JEEClass 11-12

Algebra Formulas for JEE

All Algebra formulas for JEE Maths in one place — Quadratics, Progressions, Permutations & Combinations, Binomial Theorem, Complex Numbers, and Logarithms.

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Quadratic Equations Arithmetic & Geometric Progressions Permutations & Combinations Binomial Theorem Complex Numbers Logarithms & Inequalities

Quadratic Equations

Rootsx = (−b ± √(b²−4ac)) / 2a

DiscriminantD = b² − 4ac

Sum of rootsα + β = −b/a

Product of rootsαβ = c/a

α² + β²α² + β² = (α+β)² − 2αβ

|α − β||α − β| = √D / |a|

Arithmetic & Geometric Progressions

AP: nth termaₙ = a + (n−1)d

AP: sumSₙ = n/2[2a + (n−1)d]

GP: nth termaₙ = arⁿ⁻¹

GP: sum (finite)Sₙ = a(rⁿ − 1)/(r − 1)

GP: sum (infinite)S∞ = a/(1−r), |r| < 1

AM ≥ GM(a+b)/2 ≥ √(ab) for a,b > 0

Special: ΣkΣk = n(n+1)/2

Special: Σk²Σk² = n(n+1)(2n+1)/6

Special: Σk³Σk³ = [n(n+1)/2]²

Permutations & Combinations

PermutationnPr = n!/(n−r)!

CombinationnCr = n!/(r!(n−r)!)

Circular perm.(n−1)!

With repetitionn^r (r items from n with repetition)

DerangementD(n) = n!(1 − 1/1! + 1/2! − ... + (−1)ⁿ/n!)

nCr identitynCr = nC(n−r)

Pascal's rulenCr = (n−1)C(r−1) + (n−1)Cr

Binomial Theorem

Expansion(a+b)ⁿ = Σ nCr · aⁿ⁻ʳ · bʳ

General termTᵣ₊₁ = nCr · aⁿ⁻ʳ · bʳ

Middle term (n even)Tₙ₂₊₁ (single middle term)

Sum of coefficients(1+1)ⁿ = 2ⁿ

Sum of binomial coeff.nC0 + nC1 + ... + nCn = 2ⁿ

Complex Numbers

Modulus|z| = √(a² + b²) for z = a + ib

Conjugatez̅ = a − ib, z·z̅ = |z|²

Euler's forme^{iθ} = cosθ + isinθ

De Moivre(cosθ + isinθ)ⁿ = cos(nθ) + isin(nθ)

Cube roots of unity1 + ω + ω² = 0, ω³ = 1

Triangle inequality||z₁| − |z₂|| ≤ |z₁ + z₂| ≤ |z₁| + |z₂|

Logarithms & Inequalities

log definitionlog_a(x) = y ⇔ aʸ = x

log productlog(ab) = log(a) + log(b)

log quotientlog(a/b) = log(a) − log(b)

log powerlog(aⁿ) = n·log(a)

Change of baselog_a(b) = log_c(b) / log_c(a)

AM-GM-HMAM ≥ GM ≥ HM (positive numbers)

JEE Tips

Tip 1:Vieta's formulas save time — avoid finding individual roots when you only need their sum/product
Tip 2:For binomial theorem, the general term formula solves 90% of JEE problems
Tip 3:AM-GM inequality is a powerful tool for optimization without calculus