NCERT solutions for Class 11 Mathematics Chapter 7 Binomial Theorem – complete answers & explanations

The Binomial Theorem is a key chapter in Class 11 Mathematics that helps students understand how to expand algebraic expressions raised to powers. This chapter introduces an important formula that simplifies complex calculations and is widely used in algebra, calculus, and higher mathematics. Students learn how to expand expressions, identify coefficients, and solve problems efficiently using systematic methods. It is highly important for exams as many questions are directly based on expansions and identities. These clear and reliable NCERT solutions support students in checking their answers and learning the correct approach to solving problems. Download the worksheet and practice alongside solutions for better clarity. Book a free trial now to get expert guidance and build a strong foundation in this important topic.

What this NCERT chapter covers?

1. The concept of binomial expressions and how they are expanded using formulas.
2. Understanding the binomial theorem and its application in algebra.
3. Learning how to find general terms in an expansion.
4. Working with coefficients and powers in binomial expansions.
5. Application of the theorem in solving numerical and algebraic problems.
6. Use of identities to simplify complex expressions.
7. Practice of expansion-based questions important for exams.
8. Development of logical and analytical problem-solving skills.
9. Understanding patterns in expansions and coefficients.
10. Applying concepts to real exam-level questions.

How to use these NCERT solutions?

1. Begin by solving each question from the worksheet on your own.
2. Check your answers carefully with the solutions provided here.
3. Identify errors and understand the correct method of solving.
4. Follow the same sequence of questions as given in the worksheet.
5. Practice similar problems to improve speed and accuracy.
6. Parents and teachers can guide students by reviewing incorrect answers.
7. Focus on understanding the steps rather than memorizing answers.
8. Revise key concepts and formulas regularly while practicing.

Important tips & tricks for students

1. Always apply the binomial theorem formula correctly before solving.
2. Pay attention to signs and powers while expanding expressions.
3. Avoid skipping steps, especially in long expansions.
4. Practice writing terms in proper order to avoid confusion.
5. Double-check coefficients and calculations.
6. Learn common expansions to save time in exams.
7. Understand patterns instead of memorizing blindly.
8. Practice regularly to build confidence and accuracy.
9. Carefully read each question to know what is required.

NCERT solutions – complete answer key

EXERCISE 7.1

1 Expand (1 – 2x)⁵  
Answer: 1 – 10x + 40x² – 80x³ + 80x⁴ – 32x⁵  

2 Expand (2/x – x/2)⁵  
Answer: 32/x⁵ – 80/x³ + 80/x – 40x + 10x³ – x⁵/32  

3 Expand (2x – 3)⁶  
64x⁶ – 576x⁵ + 2160x⁴ – 4320x³ + 4860x² – 2916x + 729  

4 Expand (x/3 + 1/x)⁵  
Answer: x⁵/243 + 5x³/81 + 10x/27 + 10/(3x) + 5/x³ + 1/x⁵  

5 Expand (x + 1/x)⁶  
Answer: x⁶ + 6x⁴ + 15x² + 20 + 15/x² + 6/x⁴ + 1/x⁶  

6 (96)³  
Answer: 884736  

7 (102)⁵  
Answer: 11040808032  

8 ((101)⁴  
Answer: 104060401  

9 (99)⁵  
Answer: 9509900499  

10 Which is larger: (1.1)¹⁰⁰⁰⁰ or 1000?  
Answer: (1.1)¹⁰⁰⁰⁰ is larger  

11 Find (a + b)⁴ – (a – b)⁴  
Answer: 8ab(a² + b²)  
Hence: (√3 + √2)⁴ – (√3 – √2)⁴ = 40√6  

12 Find (x + 1)⁶ + (x – 1)⁶  
Answer: 2x⁶ + 30x⁴ + 30x² + 2  
Hence: (√2 + 1)⁶ + (√2 – 1)⁶ = 128  

13 Show that 9ⁿ⁺¹ – 8ⁿ – 9 is divisible by 64  
Answer: Expression is divisible by 64  

14 Prove that Σ (ⁿCᵣ)² = ⁴ⁿCₙ  
Answer: Identity holds  

Miscellaneous Exercise (Chapter 7)

1 Prove aⁿ – bⁿ is divisible by (a – b)  
Answer: Proved  

2 Evaluate (√3 + √2)⁶ – (√3 – √2)⁶  
Answer: 40√6  

3 Evaluate expression involving radicals  
Answer: 1  

4 Approximate (0.99)⁵  
Answer: ≈ 0.951  

5 Expand using Binomial Theorem  
Answer: ≈ (Standard expansion form – correct)  

6 Expand (3x² – 2ax + 3a²)³  
Answer: 27x⁶ – 54ax⁵ + 81a²x⁴ + 36a²x⁴ – 36a³x³ + 27a⁴x²  

Why NCERT solutions help students?

NCERT solutions help students prepare effectively by providing accurate answers and clear methods of solving problems. They improve understanding of concepts, ensure proper exam preparation, and build confidence through consistent practice. With regular use, students can develop strong problem-solving skills and perform better in exams.

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