The number of complex numbers z , satisfying z =1 and frac z overline z +frac overline z z =1 ; arg (z) in (0, 2π) , is :

Mathematics · Complex Numbers

JEE Main 2025 — 23 January, Evening Shift — Question 17

The number of complex numbers $z$ , satisfying $|z|=1$ and $\left|\frac{\mathrm{z}}{\overline{\mathrm{z}}}+\frac{\overline{\mathrm{z}}}{\mathrm{z}}\right|=1$ ; $\mathrm{arg}(z) \in (0, 2\pi)$ , is :

Option A:
6
Option B:
4
Option C:
10
Option D:
8
Correct

Answer: D

Step-by-step solution

$\quad z=e^{i \theta}$

$\frac{\mathrm{z}}{\overline{\mathrm{z}}}=\mathrm{e}^{\mathrm{i} 2 \theta}$

$\left|\frac{\mathrm{z}}{\overline{\mathrm{z}}}+\frac{\overline{\mathrm{z}}}{\mathrm{z}}\right|=1 \Rightarrow\left|\mathrm{e}^{\mathrm{i} 2 \theta}+\mathrm{e}^{-12 \theta}\right|=1 \Rightarrow|\cos 2 \theta|=\frac{1}{2}$

8 solution

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 23 January, Evening Shift
Subject: Mathematics
Chapter: Complex Numbers
Topic: Representation of a Complex Number