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NCERT Solutions for Class 11 Maths Chapter 8 - Binomial Theorem

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NCERT Solutions for Class 11 Maths Chapter 8 - Free PDF Download

By referring to NCERT Solutions for Class 11 Maths Chapter 8, students are able to comprehend the topics covered in this chapter in detail. The binomial theorem is defined as the process of algebraically expanding the power of sums of two or more binomials. Coefficients of binomial terms in the process of expansion are referred to as binomial coefficients. Our subject experts at Vedantu have solved all the sums and explained all the topics covered in this chapter according to the CBSE guidelines in these NCERT Solutions Class 11 Maths Chapter 8. You can download these solutions from Vedantu for free. 


Class:

NCERT Solutions for Class 11

Subject:

Class 11 Maths

Chapter Name:

Chapter 8 - Binomial Theorem

Content-Type:

Text, Videos, Images and PDF Format

Academic Year:

2024-25

Medium:

English and Hindi

Available Materials:

  • Chapter Wise

  • Exercise Wise

Other Materials

  • Important Questions

  • Revision Notes



It is a segment of basic algebra that students are required to study in Class 11. A proper comprehension of the basic concepts of this chapter will help students before they proceed to study it in detail in their higher classes.



Binomial Theorem Chapter at a Glance - Class 11 NCERT Solutions

  • The expansion of a binomial for any positive integral $n$ is given by Binomial Theorem, which is

  • $(a+b)^n={ }^n C_0 a^n+{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2+\ldots+{ }^n C_{n-1} a b^{n-1}+{ }^n C_n b^n$.

  • The coefficients of the expansions are arranged in an array. This array is called Pascal 's triangle.

  • The general term of an expansion $(a+b)^n$ is $T_{r+1}={ }^n C_r a^{n-r}, b^r$.

  • In the expansion $(a+b)^n$, if $n$ is even, then the middle term is the $\left(\frac{n}{2}+1\right)^{\text {th }}$ term. If $n$ is odd, then the middle terms are $\left(\frac{n+1}{2}\right)^{h t}$ and $\left(\frac{n+1}{2}+1\right)^{n t}$ terms. 

Competitive Exams after 12th Science

Exercises under NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem

Exercise 8.1: This exercise consists of 13 questions and introduces students to the concept of binomial theorem. The questions are based on various concepts such as expanding binomial expressions, finding the middle terms of a binomial expansion, and solving problems related to binomial theorem. Students will learn how to expand binomial expressions using binomial theorem.

Exercise 8.2: This exercise consists of 8 questions and is focused on solving problems related to the application of binomial theorem. The questions are based on various concepts such as finding the coefficient of a term in a binomial expansion, solving problems related to the probability of an event, and solving problems related to the number of ways to select objects. Students will learn how to apply the concepts of binomial theorem to solve problems related to various scenarios.

Miscellaneous Exercise: This exercise consists of 10 questions and covers a variety of topics related to binomial theorem. The questions are based on various concepts such as solving problems related to the number of terms in a binomial expansion, finding the sum of the terms in a binomial expansion, and solving word problems related to binomial theorem. This exercise will help students to revise and reinforce the concepts learned in the previous exercises.


Overall, the exercises in NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem are designed to help students understand and apply the concepts of binomial theorem in various scenarios. The solutions to these exercises are provided in the textbook, which will help students to check their answers and understand the concepts better.


NCERT Solutions for Class 11 Maths Chapter 8 - Binomial Theorem

Class 11 Maths Chapter 8 Solutions- Free PDF Download

Solving equations from NCERT Solutions for Class 11 Maths Chapter 8 will help students comprehend the basic steps they need to follow to solve these kinds of sums. Without having a clear understanding of binomial theorems, students may have a hard time-solving the sums given in this chapter. 

With proper practice of the Binomial Theorem Class 11 NCERT Solutions, students will be able to score well in their upcoming examinations. The array of examples provided in these NCERT Solutions will allow students to understand how they are supposed to proceed with equations covered in this chapter. 


