NCERT Solutions for Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations
There is a moment in every student's mathematical journey when they are told the square root of a negative number does not exist — and then, in Class 11, they are told it does. That shift is the entry point into Chapter 5, which introduces the number system's most elegant extension: complex numbers.
The problem that motivated complex numbers is simple. The equation x² + 1 = 0 has no real solution because no real number squared gives –1. Mathematicians resolved this by defining an entirely new type of number where √(–1) = i, called the imaginary unit. From this single definition, an entire algebra is built — and it turns out to be not just mathematically consistent but deeply useful in electrical engineering, quantum physics, and signal processing. For all Chapters must, read NCERT Solutions for Class 11 Maths and subject-wise NCERT Solutions for Class 11.
NCERT solutions Chapter 5 covers the definition of complex numbers, their arithmetic (addition, subtraction, multiplication, and division), the Argand plane for geometric representation, the modulus and argument of a complex number, and the polar form. It also revisits quadratic equations, now allowing complex roots, and introduces the fundamental theorem of algebra in an accessible way. These solutions are written to ensure that every division of complex numbers is done using the conjugate method — a technique that CBSE requires explicitly.
Download PDF – All Exercises of NCERT Solutions for Class 11 Maths Chapter 5 Complex Numbers
| Exercise | Topic Covered | Number of Questions |
|---|---|---|
| Exercise 5.1 | Algebra of Complex Numbers (Basic Operations) | 14 Questions |
| Exercise 5.2 | Modulus and Argument; Polar Form | 8 Questions |
| Exercise 5.3 | Quadratic Equations with Complex Roots | 10 Questions |
| Miscellaneous | Advanced Problems on Complex Numbers | 20 Questions |
Chapter 5 – Complex Numbers: Concepts, Explanation and Key Tables
The Imaginary Unit and Standard Form
Every complex number is written as z = a + ib, where a is the real part and b is the imaginary part, both of which are real numbers. The imaginary unit i satisfies i² = –1.
This gives us: i¹ = i, i² = –1, i³ = –i, i⁴ = 1, and the cycle repeats every four powers. To find iⁿ for any large n, divide n by 4 and use the remainder: iⁿ = i^(n mod 4).
Algebra of Complex Numbers
| Operation | Rule | Example |
|---|---|---|
| Addition | (a+ib) + (c+id) = (a+c) + i(b+d) | (2+3i) + (1–i) = 3+2i |
| Subtraction | (a+ib) – (c+id) = (a–c) + i(b–d) | (5+2i) – (3+4i) = 2–2i |
| Multiplication | (a+ib)(c+id) = (ac–bd) + i(ad+bc) | (1+i)(2+3i) = –1+5i |
| Division | Multiply numerator and denominator by conjugate of denominator | (2+3i)/(1–i): multiply by (1+i)/(1+i) |
| Conjugate | Conjugate of a+ib is a–ib | Conj(3+4i) = 3–4i |
| Modulus | z |
Properties of Modulus and Conjugate
| Property | Statement |
|---|---|
| z₁ × z₂ | |
| z₁/z₂ | |
| z × z̄ = | z |
| (z̄₁ + z̄₂) = (z₁ + z₂)⁻ (overline) | Conjugate of sum = sum of conjugates |
| Re(z) = (z + z̄)/2 | Real part from conjugate |
| Im(z) = (z – z̄)/(2i) | Imaginary part from conjugate |
Argand Plane and Polar Form
A complex number z = a + ib is represented as the point (a, b) on the Argand plane, where the x-axis is the real axis and the y-axis is the imaginary axis. The modulus |z| is the distance from the origin to this point, and the argument (arg z) is the angle this line makes with the positive real axis.
The polar form is: z = r(cos θ + i sin θ), where r = |z| and θ = arg z.
The argument θ is called the principal argument when θ ∈ (–π, π].
Finding the Argument in Each Quadrant
| Quadrant of (a, b) | a, b signs | Argument Formula |
|---|---|---|
| First | a > 0, b > 0 | θ = arctan(b/a) |
| Second | a < 0, b > 0 | θ = π – arctan(b/ |
| Third | a < 0, b < 0 | θ = –π + arctan(b/a) |
| Fourth | a > 0, b < 0 | θ = –arctan( |
Quadratic Equations with Complex Roots
When the discriminant (D = b² – 4ac) of a quadratic ax² + bx + c = 0 is negative, the roots are complex and conjugate pairs.
Using the quadratic formula: x = (–b ± √D) / (2a), where √D = i√|D| when D < 0.
For example, x² + x + 1 = 0 → D = 1 – 4 = –3 → x = (–1 ± i√3)/2.
These are the complex cube roots of unity, denoted ω and ω², and they carry a famous property: 1 + ω + ω² = 0 and ω³ = 1.
Study Tips for Chapter 5
- Always convert the denominator to real form using the conjugate before division — writing a final answer with i in the denominator is an incomplete answer in CBSE.
- For modulus problems, remember that |z|² = a² + b² — you do not need to take the square root if the question asks for |z|².
- The Argand plane diagram is almost always expected in Exercise 5.2 answers — draw it and mark the point clearly.
- When the question says "express in the form a + ib", every step must end with a clearly separated real and imaginary part.