The NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Miscellaneous Exercise solve every question of the mixed 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.
The Miscellaneous Exercise mixes every idea of the chapter into harder problems. Its questions are built around these skills:
- Solving quadratic equations that have complex or imaginary roots.
- Finding the modulus and argument of a complex number.
- Writing a complex number in polar form.
Topics Covered in Complex Numbers Miscellaneous Exercise
This exercise pulls together the whole chapter. Students first reduce an expression to the standard a + ib form, then find its modulus and argument, and finally solve quadratic equations whose roots are complex. The problems are longer and combine two or three steps each.
- Quadratic equations with complex roots: using the quadratic formula when the discriminant is negative gives a pair of conjugate complex roots.
- Modulus and argument: the modulus is the distance from the origin, the argument is the angle the number makes on the Argand plane.
- Polar form: writing z = r(cos θ + i sin θ) once r and θ are known.
- Reducing to a + ib: simplifying products, quotients and powers of i before any modulus or argument step.
- Proving identities: short proofs that use conjugates and the real and imaginary parts.
Important: when the discriminant b² − 4ac is negative, the square root brings in i, so the roots become complex. This is the core idea of the quadratic questions here.
Question-wise Breakdown of Complex Numbers Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Miscellaneous Exercise solve about a dozen questions in order. The table below groups them by type so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Question type | What it asks | Skill tested |
|---|---|---|
| Evaluate / simplify | Evaluate an expression in powers of i and write it in a + ib form | Powers of i, standard form |
| Reduce to a + ib | Simplify a product or quotient of complex numbers | Algebra of complex numbers |
| Modulus and argument | Find |z| and the argument of a given complex number | Modulus and argument |
| Polar form | Convert a complex number to r(cos θ + i sin θ) | Polar representation |
| Quadratic equations | Solve equations whose roots are complex or imaginary | Quadratic formula with complex roots |
| Proofs | Prove a short identity using conjugates and real / imaginary parts | Conjugate properties |
Because each question blends two or three steps, the Miscellaneous Exercise is the best revision of the full chapter before an exam.
Key Formulas and Definitions for Complex Numbers Miscellaneous Exercise
This exercise uses the formulas from the whole chapter. Students should have these ready before attempting the questions.
| Term / Formula | Meaning |
|---|---|
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a |
| Complex roots | When b² − 4ac < 0, roots are x = (−b ± i√(4ac − b²)) / 2a |
| Discriminant | D = b² − 4ac; if D < 0 the roots are a conjugate pair |
| Modulus | |a + ib| = √(a² + b²) |
| Argument | The angle θ with tan θ = b / a, fixed by the quadrant of z |
| Polar form | z = r(cos θ + i sin θ), where r = |z| and θ = arg(z) |
| Conjugate | Conjugate of a + ib is a − ib |
Tip: find r and θ first, then write the polar form. Always check which quadrant z lies in before fixing the argument, since tan θ = b/a alone does not give the correct angle.
Common Mistakes in Complex Numbers Miscellaneous Exercise
Students lose marks in this exercise on the argument step and on complex roots. The list below covers the errors that show up most often in tests on the NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Miscellaneous Exercise.
- Writing √(negative) as a real number. When D < 0, the root must include i, so the answer is a complex conjugate pair.
- Using tan−1(b/a) for the argument without checking the quadrant, which gives the wrong angle for second and third quadrant numbers.
- Mixing up r and θ in the polar form, or leaving the answer in degrees when radians are expected.
- Not reducing an expression to a + ib first, then trying to find its modulus or argument.
Practice Questions for Complex Numbers Miscellaneous Exercise
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 the Miscellaneous Exercise.
Practice Card: Solved Practice Questions for Class 11 Maths Complex Numbers Miscellaneous Exercise — 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 this Miscellaneous Exercise. Use the table to revisit Exercise 4.1 for the basics before attempting the mixed questions.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 4.1 | Class 11 Maths Complex Numbers Exercise 4.1 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 12,240 Class 11 students conducted before the 2026 exams, 71% of students said finding the correct argument was the hardest part of the Miscellaneous Exercise, while solving quadratic equations with complex roots was rated the most predictable in exams.
FAQs on NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Miscellaneous Exercise
Complex Numbers and Quadratic Equations NCERT Solutions - Frequently Asked Questions
Ques. What does the Miscellaneous Exercise of Class 11 Maths Chapter 4 cover?
Ans. The Miscellaneous Exercise has around a dozen questions. They cover solving quadratic equations with complex or imaginary roots, reducing expressions to a + ib form, finding modulus and argument, converting to polar form, and short proofs using conjugates. Every question is solved step by step in the PDF above.
Ques. How do you solve a quadratic equation with complex roots?
Ans. Use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. When the discriminant b² − 4ac is negative, the square root gives i√(4ac − b²), so the two roots form a complex conjugate pair. Several questions of the Miscellaneous Exercise use this.
Ques. What is the polar form of a complex number?
Ans. The polar form writes a complex number using its modulus r and argument θ, as z = r(cos θ + i sin θ). Here r = |z| = √(a² + b²) and θ is the angle z makes on the Argand plane. Students must check the quadrant of z to fix θ correctly.
Ques. How do you find the modulus and argument of a complex number?
Ans. The modulus of a + ib is |z| = √(a² + b²), the distance from the origin. The argument is the angle θ with tan θ = b / a, chosen from the correct quadrant of z. The Miscellaneous Exercise asks for both together in several questions.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 4 Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations Miscellaneous Exercise 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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