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: Use the general identity
For the two non-real cube roots, $\alpha+\beta=-1$ and $\alpha\beta=1$.

Step 2: First expression
$\alpha^{2}+\beta^{2}+\alpha\beta=(\alpha+\beta)^{2}-\alpha\beta=1-1=0$

Step 3: Second expression
$\alpha^{4}=\alpha$ and $\beta^{4}=\beta$, so the value is $\alpha+\beta+1=0$.
The difference is $0$.

Final Answer:
Both expressions are zero, so their difference is zero. \[ \boxed{\text{(D) }0} \]