$z = {\pi \over 4}$$

Column II

(p) Re$${z^2}$$ = 0
(q) Im$${z^2}$$ = 0
(r) Re$${z^2}$$ = Im$${z^2}$$