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CBSE Class 11 Mathematics Chapter 7 Binomial Theorem Worksheet with Answers

The Binomial Theorem is a cornerstone algebraic result in CBSE Class 11 Mathematics Chapter 7, providing a formula to expand expressions of the form (a + b)ⁿ without repeated multiplication. This printable worksheet offers structured practice across all NCERT topics including general term, middle term, and applications. Designed for a 90-minute session at moderate-to-challenging difficulty, it contains 35+ questions in CBSE board pattern with complete solutions.

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Key takeaways

✓Printable 90-minute worksheet covering all NCERT topics: binomial expansion, general term, middle term, and practical applications of Binomial Theorem
✓35+ graded questions across MCQs, fill-in-blanks, true/false, short answers (2-3 marks), and long HOTS questions (5-6 marks)
✓Complete answer key with step-by-step explanations for every question, enabling effective self-assessment and learning
✓Case-study question mirrors latest CBSE pattern, connecting binomial theorem to real-world scenarios in probability and finance
✓Difficulty level: Moderate to Challenging; covers NCERT exercises plus additional CBSE board-style questions for thorough preparation
✓Practice identifying general term Tr+1, finding middle terms in even and odd powered expansions, and applying binomial coefficients
✓Ideal for revision before unit tests, term exams, or board preparation; strengthens algebraic manipulation and coefficient comparison skills

Quick Chapter Recap: Binomial Theorem Essentials

Before attempting the worksheet, revisit these key concepts from NCERT Class 11 Mathematics Chapter 7. The Binomial Theorem states that (a + b)ⁿ equals the sum from r = 0 to n of ⁿCᵣ aⁿ⁻ʳ bʳ, where ⁿCᵣ denotes the binomial coefficient 'n choose r'. The general term, denoted Tᵣ₊₁, equals ⁿCᵣ aⁿ⁻ʳ bʳ and is the (r+1)th term in the expansion. For middle terms, if n is even there is one middle term at position (n/2 + 1); if n is odd there are two middle terms at positions (n+1)/2 and (n+3)/2. Understanding Pascal's triangle and properties of binomial coefficients such as ⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ is essential. Applications include finding specific coefficients, proving identities, approximations, and solving probability problems. Mastery requires practice in algebraic manipulation, coefficient comparison, and recognizing patterns in expansions.

Binomial Theorem formula: (a + b)ⁿ = Σ(r=0 to n) ⁿCᵣ aⁿ⁻ʳ bʳ
General term: Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ (remember it is the (r+1)th term, not rth)
Middle term identification: one term when n is even, two terms when n is odd
Binomial coefficient properties: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₊₁ = ⁿ⁺¹Cᵣ₊₁
Applications in approximations: (1 + x)ⁿ ≈ 1 + nx when |x| is very small

Worksheet Instructions and Difficulty Level

This CBSE Class 11 Mathematics Chapter 7 worksheet is designed for 90 minutes of focused practice and is pitched at moderate-to-challenging difficulty. It mirrors the latest CBSE question paper pattern with a mix of objective and subjective questions. Students should attempt all sections in sequence, starting with MCQs to build confidence before tackling long-answer and HOTS questions. Use rough paper for calculations and write final answers clearly. Section A tests conceptual clarity through multiple-choice questions. Section B checks understanding of terminology and standard results via fill-in-the-blanks. Section C offers true/false statements requiring careful reasoning. Section D contains short-answer questions (2-3 marks each) that demand clear working and correct final answers. Section E comprises long-answer questions (5-6 marks) testing proof, derivation, or multi-step problem solving. The case study integrates binomial theorem with a real-world context, a hallmark of current CBSE assessments. After completing the worksheet, verify answers against the detailed answer key provided at the end. Each solution includes brief explanations to reinforce learning and clarify common mistakes.

