JEE Advanced 2023 Paper 1 · Q17 · Complex Numbers

Question

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 non-zero. Match each entry in List-I to the correct entries in List-II.
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) $7$

SolveKar AI · Accepted Answer

Let $z = r(\cos	heta + i\sin	heta)$. Then $r^3 + 2r^2 e^{2i	heta} + 4re^{-i	heta} - 8 = 0$. Imaginary part: $4r^2\sin	heta\cos	heta = 4r\sin	heta \Rightarrow r\cos	heta = 1$. Substitute into real part: $r^3 + 2(2 - r^2) + 4 - 8 = 0 \Rightarrow r = 2$. So $|z|^2 = 4$ — (2). $|z-\bar{z}|^2 = 4r^2\sin^2	heta = 4(r^2 - r^2\cos^2	heta) = 16 - 4 = 12$ — (1). $|z+\bar{z}|^2 = 4r^2\cos^2	heta = 4$; so $|z|^2 + |z+\bar{z}|^2 = 4 + 4 = 8$ — (3). $|z+1|^2 = z\bar{z} + z + \bar{z} + 1 = 4 + 2(r\cos	heta) + 1 = 7$ — (5).

JEE Advanced 2023 Paper 1 · Q17 · Complex Numbers

Solution