Question

If $\omega$ and ${\omega }^2$ are the two imaginary cube root unity, then the equation whose roots are $a{\omega }^{317}$ and $a{\omega }^{382}$ is:

• A. $x^2+ax-a^2=0$
• B. $x^2+a^{2}x+a=0$
• C. $x^2+ax+a^2=0$
• D. $x^2-a^{2}x+a=0$

Answer 1

Answer: (C)

If $\omega$ and ${\omega }^2$ are two imaginary cube roots of unity, then
$1+\omega +{\omega }^2=0$
$\Rightarrow$ $\omega +{\omega }^2=-1$ ...(i)
The sum of roots $=a{\omega }^{317}+a{\omega }^{382}$
$=a({\omega }^{317}+{\omega }^{382})$
$=a({\omega }^2+\omega )=-a$ [from (i)]
The product of roots
$=a{\omega }^{317}\times a{\omega }^{382}=a^2{\omega }^{699}=a^2$
Therefore, the required equation is
$x^2-$ (Sum of roots) x+ (Product of roots) = 0
$\Rightarrow$ $x^2+ax+a^2=0.$
Note: Cube roots of $-1$ are $-1,-\omega ,-{\omega }^2.$