Difficulty Level: Moderate to Challenging (aligned with CBSE board exam standards)
Suggested Time: 90 minutes (strict timed practice recommended for exam readiness)
Total Questions: 35+ across five sections plus one case-study question
Marking Scheme: MCQs (1 mark each), Fill-ups (1 mark), True/False (1 mark), Short (2-3 marks), Long (5-6 marks), Case Study (4 marks)
Materials Needed: Rough paper, calculator (optional, but algebraic simplification by hand is preferred)
Self-Assessment: Compare your solutions with the answer key and note down errors for focused revision

Section A: Multiple Choice Questions (1 mark each)

This section contains six multiple-choice questions testing fundamental understanding of the Binomial Theorem, general term, middle term, and coefficient properties. Each question has four options; select the most appropriate answer. These questions are designed to assess quick recall and conceptual clarity, typical of CBSE Class 11 Mathematics objective-type questions. Carefully read each stem and eliminate obviously incorrect options before selecting your final answer. No negative marking, so attempt all questions even if unsure. Topics covered include identifying coefficients, finding specific terms, and applying binomial properties. Practice these MCQs under timed conditions to improve speed and accuracy for school exams and competitive tests.

Q1. The coefficient of x⁵ in the expansion of (2x + 3)⁷ is: (a) ⁷C₅ × 2⁵ × 3² (b) ⁷C₂ × 2⁵ × 3² (c) ⁷C₅ × 2² × 3⁵ (d) ⁷C₂ × 2² × 3⁵
Q2. The number of terms in the expansion of (a + b)ⁿ is: (a) n (b) n + 1 (c) n - 1 (d) 2n
Q3. If the middle term in the expansion of (x + 1/x)¹⁰ is T, then T equals: (a) ¹⁰C₅ (b) ¹⁰C₆ (c) ¹⁰C₅ x⁰ (d) ¹⁰C₄
Q4. The sum ⁿC₀ + ⁿC₁ + ⁿC₂ + … + ⁿCₙ equals: (a) n² (b) 2ⁿ (c) 2ⁿ⁻¹ (d) n!
Q5. In the expansion of (1 - 2x)⁶, the term independent of x is: (a) ⁶C₀ (b) -⁶C₁ × 2 (c) ⁶C₃ × 8 (d) ⁶C₆
Q6. The general term Tᵣ₊₁ in the expansion of (a + b)ⁿ is given by: (a) ⁿCᵣ aʳ bⁿ⁻ʳ (b) ⁿCᵣ aⁿ⁻ʳ bʳ (c) ⁿCᵣ₊₁ aⁿ⁻ʳ bʳ (d) ⁿCᵣ₋₁ aⁿ⁻ʳ bʳ

Section B: Fill in the Blanks (1 mark each)

This section comprises five fill-in-the-blank questions that test your recall of definitions, standard results, and formula from NCERT Class 11 Mathematics Chapter 7. Write the exact word, expression, or number that correctly completes each statement. Pay attention to mathematical notation and ensure algebraic expressions are simplified. These questions cover binomial coefficients, properties, middle terms, and general term formulae. Precision is key; partial answers or incorrect notation will not earn marks. This format is common in CBSE board exams to assess direct knowledge without providing answer choices. Review NCERT textbook formulas and solved examples before attempting this section.

Q7. The binomial expansion of (x + y)ⁿ contains __________ terms.
Q8. In the expansion of (a + b)⁹, the middle term is the __________ term.
Q9. The value of ⁿC₀ - ⁿC₁ + ⁿC₂ - ⁿC₃ + … + (-1)ⁿ ⁿCₙ equals __________.
Q10. The general term in the binomial expansion of (2 - 3x)⁷ is Tᵣ₊₁ = __________.
Q11. If the coefficient of x⁷ and x⁸ in the expansion of (2 + x/3)ⁿ are equal, then n = __________.

Section C: True or False (1 mark each)

This section contains five statements about the Binomial Theorem. For each statement, write 'True' if it is correct or 'False' if it is incorrect. These questions require careful reasoning and understanding of subtle properties and exceptions. Some statements may appear correct at first glance but contain hidden errors or special cases. Always verify with a counterexample or reference to NCERT definitions before finalizing your answer. This format tests deeper conceptual understanding beyond rote memorization. Commonly tested areas include coefficient symmetry, middle term conditions, and applicability of the theorem. Expect at least one tricky statement designed to catch common misconceptions among Class 11 students.

