Question

$\alpha ,\beta$ are roots of the equation $x^2-x+1=0$ then ${\alpha }^{1027}+{\beta }^{1027}$ equals

• A. $1$
• B. $0$
• C. $-1$
• D. $1027$

Answer 1

Answer: (A)

$1$

the roots of given equation are -$\omega$ and -${\omega }^2$ and as we know that ${\omega }^3=$1
therefore, ${\alpha }^{1027}+{\beta }^{1027}=(-\omega {)}^{1027}+(-{\omega }^2{)}^{1027}$
$=-{\omega }^{1027}-{\omega }^{2054}$
$=-({\omega }^3{)}^{342}\ast \omega -({\omega }^3{)}^{648}\ast {\omega }^2$
$=-\omega -{\omega }^2$
$=1$ (As $1+\omega +{\omega }^2=0$)