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

$${\rm{z }} \ne {\rm{0}}$$ 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$${z^2}$$

If $$a,\,b,\,c$$ and $$u,\,v,\,w$$ are complex numbers representing the vertics of two triangles such that $$c = \left( {1 - r} \right)a + rb$$ and $$w = \left( {1 - r} \right)u + rv,$$ where $$w = \left( {1 - r} \right)u + rv,$$ is a complex number, then the two triangles

If $$z = x + iy$$ and $$\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$$ then $$\,\left| \omega \right| = 1$$ implies that, in the complex plane,