# NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem

NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem: You have studied the expansion of expressions like (a-b)2 and (a-b)in the previous classes. So you can calculate numbers like (96)3. If the power is high, it will be difficult to use normal multiplication. How will you process in such cases? In the NCERT solutions for class 11 maths chapter 8 binomial theorem, you will get the answer to the above question. In this chapter, you will study the expansion of (a+b)n, the general terms of the expansion, the middle term of the expansion, and the pascal triangle. In solutions of NCERT for class 11 chapter 8 binomial theorem, you will get questions related to these topics. This chapter covers the binomial theorem for positive integral indices only. The concepts of a binomial theorem are not only useful in solving problems of mathematics, but in various fields of science too. In this chapter, there are  26 problems in 2 exercises. All these questions are prepared in NCERT solutions for class 11 maths chapter 8 binomial theorem in a detailed manner. It will be very easy for you to understand the concepts. Check all NCERT solutions from class 6 to 12 to learn science and maths. There are 2 exercise and a miscellaneous exercise in this chapter which are explained below.

Exercise:8.1

Exercise:8.2

Miscellaneous Exercise

## The main content headings of NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem are listed below:

8.1 Introduction

8.2 Binomial Theorem for Positive Integral Indices

8.3 General and Middle Terms

The concepts of NCERT Class 11 Maths Chapter 8 Binomial Theorem can be used to find the approximate value of the power of a small number. For example, find the approximate value of 0.996 using the first three terms of expansion? This can be solved by rewriting 0.996 as (1-0.01)6 and expanding using the Binomial Theorem.

## NCERT solutions for class 11 maths chapter 8 binomial theorem-Exercise: 8.1

Question:1 Expand the expression.

Given,

The Expression:

the expansion of this Expression is,

Question:2 Expand the expression.

Given,

The Expression:

the expansion of this Expression is,

Question:3 Expand the expression.

Given,

The Expression:

the expansion of this Expression is,

Question:4 Expand the expression.

Given,

The Expression:

the expansion of this Expression is,

Question:5 Expand the expression.

Given,

The Expression:

the expansion of this Expression is,

As 96 can be written as (100-4);

As we can write 102 in the form 100+2

As we can write 101 in the form 100+1

As we can write 99 in the form 100-1

AS we can write 1.1 as 1 + 0.1,

Hence,

Question:11 Find . Hence, evaluate .

Using Binomial Theorem, the expressions  and  can be expressed as

From Here,

Now, Using this, we get

Question:12 Find . Hence or otherwise evaluate .

Using Binomial Theorem, the expressions  and  can be expressed as ,

From Here,

Now, Using this, we get

If we want to prove that  is divisible by 64, then we have to prove that

As we know, from binomial theorem,

Here putting x = 8 and replacing m by n+1, we get,

Now, Using This,

Hence

is divisible by 64.

Question:14 Prove that

As we know from Binomial Theorem,

Here putting a = 3, we get,

Hence Proved.

## Question:1 Find the coefficient of

in

As we know that the  term   in the binomial expansion of    is given by

Now let's assume  happens in the  term of the binomial expansion of

So,

On comparing the indices of x we get,

Hence the coefficient of the   in  is

Question:2 Find the coefficient of    in

As we know that the  term   in the binomial expansion of    is given by

Now let's assume  happens in the  term of the binomial expansion of

So,

On comparing the indices of x we get,

Hence the coefficient of the    in  is

As we know that the general   term   in the binomial expansion of    is given by

So the general term of the expansion of   :

.

As we know that the general   term   in the binomial expansion of    is given by

So the general term of the expansion of  is

.

Question:5 Find the 4th term in the expansion of  .

As we know that the general   term   in the binomial expansion of    is given by

So the  term of the expansion of  is

.

Question:6 Find the 13th term in the expansion of

As we know that the general   term   in the binomial expansion of    is given by

So the  term of the expansion of        is

As we know that the middle  terms in the expansion of   when n is odd are,

Hence the middle term of the expansion       are

Which are

Now,

As we know that the general   term   in the binomial expansion of    is given by

So the  term of the expansion of    is

And the  Term of the expansion of     is,

Hence the middle terms of the expansion of given expression are

As we know that the middle term in the expansion of   when n is even is,

,

Hence the middle term of the expansion        is,

Which is

Now,

As we know that the general   term   in the binomial expansion of    is given by

So the  term of the expansion of    is

Hence the middle term of the expansion of   is nbsp; .

As we know that the general   term   in the binomial expansion of    is given by

So, the general  term   in the binomial expansion of    is given by

Now, as we can see  will come when  and  will come when

So,

Coefficient of  :

CoeficientCoefficient of  :

As we can see .

Hence it is proved that the coefficients of  and  are equal.

