Question

If arg $z^{3/8}=\frac{1}{2}\arg (z^2+\overline{z}\sqrt{z})$, then $\left|z\right|=$

• A. 1
• B.
• C.
• D.

Answer 1

The correct option is A 1

arg $(z^2+\overline{z}\sqrt{z})=2\arg z^{3/8}=\arg z^{3/4}$
arg $arg\left(\frac{z^2+\overline{z}\sqrt{z}}{z^{3/4}}\right)=0,z^{5/4}+\overline{z}.z^{-1/4}=\lambda \in R$
with z = ${re}^{i\theta },the$imaginary part gives
$r^{5/4}\sin \frac{5\theta }{4}-r^{\frac{3}{4}}sin\frac{5\theta }{4}=0$
$⟹r=1$