The complex number $z$, satisfying the equation $z^3=\stackrel{-}{z}$ and $\arg \left(z+1\right)=\frac{\pi }{4}$ simultaneously, is (where, $i^2=-1$)

• Correct answer
  A

  $i$
• B

  $1+2i$
• C

  $2+3i$
• D

  $3+4i$

Solution

$z^3=\stackrel{-}{z}\mathrm{}\Rightarrow \left|z\right|=0$ or $1$
$\left|z\right|=0\mathrm{}\Rightarrow z=0$
$\left|z\right|=1\mathrm{}\Rightarrow z^4=1$
$\Rightarrow z=\pm 1,\pm 1$
Only $z=i$ satisfies $\arg \left(z+1\right)=\frac{\pi }{4}$

Hence, the correct answer is option: 'A'.