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Edited By Ramraj Saini | Updated on Sep 23, 2023 06:17 PM IST

**NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem **are provided here. These NCERT solutions are prepared by subjects matter experts considering the latest syllabus and pattern of CBSE 2023-24. 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 class 11 maths chapter 8 NCERT solutions, you will get the answer to the above question. In this NCERT Book 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.

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This Story also Contains

- Binomial Theorem Class 11 Questions And Answers
- Binomial Theorem Class 11 Questions And Answers PDF Free Download
- Binomial Theorem Class 11 Solutions - Important Formulae
- Binomial Theorem Class 11 NCERT Solutions (Intext Questions and Exercise)
- Highlights of NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem
- Key Features of NCERT Solutions for Class 11 Maths Chapter 8 – Binomial Theorem

In NCERT solutions for class 11 chapter 8 binomial theorem, you will get questions related to these topics. NCERT Book 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 the NCERT syllabus of this chapter, there are 26 problems in 2 exercises. All these questions are prepared in binomial theorem class 11 NCERT solutions 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. Here you will get NCERT solutions for class 11 also.

**Also read:**

Binomial Theorem:

The Binomial Theorem provides the expansion of a binomial (a + b) raised to any positive integer n.

The expansion of (a + b)^{n} is given by: (a + b)^{n} = nC0 * a^{n} + nC1 * a^{(n-1)} * b + nC2 * a^{(n-2)} * b^{2} + … + nCn-1 * a * b^{(n-1)} + nCn * b^{n}.

Special Cases from the Binomial Theorem:

(x - y)^{n} = nC0 * x^{n} - nC1 * x^{(n-1)} * y + nC2 * x^{(n-2)} * y^{2} + … + (-1)^{n} * nCn * x.

(1 - x)^{n} = nC0 - nC1 * x + nC2 * x^{2} - …. + (-1)^{n} * nCn * x^{n}.

The coefficients nC0 and nCn are always equal to 1.

Pascal’s Triangle:

The coefficients of the expansions in the Binomial Theorem are arranged in an array called Pascal’s triangle.

General Term of Expansions:

For (a + b)^{n}, the general term is T_{r+1} = nCr * a^{(n-r)} * b^r.

For (a - b)^{n}, the general term is (-1)r * nCr * a^{(n-r)} * b^r.

For (1 + x)^{n}, the general term is nCr * x^{r}.

For (1 - x)^{n}, the general term is (-1)^r * nCn * x^{n}.

Middle Terms:

In the expansion (a + b)^{n}, if n is even, then the middle term is the (n/2 + 1)-th term.

If n is odd, then the middle terms are the (n/2 + 1)-th and ((n+1)/2 + 1)-th terms.

Free download **NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem **for CBSE Exam.

**Binomial theorem class 11 solutions ****- Exercise 8.1**

**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

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

** Answer: **

As we can write 101 in the form 100+1

** 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.

** Binomial theorem class 11 solutions - Exercise: 8.2 **

** 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

** 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; ** .

** Question:9 ** In the expansion of , prove that coefficients of and 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 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.

** 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,

** 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 .

** 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.

**Class 11 maths chapter 8 question answer ****- Miscellaneous Exercise **

** 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,

** 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.

** 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 .

** Question:5 ** Evaluate .

** Answer: **

First let's simplify the expression using binomial theorem,

So,

And

Now,

Now, Putting we get

** Question:6 ** Find the value of

** Answer: **

First, lets simplify the expression using binomial expansion,

And

Now,

Now, Putting we get,

** 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.

** Question:9 ** Expand using Binomial Theorem

** Answer: **

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 .

** Answer: **

Given

By Binomial Theorem It can also be written as

Now, Again By Binomial Theorem,

From (1) and (2) we get,

8.1 Introduction

8.2 Binomial Theorem for Positive Integral Indices

- Pascal’s Triangle

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Download EBook8.2.1 Binomial theorem for any positive integer n

8.2.2 Some special cases

8.3 General and Middle Term

If you are interested in Binomial Theorem class 11 exercise solutions then these are listed below.

Binomial Theorem class 11 exercise 8.1

Binomial Theorem class 11 exercise 8.2

Binomial Theorem class 11 exercise miscellaneous exercise

chapter-1 | Sets |

chapter-2 | Relations and Functions |

chapter-3 | Trigonometric Functions |

chapter-4 | Principle of Mathematical Induction |

chapter-5 | Complex Numbers and Quadratic equations |

chapter-6 | Linear Inequalities |

chapter-7 | Permutation and Combinations |

chapter-8 | Binomial Theorem |

chapter-9 | Sequences and Series |

chapter-10 | Straight Lines |

chapter-11 | Conic Section |

chapter-12 | Introduction to Three Dimensional Geometry |

chapter-13 | Limits and Derivatives |

chapter-14 | Mathematical Reasoning |

chapter-15 | Statistics |

chapter-16 | Probability |

**Comprehensive explanations:** The solutions of maths chapter 8 class 11 provide step-by-step explanations to all the questions in the chapter, helping students to understand the concepts thoroughly.

**NCERT-based:** The solutions of ch 8 maths class 11 are strictly based on the latest NCERT syllabus and follow the guidelines set by the board.

**Easy language:** The class 11 maths ch 8 question answer are written in simple and easy-to-understand language, making it easy for students to grasp the concepts.

NCERT solutions for class 11 biology |

NCERT solutions for class 11 maths |

NCERT solutions for class 11 chemistry |

NCERT solutions for Class 11 physics |

-> 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 with miscellaneous exercise. To get command on this chapter, you need to solve miscellaneous exercises too. In NCERT solutions for class 11 maths chapter 8 binomial theorem, you will get solutions to miscellaneous exercises too.

**Happy Reading !!! **

1. What are important topics of the chapter Chapter 8 Binomial Theorem ?

The basic concept of the binomial theorem, binomial theorem for positive integral indices, and general and middle terms are covered in this chapter. students can go through the NCERT syllabus, all the important topics are mentioned there. practicing these topics covered in binomial theorem ncert solutions is crucial for commanding the concepts.

2. How does the NCERT solutions are helpful ?

NCERT solutions are helpful to the students while solving the NCERT problems. After solving NCERT book problems students can acquire command in concepts that will help greatly during premier exams. If they are stuck while solving, they can take help with binomial theorem class 11 NCERT solutions provided in a detailed manner. for ease students can study class 11 maths chapter 8 solutions pdf both online and offline mode.

3. Explain the concept of the Binomial Theorem covered in NCERT solutions for class 11th maths chapter 8.

The Binomial Theorem refers to the method of expanding the power of binomials that involve the addition of two or more terms. The coefficients of the terms in the expansion are known as binomial coefficients. This chapter provides essential definitions that are relevant for examinations. With the NCERT Solutions available in PDF format, students can stay up-to-date with the latest CBSE Board syllabus.

4. Where can I find the complete solutions of NCERT for class 11 maths ?

Here you will get the detailed NCERT solutions for class 11 maths by clicking on the link. if anyone facing problems to find complete solutions of NCERT Book can web through official website of Careers360 or these are listed above in the article according to topic and subject wise.

5. In the Binomial Theorem, explain the properties of positive integers covered in the class 11 maths NCERT solutions chapter 8.

The NCERT Solutions for Class 11 Maths Chapter 8 provide over 10 properties related to positive integers that students can learn. These properties are crucial in comprehending the efficient solution of equations. During the annual examination, the question paper may focus on simple chapters that are challenging to solve. Hence, it is recommended that students go through these NCERT Solutions to secure good marks in the examination.

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