Question

Let $z = (\frac{\sqrt{3}}{2} + \frac{i}{2})^5 + (\frac{\sqrt{3}}{2} - \frac{i}{2})^5$. If R(z) and I(z) respectively denote the real and imaginary parts of z, then

• A. R(z) > 0 and I(z) > 0
• B. R(z) < 0 and I(z) > 0
• C. R(z) = -3
• D. I(z) = 0

Answer 1

Answer: (D)

I(z) = 0

Answer 2

The complex numbers $\frac{\sqrt{3}}{2} + \frac{i}{2}$ and $\frac{\sqrt{3}}{2} - \frac{i}{2}$ can be represented in polar form as $e^{i\pi/6}$ and $e^{-i\pi/6}$ respectively. Using De Moivre's theorem, their fifth powers are $e^{i5\pi/6}$ and $e^{-i5\pi/6}$. The sum $z$ is given by $z = e^{i5\pi/6} + e^{-i5\pi/6}$. Using the identity $e^{i\theta} + e^{-i\theta} = 2\cos\theta$, we get $z = 2\cos(5\pi/6)$. Evaluating $\cos(5\pi/6) = -\frac{\sqrt{3}}{2}$, we find $z = 2(-\frac{\sqrt{3}}{2}) = -\sqrt{3}$. Therefore, the real part of $z$ is $R(z) = -\sqrt{3}$ and the imaginary part is $I(z) = 0$. Option (4) states $I(z)=0$, which is true.