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CBSE Notes Class 11 Maths Binomial Theorem and Mathematical Induction

by aglasem
March 14, 2023
in 11th Class

CBSE Class 11 Maths Binomial Theorem and Mathematical Induction – Get here the Notes for Class 11 Binomial Theorem and Mathematical Induction. Candidates who are ambitious to qualify the Class 11 with good score can check this article for Notes. This is possible only when you have the best CBSE Class 11 Maths study material and a smart preparation plan. To assist you with that, we are here with notes. Hope these notes will helps you understand the important topics and remember the key points for exam point of view. Below we provided the Notes of CBSE Class 11 Maths for topic Binomial Theorem and Mathematical Induction.

  • Class: 11th
  • Subject: Maths
  • Topic: Binomial Theorem and Mathematical Induction
  • Resource: Notes

CBSE Notes Class 11 Maths Binomial Theorem and Mathematical Induction

Candidates who are pursuing in CBSE Class 11 Maths are advised to revise the notes from this post. With the help of Notes, candidates can plan their Strategy for particular weaker section of the subject and study hard. So, go ahead and check the Important Notes for CBSE Class 11 Maths Binomial Theorem and Mathematical Induction from this article.

Binomial Theorem for Positive Integer

If n is any positive integer, then

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

This is called binomial theorem.

Here, nC0, nC1, nC2, … , nno are called binomial coefficients and

nCr = n! / r!(n – r)! for 0 ≤ r ≤ n.

Properties of Binomial Theorem for Positive Integer

(i) Total number of terms in the expansion of (x + a)n is (n + 1).

(ii) The sum of the indices of x and a in each term is n.

(iii) The above expansion is also true when x and a are complex numbers.

(iv) The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and

nCr = nCn – r, r = 0,1,2,…,n.

(v) General term in the expansion of (x + c)n is given by

Tr + 1 = nCrxn – r ar.

(vi) The values of the binomial coefficients steadily increase to maximum and then steadily decrease .

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(vii)

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(viii)

(ix) The coefficient of xr in the expansion of (1+ x)n is nCr.

(x)

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(xi) (a)

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(b)

(xii) (a) If n is odd, then (x + a)n + (x – a)n and (x + a)n – (x – a)n both have the same number of terms equal to (n +1 / 2).

(b) If n is even, then (x + a)n + (x – a)n has (n +1 / 2) terms. and (x + a)n – (x – a)n has (n / 2) terms.

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(xiii) In the binomial expansion of (x + a)n, the r th term from the end is (n – r + 2)th term from the beginning.

(xiv) If n is a positive integer, then number of terms in (x + y + z)n is (n + l)(n + 2) / 2.

Middle term in the Expansion of (1 + x)n

(i) It n is even, then in the expansion of (x + a)n, the middle term is (n/2 + 1)th terms.

(ii) If n is odd, then in the expansion of (x + a)n, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.

Greatest Coefficient

(i) If n is even, then in (x + a)n, the greatest coefficient is nCn / 2

(ii) Ifn is odd, then in (x + a)n, the greatest coefficient is nCn – 1 / 2 or nCn + 1 / 2 both being equal.

Greatest Term

In the expansion of (x + a)n

(i) If n + 1 / x/a + 1 is an integer = p (say), then greatest term is Tp == Tp + 1.

(ii) If n + 1 / x/a + 1 is not an integer with m as integral part of n + 1 / x/a + 1, then Tm + 1. is the greatest term.

Important Results on Binomial Coefficients

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

Divisibility Problems

From the expansion, (1+ x)n = 1+ nC1x + nC1x2+ … +nCnxn

We can conclude that,

(i) (1+ x)n – 1 = nC1x + nC1x2+ … +nCnxn is divisible by x i.e., it is multiple of x.

(1+ x)n – 1 = M(x)

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(ii)

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(iii)

Multinomial theorem

For any n ∈ N,

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(i)

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(ii)

(iii) The general term in the above expansion is

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(iv)The greatest coefficient in the expansion of (x1 + x2 + … + xm)n is where q and r are the quotient and remainder respectively, when n is divided by m.

(v) Number of non-negative integral solutions of x1 + x2 + … + xn = n is n + r – 1Cr – 1

R-f Factor Relations

Here, we are going to discuss problem involving (√A + B)sup>n = I + f, Where I and n are positive integers.

0 le; f le; 1, |A – B2| = k and |√A – B| < 1

Binomial Theorem for any Index

If n is any rational number, then

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(i) If in the above expansion, n is any positive integer, then the series in RHS is finite otherwise infinite.

