Question

Let $\alpha $ and $\beta $ be the roots of $x^{2}+x+1=0,$ then the equation whose roots are $\alpha ^{2020}$ and $\beta ^{2020}$ is

• A. $x^{2}+x+1=0$
• B. $x^{2}-x-1=0$
• C. $x^{2}+x-1=0$
• D. $x^{2}-x+1=0$

Answer 1

Answer: (A)

$x^{2}+x+1=0$
$\Rightarrow x=\omega, \omega^{2}$
$\Rightarrow \alpha=\omega$ and $\Rightarrow \beta=\omega^{2}$
$\Rightarrow \alpha^{2020}=\omega^{2020}=\left(\omega^{3}\right)^{673} \cdot \omega=\omega$
$\Rightarrow \beta^{2020}=\left(\omega^{2}\right)^{2020}=\left(\omega^{3}\right)^{2 \times 673} \times \omega^{2}=\omega^{2}$
$\Rightarrow $ The required equation is $x^{2}+x+1=0$