NCERT Solutions for Class 11 Maths Chapters

 

NCERT Solution Class 11 Maths of Chapter 8 Exercises

  • Exercise 8.1 – Introduction

In this segment of Ch 8 Maths Class 11, students are introduced to the basic concept of the Binomial Theorem. It starts off with some examples and definitions, allowing students to learn how to solve simple equations before they proceed with complex sums. 

  • Exercise 8.2 – Binomial Theorem for Positive Integral Indices

In this section of the Binomial Theorem NCERT Solutions, students learn of the different forms of expansion. This section covers four important expansions which include (l + m) to the power 2, (l – m) to the power 2, (l + m) to the power 3, and (l – m) to the power 3. Students are to keep in mind certain simple steps for the process of expansion of these equations. 

  • Exercise 8.2.1 – Binomial Theorem for any Positive Integer n 

With this section of Chapter 8 Class 11 Maths, students are to understand the positive integral indices. Properties of the binomial theorem for any positive integers in binomial theorem include information on how the number of terms after the expansion of (x + a) to the power n is (n+1). 

Other properties of the binomial theorem for any positive integer include information on how n is the total sum of indices in a and x. Expansion of x and a as explained in the first property is also true is both a and x are complex numbers. 

  • Exercise 8.2.2 – Some Special Cases

In this section of the Binomial Theorem Class 11 Solutions PDF, students learn about the usage of different formulas. By referring to examples mentioned here, it helps students comprehend the lesson well. With Binomial Theorem Class 11th Solutions, students are able to comprehend the conceptual basis of each exercise with ease. 

  • Exercise 8.3 – General and Middle Terms

Referring to Binomial Theorem Class 11 solutions will help students understand the difference between middle terms and general terms in binomial theorem. Within this section, students will be provided with examples and exercises that will help them understand exactly how they are required to solve these types of equations. 


All Important Formulae to Solve Exercises of Chapter 8 — Class 11 NCERT Maths 

To solve Exercises 8.1, 8.2 and Miscellaneous Exercise of Class 11 NCERT Mathematics, you must remember the below given Binomial theorem formulae. 

  1. (a + b)n = nC0 an + nC1 an-1 b + nC2 an-2 b2 + nC3 an-3 b3 + ….. + nCn-1 a bn-1 + nCn bn

  2. (a – b)n = nC0 an nC1 an-1 b + nC2 an-2 b2nC3 an-3 b3 + …..  + (-1)n nCn bn

  3. (1 + b)n = nC0 + nC1 b + nC2 b2 + nC3 b3 + ….. + nCn-1 bn-1 + nCn bn

  4. (1 - b)n = nC0 - nC1 b + nC2 b2 - nC3 b3 + …..  + (-1)n nCn bn

  5. When we expand (a + b)n, then the middle term will be 

  • If n is even, middle term = (\[\frac{n}{2}\] + 1 )th term

  • If n is odd, middle term = (\[\frac{n+1}{2}\])th term, (\[\frac{n+1}{2}\] + 1)th term

(In case n is odd, there will be two middle terms.)


Benefits of NCERT Solutions for Class 11 Maths Chapter 8

By referring to the NCERT Solutions Class 11 Maths Chapter 8 from Vedantu, students will be able to avail of the following benefits.

  • Examples available in these NCERT Solutions allow students to comprehend the application of formulas within this chapter. 

  • The solutions are sorted in such a manner that makes it convenient for the students to understand the concepts. 

  • The sums in these exercises are solved in a precise and step-by-step format, facilitating easy understanding. 

With Class 11 Maths Binomial Theorem Solutions from Vedantu, students can now clear their doubts right away and prepare effectively for their examination. 

