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CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem

Q: What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand the powers of a binomial expression.

Q: What is Pascal's triangle?

Pascal's triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. It helps determine binomial coefficients.

Q: How do you prove the binomial theorem?

The binomial theorem can be proved using mathematical induction or combinatorial arguments. One common approach is to use Pascal's triangle to demonstrate the patterns in binomial coefficients and then generalize them to derive the theorem.

Q: What is the significance of the middle term in a binomial expansion?

The middle term in a binomial expansion (also known as the central term) holds special significance because it represents the term with the highest coefficient, which occurs when the binomial coefficients are symmetrically distributed.

Here, we have provided CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem. Students can view these CBSE Class 11 Maths Notes Chapter 8 before exams for better understanding of the chapter.

Ananya Gupta7 May, 2024

CBSE Class 11 Maths Notes Chapter 8: Chapter 8 of CBSE Class 11 Maths focuses on the Binomial Theorem. In this chapter we will learn how to expand binomials raised to positive integer powers.

It introduces binomial coefficients and Pascal's triangle. Understanding this theorem helps simplify complex algebraic expressions and solve problems in areas like probability, algebra, and calculus. The notes for this chapter provide a clear explanation of binomial expansion, making it easier for students to understand and apply these concepts in their studies and daily life.

CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem PDF

You can find the study notes for Chapter 8 on the Binomial Theorem in CBSE Class 11 Maths through the PDF link provided below. These notes help you understand the topics covered in this chapter better. They explain things like binomial expansion and coefficients in a straightforward way. Whether you are preparing for exams or just want to improve your understanding of the binomial theorem, these notes can be really helpful.

CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem PDF

Binomial Theorem Class 11 Topics

In the study of binomial theorem in Class 11, several key topics and sub-topics are covered to provide a comprehensive understanding. These include:

Introduction: An overview of the binomial theorem and its significance in expanding binomial expressions.

Binomial theorem for positive integral indices: How to expand binomial expressions raised to positive integer powers using Pascal's triangle and combinatorial methods.

Binomial theorem for any positive integer n: Understanding the expansion of binomials for any positive integer exponent, including the formula and its application.

Special Cases: Investigating special cases such as when the exponent is a negative integer or a fractional number, and understanding the implications.

General and Middle Term: Learning about the general term in the expansion of a binomial expression and finding the middle term in even and odd expansions.

CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem

The solutions to Chapter 8 on the Binomial Theorem in CBSE Class 11 Maths are available below. These solutions are a valuable resource for students to enhance their understanding of various concepts covered in this chapter. From binomial expansion to understanding the coefficients, these solutions provide clear explanations and step-by-step methods to solve problems.

Binomial Theorem

The binomial theorem is a fundamental concept in algebra that provides a way to expand expressions of the form (a + b)^n, where "a" and "b" are numbers or variables, and "n" is a positive integer or a rational number. The theorem states that such an expression can be expanded into a sum of terms, where each term is of the form

C (n, k) \cdot a^{n - k} \cdot b^{k}

, where

C (n, k)

represents the binomial coefficient, also known as "n choose k," which is the number of ways to choose k elements from a set of n elements. The binomial theorem has numerous applications in various branches of mathematics, including combinatorics, calculus, and probability theory. It is used extensively in solving problems involving permutations, combinations, and probability distributions. Moreover, it forms the basis for many other mathematical concepts and techniques, making it an essential topic for study in mathematics.

General and Middle terms

In the binomial expansion

(a + b)^{n} = (0 n) a^{n} + (1 n) a^{n - 1} b + (2 n) a^{n - 2} b^{2} + \dots + (n - 1 n) a b^{n - 1} + (n n) b^{n}

, Each term can be expressed as follows: First term =

(0 n) a^{n}

Second term =

(1 n) a^{n - 1} b

Third term =

(2 n) a^{n - 2} b^{2}

Similarly, the

(r + 1)

th term can be represented as:

