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 $x^2+x+1=0$