Topics Covered under Chapter 8 of NCERT Solutions Class 11 Maths 

  • Introduction to Binomial Theorem 

  • Binomial Theorem for Positive Integral Indices 

  • Pascal’s Triangle 

  • Binomial theorem for any positive integer n

  • Some special cases

  • General and Middle Terms 

FAQs on NCERT Solutions for Class 11 Maths Chapter 8 - Binomial Theorem

1. What is the Binomial Theorem?

The binomial theorem is defined as the process of algebraically expanding the power of sums of two or more binomials. Coefficients of binomial terms in the process of expansion are referred to as binomial coefficients. The introductory parts of these chapters consist of proper definitions of different aspects of the binomial theorem. 

With the Binomial Theorem Class 11 NCERT Solutions PDF, students can now study with ease and be updated with all information that might appear in their examinations. Learning the concepts of the Binomial Theorem will be easier with the NCERT Solutions available on Vedantu. 

2. How will the NCERT Solutions Class 11 Maths Chapter 8 Binomial Theorem help in understanding the concepts of the Binomial Theorem?

To learn about the expansion procedure, refer to the examples available in Class 11 Maths Ch 8 NCERT Solutions. These examples have been solved in a step-by-step format that will help students understand the concepts better. 

Students are required to be well-versed in solving these equations if they wish to score well in examinations. There are also exercises that are given in this segment that students can solve on their own. This will allow them to practice what they have learned and clear any doubt that they might have related to the binomial theorem. 

3. What are the properties of positive integers in the Binomial Theorem? 

There are more than 10 properties that are listed under positive integers that students can learn when studying NCERT Solutions Class 11 Maths Chapter 8 Binomial Theorem. Students are required to study these properties to understand the basic concept of solving such equations. 

Examination papers may target aspects of chapters that seem simple to students but may be tricky to solve, which is why students will need to go through these NCERT Solutions thoroughly if they wish to score well in their upcoming examinations. 

4. Explain the concept of the Binomial Theorem covered in Chapter 8 of NCERT Solutions for Class 11 Maths.

The Binomial theorem states, for positive integer n, whenever you add any two numbers, say a and b, the result raised to the power of n can also be written as the sum of (n+10 terms). The coefficients involved are expressed as binomial coefficients. The NCERT solutions of Class 11 Maths Chapter 8 can be accessed on the Vedantu website and the app. Practice all of these judiciously if you want to score well in them and in the other topics related to them.

5. What Chapter is Binomial Theorem Class 11?

The Binomial Theorem is Chapter 8 of the NCERT Mathematics book. It explains in detail the Binomial Theorem and also provides the necessary exercises for a better understanding of the concepts by the students. The solutions for this chapter can easily be found on the Vedantu site (vedantu.com). The students are advised to go through all topics to efficiently grasp the content and score well in their exams.

6. Which is the best Solution book for NCERT Class 11 Chapter 8 Maths?

The best solution book for Chapter 8-Binomial Theorem is available on Vedantu. NCERT Class 11 Maths Chapter 8 Solutions may be obtained by visiting the Vedantu website. Aside from that, you may access a range of modules that will help you achieve excellent grades in math exams. The link to the solution to the exercise is provided below. Visit the page NCERT Solutions for Class 11 Chapter 8 to download the PDF file free of cost.

7. How can I master Class 11 Maths Chapter 8?

There is no easy or singular motto for performing well or ranking high in any topic. To do well, one must constantly be diligent about the core ideas. It is important to practice as many questions as possible thoroughly so that they have a good understanding of the concepts. Proper practise is critical for getting high grades and gaining deeper knowledge. Those interested in accessing NCERT solutions, revision notes and important questions of this chapter, visit the Vedantu website or download the Vedantu app.

8. Do I need to practice all the questions provided in Class 11 Maths Chapter 8 NCERT Solutions?

Yes, it is critical to practice and answer all questions since they cover a variety of subjects and concepts which will give you a good understanding of the kind of questions that might be set from those areas. These questions also help you learn how different questions from the same topic may be set. Each exercise should be thoroughly practised. You can discover modules on the Vedantu site or in the Vedantu app that are relevant to this topic or other topics in this chapter.