Explanation

(d)
We have, $a = \cos{\left( \frac{2\pi}{7} \right) + i\sin\left( \frac{2\pi}{7} \right)}$
$\Rightarrow \ a^{7} = \left\lbrack \cos{\left( \frac{2\pi}{7} \right) + i\sin\left( \frac{2\pi}{7} \right)} \right\rbrack^{- 7}$
$= \cos{2\pi + i\sin{2\pi = 1}}\$ …(i)
Let $S = \alpha + \beta = \left( a + a^{2} + a^{4} \right) + (a^{3} + a^{5} + a^{6})$
$\lbrack\because\ \alpha = a + a^{2} + a^{4},\ \beta = a^{3} + a^{5} + a^{6}\rbrack$
$\Rightarrow S = a + a^{2} + a^{3} + a^{4} + a^{5} + a^{6} = \frac{a(1 - a^{6})}{1 - a}$
$\Rightarrow \ \ S = \frac{a - a^{7}}{1 - a} = \frac{a - 1}{1 - a} = - 1$ …(ii)
Let $P = \alpha\beta = \left( a + a^{2} + a^{4} \right)\left( a^{3} + a^{5} + a^{6} \right)$
$= a^{4} + a^{6} + a^{7} + a^{5} + a^{7} + a^{8} + a^{7} + a^{9} + a^{10}$
$= a^{4} + a^{6} + 1 + a^{5} + 1 + a + 1 + a^{2} + a^{3}$ [from Eq. (i)]
$= 3 + \left( a + a^{2} + a^{3} + a^{4} + a^{5} + a^{6} \right) = 3 + S$
$= 3 - 1 = 2\$ [from Eq.(ii)]
Required equation is, $x^{2} - Sx + P = 0$
$\Rightarrow \ \ x^{2} + x + 2 = 0$