NCERT Solutions: Exercise Miscellaneous - Binomial Theorem
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
