1. Tardigrade
  2. Question
  3. Mathematics
  4. Let z=1+i and z1=(1+i barz/ barz(1-z)+(1)z). Then (12/π) arg (z1) is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Let $z=1+i$ and $z_1=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$. Then $\frac{12}{\pi} \arg \left(z_1\right)$ is equal to________

JEE MainJEE Main 2023Complex Numbers and Quadratic Equations

Solution:

$ z-1+i $
$ z_1=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$
$ z_1=\frac{1+i(1 \quad i)}{(1-i)(1-1-i)+\frac{1}{1+i}} $
$=\frac{1+i-i^2}{(1-i)(-i)+\frac{1-i}{2}} $
$=\frac{2+i}{-3 i-1}=\frac{4+2 i}{-3 i-1}$
$ =\frac{-(4+2 i)(3 i-1)}{(3 i)^2-(1)^2} $
$\text{Arg}\left(z_1\right)=\frac{3 \pi}{4}$
$\therefore \frac{12}{\pi} \arg \left( z _1\right)=\frac{12}{\pi} \times \frac{3 \pi}{4}=9$