Complex Numbers: If (sqrt(3)+i)^100=2^99(p+iq) , then p and q are roots of the… | JEE Main 2021

JEE Main 2021 (Online) 26th August Evening ShiftQuestion 1

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

Complex Numbers

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

Solution

$(\sqrt3+i)^{100}=2^{100}e^{i\,100\pi/6}=2^{100}e^{i(16\pi+2\pi/3)}=2^{99}(-1+\sqrt3 i)$. $p=-1,q=\sqrt3$: sum $\sqrt3-1$, product $-\sqrt3$. $x^2-(\sqrt3-1)x-\sqrt3=0$. Option (B).

Complex Numbers - JEE Mathematics question (JEE Main 2021 (Online) 26th August Evening Shift)

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