As we know that the general   term   in the binomial expansion of    is given by

So,

Term in  the expansion of  :

Term in  the expansion of  :

Term in  the expansion of  :

Now, As given in the question,

From here, we get ,

Which can be written as

From these equations we get,

As we know that the general   term   in the binomial expansion of    is given by

So, general   term   in the binomial expansion of   is,

will come when ,

So, Coefficient of  in the binomial expansion of   is,

Now,

the general   term   in the binomial expansion of   is,

Here also  will come when ,

So, Coefficient of  in the binomial expansion of   is,

Now, As we can see

Hence, the coefficient of in the expansion of  is twice the coefficient of  in the expansion of .

As we know that the general   term   in the binomial expansion of    is given by

So, the general   term   in the binomial expansion of    is

will come when . So,

The coeficient of   in the binomial expansion of    = 6

Hence the positive value of m for which the coefficient of  in the expansion is 6, is 4.

CBSE NCERT solutions for class 11 maths chapter 8 binomial theorem-Miscellaneous Exercise

As we know the Binomial expansion of  is given by

Given in the question,

Now, dividing (1) by (2) we get,

Now, Dividing (2) by (3) we get,

Now, From (4) and (5), we get,

As we know that the general   term   in the binomial expansion of    is given by

So, the general   term   in the binomial expansion of    is

Now,  will come when  and   will come when

So, the coefficient of  is

And the coefficient of  is

Now, Given in the question,

Hence the value of a is 9/7.

First, lets expand both expressions individually,

So,

And

Now,

Now, for the coefficient of , we multiply and add those terms whose product gives .So,

The term which contain are,

Hence the coefficient of  is 171.

we need to prove,

where k is some natural number.

Now let's add and subtract b from a so that we can prove the above result,

Hence, is a factor of .

Question:5 Evaluate .

First let's simplify the expression  using binomial theorem,

So,

And

Now,

Now, Putting  we get

Question:6 Find the value of

First, lets simplify the expression  using binomial expansion,

And

Now,

Now, Putting  we get,

As we can write 0.99 as 1-0.01,

Hence the value of     is 0.951 approximately.

Given, the expression

Fifth term from the beginning  is

And Fifth term from the end is,

Now, As given in the question,

So,

From Here ,

From here,

Hence the value of n is 10.

Question:9 Expand using Binomial Theorem

Given the expression,

Binomial expansion of this expression is

Now Applying Binomial Theorem again,

And

Now, From (1), (2) and (3) we get,

Question:10  Find the expansion of using binomial theorem.

Given

By Binomial Theorem It can also be written as

Now, Again By Binomial Theorem,

From (1) and (2) we get,

## NCERT solutions for class 11 mathematics

 chapter-1 NCERT solutions for class 11 maths chapter 1 Sets chapter-2 Solutions of NCERT for class 11 chapter 2 Relations and Functions chapter-3 CBSE NCERT solutions for class 11 chapter 3 Trigonometric Functions chapter-4 NCERT solutions for class 11 chapter 4 Principle of Mathematical Induction chapter-5 Solutions of NCERT for class 11 chapter 5 Complex Numbers and Quadratic equations chapter-6 CBSE NCERT solutions for class 11 maths chapter 6 Linear Inequalities chapter-7 NCERT solutions for class 11 maths chapter 7 Permutation and Combinations chapter-8 NCERT solutions for class 11 maths chapter 8 Binomial Theorem chapter-9 CBSE NCERT solutions for class 11 maths chapter 9 Sequences and Series chapter-10 NCERT solutions for class 11 maths chapter 10 Straight Lines chapter-11 Solutions of NCERT for class 11 maths chapter 11 Conic Section chapter-12 CBSE NCERT solutions for class 11 maths chapter 12 Introduction to Three Dimensional Geometry chapter-13 NCERT solutions for class 11 maths chapter 13 Limits and Derivatives chapter-14 Solutions of NCERT for class 11 maths chapter 14 Mathematical Reasoning chapter-15 CBSE NCERT solutions for class 11 maths chapter 15 Statistics chapter-16 NCERT solutions for class 11 maths chapter 16 Probability

## NCERT solutions for class 11- Subject wise

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In NCERT solutions for class 11 maths chapter 8 binomial theorem, there are some important formulas to be remembered which are mentioned below.

The binomial theorem for a positive integer n

-> binomial coefficients.

Some special cases

Put a=1, b=x

Put x=1

Put a=1,b=-x

Put x=1

There are 10 problems in miscellaneous exercise. To get command on this chapter, you need to solve miscellaneous exercise too. In NCERT solutions for class 11 maths chapter 8 binomial theorem, you will get solutions to miscellaneous exercise too.