If $\mathrm{Z}=x+i y$ is a complex number, then the number of distinct solution of the equation…
If $\mathrm{Z}=x+i y$ is a complex number, then the number of distinct solution of the equation $z^3+\bar{z}=0$ is
$1$
$3$
Infinite
$5$
Solution
Given, $z^3+\bar{z}=0$, where $z=x+i y$ is a complex number
$\begin{aligned} & \Rightarrow z^3=-\bar{z} \Rightarrow\left|z^3\right|=|-\bar{z}| \Rightarrow|z|^3=|z| \\ & \Rightarrow|z|^3-|z|=0 \Rightarrow|z|\left(|z|^2-1\right)=0\end{aligned}$
$\begin{aligned} & \Rightarrow|z|=0 \text { or }|z|^2=1 \Rightarrow z \bar{z}=1 \Rightarrow \bar{z}=\frac{1}{z} \\ & \because z=0 \text { or } z^3+\frac{1}{z}=0 \Rightarrow z=0 \text { or } z^4+1=0\end{aligned}$
So, total number of distinct solution of given equation is 5 .