CBSE · Class 12
Maths Formulas for Class 12
All important Maths formulas chapter-wise for CBSE Class 12 (2026-27) — Calculus, Vectors, 3D Geometry, Matrices, Determinants and Probability.
For reference only. Always verify formulas and information against your official NCERT textbooks or board-prescribed materials before use in exams. Learnizo does not guarantee absolute accuracy.
Ch 1.Relations and Functions
| Formula / Relation | Expression |
|---|---|
| Reflexive relation | (a, a) ∈ R for all a ∈ A |
| Symmetric relation | (a,b) ∈ R ⇒ (b,a) ∈ R |
| Transitive relation | (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R |
| Composition of functions | (g∘f)(x) = g(f(x)) |
| Bijection condition | f is bijective ⟺ f is injective AND surjective |
Ch 2.Inverse Trigonometric Functions
| Formula / Relation | Expression |
|---|---|
| sin⁻¹(−x) | = −sin⁻¹(x) |
| cos⁻¹(−x) | = π − cos⁻¹(x) |
| sin⁻¹x + cos⁻¹x | = π/2 |
| tan⁻¹x + cot⁻¹x | = π/2 |
| tan⁻¹x + tan⁻¹y | = tan⁻¹[(x+y)/(1−xy)], xy < 1 |
| 2tan⁻¹x | = sin⁻¹(2x/(1+x²)) = cos⁻¹((1−x²)/(1+x²)) |
Ch 3.Matrices
| Formula / Relation | Expression |
|---|---|
| Order of product AB | If A is m×n and B is n×p, AB is m×p |
| Transpose property | (AB)ᵀ = BᵀAᵀ |
| Symmetric matrix | Aᵀ = A |
| Skew-symmetric matrix | Aᵀ = −A; diagonal elements = 0 |
| Inverse of 2×2 matrix | A⁻¹ = (1/|A|) × adj(A) |
Ch 4.Determinants
| Formula / Relation | Expression |
|---|---|
| 2×2 Determinant | |A| = ad − bc for A = [[a,b],[c,d]] |
| Area of triangle | Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| |
| Adjoint | adj(A) = transpose of cofactor matrix |
| A · adj(A) | = |A| · I |
| Inverse via determinant | A⁻¹ = adj(A) / |A|, if |A| ≠ 0 |
| Cramer's Rule | x = D₁/D, y = D₂/D, z = D₃/D |
Ch 5.Continuity and Differentiability
| Formula / Relation | Expression |
|---|---|
| Continuity at x=a | lim(x→a) f(x) = f(a) |
| Chain rule | d/dx[f(g(x))] = f'(g(x)) · g'(x) |
| Product rule | d/dx(uv) = u'v + uv' |
| Quotient rule | d/dx(u/v) = (u'v − uv') / v² |
| 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(sin⁻¹x) | = 1/√(1−x²) |
| d/dx(cos⁻¹x) | = −1/√(1−x²) |
| d/dx(tan⁻¹x) | = 1/(1+x²) |
| Rolle's Theorem | If f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⇒ ∃c: f'(c)=0 |
| Mean Value Theorem | f'(c) = [f(b)−f(a)] / (b−a) |
Ch 6.Applications of Derivatives
| Formula / Relation | Expression |
|---|---|
| Rate of change | dy/dx at x=a is the rate of change of y w.r.t. x at a |
| Equation of tangent | y − y₁ = m(x − x₁), m = dy/dx at (x₁,y₁) |
| Equation of normal | y − y₁ = −(1/m)(x − x₁) |
| Increasing function | f'(x) > 0 on (a,b) |
| Decreasing function | f'(x) < 0 on (a,b) |
| Second derivative test | f''(c) < 0 → local max; f''(c) > 0 → local min |
| Maxima/Minima (first derivative) | f'(c) = 0 and sign changes → critical point |
Ch 7.Integrals
| Formula / Relation | Expression |
|---|---|
| ∫xⁿ dx | = xⁿ⁺¹/(n+1) + C, n ≠ −1 |
| ∫eˣ dx | = eˣ + C |
