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.
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
209 videos443 docs143 tests

1. What is the Binomial Theorem? 
2. How is the Binomial Theorem used in mathematics? 
3. Can the Binomial Theorem be applied to negative exponents? 
4. What are some reallife applications of the Binomial Theorem? 
5. How can one remember and apply the Binomial Theorem effectively in exams? 
209 videos443 docs143 tests


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