Question

If $1,\omega ,{\omega }^2$ are the three cube roots of unity, then for $\alpha ,\beta ,\gamma ,\delta ϵR$, the expression $\left(\frac{\alpha +\beta \omega +\gamma {\omega }^2+\delta {\omega }^2}{\beta +\alpha {\omega }^2+\gamma \omega +\delta \omega }\right)$ is

• A. 1
• B. $\omega$
• C. $-\omega$
• D. ${\omega }^{-1}$

Answer 1

The correct option is B

$\omega$

$\left(\frac{\alpha +\beta \omega +\gamma {\omega }^2+\delta {\omega }^2}{\beta +\alpha {\omega }^2+\gamma \omega +\delta \omega }\right)$

$=\frac{\omega (\alpha +\beta \omega +\gamma {\omega }^2+\delta {\omega }^2)}{\beta \omega +\alpha {\omega }^3+\gamma {\omega }^2+\delta {\omega }^2}$

$=\omega (\because {\omega }^3=1)$