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Question:

Let z = (√3/2 + i/2)^5 + (√3/2 - i/2)^5. If R(z) and I(z) respectively denote the real and imaginary parts of z, then:

R(z)>0 and I(z)>0

I(z)=0

R(z)<0 and I(z)>0

R(z)=3

Solution:

The correct option is D
I(z) = 0
z = (√3/2 + i/2)^5 + (√3/2 - i/2)^5
z = (e^(iπ/6))^5 + (e^(-iπ/6))^5
= e^(i5π/6) + e^(-i5π/6)
= cos(5π/6) + isin(5π/6) + cos(-5π/6) + isin(-5π/6)
= 2cos(5π/6) < 0
I(z) = 0 and Re(z) < 0