NCERT Solutions Class 11 Maths Chapter 7 helps you understand the Binomial Theorem through clear solutions to all 20 questions across 2 exercises. You’ll learn how to expand binomial expressions using Pascal’s Triangle, apply the general and middle term formulas, and solve problems involving binomial coefficients. These techniques are essential for calculus, probability, and competitive exams like JEE, making this chapter crucial for your mathematical foundation.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 11 Maths Chapter 7 Binomial Theorem – Complete Guide
NCERT Class 11 Chapter 7 on Binomial Theorem introduces you to one of the most powerful tools in algebra that simplifies the expansion of binomial expressions raised to any positive integer power. You’ll discover how Pascal’s Triangle connects to binomial coefficients and learn to use the general term formula to find any specific term without expanding the entire expression.
📊 CBSE Class 11 Maths Chapter 7 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 5 Marks |
| Unit Name | Algebra |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
This chapter holds medium importance in CBSE Class 11 Mathematics with a weightage of approximately 5 marks in board exams. You’ll explore key concepts including binomial coefficients using combination notation (ⁿCᵣ), properties of binomial expansion, finding middle terms in odd and even powers, and identifying the greatest coefficient or term. The chapter also covers practical applications in probability, series, and approximations that you’ll encounter in higher mathematics.
For CBSE board exams, you can expect 2-3 questions from this chapter, typically including one MCQ (1 mark), one short answer question (2 marks), and occasionally a long answer problem (4 marks). Common question types involve finding specific terms in expansions, determining coefficients of particular powers, solving for unknown values using binomial properties, and proving identities using the theorem.
Quick Facts – Class 11 Chapter 7
| 📖 Chapter Number | Chapter 7 |
| 📚 Chapter Name | Binomial Theorem |
| ✏️ Total Exercises | 2 Exercises |
| ❓ Total Questions | 20 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering the Binomial Theorem will not only strengthen your algebraic manipulation skills but also prepare you for advanced topics in probability, sequences and series, and calculus. The systematic approach you’ll learn here forms the foundation for understanding mathematical induction and combinatorics. With regular practice of NCERT solutions and previous year questions, you’ll confidently tackle any binomial theorem problem in your exams and competitive tests like JEE and other entrance examinations.
NCERT Solutions Class 11 Maths Chapter 7 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 7.1 | Complete step-by-step solutions for 14 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 7 | Complete step-by-step solutions for 6 questions | 📥 Download PDF |
Binomial Theorem – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Binomial Theorem (General Form) \((a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\) | Expands (a + b) raised to the power of n, where n is a positive integer. Note: Remember that \( {n \choose k} = \frac{n!}{k!(n-k)!} \) and \( {n \choose 0} = {n \choose n} = 1 \). Make sure to correctly identify ‘a’ and ‘b’. | Expanding binomial expressions with integer powers, finding specific terms in the expansion. |
| Binomial Coefficient Formula \({n \choose k} = \frac{n!}{k!(n-k)!}\) | Calculates the binomial coefficient (n choose k), also written as nCk or \(^nC_k\). Note: n! means n factorial (n x (n-1) x (n-2) x … x 2 x 1). 0! = 1 by definition. \( {n \choose k} = {n \choose n-k} \). | Finding the coefficient of a specific term in the binomial expansion. |
| General Term in Binomial Expansion \(T_{k+1} = {n \choose k} a^{n-k} b^k\) | Finds the (k+1)th term in the expansion of (a + b)^n. Note: The term number is always one more than the value of k. So, to find the 5th term, use k = 4. | Finding a specific term (e.g., the 5th term) without expanding the entire binomial. |
| Middle Term(s) in Binomial Expansion If n is even: Middle term = \(T_{\frac{n}{2}+1}\); If n is odd: Middle terms = \(T_{\frac{n+1}{2}}\) and \(T_{\frac{n+3}{2}}\) | Identifies the middle term(s) in the binomial expansion. Note: Remember to calculate the *term number* first, then use the general term formula. When n is odd, there are two middle terms. | Finding the middle term when you are asked for it specifically. |
| Term Independent of x Find k such that the power of x in \(T_{k+1}\) is 0. | Finds the term in the expansion that does not contain x (i.e., x^0). Note: Set the exponent of x in the general term equal to zero and solve for k. Then, substitute k back into the general term formula. | Problems asking for the constant term or the term independent of x. |
| Sum of Binomial Coefficients \(\sum_{k=0}^{n} {n \choose k} = 2^n\) | The sum of all the binomial coefficients in the expansion of (a + b)^n is equal to 2^n. Note: This is a direct application of the binomial theorem where a = 1 and b = 1. | When you need to find the sum of all the coefficients in a binomial expansion quickly. Can also be applied to certain combinatorial problems. |
| Sum of Odd/Even Binomial Coefficients \(\sum_{k=0}^{\lfloor n/2 \rfloor} {n \choose 2k} = \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} {n \choose 2k+1} = 2^{n-1}\) | The sum of the odd-numbered binomial coefficients equals the sum of the even-numbered binomial coefficients, and both equal 2^(n-1). Note: \(\lfloor x \rfloor\) represents the floor function (greatest integer less than or equal to x). Remember to distinguish between even and odd numbered coefficients. | When a problem asks for the sum of only the even or only the odd binomial coefficients. |
| Binomial Theorem for Negative/Fractional Index \((1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + …\) | Expands (1 + x)^n when n is a negative integer or a fraction. Note: This is an infinite series. The condition |x| < 1 is crucial for convergence. Only use a few terms for approximation. | Expanding expressions like (1 + x)^(-1), (1 + x)^(1/2) for approximations. |x| < 1 is required for convergence. |
| Approximation using Binomial Theorem \((1+x)^n \approx 1 + nx\) (for small x) | Approximates (1 + x)^n when x is very small (close to 0). Note: This approximation is only valid when |x| is significantly smaller than 1. The smaller x is, the more accurate the approximation. | Estimating values like \(\sqrt{1.02}\) or \(\frac{1}{1.01}\) without a calculator. |
Frequently Asked Questions – NCERT Class 11 Maths Chapter 7
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