Complex Numbers 3 Question 2

2. If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg \frac{1+z}{1+\bar{z}}$ is equal to

(2013 Main)

(a) $-\theta$

(b) $\frac{\pi}{2}-\theta$

(c) $\theta$

(d) $\pi-\theta$

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Answer:

Correct Answer: 2. (c)

Solution:

  1. Given, $|z|=1$, $\arg z=\theta \therefore z=e^{i \theta}$

$$ \begin{aligned} & \therefore \quad \bar{z}=e^{-i \theta} \Rightarrow \bar{z}=\frac{1}{z} \\ & \therefore \quad \arg \frac{1+z}{1+\bar{z}}=\arg \frac{1+z}{1+\frac{1}{z}}=\arg (z)=\theta \end{aligned} $$



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