Question 1: If a and b are distinct integers, prove that a – b is a factor of a^{n} – b^{n}, whenever n is a positive integer.
[Hint: write a^{n} = (a – b + b)^{n} and expand]
ANSWER :  In order to prove that (a – b) is a factor of (a^{n} – b^{n}), it has to be proved that
a^{n} – b^{n} = 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 (a^{n} – b^{n}), 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
75 videos238 docs91 tests


Explore Courses for Commerce exam
