Question

If z =$\left( \frac{\sqrt{3} + i}{2} \right)^{5} + \left( \frac{\sqrt{3} - i}{2} \right)^{5},\text{ then}$

• A. Re(z) = 0
• B. I_m(z) = 0
• C. Re(z), I_m (z) > 0
• D. Re(z) > 0, I_m(z) < 0

Answer 1

Answer: (B)

I_m(z) = 0

Answer 2

Sol. $z = \left( \frac{\sqrt{3} + i}{2} \right)^{5} + \left( \frac{\sqrt{3} - i}{2} \right)^{5}$=$\left( e^{i\pi/6} \right)^{5} + \left( e^{- i\pi/6} \right)^{5}$=$e^{i5\pi/6} + e^{- i5\pi/6}$

= 2cos$\frac{5\pi}{6}$= –2 cos$\frac{\pi}{6}$ = –$\sqrt{3}$.

Thus Im(z) = 0