(B) Let $z = x + iy$. Given $|z| + z = 3 + i$. Substituting $z = x + iy$,we get $\sqrt{x^2 + y^2} + x + iy = 3 + i$. Comparing the imaginary parts,we

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(B) Let $z = x + iy$. Given $|z| + z = 3 + i$. Substituting $z = x + iy$,we get $\sqrt{x^2 + y^2} + x + iy = 3 + i$. Comparing the imaginary parts,we get $y = 1$. Comparing the real parts,we get $\sqrt{x^2 + 1} + x = 3$. $\sqrt{x^2 + 1} = 3 - x$. Squaring both sides,$x^2 + 1 = (3 - x)^2 = 9 - 6x + x^2$. $1 = 9 - 6x$,which implies $6x = 8$,so $x = \frac{4}{3}$. Now,$|z| = \sqrt{x^2 + y^2} = \sqrt{(\frac{4}{3})^2 + 1^2} = \sqrt{\frac{16}{9} + 1} = \sqrt{\frac{25}{9}} = \frac{5}{3}$.

Class 11 Mathematics 4-1.Complex numbers Conjugate and Modulus of complex numbers

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Let $z$ be a complex number such that $|z| + z = 3 + i$ (where $i = \sqrt{-1}$). Then $|z|$ is equal to

JEE MAIN 2019 All JEE MAIN

A
$\frac{\sqrt{34}}{3}$
B
$\frac{5}{3}$
C
$\frac{\sqrt{41}}{4}$
D
$\frac{5}{4}$

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If $Z_1$ and $Z_2$ are conjugate complex numbers, match the items in Column-$I$ with Column-$II$:
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$B. Z_1 + Z_2 = 0$ $2. \text{Im}(-Z_2)$
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Column-$I$	Column-$II$
$A. Z_1 Z_2$	$1. \text{imaginary axis}$
$B. Z_1 + Z_2 = 0$	$2. \text{Im}(-Z_2)$
$C. \text{Im}(Z_1)$	$3. \|Z_1\|^2$
$D. \text{Re}(Z_1)$	$4. \text{Re}(Z_2)$

Let $z$ be a complex number such that $|z| + z = 3 + i$ (where $i = \sqrt{-1}$). Then $|z|$ is equal to

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If $Z_1$ and $Z_2$ are conjugate complex numbers, match the items in Column-$I$ with Column-$II$:
Column-$I$ Column-$II$
$A. Z_1 Z_2$ $1. \text{imaginary axis}$
$B. Z_1 + Z_2 = 0$ $2. \text{Im}(-Z_2)$
$C. \text{Im}(Z_1)$ $3. |Z_1|^2$
$D. \text{Re}(Z_1)$ $4. \text{Re}(Z_2)$