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# Modulus of complex number and its Properties - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Modulus of complex number and its Properties is considered one the most difficult concept.

• 39 Questions around this concept.

## Solve by difficulty

If $\mathrm{S}=\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-\mathrm{i}|=|\mathrm{z}+\mathrm{i}|=|\mathrm{z}-1|\}$, then, $\mathrm{n}(\mathrm{S})$ is :

## Concepts Covered - 1

Modulus of complex number and its Properties

If z = x + iy is a complex number. Then, the modulus of z, denoted by | z |, is the distance of z from origin in Argand plane, and it is a non-negative real number equal to $\\\mathrm{\sqrt{x^2+y^2}}$

i.e. |z| =$\\\mathrm{\sqrt{x^2+y^2}}$

Every complex number can be represented as a point in the argand plane with x-axis as real axis and y-axis as imaginary axis.

$\left | z \right |=\sqrt{x^2+y^2}=r$ (length r from origin to point (x,y))

Properties of Modulus

i)  |z| ≥ 0

ii) |z| = 0, iff z = 0 and |z| >0, iff z ≠ 0

iii) -|z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z|

iv) $\left | z \right |=\left | \overline{z} \right | = \left | -z \right | = \left | -\overline{z} \right |$

v) $z\overline{z}=\left | z \right |^2$

vi) $\left | z_1 z_2 \right | = \left | z_1 \right |\left | z_2 \right |. \,\,\,\,\,\,Thus,\left | z^n \right | = \left | z \right |^n$

vii) $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$

viii) $\left | z_1\pm z_2 \right | \leq \left | z_1 \right | + \left | z_2 \right |$ (Triangle inequality) this can be generalised for n complex numbers.

ix) $\left | z_1\pm z_2 \right | \geq \left | \left | z_1 \right | - \left | z_2 \right | \right |$

## Study it with Videos

Modulus of complex number and its Properties

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