Q. If $\alpha,\ \beta$ be the roots of the quadratic equation $x^{2} + x + 1 = 0$, then the equation whose roots are $\alpha^{19},\ \beta^{7}$ is
A) $x^{2} - x + 1 = 0$
B) $x^{2} - x - 1 = 0$
C) $x^{2} + x - 1 = 0$
D) $x^{2} + x + 1 = 0$
Explanation
(d)
$\alpha = \omega,\ \beta = \omega^{2}$ will satisfy the given equation
Now, $\alpha^{19} = \omega^{19} = \omega$
$\beta^{7} = \omega^{14} = \omega^{2}$
$\Rightarrow$ Required equation is
$x^{2} - \left( \omega + \omega^{2} \right)x + \omega^{2} = 0$
$\Rightarrow \ \ \ x^{2} + x + 1 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.