JEE MAIN - Mathematics (2025 - 23rd January Evening Shift - No. 2)
The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :
8
10
4
6
Explanation
$$\begin{aligned} & \mathrm{z}=\mathrm{e}^{\mathrm{i} \theta} \\ & \frac{\mathrm{z}}{\overline{\mathrm{Z}}}=\mathrm{e}^{i 2 \theta} \\ & \left|\frac{\mathrm{z}}{\overline{\mathrm{Z}}}+\frac{\overline{\mathrm{z}}}{\mathrm{z}}\right|=1 \Rightarrow\left|\mathrm{e}^{\mathrm{i} 2 \theta}+\mathrm{e}^{-12 \theta}\right|=1 \Rightarrow|\cos 2 \theta|=\frac{1}{2} \end{aligned}$$
8 solution in $[0,2\pi)$
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