Question

If \(\alpha\) and \(\beta\) are complex cube roots of unity, then the difference between the values of \(\alpha ^{2}+\beta ^{2}+\alpha \beta\) and \(\alpha ^{4}+\beta ^{4}+\alpha ^{-1}\beta ^{-1}\) is

• A. \(2\)
• B. \(-2\)
• C. \(4\)
• D. \(0\)

Hint:

Take alpha = omega and beta = omega squared, and use 1 + omega + omega squared = 0 and omega cubed = 1.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Let \(\alpha=\omega\) and \(\beta=\omega^{2}\) with \(\omega^{3}=1\) and \(1+\omega+\omega^{2}=0\).

Step 2: First expression
\(\alpha^{2}+\beta^{2}+\alpha\beta=\omega^{2}+\omega^{4}+\omega^{3}=\omega^{2}+\omega+1=0\)

Step 3: Second expression
\(\alpha^{4}+\beta^{4}+\alpha^{-1}\beta^{-1}=\omega^{4}+\omega^{8}+\dfrac{1}{\omega^{3}}=\omega+\omega^{2}+1=0\)

Step 4: Difference
\(0-0=0\)

Final Answer:
The difference is zero. \[ \boxed{\text{(D) }0} \]