Mathematics · Complex Numbers

JEE Advanced 2022 — Paper 1 — Question 4

Let z be a complex number with non-zero imaginary part. If $\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}$ is a real number,

then the value of $|z|^{2}$ is

Answer: 0.5

Numerical answer — enter this value.

$\frac{2+3 z+4 z^{2}}{2-3 z+4 z^{2}}$ is a real number

$\Rightarrow 1+\frac{6}{\frac{2}{z}-3+4 z} \in R$

$\Rightarrow 2 z+\frac{1}{z} \in R$

$\Rightarrow 2 z+\frac{1}{z}=2 \bar{z}+\frac{1}{\bar{z}}$

$\Rightarrow(2 \overline{\mathrm{z}} \mathrm{z}-1)(\mathrm{z}-\overline{\mathrm{z}})=0$

$\Rightarrow|z|^{2}=\frac{1}{2}=0.50$

Answer key and solution verified before publishing.

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