NCERT Solutions: Binomial Theorem

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)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)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 – (ab)4. Hence, evaluate
Ans: Using Binomial Theorem, the expressions, (a +  b)4 and (ab)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 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

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

Question 2: Evaluate .

ANSWER : - Firstly, the expression (a  + b)6 – (ab)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  (xy)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

The document NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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Mathematics (Maths) for JEE Main & Advanced

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FAQs on NCERT Solutions Class 11 Maths Chapter 7 - Binomial Theorem

 1. What is the Binomial Theorem?
Ans. The Binomial Theorem is a mathematical formula that provides a way to expand expressions of the form (a + b)^n, where 'n' is a positive integer.
 2. How is the Binomial Theorem used in mathematics?
Ans. The Binomial Theorem is used to simplify and expand expressions involving binomials, making it easier to calculate powers of binomials and coefficients in algebraic expressions.
 3. Can the Binomial Theorem be applied to negative exponents?
Ans. Yes, the Binomial Theorem can be applied to negative exponents by using the formula (a + b)^-n = 1/(a + b)^n for any positive integer 'n'.
 4. What are some real-life applications of the Binomial Theorem?
Ans. The Binomial Theorem is used in fields such as probability theory, statistics, finance, and engineering to model and solve various real-world problems involving combinations and permutations.
 5. How can one remember and apply the Binomial Theorem effectively in exams?
Ans. To remember and apply the Binomial Theorem effectively in exams, it is recommended to practice a variety of problems, understand the pattern of coefficients in Pascal's Triangle, and memorize key formulas and concepts related to binomial expansions.

Mathematics (Maths) for JEE Main & Advanced

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