JEE Advanced 2023 · Paper 1 · Mathematics Q17

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MathematicsList match+3 / -1Question 17 of 51

Let $z$ be a complex number satisfying $|z|^3 + 2z^2 + 4\bar z - 8 = 0$ , where $\bar z$ denotes the complex conjugate of $z$ . Let the imaginary part of $z$ be nonzero.

Match each entry in List-I to the correct entries in List-II.

The correct option is:

List-I

P
$|z|^2$ is equal to
Q
$|z - \bar z|^2$ is equal to
R
$|z|^2 + |z + \bar z|^2$ is equal to
S
$|z + 1|^2$ is equal to

List-II

1
$12$
2
$4$
3
$8$
4
$10$
5
$7$

P → 1; Q → 3; R → 5; S → 4

P → 2; Q → 1; R → 3; S → 5

P → 2; Q → 4; R → 5; S → 1

P → 2; Q → 3; R → 5; S → 4

Answer and worked solution

Official answer: B

Let $z=x+iy$ with $y\ne0$ and $r=|z|$ .

Imaginary part. $2z^2$ contributes $4xy$ and $4\bar z$ contributes $-4y$ :
$4xy-4y=0\implies x=1.$

Real part.
$r^3+2(x^2-y^2)+4x-8=0\implies r^3+2-2y^2-4=0.$
With $y^2=r^2-1$ this becomes $r^3-2r^2=0$ , so $r=2$ and $y^2=3$ .

Entry	Value	Match
(P) $\|z\|^2$	4	2
(Q) $\|z-\bar z\|^2=4y^2$	12	1
(R) $\|z\|^2+\|z+\bar z\|^2=4+4$	8	3
(S) $\|z+1\|^2=2^2+3$	7	5

P→2, Q→1, R→3, S→5.

Topics:Modulus and argument Algebra of complex numbers

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JEE Advanced 2023 · Paper 1, Mathematics question 17