Complex Numbers And Quadratic Equations question 663
Question: Number of solutions of the equation, $ z^{3}+\frac{3{{| z |}^{2}}}{z}=0, $ where z is a complex number and $ |z|=\sqrt{3} $ is
Options:
A) 2
B) 3
C) 6
D) 4
Show Answer
Answer:
Correct Answer: D
Solution:
$ z^{3}+\frac{3{{| z |}^{2}}}{z}=0,\Rightarrow z^{3}+\frac{3z.\bar{z}}{z}=0 $
$ \Rightarrow z^{3}+3\bar{z}=0 $ Let $ z=r{e^{i\theta }} $
$ \Rightarrow r^{3}{e^{i3\theta }}+3r{e^{-i\theta }}=0 $
$ \Rightarrow {e^{i4\theta }}=-1\ [\because r=\sqrt{3}] $
$ \Rightarrow cos4\theta +i sin 4\theta = -1 $
$ \Rightarrow cos4\theta =-1 $ … (i) Now $ 0 \le \theta < 2\pi \Rightarrow 0 \le 4\theta < 8\pi $
$ \therefore \theta =\pi ,3\pi ,5\pi ,7\pi $
BETA