Q12. The coefficients equidistant from the beginning and end in the expansion of (a + b)ⁿ are equal. (True / False)
Q13. The expansion of (1 + x)ⁿ has two middle terms when n is even. (True / False)
Q14. In the binomial expansion of (x - 1/x)⁸, the middle term is free of x. (True / False)
Q15. The coefficient of xⁿ in the expansion of (1 + x)²ⁿ is ²ⁿCₙ. (True / False)
Q16. For any positive integer n, ⁿC₀ + ⁿC₂ + ⁿC₄ + … = 2ⁿ⁻¹. (True / False)

Section D: Short Answer Questions (2-3 marks each)

This section presents five short-answer questions requiring concise working and clear final answers. Each question carries 2 to 3 marks and typically involves one or two calculation steps, application of the general term formula, or proving a simple identity. Write intermediate steps neatly; marks are awarded for method as well as correct answers. Common question types include finding a specific term, determining coefficients, computing middle terms, and using binomial properties to evaluate sums. These mirror typical CBSE Class 11 board exam short-answer questions and are essential for scoring well in Mathematics. Allocate roughly 3 to 4 minutes per question. Show all algebraic manipulation clearly to earn full marks even if the final numerical answer is slightly incorrect.

Q17. Find the 7th term in the expansion of (3x² - 2/x)¹⁰. (2 marks)
Q18. Determine the coefficient of x⁶y³ in the expansion of (x + 2y)⁹. (2 marks)
Q19. If the coefficients of the 5th, 6th, and 7th terms in the expansion of (1 + x)ⁿ are in arithmetic progression, find n. (3 marks)
Q20. Find the middle term(s) in the expansion of (2x - 3y)⁷. (3 marks)
Q21. Prove that ⁿC₀ + 2·ⁿC₁ + 2²·ⁿC₂ + … + 2ⁿ·ⁿCₙ = 3ⁿ. (3 marks)

Section E: Long Answer and HOTS Questions (5-6 marks each)

This section comprises three long-answer or higher-order thinking questions, each worth 5 to 6 marks. These questions demand multi-step reasoning, proof techniques, or integration of concepts from different parts of the Binomial Theorem chapter. Expect questions that require you to derive results, solve for unknowns given multiple conditions, or apply the theorem in non-standard contexts. Clearly label each step, write assumptions where necessary, and present a logical flow from given data to required result. Examiners award partial marks for correct method even if final answer is incomplete. These questions are similar to CBSE board long-answer questions and test depth of understanding. Allocate 8 to 10 minutes per question and check for algebraic errors before moving to the next question.

Q22. If the coefficients of x⁷ in (ax² + 1/bx)¹¹ and x⁻⁷ in (ax - 1/bx²)¹¹ are equal, find the relation between a and b. (5 marks)
Q23. Show that the coefficient of the middle term in the expansion of (1 + x)²ⁿ is equal to the sum of the coefficients of two middle terms in the expansion of (1 + x)²ⁿ⁻¹. (6 marks)
Q24. Using binomial theorem, evaluate (0.99)⁵ correct to four decimal places. (5 marks)

Case Study Question (4 marks)

This case-study question integrates the Binomial Theorem with a real-world scenario, reflecting the latest CBSE assessment pattern. Read the passage carefully, extract relevant numerical data, and answer the sub-questions that follow. Case studies test your ability to apply mathematical concepts to practical situations such as finance, probability, physics, or data science. Each sub-question typically carries 1 mark and may be objective or very-short-answer type. This format has become standard in CBSE Class 11 and 12 board exams since 2021. The scenario below involves quality control in manufacturing, where binomial expansions model defect probabilities. Work methodically through each sub-question, referring back to the passage for data, and use binomial coefficient properties or approximations as needed.

Case Study Passage: A pharmaceutical company tests a new batch of tablets. The probability that a tablet is defective is modelled using the binomial expansion of (0.95 + 0.05)¹⁰, where 0.95 represents the probability of a non-defective tablet and 0.05 represents a defective one. Quality engineers use specific terms of this expansion to compute probabilities of different defect counts in a sample of 10 tablets.
Sub-Q (i): How many terms are there in the expansion of (0.95 + 0.05)¹⁰? (1 mark)
Sub-Q (ii): Write the general term Tᵣ₊₁ in the above expansion. (1 mark)
Sub-Q (iii): Which term represents the probability of exactly 2 defective tablets in the sample? (1 mark)
Sub-Q (iv): What is the sum of all coefficients in the expansion, and what does it signify in this context? (1 mark)

Answer Key with Explanations

Below is the complete answer key for all sections. Each answer includes the correct response and a brief explanation or working to help you understand the solution method. Compare your answers carefully and note any mistakes. If you scored below 70 percent, revisit the corresponding NCERT chapter sections and attempt the worksheet again after a few days. Regular practice with worksheets like this one significantly improves performance in CBSE board exams. For persistent doubts, consider using CBSETUTOR.ai, where you can upload this worksheet via photo and get instant step-by-step solutions from an AI tutor available 24×7 at just ₹999 per month for any class from 6 to 12, with a 3-day free trial.

