Binomial theorem class 11

Transcript Ex 7.1, 1 Expand the expression (1 – 2x)5 (1 – 2x)5 We know that (a + b)n = nC0 an + nC1 an – 1 b1 + nC2 an – 2 b2 + ….…. + nCn – 1 a1 bn – 1 + nCn bn Hence (a + b)5 = = 5!/0!( 5 − 0)! a5 + 5!/1!( 5 − 1)! a4 b1 + 5!/2!( 5 − 2)! a3 b2 + 5!/3!( 5 − 3)! a2b3 + 5!/4!( 5 − 4)! a b4 + 5!/5!( 5 −5)! b5 = 5!/(0! × 5!) a5 + 5!/(1! × 4!) a4 b + 5!/(2! 3!) a3 b2 + 5!/(3! 2!) a2b3 + 5!/(4! 1!) a b4 + 5!/(5! 0!) b5 = 5!/5! a5 + (5 × 4!)/4! a4 b + (5 × 4 × 3!)/(2! 3!) a3 b2 + (5 × 4 × 3!)/(2 × 1 ×3!) a3b2 + (5 × 4 × 3!)/(3! ×1 ×3!) a2b3 + (5 × 4!)/4! ab4 + 5!/(5! ) b5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 Thus, (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5 Putting a = 1 & b = (– 2x) (1 – 2x)5 = (1)5 + 5(1)4 (–2x) + 10 (1)3 (–2x)3 + 10 (1)2 (–2x)3 + 5 (1) (–2x)4 + (–2x)5 = 1 – 10x + 10(4x2) + 10 (–8x3) + 5 (16x4) + (–32x5) = 1 – 10x + 40x2 – 80x3 + 80x4 – 32x5 Show More

Class 11 Latest NCERT Maths Textbook: Recently, there have been a lot of changes taking place in the syllabus of educational boards in regard to the latest National Education Policy. Due to changes introduced in the new syllabus, old textbooks are not the correct source to study. Because of this reason, NCERT (The National Council of Educational Research and Training) have modified its textbooks for senior and sub-senior classes. As per the list of rationalised content released by NCERT, classes 6 to 12 textbooks have faced changes in the content. Now the textbooks are following 2023-24 syllabus and are perfect to study for this academic year. Along with the topics, NCERT has also dropped exercise questions and examples related to deleted topics. The 2023-24 examination will be based on the latest syllabus thus, referring to new NCERT books will be the right choice. Even if you have old textbooks that you might have borrowed from your seniors, it is better to crosscheck them with the new NCERT books. Here, you will find the free downloadable pdfs for NCERT Class 11 Maths/Ganit textbook 2023-24. Read and download the chapter-wise pdfs. • CBSE Class 11 Syllabus 2023-24 (All Subjects ) PDF • CBSE Class 11 Deleted Syllabus (All Subjects) PDF • NCERT Class 11 Rationalised Content (All Subjects) PDF NCERT Book for Class 11 Maths: In English Chapter No. Chapter Name Chapter PDF 1 Sets Download PDF 2 Relations and Functions Download PDF 3 Trignometric Fucntions Download PDF 4 Comp...

CBSE Class 11 Maths Notes Chapter 8 Binomial Theorem Binomial Expression An expression consisting of two terms, connected by + or – sign is called binomial expression. Binomial Theorem If a and b are real numbers and n is a positive integer, then The general term of (r + 1) th term in the expression is given by T r+1 = nC r a n-r b r Some Important Observations from the Binomial Theorem The total number of terms in the binomial expansion of (a + b) n is n + 1. The sum of the indices of a and b in each term is n. The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficient and nC r = nC n-r, r = 0, 1, 2, 3,…, n The values of the binomial coefficient steadily increase to a maximum and then steadily decrease. The coefficient of x r in the expansion of (1 + x) n is nC r. In the binomial expansion (a + b) n, the r th term from the end is (n – r + 2) th term from the beginning. Middle Term in the Expansion of (a + b) n If n is even, then in the expansion of (a + b) n, the middle term is (\(\frac \))th term.

