A) 2
B) 3
C) 6
D) 4
Correct Answer: D
Solution :
\[{{z}^{3}}+\frac{3{{\left| z \right|}^{2}}}{z}=0,\,\,\,\Rightarrow {{z}^{3}}+\frac{3z\,.\,\bar{z}}{z}=0\] \[\Rightarrow \,\,\,{{z}^{3}}+3\bar{z}=0\] 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 \[0 \le \theta < 2\pi \Rightarrow 0 \le 4\theta < 8\pi \] \[\therefore \,\,\,\theta =\pi ,\,\,3\pi ,\,\,5\pi ,\,\,7\pi \]
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