Section A Answers: (1) b, (2) b, (3) c, (4) b, (5) a, (6) b
Section B Answers: (7) n+1, (8) 5th, (9) 0, (10) ⁷Cᵣ·2⁷⁻ʳ·(-3x)ʳ, (11) 15
Section C Answers: (12) True, (13) False, (14) True, (15) True, (16) True
Section D: Detailed solutions provided below in separate subsections
Section E: Detailed solutions provided below in separate subsections
Case Study Answers: (i) 11 terms, (ii) ¹⁰Cᵣ(0.95)¹⁰⁻ʳ(0.05)ʳ, (iii) T₃ or ¹⁰C₂(0.95)⁸(0.05)², (iv) Sum = 1, signifying total probability across all outcomes

Detailed Solutions: Section D Short Answers

Q17. Find the 7th term in (3x²-2/x)¹⁰. General term Tᵣ₊₁ = ¹⁰Cᵣ(3x²)¹⁰⁻ʳ(-2/x)ʳ = ¹⁰Cᵣ·3¹⁰⁻ʳ·(-2)ʳ·x²⁰⁻²ʳ⁻ʳ = ¹⁰Cᵣ·3¹⁰⁻ʳ·(-2)ʳ·x²⁰⁻³ʳ. For 7th term, r=6. T₇ = ¹⁰C₆·3⁴·(-2)⁶·x²⁰⁻¹⁸ = 210·81·64·x² = 1088640x². Q18. Coefficient of x⁶y³ in (x+2y)⁹. General term ⁹Cᵣ·x⁹⁻ʳ·(2y)ʳ = ⁹Cᵣ·2ʳ·x⁹⁻ʳ·yʳ. For x⁶y³, 9-r=6 and r=3. Coefficient = ⁹C₃·2³ = 84·8 = 672. Q19. Coefficients of T₅, T₆, T₇ in (1+x)ⁿ are ⁿC₄, ⁿC₅, ⁿC₆ in AP. So 2·ⁿC₅ = ⁿC₄ + ⁿC₆. Using ⁿCᵣ = n!/(r!(n-r)!), simplify: 2·n!/(5!(n-5)!) = n!/(4!(n-4)!) + n!/(6!(n-6)!). Cancel n! and simplify to get n=14. Q20. In (2x-3y)⁷, n=7 (odd), so two middle terms T₄ and T₅. T₄ = ⁷C₃(2x)⁴(-3y)³ = 35·16·(-27)x⁴y³ = -15120x⁴y³. T₅ = ⁷C₄(2x)³(-3y)⁴ = 35·8·81x³y⁴ = 22680x³y⁴. Q21. Put x=2 in (1+x)ⁿ: (1+2)ⁿ = ⁿC₀ + 2·ⁿC₁ + 4·ⁿC₂ +...+ 2ⁿ·ⁿCₙ = 3ⁿ. Hence proved.

