Question

If $iz^{3}+z^{2}-z+i=0$, then $|z|$ equals

• A. $2$
• B. $1$
• C. $0$
• D. None of these

Answer 1

Answer: (B)

Given $iz^{3}+z^{2}-z+i=0$
$\Rightarrow \, iz^{2}\left(z-i\right)-1\left(z-i\right)=0$
$\Rightarrow \, \left(z-i\right)\left(iz^{2}-1\right)=0$
$\Rightarrow \, z=i$ or $iz^{2}=1$
$\Rightarrow \,\left|z\right|=1$ or $iz^{2}=1\quad...\left(i\right)$
$\Rightarrow \, -i \bar{z}^{2}=1$ (By taking conjugate on both sides) $\quad...\left(ii\right)$
$\therefore \,$ Multiplying $\left(i\right)$ and $\left(ii\right)$, we get
$\left(z \bar{z}\right)^{2}=1$
$\Rightarrow \, \left|z\right|^{2}=1$
$\Rightarrow \, \left|z\right|=1$