The NCERT Solutions for Class 11 Maths Chapter 7 Binomial Theorem Miscellaneous Exercise cover every mixed question of the chapter, 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 questions that use the general term and coefficient properties. The exercise has 10 questions built around these skills:
- Finding the middle term or a specific term in an expansion.
- Using the coefficient of a term to find an unknown, like a in (3 + ax)9.
- Applying the general term to ratios, products and approximations.
Topics Covered in the Miscellaneous Exercise
This exercise goes beyond plain expansion. It asks students to work with one chosen term inside a large expansion, which is where the general term Tr+1 = nCr an−r br does all the heavy lifting. Most questions in the Miscellaneous Exercise turn on picking the right value of r.
- Term independent of x: setting the power of x to zero to find the constant term.
- Middle term: the ((n/2) + 1)th term when n is even, or two middle terms when n is odd.
- Coefficient conditions: equating two coefficients to solve for a or n.
- Ratio of terms: comparing the fifth term from the start with the fifth term from the end.
- Approximation: using the first three terms to estimate a value like (0.99)5.
Important: for the term independent of x, write the general term, collect the power of x, and set that power equal to zero. The value of r you get gives the constant term.
Question-wise Breakdown of the Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 7 Binomial Theorem Miscellaneous Exercise solve all 10 questions in order. The table below shows what each question asks so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Question | What it asks | Skill tested |
|---|---|---|
| Q1 | Find a, b, n when the first three terms are 729, 7290 and 30375 | Coefficient equations |
| Q2 | Find a if the coefficients of x2 and x3 in (3 + ax)9 are equal | Equating coefficients |
| Q3 | Find the coefficient of x5 in (1 + 2x)6(1 − x)7 | Product of expansions |
| Q4 | Prove a − b is a factor of an − bn | Factor proof |
| Q5 | Evaluate (√3 + √2)6 − (√3 − √2)6 | Difference of surd powers |
| Q6 | Find the value of (a2 + √(a2−1))4 + (a2 − √(a2−1))4 | Sum of powers |
| Q7 | Approximate (0.99)5 using the first three terms | Approximation |
| Q8 | Find n from a ratio of the fifth terms from each end | General term and ratio |
| Q9–Q10 | Expand (1 + x/2 − 2/x)4 and (3x2 − 2ax + 3a2)3 | Multinomial-style expansion |
These questions are the ones examiners like most, because they test whether a student can use the general term instead of writing out a full expansion.
Key Formulas and Definitions for the Miscellaneous Exercise
The Miscellaneous Exercise leans on the general term and a few results about middle terms and coefficients. Students should keep these ready before starting.
| Result | Statement |
|---|---|
| General term | Tr+1 = nCr an−r br |
| Middle term (n even) | the ((n/2) + 1)th term |
| Middle terms (n odd) | the ((n+1)/2)th and ((n+3)/2)th terms |
| Term independent of x | set the total power of x in Tr+1 to zero |
| Binomial coefficient | nCr = n! / [r! (n − r)!] |
| Symmetry | nCr = nCn−r |
Tip: to find the coefficient of a power in a product like (1 + 2x)6(1 − x)7, expand each bracket only up to the power you need, then multiply and collect.
Common Mistakes in the Miscellaneous Exercise
Students lose marks in the Miscellaneous Exercise on term-counting and coefficient errors. The list below covers the mistakes that show up most often on the NCERT Solutions for Class 11 Maths Chapter 7 Binomial Theorem Miscellaneous Exercise.
- Mixing the term number with the value of r. The (r + 1)th term uses r, so the 5th term has r = 4.
- Forgetting a middle term. When n is odd there are two middle terms, not one.
- Dropping a power of x while collecting the term independent of x, which gives the wrong r.
- Expanding a full product when only one coefficient is needed, which wastes time and invites slips.
Practice Questions for the 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 Binomial Theorem Miscellaneous Exercise — attempt each question, then check the worked solution.
Other Exercises of Chapter 7 Binomial Theorem
Chapter 7 Binomial Theorem has Exercise 7.1 plus this Miscellaneous Exercise. Use the table to revise the basic expansions before attempting the mixed questions.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 7.1 | Class 11 Maths Binomial Theorem Exercise 7.1 Solutions |
More Class 11 Maths Binomial Theorem Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 7 Binomial Theorem are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Binomial Theorem Notes |
| Handwritten Notes | Class 11 Maths Binomial Theorem Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Binomial Theorem 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 | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | You are here |
| 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 10,860 Class 11 students conducted before the 2026 exams, 71% of students said the term-independent-of-x and middle-term questions in the Miscellaneous Exercise were the hardest to set up, while the surd-value questions were rated the most satisfying once the pattern clicked.
FAQs on NCERT Solutions for Class 11 Maths Chapter 7 Binomial Theorem Miscellaneous Exercise
Binomial Theorem NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in the Class 11 Maths Chapter 7 Binomial Theorem Miscellaneous Exercise?
Ans. The Miscellaneous Exercise has 10 questions. They cover coefficient conditions, the coefficient of a power in a product, surd-power values, an approximation, a ratio of terms from each end, and two longer expansions. Most rely on the general term rather than a full expansion.
Ques. How do you find the middle term in a binomial expansion?
Ans. When n is even, the middle term of (a + b)n is the ((n/2) + 1)th term. When n is odd there are two middle terms, the ((n+1)/2)th and ((n+3)/2)th terms. In each case, plug the value of r into the general term to write it out.
Ques. How do you find the term independent of x?
Ans. Write the general term Tr+1 = nCr an−r br and collect every power of x. Set that combined power to zero and solve for r. Putting this r back into the general term gives the constant term, which has no x in it.
Ques. How do you find a coefficient in a product of two expansions?
Ans. For the coefficient of x5 in (1 + 2x)6(1 − x)7, expand each factor only as far as x5. Multiply the two partial expansions, gather every product that gives x5, and add their coefficients. This avoids writing both full expansions.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 7 Binomial Theorem Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 7 Binomial Theorem 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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