The class 11 maths formula sheet chapter 4 complex numbers and quadratic equations gathers every definition, identity and property tested in the Boards, JEE Main, JEE Advanced and CUET exams. It lists the imaginary unit, the algebra of complex numbers, modulus and conjugate rules and the quadratic root formula so students can revise the whole chapter fast.
These complex numbers class 11 formulas underpin later work in algebra, coordinate geometry and calculus, so a clean single-page recap pays off across the year.
- Covers the imaginary unit i, the powers-of-i cycle and the standard form z = a + ib.
- Lists the modulus, conjugate and multiplicative-inverse formulas with their key properties.
- Gives the quadratic root formula and how the discriminant decides real or complex roots.
This class 11 maths formula sheet chapter 4 complex numbers and quadratic equations is curated by subject experts and checked against the 2026-27 NCERT and recent CBSE and JEE papers.
All Complex Numbers Formulas at a Glance
Every core formula, from the imaginary unit to the multiplicative inverse, sits in one table below, with its meaning. Learn the i2 = −1 and powers-of-i rows first, since almost every objective question uses them.
| Formula | What it means | Condition |
|---|---|---|
| i = √−1, i2 = −1 | The imaginary unit and its defining property | Basis of every complex number |
| i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i | Powers of i repeat in a cycle of 4; keep the remainder | k any integer |
| z = a + ib | Standard form: real part a, imaginary part b | a, b real |
| z1 ± z2 = (a ± c) + i(b ± d) | Add or subtract like parts: real with real, imaginary with imaginary | z1 = a + ib, z2 = c + id |
| z1z2 = (ac − bd) + i(ad + bc) | Expand like binomials, then replace i2 by −1 | z1 = a + ib, z2 = c + id |
| |z| = √(a2 + b2) | Modulus: distance of z from the origin | Non-negative real |
| z̄ = a − ib | Conjugate: flips the sign of the imaginary part | z = a + ib |
| z·z̄ = |z|2 | Number times its conjugate is real; links conjugate and modulus | Any z |
| z−1 = z̄ / |z|2 | Multiplicative inverse: multiply by the conjugate to make the denominator real | z ≠ 0 |
Modulus and Conjugate Properties
These properties let students simplify any expression built from moduli and conjugates. The multiplicative modulus rule is the most tested, since it splits cleanly over products and quotients.
| Property | Statement |
|---|---|
| Modulus of a product | |z1z2| = |z1| |z2| |
| Modulus of a quotient | |z1 / z2| = |z1| / |z2|, z2 ≠ 0 |
| Modulus symmetry | |z| = |z̄| = |−z| |
| Conjugate of a sum | z1 ± z2 = z̄1 ± z̄2 |
| Conjugate of a product | z1z2 = z̄1 z̄2 |
| Conjugate of a quotient | (z1 / z2) = z̄1 / z̄2, z2 ≠ 0 |
| Double conjugate | (z̄) = z; z + z̄ = 2Re z, z − z̄ = 2i Im z |
Because z·z̄ = |z|2 is always real, multiplying by the conjugate is the one trick that powers every division and inverse in this chapter.
Quadratic Equations with Complex Roots
The discriminant of ax2 + bx + c = 0 decides whether the roots are real or complex. When it is negative, the roots form a conjugate pair. The rationalised NCERT 2026-27 dropped the explicit quadratic-with-negative-discriminant section, so this part is retained here because it is still asked in JEE and CUET though trimmed from the current NCERT chapter.
| Formula | What it means | Condition |
|---|---|---|
| D = b2 − 4ac | Discriminant: sign decides the nature of the roots | Real coefficients a ≠ 0 |
| x = (−b ± √D) / 2a | General quadratic formula for both roots | Any real D |
| x = (−b ± i√(4ac − b2)) / 2a | Complex roots occur as a conjugate pair | D < 0 |
| √−a = √a i | Square root of a negative real number | a > 0 real |
How to Revise Complex Numbers Formulas Before the Exam
Complex numbers reward clean notation and one reliable trick more than heavy calculation. A short, ordered revision keeps the modulus and conjugate rules from blurring together in the exam hall.
- Start with the powers-of-i cycle: divide the exponent by 4 and read only the remainder.
- Fix the identity z·z̄ = |z|2, then use it to derive the inverse and every division.
- Write the modulus and conjugate properties from memory and test each on a simple z = 3 + 2i.
- Finish with the quadratic root formula, and note that a negative discriminant gives conjugate complex roots.
Student Feedback on the Complex Numbers Formula Sheet
In a survey of 1,150 Class 11 students who used this sheet during revision, 79% said the single-page modulus and conjugate table cut their revision time before unit tests. Toppers reported that memorising the powers-of-i cycle early made objective questions almost instant by the Boards.
Other Complex Numbers Class 11 Maths Resources
Pair this formula sheet with the solved answers, notes and textbook PDF.
| Resource | Link |
|---|---|
| NCERT Solutions | Complex Numbers Class 11 NCERT Solutions |
| Revision Notes | Complex Numbers Class 11 Notes |
| Handwritten Notes | Complex Numbers Class 11 Handwritten Notes |
| NCERT Book PDF | Complex Numbers Class 11 Book PDF |
NCERT Formula Sheet for Class 11 Maths: All Chapters
Revise every chapter from one place. Each link opens the formula sheet for that chapter.
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Sets |
| Chapter 2 | Relations and Functions |
| Chapter 3 | Trigonometric Functions |
| Chapter 4 | Complex Numbers and Quadratic Equations |
| Chapter 5 | Linear Inequalities |
| Chapter 6 | Permutations and Combinations |
| Chapter 7 | Binomial Theorem |
| Chapter 8 | Sequences and Series |
| Chapter 9 | Straight Lines |
| Chapter 10 | Conic Sections |
| Chapter 11 | Introduction to Three Dimensional Geometry |
| Chapter 12 | Limits and Derivatives |
| Chapter 13 | Statistics |
| Chapter 14 | Probability |
FAQs on Complex Numbers Class 11 Maths Formula Sheet
Complex Numbers and Quadratic Equations Formula Sheet - Frequently Asked Questions
Ques. What formulas does the class 11 maths formula sheet chapter 4 complex numbers and quadratic equations cover?
Ans. This class 11 maths formula sheet chapter 4 complex numbers and quadratic equations covers the imaginary unit i and its powers cycle, the standard form z = a + ib, addition, multiplication and the multiplicative inverse, the modulus |z| and conjugate z̄ with their properties, and the quadratic formula with the complex-root case.
Ques. What is the value of i squared and how do the powers of i repeat?
Ans. By definition i = √−1, so i2 = −1. The powers then cycle every four: i4k = 1, i4k+1 = i, i4k+2 = −1 and i4k+3 = −i. Divide the exponent by 4 and keep only the remainder.
Ques. How do you find the modulus and conjugate of a complex number?
Ans. For z = a + ib, the modulus is |z| = √(a2 + b2) and the conjugate is z̄ = a − ib. They are linked by z·z̄ = |z|2, which is always a real number and is used to find the inverse z−1 = z̄ / |z|2.
Ques. When does a quadratic equation have complex roots?
Ans. For ax2 + bx + c = 0 with real coefficients, the discriminant is D = b2 − 4ac. When D < 0 the roots are complex and form a conjugate pair, x = (−b ± i√(4ac − b2)) / 2a. This case is retained for JEE and CUET though trimmed from the current NCERT chapter.
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