Question

x and y are two complex numbers such that \(|x| = |y| = 1\). If \( \text{Arg}(x) = 2\alpha \), \( \text{Arg}(y) = 3\beta \) and \( \alpha + \beta = \frac{\pi}{36} \), then \( x^6 y^4 + \frac{1}{x^6 y^4} \) is:

• A. \(0\)
• B. \(-1\)
• C. \(1\)
• D. \(\frac{1}{2}\)

Hint:

Utilize Euler's formula \( e^{i\theta} = \cos\theta + i\sin\theta \) for simplifying complex exponentials.

Answer 1

The Correct Option is C

Step 1: Expressing in exponential form.
Since \(|x| = |y| = 1\), we express them as \( x = e^{i 2\alpha} \) and \( y = e^{i 3\beta} \). 

Step 2: Evaluating \( x^6 y^4 \).
\[ x^6 y^4 = e^{i(12\alpha + 12\beta)} = e^{i 12 (\alpha + \beta)} = e^{i 12 \frac{\pi}{36}} = e^{i \frac{\pi}{3}} \] \[ \frac{1}{x^6 y^4} = e^{-i \frac{\pi}{3}} \] \[ x^6 y^4 + \frac{1}{x^6 y^4} = e^{i \frac{\pi}{3}} + e^{-i \frac{\pi}{3}} = 2\cos \frac{\pi}{3} = 1 \] 

Step 3: Conclusion.
Thus, \( x^6 y^4 + \frac{1}{x^6 y^4} = 1 \).