Class 12 Maths Formulas (PDF) — All Important Formulas Free
Free class 12 maths formulas — all 63 key formulas on one page, downloadable as PDF for fast revision before CBSE board exams, JEE & NEET. No signup.
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Comprehensive Class 12 Maths formulas covering Inverse Trigonometry, Matrices & Determinants, Continuity & Differentiability, Derivatives, Integration, Differential Equations, Vectors, 3D Geometry, and Probability. Essential for board exams and competitive entrance tests.
Inverse Trigonometric Functions
| Formula | Expression |
|---|---|
| Domain and range of inverse trig functions | sin⁻¹: [-1, 1] → [-π/2, π/2]; cos⁻¹: [-1, 1] → [0, π]; tan⁻¹: ℝ → (-π/2, π/2) — cot⁻¹: ℝ → (0, π); sec⁻¹: ℝ∖(-1,1) → [0,π]∖{π/2}; cosec⁻¹: ℝ∖(-1,1) → [-π/2,π/2]∖{0} |
| Complementary inverse trig relations | sin⁻¹ x + cos⁻¹ x = π/2; tan⁻¹ x + cot⁻¹ x = π/2; sec⁻¹ x + cosec⁻¹ x = π/2 — For x in appropriate domains |
| Inverse trig sum formulas | tan⁻¹ x + tan⁻¹ y = tan⁻¹((x + y)/(1 - xy)) if xy < 1; = π + tan⁻¹((x + y)/(1 - xy)) if xy > 1, x > 0 |
| sin⁻¹ x + sin⁻¹ y formula | sin⁻¹ x + sin⁻¹ y = sin⁻¹(x√(1-y²) + y√(1-x²)) if x² + y² ≤ 1 |
| Derivatives of inverse trig functions | d/dx(sin⁻¹ x) = 1/√(1-x²); d/dx(cos⁻¹ x) = -1/√(1-x²); d/dx(tan⁻¹ x) = 1/(1+x²) — d/dx(cot⁻¹ x) = -1/(1+x²); d/dx(sec⁻¹ x) = 1/(|x|√(x²-1)); d/dx(cosec⁻¹ x) = -1/(|x|√(x²-1)) |
Matrices & Determinants
| Formula | Expression |
|---|---|
| Matrix addition and scalar multiplication | (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ; (kA)ᵢⱼ = kAᵢⱼ — Commutative: A + B = B + A; Associative: (A + B) + C = A + (B + C) |
| Matrix multiplication | (AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ — Not commutative: AB ≠ BA in general; Associative: (AB)C = A(BC) |
| Transpose of a matrix | (Aᵀ)ᵢⱼ = Aⱼᵢ — (AB)ᵀ = BᵀAᵀ; (A + B)ᵀ = Aᵀ + Bᵀ |
| Symmetric and skew-symmetric matrices | Symmetric: Aᵀ = A; Skew-symmetric: Aᵀ = -A — Any matrix A = (A + Aᵀ)/2 + (A - Aᵀ)/2 (sum of symmetric and skew-symmetric) |
| Determinant of 2×2 matrix | det([a b; c d]) = ad - bc |
| Determinant of 3×3 matrix (expansion) | det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) — Expansion along first row of [a b c; d e f; g h i] |
| Adjugate and inverse | Inverse: A⁻¹ = adj(A)/det(A); Cofactor Cᵢⱼ = (-1)^(i+j) × minor Mᵢⱼ; adj(A) = (Cofactor matrix)ᵀ — A exists only if det(A) ≠ 0 |
| Determinant properties | det(AB) = det(A) × det(B); det(Aᵀ) = det(A); det(kA) = kⁿ det(A) for n×n matrix — Swapping rows changes sign; Same rows/columns → det = 0 |
Continuity & Differentiability
| Formula | Expression |
|---|---|
| Continuity at a point | f is continuous at x = a if lim(x→a) f(x) = f(a) — Three conditions: (1) f(a) defined, (2) limit exists, (3) they're equal |
| Differentiability at a point | f'(a) = lim(h→0) [f(a+h) - f(a)]/h (right derivative = left derivative) — If differentiable at a, then continuous at a. Converse is not always true. |
| Algebra of derivatives | (u + v)' = u' + v'; (uv)' = u'v + uv'; (u/v)' = (u'v - uv')/v² — Product rule and quotient rule |
| Chain rule | dy/dx = (dy/du) × (du/dx) — For composite functions; if y = f(u) and u = g(x) |
| Parametric differentiation | dy/dx = (dy/dt)/(dx/dt) — When x = f(t) and y = g(t) |
| Implicit differentiation | Differentiate both sides w.r.t. x and solve for dy/dx — When y is not explicitly given as function of x |
| Logarithmic differentiation | For y = u^v, take ln: ln y = v ln u, then differentiate — Useful for functions like y = x^x or y = (sin x)^x |
Derivatives of Standard Functions
| Formula | Expression |
|---|---|
| Power rule | d/dx(xⁿ) = nxⁿ⁻¹ |
| Exponential and logarithmic | d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ ln a; d/dx(ln x) = 1/x; d/dx(log_a x) = 1/(x ln a) |
