Complex Numbers and Quadratic Equations – Class 11 Maths (NCERT Based)
The chapter Complex Numbers and Quadratic Equations introduces a new number system that helps solve equations which do not have real solutions. This chapter is very important for higher algebra, coordinate geometry, and competitive examinations.
📖 Part A: Complex Numbers
📌 What is a Complex Number?
A complex number is a number of the form:
z = a + ib
where:
- a is the real part
- b is the imaginary part
- i is the imaginary unit such that i² = −1
✍️ Types of Complex Numbers
- Purely Real Number: b = 0
- Purely Imaginary Number: a = 0
- Zero Complex Number: a = 0 and b = 0
🔢 Algebra of Complex Numbers
For two complex numbers
z₁ = a + ib and z₂ = c + id:
- Addition: (a + c) + i(b + d)
- Subtraction: (a − c) + i(b − d)
- Multiplication: (ac − bd) + i(ad + bc)
🔄 Conjugate of a Complex Number
The conjugate of z = a + ib is:
z̄ = a − ib
Important Property:
z × z̄ = a² + b²
📐 Modulus of a Complex Number
The modulus of z = a + ib is:
|z| = √(a² + b²)
It represents the distance of the complex number from the origin in the complex plane.
📖 Part B: Quadratic Equations
📌 What is a Quadratic Equation?
A quadratic equation is an equation of the form:
ax² + bx + c = 0, where a ≠ 0
✍️ Solution of Quadratic Equations
Quadratic equations can be solved using the quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
🔍 Discriminant and Nature of Roots
The expression D = b² − 4ac is called the discriminant.
- D > 0 → Two distinct real roots
- D = 0 → Two equal real roots
- D < 0 → Two complex roots
🔗 Relation Between Roots and Coefficients
If α and β are the roots, then:
- α + β = −b / a
- αβ = c / a