Question

Which of the following complex numbers is conjugate to its square?

• A. \(1 - i\sqrt{3}\)
• B. \(-1 - i\sqrt{3}\)
• C. \(\frac{1}{2} - \frac{i\sqrt{3}}{2}\)
• D. \(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\)

Answer 1

The Correct Option is D

Let \(z = -\frac{1}{2} + \frac{i\sqrt{3}}{2}\), then \[ z^2 = \left(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\right)^2 = \frac{1}{4} - i\sqrt{3} \cdot \frac{1}{2} - \frac{3}{4} = -\frac{1}{2} - \frac{i\sqrt{3}}{2} \] Now, \[ \overline{z} = -\frac{1}{2} - \frac{i\sqrt{3}}{2} = z^2 \] So, \(z\) is conjugate of \(z^2\).