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.
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}sin⁻¹ x + cos⁻¹ x = π/2; tan⁻¹ x + cot⁻¹ x = π/2; sec⁻¹ x + cosec⁻¹ x = π/2For x in appropriate domainstan⁻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 > 0sin⁻¹ x + sin⁻¹ y = sin⁻¹(x√(1-y²) + y√(1-x²)) if x² + y² ≤ 1d/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))(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ; (kA)ᵢⱼ = kAᵢⱼCommutative: A + B = B + A; Associative: (A + B) + C = A + (B + C)(AB)ᵢⱼ = Σₖ Aᵢₖ BₖⱼNot commutative: AB ≠ BA in general; Associative: (AB)C = A(BC)(Aᵀ)ᵢⱼ = Aⱼᵢ(AB)ᵀ = BᵀAᵀ; (A + B)ᵀ = Aᵀ + BᵀSymmetric: Aᵀ = A; Skew-symmetric: Aᵀ = -AAny matrix A = (A + Aᵀ)/2 + (A - Aᵀ)/2 (sum of symmetric and skew-symmetric)det([a b; c d]) = ad - bcdet(A) = a(ei - fh) - b(di - fg) + c(dh - eg)Expansion along first row of [a b c; d e f; g h i]Inverse: A⁻¹ = adj(A)/det(A); Cofactor Cᵢⱼ = (-1)^(i+j) × minor Mᵢⱼ; adj(A) = (Cofactor matrix)ᵀA exists only if det(A) ≠ 0det(AB) = det(A) × det(B); det(Aᵀ) = det(A); det(kA) = kⁿ det(A) for n×n matrixSwapping rows changes sign; Same rows/columns → det = 0f 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 equalf'(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.(u + v)' = u' + v'; (uv)' = u'v + uv'; (u/v)' = (u'v - uv')/v²Product rule and quotient ruledy/dx = (dy/du) × (du/dx)For composite functions; if y = f(u) and u = g(x)dy/dx = (dy/dt)/(dx/dt)When x = f(t) and y = g(t)Differentiate both sides w.r.t. x and solve for dy/dxWhen y is not explicitly given as function of xFor y = u^v, take ln: ln y = v ln u, then differentiateUseful for functions like y = x^x or y = (sin x)^xd/dx(xⁿ) = nxⁿ⁻¹d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ ln a; d/dx(ln x) = 1/x; d/dx(log_a x) = 1/(x ln a)d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(tan x) = sec² xd/dx(cot x) = -cosec² x; d/dx(sec x) = sec x tan x; d/dx(cosec x) = -cosec x cot xd/dx(sin⁻¹ x) = 1/√(1-x²); d/dx(tan⁻¹ x) = 1/(1+x²); d/dx(sec⁻¹ x) = 1/(|x|√(x²-1))d/dx(sinh x) = cosh x; d/dx(cosh x) = sinh x; d/dx(tanh x) = sech² xsinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2Tangent: y - y₁ = f'(x₁)(x - x₁); Normal: y - y₁ = -(1/f'(x₁))(x - x₁)At point (x₁, y₁) on curve y = f(x)dy/dt = (dy/dx) × (dx/dt)Related rates problemsf'(x) > 0 ⟹ f increasing; f'(x) < 0 ⟹ f decreasingFirst derivative testCritical points: f'(x) = 0. Second derivative test: f''(x) > 0 (minimum), f''(x) < 0 (maximum)If f''(x) = 0, test furtherf(x + Δx) ≈ f(x) + f'(x) × Δxdy = f'(x) dxIf f continuous on [a, b] and differentiable on (a, b), then ∃ c ∈ (a, b): f'(c) = [f(b) - f(a)]/(b - a)∫ f(x) dx = F(x) + C if F'(x) = f(x)F(x) is antiderivative, C is constant of integration∫ 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∫ 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∫ 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∫ f(g(x)) g'(x) dx = ∫ f(u) du where u = g(x)Change of variable∫ u dv = uv - ∫ v duUse ILATE rule: Inverse, Logarithm, Algebraic, Trigonometric, ExponentialDecompose rational function into partial fractions before integratingFor P(x)/Q(x) where deg(P) < deg(Q)∫ₐᵇ f(x) dx = F(b) - F(a) where F is antiderivativeFundamental theorem of calculus∫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) dxLinearity and additivity propertiesOrder = highest derivative present; Degree = power of highest derivativeFor (d³y/dx³)² + (d²y/dx²) - 3dy/dx + y = 0: order 3, degree 2dy/dx = f(x)/g(y) ⟹ g(y) dy = f(x) dx ⟹ integrate both sidesVariables can be separateddy/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)dy/dx = f(y/x); substitute v = y/x ⟹ y = vx ⟹ dy = v dx + x dvReduces to separable form|a| = √(a₁² + a₂² + a₃²); unit vector = a/|a|For vector a = (a₁, a₂, a₃)a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃Angle between vectors: cos θ = (a·b)/(|a||b|)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 rulea · (b × c) = (a × b) · c = |a₁ a₂ a₃| = volume of parallelepiped|b₁ b₂ b₃|proj_b a = (a · b)/|b| = |a| cos θScalar projection; vector projection = (a·b/|b|²) × ba ⊥ b ⟺ a · b = 0; a ∥ b ⟺ a × b = 0(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 vectorcos θ = |a₁a₂ + b₁b₂ + c₁c₂| / (√(a₁² + b₁² + c₁²) × √(a₂² + b₂² + c₂²))For direction vectors (a₁, b₁, c₁) and (a₂, b₂, c₂)A(x - x₀) + B(y - y₀) + C(z - z₀) = 0 or Ax + By + Cz + D = 0(A, B, C) is normal vectorcos θ = |A₁A₂ + B₁B₂ + C₁C₂| / (√(A₁² + B₁² + C₁²) × √(A₂² + B₂² + C₂²))Normal vectors: (A₁, B₁, C₁) and (A₂, B₂, C₂)d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)Point (x₀, y₀, z₀) to plane Ax + By + Cz + D = 0P(A|B) = P(A ∩ B) / P(B)Probability of A given B has occurredP(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)P(A) = P(A|B₁)P(B₁) + P(A|B₂)P(B₂) + ... + P(A|Bₙ)P(Bₙ)If B₁, B₂, ..., Bₙ partition sample spaceP(Bᵢ|A) = P(A|Bᵢ)P(Bᵢ) / Σ P(A|Bⱼ)P(Bⱼ)Posterior probability; used for updating beliefsA and B independent ⟺ P(A ∩ B) = P(A) × P(B) ⟺ P(A|B) = P(A)P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏX ~ B(n, p); n trials, p = success probabilityE(X) = Σ xᵢ P(xᵢ)For binomial: E(X) = npVar(X) = E(X²) - [E(X)]² = Σ (xᵢ - μ)² P(xᵢ); σ = √Var(X)For binomial: Var(X) = np(1-p)