Question

If $\alpha$ and $\beta$ are imaginary cube roots of unity and x = a+b,y = a$\alpha$ + b$\beta$, z= a$\beta$ + b$\alpha$, then xyz =

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

$\alpha$,$\beta$ are imaginary cube roots of unity.

$\therefore$ let $\alpha$ = $\omega$ and $\beta$ = ${\omega }^2$

$\therefore$ xyz = (a + b)(a$\alpha$ + b$\beta$)(a$\beta$ + b$\alpha$)

= (a + b) $\left[\right(a^2+b^2)\alpha \beta +ab({\alpha }^2+bet{a}^2\left)\right]$

= (a + b) $\left[\right(a^2+b^2){\omega }^3+ab({\omega }^2+omeg{a}^4\left)\right]$

= (a + b) $[a^2+b^2+ab({\omega }^2+omega\left)\right]$

= $(a+b)(a^2+b^2-ab)=a^3+b^3$