Question

Let α and β be the roots of the equation $x^{2} + x + 1 = 0$, 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. $x^{2} + x + 1 = 0$

Answer 1

Answer: (D)

$x^{2} + x + 1 = 0$

Answer 2

Roots of $x^{2} + x + 1 = 0$ are

$x = \frac{- 1 \pm \sqrt{1 - 4}}{2}, = \frac{- 1 \pm \sqrt{3}i}{2} = \omega,\omega^{2}$

Take $\alpha = \omega,\beta = \omega^{2}$

∴ $\alpha^{19} = w^{19} = w,\beta^{7} = (w^{2})^{7} = w^{14} = w^{2}$

∴ Required equation is $x^{2} + x + 1 = 0$