Question

The equation whose solutions are the non-zero solutions of the equation $\bar{z}=i{z}^2$ is

• A. $z^3+i=0$
• B. $z^3+z+1=0$
• C. $z^3-i=0$
• D. $z^3+iz+1=0$

Answer 1

Answer: (A)

$\bar{z}=i{z}^2$
$\Rightarrow z=-i{\bar{z}}^2$
$\Rightarrow z=-i{\left[i{z}^2\right]}^2$
$\Rightarrow z=-i{i}^2{z}^4$
$\Rightarrow z=i{z}^4$
$\Rightarrow z^4=\frac{1}{i}z$
$\Rightarrow z^4=-iz$
$\Rightarrow z^4+iz=0$
$\Rightarrow z\left(z^3+i\right)=0$
$\Rightarrow z^3+i=0(\because z\ne 0)$