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NCERT Mathematics Class 11 - Chapter 7: Binomial Theorem - Notes

द्विपद प्रमेय

Learning Objectives

  • Understand the Binomial Theorem for positive integral index
  • Learn to find general term and middle term
  • Apply binomial theorem to solve problems

Key Concepts

Binomial Theorem

(a + b)n = Σr=0n C(n,r) an-r br = C(n,0)an + C(n,1)an-1b + C(n,2)an-2b2 + ... + C(n,n)bn. The expansion has (n+1) terms. General term (Tr+1): Tr+1 = C(n,r) an-r br.

Special Cases

(1 + x)n = Σ C(n,r) xr = 1 + C(n,1)x + C(n,2)x2 + ... + xn.

(a - b)n: Alternate signs: C(n,0)an - C(n,1)an-1b + C(n,2)an-2b2 - ...

Putting x = 1: C(n,0) + C(n,1) + C(n,2) + ... + C(n,n) = 2n.

Putting x = -1: C(n,0) - C(n,1) + C(n,2) - ... = 0 (sum of even-index terms = sum of odd-index terms = 2n-1).

Middle Term

If n is even: one middle term = T(n/2)+1. If n is odd: two middle terms = T(n+1)/2 and T(n+3)/2. The middle term has the largest binomial coefficient.

Important Properties of Binomial Coefficients

C(n,0) + C(n,1) + ... + C(n,n) = 2n. C(n,0) + C(n,2) + C(n,4) + ... = 2n-1 (even-indexed terms). C(n,1) + C(n,3) + C(n,5) + ... = 2n-1 (odd-indexed terms). Greatest binomial coefficient: C(n, n/2) if n even; C(n, (n-1)/2) = C(n, (n+1)/2) if n odd.

Finding Particular Terms (JEE Important)

Term independent of x: In expansion of (xp + 1/xq)n, find r such that power of x in Tr+1 is 0. Greatest term: Use the ratio Tr+1/Tr to find which term has the greatest value.

Summary

The Binomial Theorem expands (a + b)n into (n+1) terms. The general term is C(n,r)an-rbr. Middle terms depend on whether n is even or odd. Binomial coefficients satisfy important identities useful in combinatorics and algebra.

Important Terms

  • Binomial coefficient: C(n,r) — coefficient of the (r+1)th term
  • General term: Tr+1 = C(n,r) an-r br
  • Middle term: Central term(s) in the binomial expansion
  • Pascal's triangle: Triangular array of binomial coefficients

Quick Revision

  • (a+b)n has (n+1) terms
  • General term: Tr+1 = C(n,r) an-r br
  • Middle term: T(n/2)+1 for even n
  • Sum of coefficients: put a = b = 1 → 2n
  • Sum of even-indexed = sum of odd-indexed = 2n-1
  • For (a-b)n: signs alternate starting positive
  • Term independent of x: set power of x in general term = 0
NCERT Mathematics Class 11 - Chapter 7: Binomial Theorem - Notes | EduMunch