Question

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. $x^2 + x + 1 = 0$

Answer 1

Answer: (D)

The roots of $x^{2}+x+1=0$
are $\omega, \,\omega^{2}$
Let $\alpha=\omega,\, \beta=\omega^{2}$
Now, $\alpha^{19}=\omega^{19}=\omega^{18+1}\,=\omega^{18} \omega=\omega$
and $\beta^{7}=\left(\omega^{2}\right)^{7}\,=\omega^{12} \omega^{2}=\omega^{2}$
which are the same roots.
$\therefore $ Equation $x^{2}+x+1=0$