Let z be a complex number such that |z| + z = 3 + i (where i = √(-1)). Then, |z| is equal to

Let z be a complex number such that |z| + z = 3 + i (where… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 11 Jan · Shift II

Let

z

be a complex number such that

|z| + z = 3 + i

(where

i = \sqrt{-1}

). Then,

|z|

is equal to

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

Show answer and solution

Correct option: B

Solution

Let z = x + iy. Then |z| = √(x²+y²) is real, so the imaginary part of |z|+z equals y = 1, giving y = 1.

The real part gives √(x²+y²) + x = 3, so √(x²+1) = 3-x. Squaring: x²+1 = 9-6x+x², giving 6x = 8, x = 4/3.

Therefore |z| = √((4/3)²+1²) = √(16/9 + 9/9) = √(25/9) = 5/3.

[JEE Main 2019 · 11 Jan · Shift II]

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JEE Main 2019 Mathematics question on Complex Numbers and Quadratic Equations