Question

The complex number given by [(√3/2)+i/2]^5+[(√3/2)-i/2]^5 lies, on which of the following regions?

(a) imaginary axis

(b) real axis

(c) first quadrant

(d) fourth quadrant

The question was asked during a job interview.

Question is taken from Regions in the Complex Plane in portion Complex Numbers of Engineering Mathematics

Answer 1

Right answer is (b) real axis

The explanation is: Let z=[(√3/2)+i/2]^5+[(√3/2)-i/2]^5 and ω=[(√3/2)+i/2]^5 ⇒ \(\overline{\omega}\)=[(√3/2)-i/2]^5

⇒ z=ω+\(\overline{\omega}\) ⇒ z is real (sum of conjugates is real) ⇒ z lies on real axis.