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Binomial Theorem

Telangana Open School (TOSS) · Class 12 · Mathematics

Quick revision notes for Binomial Theorem — Telangana Open School (TOSS) Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

53 questions24 flashcards5 concepts

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Key Topics to Revise

1

Binomial Theorem for Positive Integral Index

  • The binomial expansion of \((x + y)^n\) has \(n + 1\) terms.
  • The coefficients of the terms are binomial coefficients, denoted by \(^nC_r\).
  • The general term is \(T_{r+1} = ^nC_r x^{n-r} y^r\).
2

General Term and Middle Term

  • The general term in the expansion of \((x + y)^n\) is \(T_{r+1} = ^nC_r x^{n-r} y^r\).
  • When \(n\) is even, the middle term is the \(\left(\frac{n}{2} + 1\right)^{\text{th}}\) term.
  • When \(n\) is odd, there are two middle terms: \(\left(\frac{n+1}{2}\right)^{\text{th}}\) and \(\left(\frac{n+3}{2}\right)^{\text{th}}\) terms.
3

Binomial Theorem for Rational and Negative Indices

  • For rational or negative exponents, the binomial expansion has infinitely many terms.
  • The expansion \((1 + x)^r\) is valid only when \(|x| < 1\).
  • The general term is \(T_{r+1} = \frac{r(r-1)(r-2)\cdots(r-k+1)}{k!} x^k\).
4

Applications in Approximations

  • The binomial theorem can approximate values like \(\sqrt[3]{9}\), \(\sqrt{1.02}\), etc.
  • Neglect higher powers of small quantities (like \(x^2, x^3\)) when \(x\) is very small.
  • Used in physics and engineering for linear approximations.

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Full Notes

Key Concepts

For any positive integer \(n\)The \((r+1)^{\text{th}}\) term in the expansionWhen \(n\) is evenWhen the exponent \(r\) isFor small values of \(x\)

Frequently Asked Questions

What are the important topics in Binomial Theorem for Telangana Open School (TOSS) Class 12 Mathematics?
Binomial Theorem covers several key topics that are frequently asked in Telangana Open School (TOSS) Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Binomial Theorem — Telangana Open School (TOSS) Class 12 Mathematics?
Understand the core concepts first, then work through the 53 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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