NCERT Solutions for Class 11 Maths Chapter 4 - FREE PDF Download
FAQs on NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations
1. What key topics do the NCERT Solutions for Class 11 Maths Chapter 4 cover for the 2025-26 session?
The NCERT Solutions for Class 11 Maths Chapter 4, updated for the CBSE 2025-26 syllabus, provide step-by-step methods for the following topics:
- The concept of the imaginary unit (iota, i) and its powers.
- Algebraic operations on complex numbers: addition, subtraction, multiplication, and division.
- Finding the modulus and conjugate of a complex number.
- Calculating the multiplicative inverse.
- Solving quadratic equations that have complex roots (where the discriminant is negative).
- Expressing all solutions in the standard form a + ib.
2. How do the NCERT Solutions explain the method for solving quadratic equations with complex roots?
The NCERT Solutions demonstrate a clear, step-by-step method. When a quadratic equation's discriminant (D = b² - 4ac) is negative, the solutions show how to use the quadratic formula by expressing the square root of the negative discriminant in terms of 'i'. For instance, √(-D) is written as i√(D). This allows you to find the two complex roots and present them in the standard a + ib form, as required by the CBSE pattern.
3. What is the correct method used in the NCERT Solutions to find the multiplicative inverse of a complex number?
The NCERT Solutions use a standard formula to find the multiplicative inverse of a complex number z = a + ib. The inverse, denoted as z⁻¹, is calculated using its conjugate (z̄ = a - ib) and its modulus (|z| = √(a² + b²)). The step-by-step method shown is: z⁻¹ = z̄ / |z|², which simplifies to (a - ib) / (a² + b²). This ensures the denominator becomes a real number, and the final answer is presented in the a + ib form.
4. Why do the NCERT Solutions frequently use the conjugate when dividing complex numbers?
The conjugate is a critical tool used in the solutions to rationalise the denominator. When dividing one complex number by another, the denominator is a complex number. To convert it into a real number, the solutions show how to multiply both the numerator and the denominator by the conjugate of the denominator. This process, based on the property (a + ib)(a - ib) = a² + b², simplifies the expression into the standard a + ib form, making the final answer clear and correct.
5. How should a student use these NCERT Solutions to approach the Miscellaneous Exercise for Chapter 4?
The Miscellaneous Exercise contains more challenging, application-based problems. The NCERT Solutions for this section are designed to help you break down complex questions. It's best to first attempt the problems yourself, then use the solutions to:
- Understand the sequence of steps required for multi-step problems.
- Identify which properties of complex numbers (like modulus, conjugate, or distributive law) are needed.
- Verify if your final answer, expressed in the a + ib form, is correct.
6. What is the principle behind simplifying high powers of 'i' (iota) as shown in the exercise solutions?
The solutions demonstrate that any power of 'i' can be simplified by using its cyclic pattern of four values: i, -1, -i, and 1. The core principle shown is to express any high power of 'i', say iⁿ, in the form of (i⁴)ᑫ · iʳ, where 'n' is divided by 4 to get a quotient 'q' and a remainder 'r'. Since i⁴ = 1, the expression simplifies to 1ᑫ · iʳ = iʳ. The answer is then determined by the remainder (0, 1, 2, or 3), which is a method consistently applied across the solutions.
7. Which topics and questions from Chapter 4 have been deleted for the CBSE 2025-26 exams?
Yes, as per the latest CBSE syllabus for 2025-26, students should skip the following topics and corresponding questions while using the NCERT book and its solutions:
- Polar representation of complex numbers and the concept of argument.
- The square root of a complex number.
- All questions in Exercise 4.3.
- From the Miscellaneous Exercise, questions 5, 8, 9, and 13 are excluded.
The provided NCERT solutions focus only on the topics included in the current syllabus.
8. How do the step-by-step answers in the NCERT solutions help prevent common mistakes in this chapter?
The detailed solutions help prevent common errors by reinforcing correct procedures. For example, they show how to:
- Consistently substitute i² = -1 during multiplication to avoid sign errors.
- Always combine real parts with real parts and imaginary parts with imaginary parts separately.
- Properly apply the formula for the multiplicative inverse instead of just taking the reciprocal.
- Ensure the final answer is always simplified to the standard a + ib form, a common requirement in exams.
9. What is the importance of expressing the final answer in the standard 'a + ib' form, as shown in all NCERT solutions?
Expressing the answer in the standard form a + ib is crucial because it clearly distinguishes the real part (a) from the imaginary part (b). This format is a mandatory requirement in CBSE examinations for full marks. The NCERT solutions consistently follow this convention to train students to present their answers in a clear, standardized, and accurate manner, which is essential for both understanding and scoring well.
10. What are the key properties of multiplication of complex numbers demonstrated in the NCERT Solutions?
The solutions implicitly or explicitly demonstrate several key properties of complex number multiplication through solved examples. These include:
- Closure Law: The product of two complex numbers is always another complex number.
- Commutative Law: For any two complex numbers z₁ and z₂, z₁z₂ = z₂z₁.
- Associative Law: For any three complex numbers, (z₁z₂)z₃ = z₁(z₂z₃).
- Distributive Law: Multiplication distributes over addition, i.e., z₁(z₂ + z₃) = z₁z₂ + z₁z₃.
Following the solution steps helps in understanding how these properties are applied.

















