Q63 / 150 · MHT CET 2022 (10 Aug Shift 2) · Mathematics

If $\alpha$ and $\beta$ are the complex cube roots of unity, then $\alpha^3+\beta^3+\alpha^{-2} \times \beta^{-2}$ is equal to

  • A
    $1$
  • B
    $-3$
  • Correct answer
    C
    $3$
  • D
    $0$

Solution

$\begin{aligned} & \alpha^3+\beta^3+\alpha^{-2} \cdot \beta^{-2}=\alpha^3+\beta^3+\frac{1}{\alpha^2 \beta^2} \\ & =\omega^3+\omega^6+\frac{1}{\omega^2 \omega^4}=1+1+1=3\end{aligned}$

Hence, the correct answer is option: 'C'.

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