# NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem

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NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem:
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You have studied the expansion of expressions like (a-b)
^{
2
}
and (a-b)
^{
3
}
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.

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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 0.99
^{
6
}
using the first three terms of expansion? This can be solved by rewriting 0.99
^{
6
}
as (1-0.01)
^{
6
}
and expanding using the Binomial Theorem.

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The
**
**
NCERT Solutions
**
**
of this chapter are given below
**
:

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NCERT solutions for class 11 maths chapter 8 binomial theorem-Exercise: 8.1
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**
Question:1
**
Expand the expression.
**
**

**
Answer:
**

Given,

The Expression:
**
**

**
**

the expansion of this Expression is,

**
**

**
Question:2
**
Expand the expression.
**
**

**
Answer:
**

Given,

The Expression:
**
**

**
**

the expansion of this Expression is,

**
**

**
Question:3
**
Expand the expression.
**
**

**
Answer:
**

Given,

The Expression:
**
**

**
**

the expansion of this Expression is,

**
**

**
Question:4
**
Expand the expression.
**
**

**
Answer:
**

Given,

The Expression:
**
**

**
**

the expansion of this Expression is,

**
**

**
Question:5
**
Expand the expression.
**
**

**
Answer:
**

Given,

The Expression:
**
**

**
**

the expansion of this Expression is,

**
**

**
Question:7
**
Using binomial theorem, evaluate the following:
**
**

**
Answer:
**

As we can write 102 in the form 100+2

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Question:8
**
Using binomial theorem, evaluate the following:

**
Answer:
**

As we can write 101 in the form 100+1

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Question:9
**
Using binomial theorem, evaluate the following:
**
**

**
Answer:
**

As we can write 99 in the form 100-1

**
Question:10
**
Using Binomial Theorem, indicate which number is larger (1.1)
^{
10000
}
or 1000.

**
Answer:
**

AS we can write 1.1 as 1 + 0.1,

Hence,

**
Question:11
**
Find
. Hence, evaluate
.

**
Answer:
**

Using Binomial Theorem, the expressions and can be expressed as

From Here,

Now, Using this, we get

**
Question:12
**
Find
. Hence or otherwise evaluate
.

**
Answer:
**

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

From Here,

Now, Using this, we get

**
Question:13
**
Show that
is divisible by 64, whenever
*
n
*
is a positive integer.

**
Answer:
**

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

**
Answer:
**

As we know from Binomial Theorem,

Here putting a = 3, we get,

Hence Proved.

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NCERT solutions for class 11 maths chapter 8 binomial theorem-Exercise: 8.2
**

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**
Question:1
**
Find the coefficient of

**
Answer:
**

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

**
Answer:
**

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:3
**
Write the general term in the expansion of

**
Answer:
**

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

So the general term of the expansion of
**
:
**

.

**
Question:4
**
Write the general term in the expansion of

**
Answer:
**

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 4
^{
th
}
term in the expansion of
.

**
Answer:
**

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 13
^{
th
}
term in the expansion of

**
Answer:
**

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

So the
term of the expansion of
**
**
is

**
Question:7
**
Find the middle terms in the expansion of
**
**

**
Answer:
**

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

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Question:8
**
Find the middle terms in the expansion of
**
**

**
Answer:
**

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;
**
.

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Question:9
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In the expansion of
, prove that coefficients of
and
are equal

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Answer:
**

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.

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Answer:
**

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,

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Question:11
**
Prove that the coefficient of
in the expansion of
is twice the coefficient of
in the expansion of
.

**
Answer:
**

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 .

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Question:12
**
Find a positive value of
*
m
*
for which the coefficient of
in the expansion
is 6.

**
Answer:
**

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.

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

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,

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Question:2
**
Find
*
a
*
if the coefficients of
and
in the expansion of
are equal.

**
Answer:
**

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.

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Question:3
**
Find the coefficient of
in the product
using binomial theorem.

**
Answer:
**

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.

**
Answer:
**

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 .

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Question:5
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Evaluate
.

**
Answer:
**

First let's simplify the expression using binomial theorem,

So,

And

Now,

Now, Putting we get

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Question:6
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Find the value of

**
Answer:
**

First, lets simplify the expression using binomial expansion,

And

Now,

Now, Putting we get,

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Question:7
**
Find an approximation of (0.99)
^{
5
}
using the first three terms of its expansion.

**
Answer:
**

As we can write 0.99 as 1-0.01,

Hence the value of is 0.951 approximately.

**
Answer:
**

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.

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Question:9
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Expand using Binomial Theorem

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Answer:
**

Given the expression,

Binomial expansion of this expression is

Now Applying Binomial Theorem again,

And

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

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Question:10
**
Find the expansion of
using binomial theorem
.

**
Answer:
**

Given

By Binomial Theorem It can also be written as

Now, Again By Binomial Theorem,

From (1) and (2) we get,

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NCERT solutions for class 11 mathematics
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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.

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Happy Reading !!!
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