The Ultimate Mathematics Formula Sheet for JEE Main & Advanced

The Complete JEE Mathematics Formula Reference

This comprehensive formula reference is organized by high-weightage topics in JEE Main and Advanced Mathematics. Bookmark this page and review it daily.

1. Calculus

Differentiation — Key Formulas

→d/dx [xⁿ] = nxⁿ⁻¹

→d/dx [eˣ] = eˣ

→d/dx [ln x] = 1/x

→d/dx [sin x] = cos x; d/dx [cos x] = −sin x

→d/dx [tan x] = sec²x; d/dx [cot x] = −cosec²x

→Product Rule: d/dx [uv] = u'v + uv'

→Chain Rule: d/dx [f(g(x))] = f'(g(x)) · g'(x)

→L'Hôpital's Rule: If 0/0 or ∞/∞ form → lim f/g = lim f'/g'

Integration — Key Formulas

→∫xⁿ dx = xⁿ⁺¹/(n+1) + C

→∫eˣ dx = eˣ + C

→∫1/x dx = ln|x| + C

→∫sin x dx = −cos x + C; ∫cos x dx = sin x + C

→Integration by Parts: ∫u·dv = uv − ∫v·du (ILATE rule for choosing u)

Limits

→lim(x→0) sin x / x = 1

→lim(x→0) (1 + x)^(1/x) = e

→lim(x→∞) (1 + 1/x)^x = e

2. Algebra

Quadratic Equations

→Roots: x = [−b ± √(b²−4ac)] / 2a

→Sum of roots (α+β) = −b/a

→Product of roots (αβ) = c/a

→Discriminant: D = b²−4ac (D > 0: real distinct, D = 0: equal, D < 0: complex)

→Condition for both roots positive: D ≥ 0, −b/a > 0, c/a > 0

Sequences & Series

→AP: Sum = n/2 · [2a + (n−1)d]

→GP: Sum = a(rⁿ−1)/(r−1), for r ≠ 1

→GP Infinite Sum (|r| < 1): S = a/(1−r)

→Sum of first n natural numbers: n(n+1)/2

→Sum of squares: n(n+1)(2n+1)/6

→Sum of cubes: [n(n+1)/2]²

Binomial Theorem

→(a+b)ⁿ = Σ C(n,r)·aⁿ⁻ʳ·bʳ (r from 0 to n)

→General Term: T(r+1) = C(n,r)·aⁿ⁻ʳ·bʳ

→Middle term: For even n, T(n/2 + 1)

Permutations & Combinations

→nPr = n! / (n−r)!

→nCr = n! / [r!(n−r)!]

→nCr = nC(n−r)

→nC0 = nCn = 1

3. Coordinate Geometry

Straight Lines

→Slope: m = (y₂−y₁)/(x₂−x₁)

→Equation: y−y₁ = m(x−x₁)

→Distance from point (x₁,y₁) to line ax+by+c=0: |ax₁+by₁+c| / √(a²+b²)

→Angle between two lines: tan θ = |m₁−m₂| / |1+m₁m₂|

Circle

→Standard form: x²+y²=r²

→General form: x²+y²+2gx+2fy+c=0; Centre: (−g,−f), Radius: √(g²+f²−c)

→Tangent at (x₁,y₁): xx₁+yy₁=r²

Parabola, Ellipse, Hyperbola

→Parabola y²=4ax: Focus (a,0), Directrix x=−a, Latus Rectum = 4a

→Ellipse x²/a²+y²/b²=1 (a>b): e=c/a, c²=a²−b², Foci (±c,0)

→Hyperbola x²/a²−y²/b²=1: e=c/a, c²=a²+b², Asymptotes y=±(b/a)x

4. Trigonometry

→sin²θ + cos²θ = 1; 1+tan²θ = sec²θ; 1+cot²θ = cosec²θ

→sin(A±B) = sinA·cosB ± cosA·sinB

→cos(A±B) = cosA·cosB ∓ sinA·sinB

→tan(A±B) = (tanA ± tanB) / (1 ∓ tanA·tanB)

→sin 2A = 2 sinA cosA; cos 2A = cos²A − sin²A = 1 − 2sin²A

→Sine Rule: a/sinA = b/sinB = c/sinC = 2R

→Cosine Rule: a² = b² + c² − 2bc·cosA

5. Probability

→P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

→P(A|B) = P(A ∩ B) / P(B)

→Bayes' Theorem: P(Aᵢ|B) = P(B|Aᵢ)·P(Aᵢ) / Σ P(B|Aⱼ)·P(Aⱼ)

→Binomial Distribution: P(X=r) = C(n,r)·pʳ·(1−p)ⁿ⁻ʳ

How to Use This Formula Sheet

1. Print it out and keep it on your study desk.

2. Every morning, spend 10 minutes reviewing 2–3 sections.

3. After solving a problem, verify the formula you used against this sheet.

4. Test your recall by covering formulas and writing them from memory.

Use SmartTutor's Mathematics AI quizzes to practise applying these formulas in actual JEE-level problems.