Question

Statement I Let $z_1$ and $z_2$ be two complex numbers such that $\overline{z_1}+i \overline{z_2}=0$ and $\arg \left(z_1 \cdot z_2\right)=\pi$, then $\arg \left(z_1\right)$ is $\frac{3 \pi}{4}$.
Statement II $\arg \left(z_1 \cdot z_2\right)=\arg z_1+\arg z_2$.

• A. Statement I is correct
• B. Statement II is correct
• C. Both are correct
• D. Neither I nor II is correct

Answer 1

Answer: (C)

Given that, $\overline{z_1}+i \overline{z_2}=0$
$z_1=i z_2 \text {, i.e., } z_2=-i z_1 $
$\text { Thus, } \arg \left(z_1 z_2\right)=\arg z_1+\arg \left(-i z_1\right)=\pi$
$ \because \arg \left(z_1 z_2\right)=\arg z_1+\arg z_2$
$ \Rightarrow \arg \left(-i z_1^2\right)=\pi$
$ \Rightarrow \arg (-i)+2 \arg \left(z_1\right)=\pi$
$ \Rightarrow \frac{-\pi}{2}+2 \arg \left(z_1\right)=\pi $
$\Rightarrow \arg \left(z_1\right)=\frac{3 \pi}{4} $