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JEE MAIN Mathematics QUESTION #6035
Question 1

Suppose $z$ is a complex number of unit modulus with argument $\theta$. Find $\arg\!\left(\dfrac{1+z}{1+\bar{z}}\right)$.

  • $-\theta$
  • $\dfrac{\pi}{2}-\theta$
  • $\theta$✔️
  • $\pi-\theta$
Correct Answer Explanation

Let $z=e^{i\theta}=\cos\theta+i\sin\theta$. Then $\bar{z}=e^{-i\theta}$.

$\frac{1+z}{1+\bar{z}}=\frac{1+e^{i\theta}}{1+e^{-i\theta}}=\frac{e^{i\theta/2}(e^{-i\theta/2}+e^{i\theta/2})}{e^{-i\theta/2}(e^{i\theta/2}+e^{-i\theta/2})}=\frac{e^{i\theta/2}}{e^{-i\theta/2}}=e^{i\theta}$

So $\arg\!\left(\frac{1+z}{1+\bar{z}}\right)=\theta$