Number of solutions of the equation, $${z^3} + \frac{{3{{\left| z \right|}^2}}}{z} = 0,$$   where $$z$$ is a complex number and $$\left| z \right| = \sqrt 3 $$   is

Solution :
$$\eqalign{ & {z^3} + \frac{{3{{\left| z \right|}^2}}}{z} = 0, \cr & \Rightarrow \,{z^3} + \frac{{3z.\bar z}}{z} = 0 \cr & \Rightarrow \,{z^3} + 3\bar z = 0 \cr & {\text{Let, }}\,z = r{e^{i\theta }} \cr & \Rightarrow \,{r^3}{e^{i3\theta }} + 3r{e^{ - i\theta }} = 0 \cr & \Rightarrow \,{e^{i4\theta }} = - 1\,\,\,\left[ {\because \,r = \sqrt 3 } \right] \cr & \Rightarrow \,\cos \,4\theta + i\,\sin \,4\theta = - 1 \cr & \Rightarrow \,\cos \,4\theta = - 1\,\,\,\,\,\,.....\left( {\text{i}} \right) \cr & {\text{Now, }}\,0 \leqslant \theta < 2\pi \cr & \Rightarrow \,0 \leqslant 4\theta < 8\pi \cr & \therefore \,\theta = \pi ,3\pi ,5\pi ,7\pi \cr} $$