(ii) General term in the expansion of (1 + x)n is Tr + 1 = n(n – 1)(n – 2)… [n – (r – 1)] / r! * xr

(iii) Expansion of (x + a)n for any rational index

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(vii) (1 + x)– 1 = 1 – x + x2 – x3 + …∞

(viii) (1 – x)– 1 = 1 + x + x2 + x3 + …∞

(ix) (1 + x)– 2 = 1 – 2x + 3x2 – 4x3 + …∞

(x) (1 – x)– 2 = 1 + 2x + 3x2 – 4x3 + …∞

(xi) (1 + x)– 3 = 1 – 3x + 6x2 – …∞

(xii) (1 – x)– 3 = 1 + 3x + 6x2 – …∞

(xiii) (1 + x)n = 1 + nx, if x2, x3,… are all very small as compared to x.

Important Results

(i) Coefficient of xm in the expansion of (axp + b / xq)n is the coefficient of Tr + l where r = np – m / p + q

(ii) The term independent of x in the expansion of axp + b / xq)n is the coefficient of Tr + l where r = np / p + q

(iii) If the coefficient of rth, (r + l)th and (r + 2)th term of (1 + x)n are in AP, then n2 – (4r+1) n + 4r2 = 2

(iv) In the expansion of (x + a)n

Tr + 1 / Tr = n – r + 1 / r * a / x

(v) (a) The coefficient of xn – 1 in the expansion of

(x – l)(x – 2) ….(x – n) = – n (n + l) / 2

(b) The coefficient of xn – 1 in the expansion of

(x + l)(x + 2) ….(x + n) = n (n + l) / 2

(vi) If the coefficient of pth and qth terms in the expansion of (1 + x)n are equal, then p + q = n + 2

(vii) If the coefficients of xr and xr + 1 in the expansion of a + x / b)n are equal, then

n = (r + 1)(ab + 1) – 1

(viii) The number of term in the expansion of (x1 + x2 + … + xr)n is n + r – 1C r – 1.

(ix) If n is a positive integer and a1, a2, … , am ∈ C, then the coefficient of xr in the expansion of (a1 + a2x + a3x2 +… + amxm – 1)n is

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(x) For |x| < 1,

(a) 1 + x + x2 + x3+ … + ∞ = 1 / 1 – x

(b) 1 + 2x + 3x2 + … + ∞ = 1 / (1 – x)2

(xi) Total number of terms in the expansion of (a + b + c + d)n is (n + l)(n + 2)(n + 3) / 6.

Important Points to be Remembered

(i) If n is a positive integer, then (1 + x)n contains (n +1) terms i.e., a finite number of terms. When n is general exponent, then the expansion of (1 + x)n contains infinitely many terms.

(ii) When n is a positive integer, the expansion of (l + x)n is valid for all values of x. If n is general exponent, the expansion of (i + x)n is valid for the values of x satisfying the condition |x| < 1.

Class 11 Key Points, Important Questions & Practice Papers

Hope these notes helped you in your schools exam preparation. Candidates can also check out the Key Points, Important Questions & Practice Papers for various Subjects for Class 11 in both Hindi and English language form the link below.

Class 11 Physics कक्षा 11 अर्थशास्त्र
Class 11 Chemistry कक्षा 11 समाज शास्त्र
Class 11 Maths कक्षा 11 शारीरिक शिक्षा
Class 11 Biology कक्षा 11 राजनीति विज्ञान
Class 11 Business Studies कक्षा 11 इतिहास
Class 11 Accountancy कक्षा 11 भूगोल 
Class 11 Sociology 
Class 11 Physical Education 
Class 11 English 

 Class 11 NCERT Solutions

Candidates who are studying in Class 11 can also check Class 11 NCERT Solutions from here. This will help the candidates to know the solutions for all subjects covered in Class 11th. Candidates can click on the subject wise link to get the same. Class 11 Chapter-wise, detailed solutions to the questions of the NCERT textbooks are provided with the objective of helping students compare their answers with the sample answers.

  • Biology
  • Chemistry
  • Physics
  • Mathematics
  • English
  • Hindi
  • Business Studies

Class 11 Mock Test / Practice

Mock test are the practice test or you can say the blue print of the main exam. Before appearing in the main examination, candidates must try mock test as it helps the students learn from their mistakes. With the help of Class 11 Mock Test / Practice, candidates can also get an idea about the pattern and marking scheme of that examination. For the sake of the candidates we are providing Class 11 Mock Test / Practice links below.

  • Maths
  • Physics
  • Chemistry
  • Biology

Class 11 Exemplar Questions

Exemplar Questions Class 11 is a very important resource for students preparing for the Examination. Here we have provided Exemplar Problems Solutions along with NCERT Exemplar Problems Class 11. Question from very important topics is covered by Exemplar Questions for Class 11.

Physics रसायन विज्ञान
Chemistry जीव विज्ञान
Maths
Biology

CBSE Notes for Class 11 Maths Biology Notes Physics Notes Chemistry Notes

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