Binomial Theorem Class 11 Maths NCERT Solutions are extremely helpful while doing your homework. NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem All Exercises were prepared by Experienced LearnCBSE.in Teachers. Free download NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Ex 8.1, Ex 8.2, and Miscellaneous Exercise PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20. • • • • • • • • • • NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem Topics and Sub Topics in Class 11 Maths Chapter 8 Binomial Theorem: Section Name Topic Name 8 Binomial Theorem 8.1 Introduction 8.2 Binomial Theorem for Positive Integral Indices 8.3 General and Middle Terms NCERT Solutions for Class 11 Maths Chapter 8 Exercise.8.1 Ex 8.1 Class 6 Maths Question-1 Ans. Ex 8.1 Class 6 Maths Question-2 Ans. More Resources for CBSE Class 11 • • • • • • • • • • • • • Ex 8.1 Class 6 Maths Question-3 Ans. Ex 8.1 Class 6 Maths Question-4 Ans. Ex 8.1 Class 6 Maths Question-5 Ans. Ex 8.1 Class 6 Maths Question-6 Ans. Ex 8.1 Class 6 Maths Question-7 Ans. Ex 8.1 Class 6 Maths Question-8 Ans. Ex 8.1 Class 6 Maths Question-9 Ans. Ex 8.1 Class 6 Maths Question-10 Ans. Ex 8.1 Class 6 Maths Question-11 Ans. Ex 8.1 Class 6 Maths Question-12 Ans. Ex 8.1 Cla...

NCERT Solutions for Class 11 Maths Chapter 8 - Binomial Theorem Miscellaneous Exercise * According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 7. NCERT Solutions are provided to help the students understand the steps to solve mathematical problems presented in the textbook. The Miscellaneous Exercise of • Binomial Theorem for Positive Integral Indices • General and Middle Terms The solutions enhance topics with frequent, focused, engaging challenges and activities that strengthen Maths concepts. Each question of the exercises has been carefully solved in Download PDF carouselExampleControls112 Previous Next Solutions for Class 11 Maths Chapter 8 – Miscellaneous Exercise 1. Find a, b and n in the expansion of (a + b) n if the first three terms of the expansion are 729, 7290 and 30375, respectively. Solution: We know that (r + 1) th term, (T r+1) in the binomial expansion of (a + b) n is given by T r+1 = nC r a n-t b r The first three terms of the expansion are given as 729, 7290 and 30375, respectively. Then we have, T 1 = nC 0 a n-0 b 0 = a n = 729….. 1 T 2 = nC 1 a n-1 b 1 = na n-1 b = 7290…. 2 T 3 = nC 2 a n-2 b 2 = a n-2 b 2 = 30375……3 Dividing 2 by 1, we get \(\begin \) From 4 and 5, we have n. 5/3 = 10 n = 6 Substituting n = 6 in 1, we get a 6 = 729 a = 3 From 5, we have b/3 = 5/3 b = 5 Thus, a = 3, b = 5 and n = 76 2. Find a if the coefficients of x 2 and x 3 in the expansion of (3 + a x) 9 are equal. Solution: 3. Find the ...

Binomial Theorem Class 11 In binomial theorem class 11, chapter 8 provides the information regarding the introduction and basic definitions for binomial theorem in a detailed way. To score good marks in binomial theorem class 11 concepts, go through the given problems here. Solve all class 11 Maths Chapter 8 problems in the book by referring to the examples to clarify your binomial theorem concepts. Binomial Theorem Class 11 Topics The topics and sub-topics covered in binomial theorem class 11 are: • Introduction • Binomial theorem for positive integral indices • Binomial theorem for any positive integer n • Special Cases • General and Middle Term Binomial Theorem Class 11 Notes The binomial theorem states a formula for the expression of the powers of sums. The most succinct version of this formula is shown immediately below: \(\begin \) From the above representation, we can expand (a + b) n as given below: (a + b) n = nC 0 a n + nC 1 a n-1 b + nC 2 a n-2 b 2 + … + nC n-1 a b n-1 + nC n b n This is the binomial theorem formula for any positive integer n. Some special cases from the binomial theorem can be written as: • (x + y) n = nC 0 x n + nC 1 x n-1 by+ nC 2 x n-2 y 2 + … + nC n-1 x y n-1 + nC n x n • (x – y) n = nC 0 x n– nC 1 x n-1 by + nC 2 x n-2 y 2 + … + (-1) n nC n x n • (1 – x) n = nC 0– nC 1 x + nC 2 x 2– …. (-1) n nC n x n Also, nC 0 = nC n = 1 However, there will be (n + 1) terms in the expansion of (a + b) n. General and Middle ter...