Binomial Theorem
Kerala Board · Class 12 · Mathematics
Quick revision notes for Binomial Theorem — Kerala Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Basic Binomial Expansion for Positive Integer n
- For positive integer n: (x + y)ⁿ = ⁿC₀xⁿ + ⁿC₁xⁿ⁻¹y + ⁿC₂xⁿ⁻²y² + ... + ⁿCₙyⁿ
- Total number of terms in expansion = n + 1
- Sum of powers of x and y in each term = n
Middle Terms in Binomial Expansion
- When n is even: Middle term = (n/2 + 1)th term
- When n is odd: Two middle terms = (n+1)/2 th and (n+3)/2 th terms
- Middle terms have maximum binomial coefficients
Properties of Binomial Coefficients
- ⁿC₀ = ⁿCₙ = 1 (first and last terms always have coefficient 1)
- ⁿCᵣ = ⁿCₙ₋ᵣ (symmetry property)
- ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal's triangle property)
Binomial Theorem for Rational/Negative Exponents
- For rational exponent r: (1 + x)ʳ = 1 + rx + r(r-1)x²/2! + r(r-1)(r-2)x³/3! + ... (|x| < 1)
- Infinite series when r is not a positive integer
- Convergence condition: |x| < 1 is essential
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Sources & Official References
- Kerala Board of Public Examinations — keralapareekshabhavan.in
- National Education Policy 2020 — education.gov.in
Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.
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Differentiation Of Exponential and Logarithmic Functions
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