About Binomial Theorem Class 11 Chapter 8
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 Class 11
1. What is the Binomial Theorem as per the Class 11 syllabus?
The Binomial Theorem provides a formula for expanding any binomial expression raised to a positive integer power. For any positive integer 'n', the expansion of (a + b)n is given by the formula: (a + b)n = nC0an + nC1an-1b + nC2an-2b2 + ... + nCnbn. It is a fundamental tool in algebra for simplifying complex polynomial expansions.
2. What is the importance of the terms 'n' and 'r' in the general term Tr+1 of a binomial expansion?
In the general term formula, Tr+1 = nCr an-rbr, 'n' represents the total power of the binomial expansion, defining its length. The index 'r' is crucial as it specifies the position of the term. For example, to find the 5th term, we set r=4. This formula allows us to calculate any specific term without writing out the entire expansion.
3. Is the Binomial Theorem topic deleted from the CBSE Class 11 Maths syllabus for 2025-26?
No, according to the official CBSE syllabus for the 2025-26 academic session, the Binomial Theorem chapter is included in the Class 11 Maths curriculum. It covers the statement and proof for positive integral indices, Pascal's triangle, and finding the general and middle terms in the expansion.
4. How is Pascal's Triangle related to the Binomial Theorem?
Pascal's Triangle offers a visual and intuitive method to determine the binomial coefficients (nCr) for an expansion. Each row in the triangle corresponds to the coefficients of (a + b)n. For example, the row '1 4 6 4 1' provides the coefficients for the expansion of (a + b)4. It demonstrates the symmetric property of coefficients where nCr = nCn-r.
5. What are the key formulas to remember in the Binomial Theorem chapter?
For the Class 11 syllabus, the most important formulas are:
- The Expansion Formula: (a + b)n = Σ nCr an-rbr, where r ranges from 0 to n.
- The General Term: Tr+1 = nCr an-rbr.
- Middle Term for even 'n': The middle term is the ((n/2) + 1)th term.
- Middle Terms for odd 'n': There are two middle terms: the ((n+1)/2)th and the (((n+1)/2) + 1)th terms.
6. What is the main difference between the 'general term' and a 'middle term' in an expansion?
The general term (Tr+1) is a formula that acts as a template to find *any* term in the expansion by substituting the appropriate value for 'r'. In contrast, the middle term is a specific term located at the centre of the expansion. Its position is fixed and depends entirely on whether the power 'n' is even or odd. You use the general term formula to calculate the value of the middle term once its position is determined.
7. How is the Binomial Theorem applied in other subjects like Physics or Economics?
The Binomial Theorem is widely used for approximations. In Physics, for calculations involving gravity or relativity where a variable is very small, the binomial approximation (1+x)n ≈ 1+nx simplifies complex equations. In Economics, it is used to derive formulas for compound interest over multiple periods, showing how principal amounts grow exponentially.
8. Is Binomial Theorem an important topic for competitive exams like JEE?
Yes, the Binomial Theorem is a very important and high-weightage topic for JEE Main and Advanced. Questions frequently test concepts like finding the term independent of x, the greatest coefficient, the sum of coefficients, and properties of binomial coefficients. A deep understanding beyond simple expansion is crucial for these exams.
9. What kind of real-world problems can be solved using the Binomial Theorem?
Beyond academic applications, the theorem is foundational in fields that rely on probability and distribution. For instance, it can be used in:
- Finance: Calculating outcomes in financial markets and option pricing models.
- Genetics: Predicting the probability of inheriting specific genetic traits.
- Computing: In computer science for error detection and correction codes.
- National Economic Planning: Forecasting economic growth and resource distribution based on probabilistic models.
