Question

If \( z = e^{i\theta} \), then which of the following complex numbers is of unit modulus?

• A. \( \frac{1}{z} - 1 \)
• B. \( z + 1 \)
• C. \( z + \frac{1}{z} \)
• D. \( \frac{1}{z} \)

Hint:

A complex number of unit modulus satisfies \( |z| = 1 \). Use Euler's formula for simplification.

Answer 1

The Correct Option is C

Step 1: Modulus of \( z \).
Since \( z = e^{i\theta} \), the modulus of \( z \) is \( |z| = 1 \). Step 2: Evaluating the given options.
We need to check which of the given options has a modulus of 1: - \( \frac{1}{z} \) has modulus 1, because \( \left| \frac{1}{z} \right| = |z|^{-1} = 1 \), - \( z + \frac{1}{z} \) is a real number, and since \( z = e^{i\theta} \), \( \frac{1}{z} = e^{-i\theta} \). Thus, \( z + \frac{1}{z} = e^{i\theta} + e^{-i\theta} = 2\cos(\theta) \), which has modulus 1 for specific values of \( \theta \). Step 3: Conclusion.
The correct answer is (3) \( z + \frac{1}{z} \).