Binomial Theorem Class 11 formula proof and solved examples
The NCERT Solutions for Binomial Theorem Class 11 Chapter 8 can be downloaded at Vedantu without any hassle. Practicing these Binomial Theorem Class 11 solutions can help the students clear their doubts as well as solve the problems faster. Students can learn new tricks to answer a particular question in different ways giving them an edge with exam preparation.
The concepts of Binomial Theorem Class 11 covered in Chapter 8 of the Maths textbook include the study of essential topics, such as Positive Integral Indices, Pascal’s Triangle, Binomial theorem for any positive integer, and some special cases. Students can score high marks in the exams with ease by practicing the Binomial Theorem Class 11 solutions for all the questions present in the textbook. It gets pretty apparent to know the logic set behind each answer and develop a far better comprehension of the concepts.
NCERT Solutions for Binomial Theorem Class 11 Maths Chapter 8
All the Binomial Theorem Class 11 solutions Maths Chapter 8 is given here. This solution also contains questions, answers, images, and explanations for the whole chapter 8 titled with the Binomial Theorem taught in Class 11. If you are a student of Class 11th who is referring to the NCERT book to study Mathematics, then you will come across chapter 8 Binomial Theorem after you have completed studying the lesson, you also must be looking for the answers to the questions. You will get here all the Binomial Theorem Class 11 solutions for Mathematics Chapter 8 all in a single place.
Definition of Binomial Theorem
Sometimes, when the power increases, the expansion becomes lengthy and tedious to calculate. A binomial expression that has been raised to a really large power is often easily calculated with the assistance of the theorem.
Binomial Expansion
Important points to remember
The total number of the terms in the expansion for (x+y)n is (n+1)
The sum of the exponents of x and y is always n.
nC0, nC1, nC2, … .., nCn are called binomial coefficients and also are represented by C0, C1, C2, ….., Cn
The binomial coefficients which are equidistant from the beginning and from the ending are also equal i.e. nC0 = nCn, nC1 = nCn-1 , nC2 = nCn-2 ,….. etc.
Terms in the Binomial Expansion
In the binomial expansion, it is normally asked to find the middle term or the general term. Different terms in the binomial expansion that are covered are included here:
General Term
Middle Term
Independent Term
Determining a Particular Term
Numerically greatest term
The ratio of Consecutive Terms/Coefficients
What is a Binomial Expression?
A binomial is also a mathematical expression that includes two terms. Further, these two terms must be separated from each other by either addition or subtraction. To add the binomials, one should combine equal terms to get an answer. To multiply the binomials, the distributive property should be used.
Topics and SubTopics of Binomial Theorem
Binomial Theorem Formula
The Binomial Theorem is a quick method for expanding or multiplying out of a binomial expression. The expression also has been raised to some bigger power. As we know the multiplication of such expressions is always difficult with large powers. But the Binomial expansions and their formulas help us a lot in this regard. In this article, we will discuss the Binomial theorem and its Formula.
( a + b )n = k =0n(kn) ak bn-k
The upper index n is known as the exponent for the expansion; the lower index k points out which term, starting with k equals 0. For example, when n equals 5, each of the terms in the expansion for (a + b)5 will look like: a5 − kbk.
Properties of the Binomial Expansion (x + y)n
There are a total of n+1 terms.
The first term is known to be (x)n
Progressing from the first term to the last term, the exponent for x decreases by 1 from term to term. Whereas the exponent of y increases by 1 term. Also, the sum for both the exponents in each term will be n.
If the coefficient of each of the terms is multiplied by the exponent of x in that term, and the product is divided by the number of that term, we can easily get the coefficient for the next term.
Solved Questions
Expand the following terms using binomial theorem.
(1-2x)5
Using the binomial theorem, we will expand the above given expression as: (1-2x)5 = 5C0(1)5 - 5C1(1)4 (2x) + 5C2(1)3 (2x)2 - 5C3(1)2 (2x)3 + 5C4(1)1 (2x)4 - 5C5 (2x)5
= 1 - 5(2x) + 10 (4x)2 -10 (8x3) +5 (16x4) - (32 x5)
= 1 - 10x + 40x2 - 80x3 + 80 x4 -32 x5
This is the final expanded form of (1-2x)5, found using binomial theorem.
(2x-3)6
Using binomial theorem, the following term can be expanded as follows:
(2x-3)6 = 6C0 (2x)6 - 6C1 (2x)5(3)1 + 6C2 (2x)4(3)2 - 6C3 (2x)3(3)3 + 6C4 (2x)2(3)4 - 6C5 (2x)1(3)5 + 6C6(3)6
= 64 x6 - 6(32 x5) (3) + 15 (16 x4)(9) - 20(8 x3) (27) + 15(4x2)(81) - 6(2x)(243) + 729
= 64 x6 - 576 x5 + 2160 x4 - 4320 x3 + 4860 x2 - 2916 x + 729.
