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If ${(\sqrt{3} + i)}^{100} = 2^{99} (p + i q)$ then $p^{2}$ and $q^{2}$ are roots of the equation

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$x^{2} + (\sqrt{3} - 1) x - \sqrt{3} = 0$

$x^{2} - (\sqrt{3} - 1) x - \sqrt{3} = 0$

$x^{2} - 4 x + 3 = 0$

$x^{2} + 4 x + 3 = 0$

answer is C.

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Detailed Solution

${(2 c i s \frac{π}{6})}^{100} = 2^{99} (p + i q) \Rightarrow 2 c i s \frac{50 π}{3} = p + i q$

$\Rightarrow p + i q = 2 c i s \frac{2 π}{3} \Rightarrow p = - 1, q = \sqrt{3} \Rightarrow p^{2} = 1, q^{2} = 3$

Required equation is $x^{2} - 4 x + 3 = 0$

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