# NCERT Solutions | Class 11 Maths Chapter 8 | Binomial Theorem

## CBSE Solutions | Maths Class 11

Check the below NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Pdf free download. NCERT Solutions Class 11 Maths  were prepared based on the latest exam pattern. We have Provided Binomial Theorem Class 11 Maths NCERT Solutions to help students understand the concept very well.

### NCERT | Class 11 Maths

Book: National Council of Educational Research and Training (NCERT) Central Board of Secondary Education (CBSE) 11th Maths 8 Binomial Theorem English

#### Binomial Theorem | Class 11 Maths | NCERT Books Solutions

You can refer to MCQ Questions for Class 11 Maths Chapter 8 Binomial Theorem to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.

## NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Ex 8.1

Expand each of the expressions in Exercises 1 to 5.

Ex 8.1 Class 11 Maths Question 1.
$${ \left( 1-2x \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 2.
$${ \left( \frac { 2 }{ x } -\frac { x }{ 2 } \right) }^{ 5 }$$

Solution.

Ex 8.1 Class 11 Maths Question 3.
$${ \left( 2x-3 \right) }^{ 6 }$$
Solution.

Ex 8.1 Class 11 Maths Question 4.
$${ \left( \frac { x }{ 3 } +\frac { 1 }{ x } \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 5.
$${ \left( x+\frac { 1 }{ x } \right) }^{ 6 }$$
Solution.

Using binomial theorem, evaluate each of the following

Ex 8.1 Class 11 Maths Question 6.
$${ \left( 96 \right) }^{ 3 }$$
Solution.

Ex 8.1 Class 11 Maths Question 7.
$${ \left( 102 \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 8.
$${ \left( 101 \right) }^{ 4 }$$
Solution.

Ex 8.1 Class 11 Maths Question 9.
$${ \left( 99 \right) }^{ 5 }$$
Solution.

Ex 8.1 Class 11 Maths Question 10.
Using Binomial Theorem, indicate which number is larger$${ \left( 1.1\right) }^{ 10000 }$$ or 1000.
Solution.
Splitting 1.1 and using binomial theorem to write the first few terms we have

Ex 8.1 Class 11 Maths Question 11.
Find $${ \left( a+b \right) }^{ 4 }-{ \left( a-b \right) }^{ 4 }$$. Hence, evaluate $${ \left( \sqrt { 3 } +\sqrt { 2 } \right) }^{ 4 }-{ \left( \sqrt { 3 } -\sqrt { 2 } \right) }^{ 4 }$$.
Solution.
By binomial theorem, we have

Ex 8.1 Class 11 Maths Question 12.
Find $${ \left( x+1 \right) }^{ 6 }+{ \left( x-1 \right) }^{ 6 }$$. Hence or otherwise evaluate $${ \left( \sqrt { 2 } +1 \right) }^{ 6 }+{ \left( \sqrt { 2 } -1 \right) }^{ 6 }$$.
Solution.
By using binomial theorem, we have

Ex 8.1 Class 11 Maths Question 13.
Show that $${ 9 }^{ n+1 }-8n-9$$ is divisible by 64, whenever n is a positive integer.
Solution.
We have to prove that $${ 9 }^{ n+1 }-8n-9=64k$$

Ex 8.1 Class 11 Maths Question 14.
Prove that $$\sum _{ r=0 }^{ n }{ { 3 }^{ r } }$$ 8Cr = 4n
Solution.
We have,

## NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Ex 8.2

Expand binomials using the binomial expansion method step-by-step.

Ex 8.2 Class 11 Maths Question 1.
Find the coefficient of x5 in (x + 3)8
Solution.
Suppose x5 occurs in the (r + 1)th term of the expansion (x + 3)8

Ex 8.2 Class 11 Maths Question 2.
a5 b7in (a-2b)12
Solution.
Suppose a5 b7 occurs in the (r + 1)th term of the expansion (a – 2b)12.

Write the general term in the expansion of

Ex 8.2 Class 11 Maths Question 3.
(x2 – y)6
Solution.

Ex 8.2 Class 11 Maths Question 4.
(x2 – yx)12, x ≠ 0
Solution.
We have given, (x2 – yx)12 = (x2 + (- yx))12, x ≠ 0

Ex 8.2 Class 11 Maths Question 5.
Find the 4th term in the expansion of (x – 2y)12.
Solution.

Ex 8.2 Class 11 Maths Question 6.
Find the 13th term in the expansion of $${ \left( 9x-\frac { 1 }{ 3\sqrt { x } } \right) }^{ 18 }$$, x ≠ 0
Solution.

Find the middle terms in the expansions of

Ex 8.2 Class 11 Maths Question 7.
$${ \left( 3-\frac { { x }^{ 3 } }{ 6 } \right) }^{ 7 }$$
Solution.
As the exponent 7 is odd, so there will be two middle terms in the expansion

Ex 8.2 Class 11 Maths Question 8.
$${ \left( \frac { x }{ 3 } +9y \right) }^{ 10 }$$
Solution.

Ex 8.2 Class 11 Maths Question 9.
In the expansion of (1 + a)m+n, prove that coefficients of am and an are equal.
Solution.

Ex 8.2 Class 11 Maths Question 10.
The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of (x + 1 )n are in the ratio 1: 3: 5. Find n and r.
Solution.

Ex 8.2 Class 11 Maths Question 11.
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n-1.
Solution.

Ex 8.2 Class 11 Maths Question 12.
Find a positive value of m for which the coefficient of x2 in the expansion (1 + x)m is 6.
Solution.