This is the final expanded form of (2x-3)6, found using the binomial theorem.
Solve (101)4 using the binomial theorem
Any number like 101 can be expressed as the sum or difference of two numbers whose powers are easy to calculate. Using Binomial Theorem the value can be evaluated as :
101 can be written as 100+1
So 1014 = (100 +1)4
(100 +1)4 = 4C0(100)4 + 4C1(100)3 (1) + 4C2(100)2 (1)2 + 4C3(100)1 (1)3 + 4C4 (1)4
= (100)4+ 4 (100)3 + 6 (100)2 + 4(100) +1
= 104060401
Did You know?
Generally, Sir Issac Newton is credited for the work of the binomial theorem.
The binomial theorem is highly used in statistical and probability analysis.
Study Tips
Math is often considered difficult to study as compared to other subjects.
However, it should not be under-determined that after consecutive efforts and solving enough practice problems, one can score good marks in maths.
Binomial Theorem is another topic in mathematics for which minimal memorization is required
First, understand the concept and then remember the important expansion formulas.
Once the concept is being understood, then go to solve practice problems as much as possible.
There are many online websites that give solutions and questions for NCERT or any of these State Board exams.
Practicing more questions of Binomial Theorem will give more confidence in solving questions from Binomial topic
This was all about binomial theorem formulas, properties and FAQs. For more such information, access free resources available on the Vedantu website useful for the state board, CBSE, ICSE, and competitive examinations. All NCERT Solutions for all subjects are available on the Vedantu website.
FAQs on Binomial Theorem for Class 11 Maths Explained
1. What is the Binomial Theorem in Class 11 Maths?
The Binomial Theorem states that for any positive integer n, the expansion of (a + b)n is given by a specific formula involving combinations. The formula is:
(a + b)n = Σ C(n, r) an−r br, where r = 0 to n.
Here:
- C(n, r) = n! / [r!(n − r)!]
- n is a non-negative integer
- a and b are any real numbers
2. What is the formula for general term in Binomial Theorem?
The general term in the binomial expansion of (a + b)n is Tr+1 = C(n, r) an−r br. Here:
- r = 0, 1, 2, …, n
- C(n, r) = n! / [r!(n − r)!]
C(5, 3)a2b3 = 10a2b3.
3. How do you expand (x + y)4 using Binomial Theorem?
The expansion of (x + y)4 using the Binomial Theorem is x4 + 4x3y + 6x2y2 + 4xy3 + y4.
Steps:
- Use C(4, r) for r = 0 to 4
- Coefficients are 1, 4, 6, 4, 1
- Powers of x decrease from 4 to 0
- Powers of y increase from 0 to 4
4. What are binomial coefficients?
Binomial coefficients are the numerical factors in the binomial expansion, represented by C(n, r). They are calculated as:
C(n, r) = n! / [r!(n − r)!]
For example, in (a + b)5, the coefficients are 1, 5, 10, 10, 5, 1. These numbers can also be obtained from Pascal’s Triangle.
5. How many terms are there in the expansion of (a + b)n?
The expansion of (a + b)n contains n + 1 terms. For example:
- (a + b)3 has 4 terms
- (a + b)5 has 6 terms
6. How do you find the middle term in Binomial Theorem?
The middle term in the expansion of (a + b)n depends on whether n is even or odd.
- If n is even, there is one middle term: T(n/2)+1
- If n is odd, there are two middle terms: T(n+1)/2 and T(n+3)/2
7. What is Pascal’s Triangle and how is it related to Binomial Theorem?
Pascal’s Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers above it, and it gives the binomial coefficients. For example:
- Row 0: 1
- Row 1: 1 1
- Row 2: 1 2 1
- Row 3: 1 3 3 1
8. How do you find a specific term in a binomial expansion?
To find a specific term in a binomial expansion, use the general term formula Tr+1 = C(n, r)an−rbr and substitute the required value of r. Steps:
- Identify n from (a + b)n
- Determine r using term number = r + 1
- Substitute in C(n, r)an−rbr
9. What is the value of C(n, r) in Binomial Theorem?
The value of C(n, r) is given by the formula n! / [r!(n − r)!]. It represents the number of ways to choose r objects from n objects.
Important properties:
- C(n, 0) = 1
- C(n, n) = 1
- C(n, r) = C(n, n − r)
10. What are the important properties of Binomial Theorem for exams?
The important properties of the Binomial Theorem for Class 11 exams include key rules about terms, coefficients, and symmetry.
- Total number of terms = n + 1
- Sum of exponents in each term = n
- Coefficients are symmetric: C(n, r) = C(n, n − r)
- Middle term depends on whether n is even or odd
- Sum of all coefficients = 2n (put a = 1, b = 1)
