JEE Advanced 2022 · Paper 1 · Mathematics Q4

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MathematicsNumerical+3Question 4 of 54

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

Your answer

Answer and worked solution

Official answer: 0.49

Reduce to one quantity. Divide the numerator and denominator by $z$ , and put $u=4z+\frac2z$ :
$\frac{2+3z+4z^2}{2-3z+4z^2}=\frac{u+3}{u-3}.$
This is real exactly when $u$ is real.

Condition. With $z=x+iy$ ( $y\ne0$ ):
$\operatorname{Im}u=4y-\frac{2y}{|z|^2}=0\implies|z|^2=\frac12=0.5.$

Topics:Algebra of complex numbers Modulus and argument

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JEE Advanced 2022 · Paper 1, Mathematics question 4