Question

Let \(z_{1} = \frac{1}{2} +i\frac{\sqrt{3}}{2}\) and \(z_{2} = -\frac{1}{2} -i\frac{\sqrt{3}}{2}\) . If \(w = z_{1} + \bar{z}_{2}\) , then \(\overline{w} =\)

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

Hint:

\(z_1\) and \(z_2\) are cube roots of unity: \(z_1 = e^{i\pi/3}, z_2 = e^{-i2\pi/3}\). Their sum with conjugate may simplify.

Answer 1

The Correct Option is D

Step 1: Concept:
• Given: \(\bar{z}_2 = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\)
• We need to find: \[ w = z_1 + \bar{z}_2 \]

Step 2: Calculation:
• Substitute values: \[ w = \left(\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) + \left(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\right) \]
• Add real parts: \[ \frac{1}{2} - \frac{1}{2} = 0 \]
• Add imaginary parts: \[ \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \sqrt{3} \]
• So, \[ w = i\sqrt{3} \]

Step 3: Final Answer:
• Conjugate of \(w\): \[ \overline{w} = -i\sqrt{3} \]
• Correct Option: (D)