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Question:

Let w = √3 + i/2 and P = {wn : n = 1, 2, 3, ...}. Further H1 = {z ∈ C : Re(z) > 1/2} and H2 = {z ∈ C : Re(z) < -1/2}, where C is the set of all complex numbers. If z1 ∈ P ∩ H1, z2 ∈ P ∩ H2 and O represents the origin, then ∠z1Oz2 = π/6, 5π/6, π/2, 2π/3

π/6

5π/6

π/2

2π/3

Solution:

w = √3 + i/2 = eiπ/6
So wn = ei(nπ/6)
Now, for z1, cos(nπ/6) > 1/2 and for z2, cos(nπ/6) < -1/2
Possible positions of z1 are A1, A2, A3 whereas of z2 are B1, B2, B3 (as shown in the figure)
So, possible value of ∠z1Oz2 according to the given options is 2π/3 or 5π/6