Question

The equation whose solutions are the non-zero solutions of the equation $\bar{z} = iz^2$ 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}=-i z$
$ \Rightarrow z^{4}+i z =0 $
$ \Rightarrow z\left(z^{3}+i\right)=0$
$ \Rightarrow z^{3}+i =0 (\because z \neq 0) $