JEE Advanced 2022 Paper 1, Mathematics Section 1 Q4: Introduction to Complex Numbers And Their Conjugates

JEE Advanced2022Paper 1Mathematics Section 1

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 _________.

Your answer

Solution

Write $w = \frac{4 z ^{2} + 3 z + 2}{4 z ^{2} - 3 z + 2} = 1 + \frac{6 z}{4 z ^{2} - 3 z + 2} = 1 + \frac{6}{2 ( 2 z + \frac{1}{z} ) - 3}$ after dividing numerator and denominator of the second fraction by $z$ .

For $w \in R$ , the quantity $2 z + \frac{1}{z}$ must be real. Equating it to its conjugate gives $2 z + \frac{1}{z} = 2 \overset{z}{ˉ} + \frac{1}{z ˉ}$ .

Rearranging, $2 (z - \overset{z}{ˉ}) - \frac{z - z ˉ}{∣ z ∣ ^{2}} = 0$ , i.e., $(z - \overset{z}{ˉ}) (2 - \frac{1}{∣ z ∣ ^{2}}) = 0$ .

Since the imaginary part of $z$ is non-zero, $z \neq = \overset{z}{ˉ}$ . Therefore $2 - \frac{1}{∣ z ∣ ^{2}} = 0$ , which gives $∣ z ∣^{2} = \frac{1}{2} = 0.50$ .

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JEE Advanced 2022 Paper 1, Mathematics Section 1 Q4: Introduction to Complex Numbers And Their Conjugates

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