Commerce Exam  >  Commerce Notes  >  Mathematics (Maths) Class 11  >  NCERT Solutions: Exercise Miscellaneous - Binomial Theorem

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

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

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

ANSWER : - In order to prove that (ab) is a factor of (anbn), it has to be proved that

anbn = k (ab), where k is some natural number

It can be written that, a = ab +  b

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

This shows that (ab) is a factor of (anbn), where n is a positive integer.

Question 2: Evaluate NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem.

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

This can be done as

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem  

Question 3: Find the value of NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem.

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

This can be done as

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem  

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

ANSWER : - 0.99 = 1 – 0.01

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

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 NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

ANSWER : - In the expansion, NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem ,

Fifth term from the beginning NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

Fifth term from the end NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

Therefore, it is evident that in the expansion of NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem, the fifth term from the beginning is NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem and the fifth term from the end is NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem.

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

It is given that the ratio of the fifth term from the beginning to the fifth term from the end is NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem. Therefore, from (1) and (2), we obtain

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

Thus, the value of n is 10.

Question 6: Expand using Binomial Theorem NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem.

 ANSWER : - Using Binomial Theorem, the given expression NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem  can be expanded as

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

Again by using Binomial Theorem, we obtain

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

From (1), (2), and (3), we obtain

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

 

Question 7: Find the expansion of NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem using binomial theorem.

 ANSWER : - Using Binomial Theorem, the given expression  NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem can be expanded as

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

Again by using Binomial Theorem, we obtain

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

From (1) and (2), we obtain

NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

The document NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem is a part of the Commerce Course Mathematics (Maths) Class 11.
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