| Trigonometric | d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(tan x) = sec² x — d/dx(cot x) = -cosec² x; d/dx(sec x) = sec x tan x; d/dx(cosec x) = -cosec x cot x |
| Inverse trigonometric | d/dx(sin⁻¹ x) = 1/√(1-x²); d/dx(tan⁻¹ x) = 1/(1+x²); d/dx(sec⁻¹ x) = 1/(|x|√(x²-1)) |
| Hyperbolic functions | d/dx(sinh x) = cosh x; d/dx(cosh x) = sinh x; d/dx(tanh x) = sech² x — sinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2 |
Applications of Derivatives
| Formula | Expression |
|---|---|
| Tangent and normal | Tangent: y - y₁ = f'(x₁)(x - x₁); Normal: y - y₁ = -(1/f'(x₁))(x - x₁) — At point (x₁, y₁) on curve y = f(x) |
| Rate of change | dy/dt = (dy/dx) × (dx/dt) — Related rates problems |
| Increasing and decreasing functions | f'(x) > 0 ⟹ f increasing; f'(x) < 0 ⟹ f decreasing — First derivative test |
| Local maxima and minima | Critical points: f'(x) = 0. Second derivative test: f''(x) > 0 (minimum), f''(x) < 0 (maximum) — If f''(x) = 0, test further |
| Approximation (linear) | f(x + Δx) ≈ f(x) + f'(x) × Δx — dy = f'(x) dx |
| Mean Value Theorem | If f continuous on [a, b] and differentiable on (a, b), then ∃ c ∈ (a, b): f'(c) = [f(b) - f(a)]/(b - a) |
Integration
| Formula | Expression |
|---|---|
| Indefinite integral definition | ∫ f(x) dx = F(x) + C if F'(x) = f(x) — F(x) is antiderivative, C is constant of integration |
| Integration of standard functions | ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1); ∫ (1/x) dx = ln|x| + C; ∫ eˣ dx = eˣ + C — ∫ aˣ dx = aˣ/ln a + C |
| Trigonometric integrals | ∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C; ∫ sec² x dx = tan x + C; ∫ cosec² x dx = -cot x + C — ∫ sec x tan x dx = sec x + C; ∫ cosec x cot x dx = -cosec x + C |
| Inverse trigonometric integrals | ∫ 1/√(1-x²) dx = sin⁻¹ x + C; ∫ 1/(1+x²) dx = tan⁻¹ x + C; ∫ 1/(x√(x²-1)) dx = sec⁻¹|x| + C |
| Integration by substitution | ∫ f(g(x)) g'(x) dx = ∫ f(u) du where u = g(x) — Change of variable |
| Integration by parts | ∫ u dv = uv - ∫ v du — Use ILATE rule: Inverse, Logarithm, Algebraic, Trigonometric, Exponential |
| Partial fractions | Decompose rational function into partial fractions before integrating — For P(x)/Q(x) where deg(P) < deg(Q) |
| Definite integral | ∫ₐᵇ f(x) dx = F(b) - F(a) where F is antiderivative — Fundamental theorem of calculus |
| Properties of definite integrals | ∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx; ∫ₐᵃ f(x) dx = 0; ∫ₐᵇ [f(x) + g(x)] dx = ∫ₐᵇ f(x) dx + ∫ₐᵇ g(x) dx — Linearity and additivity properties |
Differential Equations
| Formula | Expression |
|---|---|
| Order and degree | Order = highest derivative present; Degree = power of highest derivative — For (d³y/dx³)² + (d²y/dx²) - 3dy/dx + y = 0: order 3, degree 2 |
| Separable differential equation | dy/dx = f(x)/g(y) ⟹ g(y) dy = f(x) dx ⟹ integrate both sides — Variables can be separated |
| Linear differential equation (first order) | dy/dx + Py = Q where P, Q are functions of x; solution: y = e^(-∫P dx) [∫Q e^(∫P dx) dx + C] — Using integrating factor IF = e^(∫P dx) |
| Homogeneous differential equation | dy/dx = f(y/x); substitute v = y/x ⟹ y = vx ⟹ dy = v dx + x dv — Reduces to separable form |
Vectors
| Formula | Expression |
|---|---|
| Vector magnitude and unit vector | |a| = √(a₁² + a₂² + a₃²); unit vector = a/|a| — For vector a = (a₁, a₂, a₃) |
| Dot product (scalar product) | a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃ — Angle between vectors: cos θ = (a·b)/(|a||b|) |
| Cross product (vector product) | a × b = |a||b| sin θ n̂ = |i j k | = (a₂b₃ - a₃b₂)i - (a₁b₃ - a₃b₁)j + (a₁b₂ - a₂b₁)k — |a × b| = area of parallelogram; direction by right-hand rule |
| Scalar triple product | a · (b × c) = (a × b) · c = |a₁ a₂ a₃| = volume of parallelepiped — |b₁ b₂ b₃| |
