Question

Let $\alpha$ be a root of the equation $x^2-x+1=0.$ Let $S={(\alpha +\frac{1}{\alpha })}^2+{({\alpha }^2+\frac{1}{{\alpha }^2})}^2+{({\alpha }^3+\frac{1}{{\alpha }^3})}^2+\cdots +{({\alpha }^{100}+\frac{1}{{\alpha }^{100}})}^2.$ Then which of the following is CORRECT?

• A. $S$ is perfect square
• B. Number of positive factors of $S$ is $9$
• C. $S$ is divisible by $10$
• D. None of these

Answer 1

The correct option is D None of these

$\alpha$ can be $-\omega \text{or}-{\omega }^2$ where $\omega$ is cube root of unity.
$S={(\alpha +\frac{1}{\alpha })}^2+{({\alpha }^2+\frac{1}{{\alpha }^2})}^2+{({\alpha }^3+\frac{1}{{\alpha }^3})}^2+\cdots +{({\alpha }^{100}+\frac{1}{{\alpha }^{100}})}^2$
$={(-\omega -{\omega }^2)}^2+{({\omega }^2+\omega )}^2+{(-{\omega }^3-\frac{1}{{\omega }^3})}^2+\cdots +{(\omega +{\omega }^2)}^2=(1{)}^2+(-1{)}^2+(-2{)}^2+\cdots =67\times 1+33\times 4=199$