NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

Exercise 7.1

Q1: Expand the expression (1– 2x)⁵
Ans: By using Binomial Theorem, the expression (1– 2x)⁵ can be expanded as

Q2:  Expand the expression 
Ans: By using Binomial Theorem, the expression   can be expanded as

Q3: Expand the expression (2x – 3)⁶
Ans: By using Binomial Theorem, the expression (2x – 3)⁶ can be expanded as

Q4: Expand the expression 
Ans: By using Binomial Theorem, the expression   can be expanded as

Q5: Expand 
Ans: By using Binomial Theorem, the expression   can be expanded as

 

Q6: Using Binomial Theorem, evaluate (96)³
Ans: 96 can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, binomial theorem can be applied.
It can be written that, 96 = 100 – 4

Q7: Using Binomial Theorem, evaluate (102)⁵
Ans: 102 can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied.
It can be written that, 102 = 100+ 2

Q8: Using Binomial Theorem, evaluate (101)⁴
Ans: 101 can be expressed as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied.
It can be written that, 101 = 100+ 1

Q9: Using Binomial Theorem, evaluate (99)⁵
Ans: 99 can be written as the sum or difference of two numbers whose powers are easier to calculate and then, Binomial Theorem can be applied.
It can be written that, 99 = 100 – 1

Q10: Using Binomial Theorem, indicate which number is larger (1.1)¹⁰⁰⁰⁰ or 1000.
Ans: By splitting 1.1 and then applying Binomial Theorem, the first few terms of (1.1)¹⁰⁰⁰⁰ can be obtained as

Q11: Find (a  + b)⁴ – (a – b)⁴. Hence, evaluate
Ans: Using Binomial Theorem, the expressions, (a +  b)⁴ and (a – b)⁴, can be expanded as

Q12: Find (x + 1)⁶ + (x – 1)⁶. Hence or otherwise evaluate .
Ans: Using Binomial Theorem, the expressions, (x + 1)⁶ and (x – 1)⁶, can be expanded as

By putting , we obtain

Q13: Show that  is divisible by 64, whenever n is a positive integer.
Ans: In order to show that  is divisible by 64, it has to be proved that,
, where k is some natural number
By Binomial Theorem,

For a = 8 and m = n  1, we obtain

Thus,  is divisible by 64, whenever n is a positive integer.

Q14: Prove that .
Ans: By Binomial Theorem,

By putting b = 3 and a = 1 in the above equation, we obtain

Hence, proved.

Exercise Miscellaneous

Question 1: If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer.

[Hint: write aⁿ = (a – b  + b)ⁿ and expand]

ANSWER : - In order to prove that (a – b) is a factor of (aⁿ – bⁿ), it has to be proved that

aⁿ – bⁿ = k (a – b), where k is some natural number

It can be written that, a = a – b +  b

This shows that (a – b) is a factor of (aⁿ – bⁿ), where n is a positive integer.

Question 2: Evaluate .

ANSWER : - Firstly, the expression (a  + b)⁶ – (a – b)⁶ is simplified by using Binomial Theorem.

This can be done as

Question 3: Find the value of .

ANSWER : - Firstly, the expression (x  + y)⁴  (x – y)⁴ is simplified by using Binomial Theorem.

This can be done as

 Question 4: Find an approximation of (0.99)⁵ using the first three terms of its expansion.

ANSWER : - 0.99 = 1 – 0.01

Thus, the value of (0.99)⁵ is approximately 0.951.

Question 5: Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of 

ANSWER : - In the expansion,  ,

Fifth term from the beginning

Fifth term from the end

Therefore, it is evident that in the expansion of , the fifth term from the beginning is  and the fifth term from the end is .

It is given that the ratio of the fifth term from the beginning to the fifth term from the end is . Therefore, from (1) and (2), we obtain

Thus, the value of n is 10.

Question 6: Expand using Binomial Theorem .

 ANSWER : - Using Binomial Theorem, the given expression   can be expanded as

Again by using Binomial Theorem, we obtain

From (1), (2), and (3), we obtain

Question 7: Find the expansion of  using binomial theorem.

 ANSWER : - Using Binomial Theorem, the given expression   can be expanded as

Again by using Binomial Theorem, we obtain

From (1) and (2), we obtain