Let $a = e^{i\frac{2\pi}{3}\ }$. Then, the equation whose roots are $a + a^{- 2}$ and $a^{2} + a^{- 4}$ is

Question

Dr. Abubacker · Accepted Answer

(d)
Here, $a = e^{i\ 2\pi/3} = \omega$
$	herefore\ \ \ a + \frac{1}{a^{2}} = \omega + \frac{1}{\omega^{2}} = \omega + \omega = 2\omega$
Similarly, $a^{2} + \frac{1}{a^{2}} = \omega^{2} + \frac{1}{\omega^{4}} = 2\omega^{2}$
$	herefore\ \ \ \ \ a + \frac{1}{a^{2}} + a^{2} + \frac{1}{a^{4}} = 2\omega + 2\omega^{2} = - 2$
And $\left( a + \frac{1}{a^{2}} ight)\left( a^{2} + \frac{1}{a^{4}} ight) = 2\omega.2\omega^{2} = 4$
$	herefore$ required equation is $x^{2} + 2x + 4 = 0$

Q. Let $a = e^{i\frac{2\pi}{3}\ }$. Then, the equation whose roots are $a + a^{- 2}$ and $a^{2} + a^{- 4}$ is

Explanation

Reviewed by Dr. Abubacker

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