T_{r + 1} = (r n) a^{n - r} b^{r}

This term is known as the middle term of the expansion

(a + b)^{n}

Binomial Theorem for Positive Integral Indices

(a+b) 0 =1

, where

a+b \neq 0

(a+b) 1 = a+b

(a+b) 2 = a 2 +2ab+ b 2

(a+b) 3 = a 3 +3 a 2 b+3a b 2 + b 3

(a+b) 4 = (a+b) 3 . (a+b) = a 4 +4 a 3 b+6 a 2 b 2 +4a b 3 + b 4

Binomial Theorem for any Positive Integer 𝑛

: (a+b)n=nC0anb0+nC1an-1b1+nC2an-2b2+.....+nCna0bn(a+b)n=nC0anb0+nC1an-1b1+nC2an-2b2+.....+nCna0bn That is, (a+b)n=nC0an+nC1an-1b1+nC2an-2b2+.....+nCnbn(a+b)n=nC0an+nC1an-1b1+nC2an-2b2+.....+nCnbn Remarks: Binomial theorem can also be written as, (a+b)n=\sumk=0nnCkan-kbk(a+b)n=\sumk=0nnCkan-kbk Where, \sumk=0nnCkan-kbk\sumk=0nnCkan-kbk represents nC0anb0+nC1an-1b1+nC2an-2b2+.....+nCna0bnnC0anb0+nC1an-1b1+nC2an-2b2+.....+nCna0bn The coefficients nCrnCr are known as binomial coefficients.

Binomial Theorem for any Index

If n is negative integer then n! cannot be defined. We state binomial theorem in another form. (a+b)n=an+n1!an-1b1+n(n−1)2!an-2b2+.....+n(n−1)(n−r+1)r!an-rbr+... Here, tr+1 = n(n−1)(n−r+1)r!an-rbr Theorem : If n is any real number, a = 1,b = x and |x|<1 then (1+x)n= 1+ nx + n(n−1)2!x2+ n(n−1)(n−2)3!x3+... Here there are infinite number of terms in the expansion, the general term is given by tr+1 = n(n−1)(n−2)(n−r+1)r!, r≥0

Note:

Expansion is valid only when -1 x 1 nCr can not be used because it is defined only for natural number, so nCr can be written as n(n−1)(n−2)(n−r+1)r. As the series never terminates, the number of terms in the series is infinite. General term of the series (1+x)−n=Tr+1→(−1)r 1+x1-x if |x|<1 General term of the series (1+x)−n→Tr+1 = ( -1)( +2)...( + -1)r!x If first term is not 1 , then make it unity in the following way. (a+x)n=an(1+xa)n if ∣∣xa∣∣<1

Remarks:

If |x|<1 and n is any real number, then (1−x)n= 1- nx + n(n−1)2!x2− n(n−1)(n−2)3!x3+... The general term is given by tr+1 = (−1)rn(n−1)(n−2)(n−r+1)r!xr If n is any real number and |b|<|a| , then (a+b)n=[a(1+ba)]n (a+b)n=an(1+ba)n

Note:

While expanding (a+b)n where n a negative integer or a fraction is, reduce the binomial to the form in which the first term is unity and the second term is numerically less than unity. Particular expansion of the binomials for negative index, |x|<1 11+x = (1+x)-1 = 1-x+x2-x3+x4-x5+.... 11-x = (1-x)-1 = 1+x+x2+x3+x4+x5+.... 1(1+x)2 = (1+x)-2 = 1-2x+3x2-4x3+.... 1(1-x)2 = (1-x)-2 = 1+2x+3x2+4x3+....

