# NCERT Class 11-Math՚S: Chapter – 5 Complex Numbers and Quadratic Equations Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**5.1. 9 Modulus of a complex number**

Let be a complex number. Then the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute value) of and it is denoted by

In the set of complex numbers are meaningless but

are meaningful because are real numbers.

**5.1. 10 Properties of modulus of a complex number**

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

In particular.

11. As stated earlier multiplicative inverse (reciprocal) of a complex number is

**5.2 Argand Plane**

A complex number can be represented by a unique point in the Cartesian plane referred to a pair of rectangular axes. The complex number represent the origin . A purely real number *a*, i.e.. , is represented by the point on axis. Therefore, axis is called real axis. A purely imaginary number , i.e.. , is represented by the point on *y*-axis. Therefore, is called imaginary axis.

Similarly, the representation of complex numbers as points in the plane is known as

Argand diagram. The plane representing complex numbers as points is called complex plane or Argand plane or Gaussian plane.

If two complex numbers and be represented by the points P and Q in the complex plane, then

**5.2. 1 Polar form of a complex number**

Let P be a point representing a non-zero complex number in the Argand plane. If makes an angle with the positive direction of *x*-axis, then is called the polar form of the complex number, where and . Here called argument or amplitude of and we write it as .

The unique value of such that is called the principal argument.

**5.2. 2 Solution of a quadratic equation**

The equations , where and are numbers (real or complex, ) is called the general quadratic equation in variable . The values of the variable satisfying the given equation are called roots of the equation.

The quadratic equation with real coefficients has two roots given by where , called the discriminant of the equation.

**Notes**:

1. When , roots of the quadratic equation are real and equal. When , roots are real and unequal.

Further, if , and is a perfect square, then the roots of the equation are rational and unequal, and if *a*, *b*, and is not a perfect square, then the roots are irrational and occur in pair.

When , roots of the quadratic equation are non-real (or complex) .

2. Let be the roots of the quadratic equation then sum of the roots

and the product of the roots

3. Let S and P be the sum of roots and product of roots, respectively, of a quadratic equation. Then the quadratic equation is given by

## 5.2 Solved Examples

### Short Answer Type

**Question 1**:

Evaluate:

**Answer**:

And

Therefore,