Question

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

• A. $x^{2}-(\sqrt{3}-1) x-\sqrt{3}=0$
• B. $x^{2}+(\sqrt{3}+1) x+\sqrt{3}=0$
• C. $x^{2}+(\sqrt{3}-1) x-\sqrt{3}=0$
• D. $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$

Answer 1

Answer: (A)

$\left(2 e ^{ i \pi / 6}\right)^{100}=2^{99}( p + iq )$
$2^{100}\left(\cos \frac{50 \pi}{3}+ i \sin \frac{50 \pi}{3}\right)$
$=2^{99}( p + iq )$
$p + iq =2\left(\cos \frac{2 \pi}{3}+ i \sin \frac{2 \pi}{3}\right)$
$p =-1, $
$q =\sqrt{3}$
$x ^{2}-(\sqrt{3}-1) x -\sqrt{3}=0$