Q. Let $x = \alpha + \beta,\ y = \alpha\omega + \beta\omega^{2},\ z = \alpha\omega^{2} + \beta\omega,\ \omega$ is an imaginary cube root of unity. The value of $xyz$ is
A) $\alpha^{2} + \beta^{2}$
B) $\alpha^{2} - \beta^{2}$
C) $\alpha^{3} + \beta^{3}$
D) $\alpha^{3} - \beta^{3}$
Explanation
(c)
$xyz = (\alpha + \beta)\left( \alpha\omega + \beta\omega^{2} \right)(\alpha\omega^{2} + \beta\omega)$
$= (\alpha + \beta)\lbrack\alpha^{2} + \alpha\beta\left( \omega^{2} + \omega \right) + \beta^{2}\rbrack$ $\left\lbrack \begin{array}{r}
\because 1 + \omega + \omega^{2} = 0 \\
and\ \ \omega^{3} = 1
\end{array} \right\rbrack\ \$
$= (\alpha + \beta)(\alpha^{2} - \alpha\beta + \beta^{2})$
$= \alpha^{3} + \beta^{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.