Binomial Theorem Class 11 Examples

Example 1: Expand: [x ² + (3/x)] ⁴ , x ≠ 0 Solution: [x ² + (3/x)] ⁴ Using binomial theorem, [x ² + (3/x)] ⁴ = ⁴ C ₀ (x ² ) ⁴ + ⁴ C ₁ (x ² ) ³ (3/x) + ⁴ C ₂ (x ² ) ² (3/x) ² + ⁴ C ₃ (x ² ) (3/x) ³ + ⁴ C ₄ (3/x) ⁴ = x ⁸ + 4 x ⁶ (3/x) + 6 x ⁴ (9/x ² ) + 4 x ² (27/x ³ ) + (81/x ⁴ ) = x ⁸ + 12x ⁵ + 54x ² + (108/x) + (81/x ⁴ ) Example 2: Compute (98) ⁵ Solution: Let us write the number 98 as the difference between the two numbers. 98 = 100 – 2 So, (98) ⁵ = (100 – 2) ⁵ Using binomial expansion, (98) ⁵ = ⁵ C ₀ (100) ⁵ – ⁵ C ₁ (100) ⁴ (2) + ⁵ C ₂ (100) ³ (2) ² – ⁵ C ₃ (100) ² (2) ³ + ⁵ C ₄ (100) (2) ⁴ – ⁵ C ₅ (2) ⁵ = 10000000000 – 5 × 100000000 × 2 + 10 × 1000000 × 4 – 10 ×10000 × 8 + 5 × 100 × 16 – 32 = 10040008000 – 1000800032 = 9039207968 Example 3: Find the coefficient of x ⁶ y ³ in the expansion (x + 2y) ⁹ . Solution: Let x ⁶ y ³ be the (r + 1)th term of the expansion (x + 2y) ⁹ . So, T _r+1 = ⁹ C _r x ^9-r (2y) ^r x ⁶ y ³ = ⁹ C _r x ^9-r 2 ^r y ^r By comparing the indices of x and y, we get r = 3. Coefficient of x ⁶ y ³ = ⁹ C ₃ (2) ³ = 84 × 8 = 672 Therefore, the coefficient of x ⁶ y ³ in the expansion (x + 2y) ⁹ is 672. Example 4: The second, third and fourth terms in the binomial expansion (x + a) ⁿ are 240, 720 and 1080, respectively. Find x, a and n. Solution: Given, Second term = T ₂ = 240 Third term = T ₃ = 720 Fourth term = T ₄ = 1080 Now, T ₂ = T ₁₊₁ = ⁿ C ₁ x ^n-1 (a) ⁿ C ₁ x ^n-1 a = 240….(i) Similarly, ⁿ C ₂ x ^n-2 a ² = 720….(ii) ⁿ C ₃ x ^n-3 a ³ = 1080….(iii) Dividing (ii) by (i), [ ⁿ C ₂ x ^n-2 a ² ]/ [ ⁿ C ₁ x ^n-1 a] = 720/240 [(n – 1)!/(n – 2)!].(a/x) = 6 (n – 1) (a/x) = 6 a/x = 6/(n – 1)….(iv) Similarly, by dividing (iii) by (ii), a/x = 9/[2(n – 2)]….(v) From (iv) and (v), 6/(n – 1) = 9/[2(n – 2)] 12(n – 2) = 9(n – 1) 12n – 24 = 9n – 9 12n – 9n = 24 – 9 3n = 15 n = 5 Subsituting n = 5 in (i), ⁵ C ₁ x ⁴ a = 240 ax ⁴ = 240/5 ax ⁴ = 48….(vi) Substituting n = 5 in (iv), a/x = 6/(5 – 1) a/x = 6/4 = 3/2 a = (3x/2) Putting this oin equ (vi), we get; (3x/2) x ⁴ = 48 x ⁵ = 32 x ⁵ = 25 ⇒ x = 2 Substituting x = 2 in a = (3x/2) a = 3(2)/2 = 3 Therefore, x = 2, a = 3 and n = 5.

Benefits of CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem

CBSE Class 11 Maths Notes for Chapter 8 on Binomial Theorem provide detailed coverage of the topic.
They provide explanations of concepts in simple language, making it easier for students to understand.
The notes include examples and illustrations to clarify key points and demonstrate problem-solving techniques.
Students can access the notes online, making them convenient for studying anytime and anywhere.
These notes provide practice questions and exercises to help students reinforce their learning.
They are a valuable resource for revision before exams, helping students to consolidate their understanding and prepare effectively.
The CBSE Class 11 Maths Notes for Binomial Theorem are beneficial for students in mastering the subject and achieving academic success.

CBSE Class 11 Notes for Maths
Chapter 1 Sets Notes	Chapter 2 Relations and Functions Notes
Chapter 3 Trigonometric Functions Notes	Chapter 4 Principle of Mathematical Induction Notes
Chapter 5 Complex Numbers and Quadratic Equations Notes	Chapter 6 Linear Inequalities Notes
Chapter 7 Permutations and Combinations Notes	Chapter 8 Binomial Theorem Notes
Chapter 9 Sequences and Series Notes	Chapter 10 Straight Lines Notes
Chapter 11 Conic Sections Notes	Chapter 12 Introduction to Three Dimensional Geometry Notes
Chapter 13 Limits and Derivatives Notes	Chapter 14 Mathematical Reasoning Notes
Chapter 15 Statistics Notes	Chapter 16 Probability Notes

CBSE Class 11 Maths Notes Chapter 8 FAQs

What is the binomial theorem?

The binomial theorem is a mathematical formula used to expand the powers of a binomial expression.

What is Pascal's triangle?

Pascal's triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. It helps determine binomial coefficients.

How do you prove the binomial theorem?

The binomial theorem can be proved using mathematical induction or combinatorial arguments. One common approach is to use Pascal's triangle to demonstrate the patterns in binomial coefficients and then generalize them to derive the theorem.

What is the significance of the middle term in a binomial expansion?

The middle term in a binomial expansion (also known as the central term) holds special significance because it represents the term with the highest coefficient, which occurs when the binomial coefficients are symmetrically distributed.

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