CBSE Class 11 Mathematics Chapter 4 Notes: Complex Numbers and Quadratic Equations

Chapter Overview

Complex Numbers and Quadratic Equations is Chapter 4 of CBSE Class 11 Mathematics and belongs to the Algebra unit (Unit II). It extends the number system beyond real numbers to solve equations like x² + 1 = 0 that have no real solutions. Complex numbers are fundamental to engineering, physics, and higher mathematics.

This guide covers every NCERT topic with definitions, algebra tables, worked examples, common mistakes, and exam tips.

Key Definitions

Algebra of Complex Numbers

Addition

(a + ib) + (c + id) = (a + c) + i(b + d)

Subtraction

(a + ib) − (c + id) = (a − c) + i(b − d)

Multiplication

(a + ib)(c + id) = (ac − bd) + i(ad + bc)

Division

(a + ib)/(c + id) = [(ac + bd) + i(bc − ad)]/(c² + d²)

Method: Multiply numerator and denominator by the conjugate of the denominator: (a + ib)/(c + id) × (c − id)/(c − id) = [(ac + bd) + i(bc − ad)]/(c² + d²)

Algebraic Operations Summary

Properties of Conjugate and Modulus

Conjugate Properties

Modulus Properties

Argand Plane and Polar Representation

Argand Plane

The Argand plane (or complex plane) represents complex numbers geometrically:

• The x-axis represents the real part (Re).
• The y-axis represents the imaginary part (Im).
• z = a + ib is the point (a, b).

Polar Form

Any complex number z = a + ib can be written as:

z = r(cos θ + i sin θ)

where:

• r = |z| = √(a² + b²) is the modulus
• θ = arg(z) = tan⁻¹(b/a) is the argument (principal value: −π < θ ≤ π)

Conversion formulas:

• a = r cos θ
• b = r sin θ

Finding the Argument

The argument θ depends on the quadrant:

Euler Form

z = r e^(iθ) = r(cos θ + i sin θ)

This is Euler's formula. It simplifies multiplication and division:

• z₁z₂ = r₁r₂ e^(i(θ₁+θ₂))
• z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂))

Quadratic Equations with Complex Roots

For the quadratic equation ax² + bx + c = 0:

• Discriminant: D = b² − 4ac
• If D > 0: Two distinct real roots
• If D = 0: One repeated real root (x = −b/2a)
• If D < 0: Two complex conjugate roots

When D < 0, the roots are:

The roots are always complex conjugates: if one root is α + iβ, the other is α − iβ.

Fundamental Theorem of Algebra

Every polynomial equation of degree n has exactly n roots in the complex number system (counting repeated roots by their multiplicity).

Important Numericals

1. Simplifying Complex Number Expressions

Simplify: (3 + 4i)(2 − i)

= 3(2) + 3(−i) + 4i(2) + 4i(−i) = 6 − 3i + 8i − 4i² = 6 + 5i + 4 (since i² = −1) = 10 + 5i

2. Division of Complex Numbers

Find: (1 + 2i)/(3 − 4i)

Multiply by conjugate of denominator: = (1 + 2i)(3 + 4i) / (3 − 4i)(3 + 4i) = (3 + 4i + 6i + 8i²) / (9 + 16) = (3 + 10i − 8) / 25 = (−5 + 10i)/25 = −1/5 + (2/5)i

3. Finding Modulus and Argument

arg(z): z is in Q-II (a = −1, b = √3). θ = π − tan⁻¹(√3/1) = π − π/3 = 2π/3

Polar form: z = 2(cos 2π/3 + i sin 2π/3)

4. Solving Quadratic with Complex Roots

Solve: x² + 3x + 5 = 0.

D = 9 − 20 = −11 < 0

x = (−3 ± √(−11))/2 = (−3 ± i√11)/2

Roots: x = (−3 + i√11)/2 and x = (−3 − i√11)/2

5. Conjugate Property Verification

z̄ = 2 − 3i z · z̄ = (2 + 3i)(2 − 3i) = 4 − 9i² = 4 + 9 = 13

6. Finding the Square Root of a Complex Number

Find √(3 + 4i).

Let √(3 + 4i) = a + ib. Then (a + ib)² = 3 + 4i → a² − b² + 2abi = 3 + 4i → a² − b² = 3 and 2ab = 4 → ab = 2

From a² − b² = 3 and a² + b² = 5: a² = 4, b² = 1 So a = 2, b = 1 (or a = −2, b = −1)

√(3 + 4i) = ±(2 + i)

Practice Problems

Problem 1

Find the multiplicative inverse of 4 − 3i.

1/z = (4 + 3i)/25 = 4/25 + (3/25)i

Problem 2

Express (1 − i)⁴ in the form a + ib.

(1 − i)² = 1 − 2i + i² = 1 − 2i − 1 = −2i (1 − i)⁴ = (−2i)² = 4i² = 4(−1) = −4

Problem 3

Problem 4

Represent z = 3 − 3i in polar form.

z is in Q-IV (a = 3, b = −3). arg(z) = −tan⁻¹(3/3) = −π/4 z = 3√2(cos(−π/4) + i sin(−π/4)) = 3√2(cos π/4 − i sin π/4)

NCERT Exercise Mapping

Common Mistakes

1. Writing i² = 1 instead of −1: The fundamental rule is i² = −1. Every complex algebra step depends on this. 2. Forgetting to multiply by the conjugate in division: (a + ib)/(c + id) is NOT (a/c) + i(b/d). You must rationalise by multiplying by the conjugate of the denominator. 3. Wrong quadrant for argument: tan⁻¹(b/a) gives the correct angle only in Q-I. For other quadrants, adjust using π or −π. 4. Not taking both signs for square roots: If √z = a + ib is a solution, then −(a + ib) is also a solution. Always write ±.

6. Missing complex conjugate roots: If a polynomial has real coefficients and one root is a + ib, then a − ib is also a root. Always write both. 7. Saying D < 0 means no solution: The equation still has solutions — they are complex, not real. Never write "no solution" when D < 0.

Exam Tips

• 1-mark questions: Define complex number, conjugate, modulus, or find Re(z) and Im(z).
• 2-mark questions: Add, subtract, multiply, or divide complex numbers. Find modulus and conjugate.
• 4-mark questions: Convert to polar form, solve quadratic equations with complex roots, or verify conjugate/modulus properties.
• 6-mark questions: Find the square root of a complex number with full working, or solve a word problem involving complex number operations.
• Always show the step i² = −1 when simplifying.
• For polar form, clearly state r and θ separately before writing z = r(cos θ + i sin θ).
• When solving quadratics, always compute the discriminant first and state its sign.
• For argument calculations, identify the quadrant first, then apply the correct formula.

Quick Revision Notes

• Complex number: z = a + ib, where i² = −1. Re(z) = a, Im(z) = b.
• Addition: add real and imaginary parts separately.
• Multiplication: expand and use i² = −1.
• Division: multiply numerator and denominator by the conjugate of the denominator.
• Conjugate: z̄ = a − ib. Key: z + z̄ = 2a, z · z̄ = |z|².
• Modulus: |z| = √(a² + b²). Always non-negative.
• Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂|.
• Argand plane: x-axis = real, y-axis = imaginary.
• Polar form: z = r(cos θ + i sin θ), r = |z|, θ = arg(z).
• Euler form: z = r e^(iθ).
• Quadratic with D < 0: roots are complex conjugates x = (−b ± i√|D|)/(2a).
• Fundamental theorem of algebra: degree n polynomial has n roots in complex numbers.