The NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Exercise 4.1 solve every question of the first exercise, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
Exercise 4.1 is the first real practice on complex numbers. The exercise has 14 questions built around these skills:
- Writing any complex number in the standard a + ib form.
- Using the powers of i and the rule i² = −1.
- Finding the multiplicative inverse of a complex number.
Topics Covered in Complex Numbers Exercise 4.1
This exercise teaches the algebra of complex numbers. A complex number is written as a + ib, where a is the real part, b is the imaginary part, and i = √−1. Exercise 4.1 keeps the focus on adding, subtracting, multiplying and inverting these numbers.
- Powers of i: using the cycle i, −1, −i, 1 to simplify high powers like i9 or i−39.
- Addition and subtraction: combining real parts with real parts and imaginary parts with imaginary parts.
- Multiplication: expanding products and cubes such as (1 − i)4, then replacing i² with −1.
- Multiplicative inverse: writing z−1 as the conjugate divided by the modulus squared.
- Modulus and conjugate: the tools that make the inverse step quick and clean.
Important: the single rule i² = −1 drives almost every question in this exercise, so it must be applied at every step.
Question-wise Breakdown of Complex Numbers Exercise 4.1
The NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Exercise 4.1 solve all 14 questions in order. The table below groups the questions by type so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Questions | What they ask | Skill tested |
|---|---|---|
| Q1 to Q2 | Simplify products and powers of i into a + ib form | Powers of i |
| Q3 | Simplify a negative power like i−39 | Reciprocal powers of i |
| Q4 to Q7 | Add and subtract complex numbers, write the result as a + ib | Addition and subtraction |
| Q8 to Q10 | Expand powers such as (1 − i)4 and cubes into a + ib | Multiplication and cubes |
| Q11 to Q13 | Find the multiplicative inverse of a given complex number | Inverse using conjugate |
| Q14 | Express a mixed expression in the standard a + ib form | Full algebra of a + ib |
Most questions ask for a final answer in a + ib form, so the exercise gives steady practice on simplifying and tidying complex expressions.
Key Formulas and Definitions for Complex Numbers Exercise 4.1
Exercise 4.1 rests on a small, fixed set of rules for the algebra of complex numbers. Students should learn these before attempting the questions.
| Term / Formula | Meaning |
|---|---|
| i = √−1 | The imaginary unit |
| i² = −1 | The rule that removes the imaginary square |
| i³ = −i, i⁴ = 1 | Powers of i repeat every four steps |
| Complex number | a + ib, with real part a and imaginary part b |
| Addition | (a + ib) + (c + id) = (a + c) + i(b + d) |
| Multiplication | (a + ib)(c + id) = (ac − bd) + i(ad + bc) |
| Conjugate | Conjugate of a + ib is a − ib |
| Modulus | |a + ib| = √(a² + b²) |
| Multiplicative inverse | (a + ib)−1 = (a − ib) / (a² + b²) |
Tip: to find an inverse, multiply the top and bottom by the conjugate. The bottom becomes a² + b², a real number, which makes the answer easy to write in a + ib form.
Common Mistakes in Complex Numbers Exercise 4.1
Students lose easy marks in Exercise 4.1 on small algebra slips. The list below covers the errors that show up most often in class tests on the NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Exercise 4.1.
- Writing √−1 × √−1 = √1 = 1. The correct value is i × i = −1, because the rule √a × √b = √(ab) fails for negative numbers.
- Getting the powers of i wrong. Reduce the power using the four-step cycle, so i9 = i and i−39 needs the reciprocal handled carefully.
- Forgetting to divide by a² + b² when finding the multiplicative inverse.
- Sign errors while expanding cubes and fourth powers, usually by missing that i² = −1 changes the sign of a term.
Practice Questions for Complex Numbers Exercise 4.1
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for Exercise 4.1.
Practice Card: Solved Practice Questions for Class 11 Maths Complex Numbers Exercise 4.1 — attempt each question, then check the worked solution.
Other Exercises of Chapter 4 Complex Numbers and Quadratic Equations
Chapter 4 has one numbered exercise plus a Miscellaneous Exercise. Use the table to move to the Miscellaneous Exercise once Exercise 4.1 is done.
| Exercise | NCERT Solutions link |
|---|---|
| Miscellaneous Exercise | Class 11 Maths Complex Numbers Miscellaneous Exercise Solutions |
More Class 11 Maths Complex Numbers Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 4 Complex Numbers and Quadratic Equations are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Complex Numbers Notes |
| Handwritten Notes | Class 11 Maths Complex Numbers Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Complex Numbers NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | Sets Solutions |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | You are here |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 11,860 Class 11 students conducted before the 2026 exams, 64% of students said the multiplicative inverse questions in Exercise 4.1 were the part they got wrong most often, while simplifying powers of i was rated the easiest to learn.
FAQs on NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Exercise 4.1
Complex Numbers and Quadratic Equations NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 4 Complex Numbers Exercise 4.1?
Ans. Exercise 4.1 has 14 questions. They test writing complex numbers in a + ib form, simplifying powers of i, adding and subtracting complex numbers, expanding cubes and fourth powers, and finding the multiplicative inverse. Every answer is solved step by step in the PDF above.
Ques. What is the standard form of a complex number?
Ans. A complex number is written as a + ib, where a is the real part, b is the imaginary part, and i = √−1. Exercise 4.1 asks students to reduce every expression to this standard a + ib form.
Ques. How do you find the multiplicative inverse of a complex number in Exercise 4.1?
Ans. The multiplicative inverse of a + ib is (a − ib) / (a² + b²). Multiply the top and bottom by the conjugate a − ib, so the bottom becomes the real number a² + b². Questions 11 to 13 of Exercise 4.1 use this method.
Ques. Why is i² equal to −1?
Ans. The imaginary unit i is defined as √−1, so squaring it gives i² = −1. This one rule is used in almost every question of Exercise 4.1 to remove imaginary squares and reach the a + ib form.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers Exercise 4.1 free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Exercise 4.1 are free to download as a PDF from this page. Every question is solved step by step, according to the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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