Let z be a complex number with non-zero imaginary part. If (2+3z+4z^2)/(2-3z+4z^2) is a real number, then the value of z^2 is ______.

Correct answer: 0.5. Let α=(4z^2+3z+2)/(4z^2-3z+2)=1+6z/(4z^2-3z+2) ⇒ α=1+6/(22z+1z-3) ∵ α∈R then 2z+1/z∈R ⇒2z+1/z=2z¯+1/(z¯) ⇒2z-z¯-z-z¯z^2=0 ⇒z-z¯2-(1z^2=0∵ z≠z¯ (given)So z^2=1)/2

Q22 / 54 · JEE Advanced 2022 (Paper 1) · Mathematics

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 ______.

Correct answer: 0.5

Solution

Let $α = \frac{4 z^{2} + 3 z + 2}{4 z^{2} - 3 z + 2} = 1 + \frac{6 z}{4 z^{2} - 3 z + 2}$
$\Rightarrow α = 1 + \frac{6}{2 (2 z + \frac{1}{z}) - 3}$
$∵ α \in R$ then $2 z + \frac{1}{z} \in R$
$\Rightarrow 2 z + \frac{1}{z} = 2 \bar{z} + \frac{1}{\bar{z}}$
$\Rightarrow 2 (z - \bar{z}) - \frac{z - \bar{z}}{{|z|}^{2}} = 0$
$\Rightarrow (z - \bar{z}) (2 - \frac{1}{{|z|}^{2}}) = 0$
$∵ z \neq \bar{z}$ (given)
So ${|z|}^{2} = \frac{1}{2}$

Hence, the correct answer is 0.5.

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Let z be a complex number with non-zero imaginary part. If 2+3z+4z22-3z+4z2 is a real number, then the value of z2 is ______.

Solution

Relevant Questions

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 ______.