| Projection of a on b | proj_b a = (a · b)/|b| = |a| cos θ — Scalar projection; vector projection = (a·b/|b|²) × b |
| Perpendicularity and parallelism | a ⊥ b ⟺ a · b = 0; a ∥ b ⟺ a × b = 0 |
3D Geometry
| Formula | Expression |
|---|---|
| Equation of line (parametric) | (x, y, z) = (x₀, y₀, z₀) + t(a, b, c) or x-x₀/a = y-y₀/b = z-z₀/c — (x₀, y₀, z₀) is point on line; (a, b, c) is direction vector |
| Angle between two lines | cos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁² + b₁² + c₁²) × √(a₂² + b₂² + c₂²)) — For direction vectors (a₁, b₁, c₁) and (a₂, b₂, c₂) |
| Equation of plane | A(x - x₀) + B(y - y₀) + C(z - z₀) = 0 or Ax + By + Cz + D = 0 — (A, B, C) is normal vector |
| Angle between two planes | cos θ = |A₁A₂ + B₁B₂ + C₁C₂| / (√(A₁² + B₁² + C₁²) × √(A₂² + B₂² + C₂²)) — Normal vectors: (A₁, B₁, C₁) and (A₂, B₂, C₂) |
| Distance from point to plane | d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²) — Point (x₀, y₀, z₀) to plane Ax + By + Cz + D = 0 |
Probability & Statistics
| Formula | Expression |
|---|---|
| Conditional probability | P(A|B) = P(A ∩ B) / P(B) — Probability of A given B has occurred |
| Multiplication rule | P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B) |
| Total probability theorem | P(A) = P(A|B₁)P(B₁) + P(A|B₂)P(B₂) + ... + P(A|Bₙ)P(Bₙ) — If B₁, B₂, ..., Bₙ partition sample space |
| Bayes' theorem | P(Bᵢ|A) = P(A|Bᵢ)P(Bᵢ) / Σ P(A|Bⱼ)P(Bⱼ) — Posterior probability; used for updating beliefs |
| Independence | A and B independent ⟺ P(A ∩ B) = P(A) × P(B) ⟺ P(A|B) = P(A) |
| Binomial probability | P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏ — X ~ B(n, p); n trials, p = success probability |
| Expected value (mean) | E(X) = Σ xᵢ P(xᵢ) — For binomial: E(X) = np |
| Variance and standard deviation | Var(X) = E(X²) - [E(X)]² = Σ (xᵢ - μ)² P(xᵢ); σ = √Var(X) — For binomial: Var(X) = np(1-p) |
FAQs
What's the difference between definite and indefinite integrals?
Indefinite integral ∫f(x)dx = F(x) + C gives a family of antiderivatives with arbitrary constant C. Definite integral ∫ₐᵇf(x)dx = F(b) - F(a) gives a specific number (area under curve from a to b). Use the Fundamental Theorem of Calculus to connect them.
When should I use integration by parts vs substitution?
Use substitution when you can identify a function and its derivative in the integrand. Use integration by parts when you have a product of functions (especially polynomial × exponential/trig). Remember ILATE: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential—choose the first function by this priority.
What's the difference between order and degree of a differential equation?
Order is the highest derivative present (e.g., d²y/dx² = order 2). Degree is the power/exponent of the highest derivative (e.g., (d²y/dx²)³ = degree 3). For example, (d²y/dx²)² + dy/dx = 0 has order 2, degree 2.
How do I find the angle between two vectors?
Use the dot product formula: cos θ = (a·b)/(|a||b|). First calculate a·b = a₁b₁ + a₂b₂ + a₃b₃, then |a| = √(a₁² + a₂² + a₃²), |b| = √(b₁² + b₂² + b₃²). Then θ = arccos of the result.
What's Bayes' theorem and when do I use it?
Bayes' theorem: P(Bᵢ|A) = P(A|Bᵢ)P(Bᵢ)/[ΣP(A|Bⱼ)P(Bⱼ)]. It updates the probability of an event based on new evidence. Use it when you know the probability of observing evidence given different causes, and want to find the probability of a cause given the observed evidence (e.g., disease diagnosis given test result).
More Class 12 Mathematics Formula Sheets
- Class 12 Mathematics: 3D Geometry Formulas
- Class 12 Mathematics: Application of Derivatives Formulas
- Class 12 Mathematics: Continuity and Differentiability Formulas
- Class 12 Mathematics: Differential Equations Formulas
- Class 12 Mathematics: Integration Formulas
- Class 12 Mathematics: Matrices and Determinants Formulas