$$\begin{gathered} \mathrm{If} {\left(\sqrt{3}+\mathrm{i}\right)}^{100}=2^{99}\left(\mathrm{p}+\mathrm{iq}\right), \mathrm{then} \mathrm{p} \mathrm{and} \mathrm{q} \mathrm{are} \\ \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} \end{gathered}$$

$\begin{array}{rcl}{\left(\sqrt{3}+\mathrm{i}\right)}^{100}& =& 2^{99}(\mathrm{p}+\mathrm{q})\end{array}$

$\begin{array}{rcl}{\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)}^{100}& =& \frac{\mathrm{p}+\mathrm{iq}}{2}\end{array}$

$\begin{array}{rcl}\frac{\mathrm{p}+\mathrm{iq}}{2}& =& {\left(\cos \frac{\mathrm{\pi }}{6}+\mathrm{i} \sin \frac{\mathrm{\pi }}{6}\right)}^{100}\end{array}$

$\begin{array}{rcl}\frac{\mathrm{p}+\mathrm{iq}}{2}& =& \left(\cos 50\frac{\mathrm{\pi }}{3}+\mathrm{i} \sin 50\frac{\mathrm{\pi }}{3}\right)\end{array}$

$\begin{array}{rcl}\frac{\mathrm{p}+\mathrm{iq}}{2}& =& \cos \left(16\mathrm{\pi }+\frac{2\mathrm{\pi }}{3}\right)+\mathrm{i} \sin \left(16\mathrm{\pi }+\frac{2\mathrm{\pi }}{3}\right)\end{array}$

$\begin{array}{rcl} & =& \cos \frac{2\mathrm{\pi }}{3}+\mathrm{i} \sin \frac{2\mathrm{\pi }}{3}\end{array}$

$\begin{array}{rcl} & =& \frac{-1}{2}+\frac{\mathrm{i}\sqrt{3}}{2}\end{array}$

$\begin{array}{rcl}\mathrm{p}& =& -1, \mathrm{q}=\sqrt{3}\end{array}$

The quadratec equation whose roots are p, q is

$$\begin{gathered} {\mathrm{x}}^2-\mathrm{x}(\mathrm{p}+\mathrm{q})+\mathrm{pq}=0 \\ {\mathrm{x}}^2-\mathrm{x}\left(\sqrt{3}-1\right)-\sqrt{3}=0 \end{gathered}$$