Question

Assertion (A): If the arguments of $\bar{z}_{1}$ and $z_{2}$ are $\frac{\pi}{5}$ and $\frac{\pi}{3}$ respectively, then $\arg \left(z_{1} z_{2}\right)$ is $\frac{2 \pi}{15}$
Reason (R): For any complex number $z, \arg \bar{z}=\frac{\pi}{2}+\arg z$
The collect option among the following is

• A. (A) is true, (R) is true and (R) is the correct explanation for (A)
• B. (A) is true, (R) is true but (R) is not the correct explanation for (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Answer 1

Answer: (C)

We have, $\arg \left(\bar{z}_{1}\right)=\frac{\pi}{5}, \arg \left(z_{2}\right)=\frac{\pi}{3}$
$\therefore \arg \left(z_{1} z_{2}\right)=\arg \left(z_{2}\right)-\arg \left(\bar{z}_{1}\right)=\frac{\pi}{3}-\frac{\pi}{5}=\frac{2 \pi}{15}$
Also, $\arg (\bar{z})=-\arg (z)$.
(A) is ture but (R) is false.