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  4. If z is a complex number of unit modulus and argument θ , then arg ((1 + z/1 + barz)) is equal to

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

JEE MainJEE Main 2013Complex Numbers and Quadratic Equations

Solution:

Given, |z | = 1, arg 2 =$\theta \therefore \, z=e^{i\theta}$
But $ \bar{z}=\frac{1}{z}$
$\therefore arg \bigg( \frac{1+z}{1+\frac{1}{2}}\bigg)=arg (z) =\theta $