Detailed Solutions: Section E Long Answers

Q22. In (ax²+1/bx)¹¹, general term ¹¹Cᵣ(ax²)¹¹⁻ʳ(1/bx)ʳ = ¹¹Cᵣ·a¹¹⁻ʳ·b⁻ʳ·x²²⁻²ʳ⁻ʳ = ¹¹Cᵣ·a¹¹⁻ʳ·b⁻ʳ·x²²⁻³ʳ. For x⁷, 22-3r=7 so r=5. Coefficient = ¹¹C₅·a⁶·b⁻⁵. In (ax-1/bx²)¹¹, general term ¹¹Cᵣ(ax)¹¹⁻ʳ(-1/bx²)ʳ = ¹¹Cᵣ·a¹¹⁻ʳ·(-1)ʳ·b⁻ʳ·x¹¹⁻ʳ⁻²ʳ = ¹¹Cᵣ·a¹¹⁻ʳ·(-1)ʳ·b⁻ʳ·x¹¹⁻³ʳ. For x⁻⁷, 11-3r=-7 so r=6. Coefficient = ¹¹C₆·a⁵·(-1)⁶·b⁻⁶ = ¹¹C₆·a⁵·b⁻⁶. Given equal: ¹¹C₅·a⁶/b⁵ = ¹¹C₆·a⁵/b⁶. Since ¹¹C₅=¹¹C₆=462, cancel: a⁶/b⁵ = a⁵/b⁶, so a⁶b⁶ = a⁵b⁵, hence a=b (assuming positive). Q23. Middle term of (1+x)²ⁿ is T₍ₙ₊₁₎ with coefficient ²ⁿCₙ. Middle terms of (1+x)²ⁿ⁻¹ are Tₙ and T₍ₙ₊₁₎ with coefficients ²ⁿ⁻¹Cₙ₋₁ and ²ⁿ⁻¹Cₙ. By Pascal's identity ²ⁿ⁻¹Cₙ₋₁ + ²ⁿ⁻¹Cₙ = ²ⁿCₙ. Hence proved. Q24. Write 0.99=1-0.01, so (0.99)⁵=(1-0.01)⁵. Use binomial: ⁵C₀ - ⁵C₁(0.01) + ⁵C₂(0.01)² - ⁵C₃(0.01)³ + ⁵C₄(0.01)⁴ - ⁵C₅(0.01)⁵ = 1 - 0.05 + 0.001 - 0.00001 + very small terms ≈ 0.9510 to four decimals.

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Frequently asked questions

What is the difficulty level and suggested time for this CBSE Class 11 Binomial Theorem worksheet?+

This worksheet is pitched at moderate-to-challenging difficulty, aligned with CBSE board exam standards. The suggested time is 90 minutes for completing all 35+ questions across five sections plus the case study. Practicing under timed conditions will help you build exam speed and accuracy.

Does this worksheet cover all topics from NCERT Class 11 Mathematics Chapter 7?+

Yes, this worksheet comprehensively covers the NCERT syllabus for Binomial Theorem including binomial expansion, general term (Tr+1), middle term identification for even and odd powers, binomial coefficients, properties, and real-world applications. It mirrors the question types seen in CBSE board exams.

How do I identify the middle term in a binomial expansion?+

If the power n is even, there is one middle term at position (n/2 + 1). If n is odd, there are two middle terms at positions (n+1)/2 and (n+3)/2. For example, in (a+b)⁸ the middle term is T₅, while in (a+b)⁷ the middle terms are T₄ and T₅.

What is the general term formula in the Binomial Theorem and how do I use it?+

The general term in the expansion of (a+b)ⁿ is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ, where r ranges from 0 to n. To find a specific term, say the 5th term, set r+1=5 so r=4 and substitute into the formula. This technique is essential for finding coefficients and specific terms quickly.

Are the answers provided with explanations in this worksheet?+

Yes, the worksheet includes a complete answer key with brief step-by-step explanations for every question. This enables effective self-assessment and helps you understand the solution method, not just the final answer. Review explanations carefully to learn from any mistakes.

Can I use this worksheet for CBSE board exam preparation?+

Absolutely. This worksheet is designed specifically for CBSE Class 11 board exam preparation. It follows the latest CBSE question paper pattern including MCQs, short answers, long answers, and a case-study question. Regular practice with such worksheets boosts confidence and exam performance.

What is the case-study question format and why is it included?+

Case-study questions integrate mathematics with real-world scenarios such as quality control, finance, or probability. Introduced by CBSE in recent years, they test application skills. This worksheet includes one case study with four sub-questions worth 4 marks, mirroring the current board exam pattern.

How can I get help if I am stuck on a question from this worksheet?+

You can use CBSETUTOR.ai to upload a photo of any question you find difficult. The AI tutor provides instant step-by-step solutions with clear explanations. It is available 24×7 for just ₹999/month for all classes and subjects, with a 3-day free trial to start.

Is it necessary to memorize Pascal's triangle for binomial expansions?+

While not mandatory, knowing the first few rows of Pascal's triangle helps you quickly write binomial coefficients for small n. For larger n or exam conditions, use the formula ⁿCᵣ = n! / (r!(n-r)!) or learn to compute combinations efficiently. NCERT emphasizes understanding over rote memorization.

What are common mistakes students make in Binomial Theorem questions?+

Common errors include confusing the general term Tᵣ with Tᵣ₊₁, incorrect power assignment in expansions with negative or fractional terms, sign errors when b is negative, and miscounting middle terms. Careful working and double-checking powers and coefficients prevent most mistakes.

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