| ∫(1/x) dx | = ln|x| + C |
| ∫sin x dx | = −cos x + C |
| ∫cos x dx | = sin x + C |
| ∫sec²x dx | = tan x + C |
| ∫cosec²x dx | = −cot x + C |
| ∫sec x tan x dx | = sec x + C |
| ∫1/√(1−x²) dx | = sin⁻¹x + C |
| ∫1/(1+x²) dx | = tan⁻¹x + C |
| Integration by parts | ∫u dv = uv − ∫v du |
| Definite integral property | ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b−x)dx |
| ∫₀²ᵃ f(x)dx | = 2∫₀ᵃ f(x)dx if f(2a−x)=f(x) |
Ch 8.Applications of Integrals
| Formula / Relation | Expression |
|---|---|
| Area between curve and x-axis | A = ∫ₐᵇ |f(x)| dx |
| Area between two curves | A = ∫ₐᵇ [f(x) − g(x)] dx (f(x) ≥ g(x)) |
Ch 9.Differential Equations
| Formula / Relation | Expression |
|---|---|
| Order and Degree | Order = highest derivative; Degree = power of highest derivative |
| Variable separable | dy/dx = f(x)g(y) ⇒ ∫dy/g(y) = ∫f(x)dx |
| Linear DE (1st order) | dy/dx + Py = Q; IF = e^(∫P dx); solution: y·IF = ∫Q·IF dx |
Ch 10.Vector Algebra
| Formula / Relation | Expression |
|---|---|
| Magnitude | |a⃗| = √(a₁²+a₂²+a₃²) |
| Unit vector | â = a⃗ / |a⃗| |
| Dot product | a⃗·b⃗ = |a⃗||b⃗|cos θ = a₁b₁+a₂b₂+a₃b₃ |
| Cross product magnitude | |a⃗×b⃗| = |a⃗||b⃗|sin θ |
| Area of parallelogram | A = |a⃗ × b⃗| |
| Area of triangle | A = ½|a⃗ × b⃗| |
| Projection of a⃗ on b⃗ | (a⃗·b⃗)/|b⃗| |
Ch 11.Three Dimensional Geometry
| Formula / Relation | Expression |
|---|---|
| Direction cosines | l² + m² + n² = 1 |
| Distance between two points | d = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²] |
| Angle between two lines | cos θ = |l₁l₂+m₁m₂+n₁n₂| |
| Equation of plane | ax + by + cz + d = 0 |
| Distance from point to plane | d = |ax₁+by₁+cz₁+d| / √(a²+b²+c²) |
| Angle between two planes | cos θ = |a₁a₂+b₁b₂+c₁c₂| / (√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)) |
Ch 12.Linear Programming
| Formula / Relation | Expression |
|---|---|
| Corner point method | Optimal value of Z = ax + by occurs at one of the corner points of feasible region |
| Feasible region | Set of all points satisfying all constraints |
Ch 13.Probability
| Formula / Relation | Expression |
|---|---|
| Conditional Probability | P(A|B) = P(A∩B) / P(B) |
| Multiplication theorem | P(A∩B) = P(A) · P(B|A) |
| Independent events | P(A∩B) = P(A) · P(B) |
| Total Probability | P(A) = ΣP(Bᵢ)·P(A|Bᵢ) |
| Bayes' Theorem | P(Bᵢ|A) = P(Bᵢ)·P(A|Bᵢ) / ΣP(Bⱼ)·P(A|Bⱼ) |
| Binomial distribution mean | μ = np |
| Binomial distribution variance | σ² = npq |
| P(X=r) Binomial | = ⁿCᵣ · pʳ · qⁿ⁻ʳ |
Frequently Asked Questions
Which chapters in Class 12 Maths have the most formulas?
Calculus chapters — Continuity & Differentiability, Integrals, and Applications of Derivatives — have the most formulas and carry ~44 marks in the CBSE board exam. Vectors and 3D Geometry together carry another ~17 marks. Focus these chapters first.
What is the integration by parts formula?
∫u dv = uv − ∫v du. Use the ILATE rule to choose u: Inverse trig, Logarithm, Algebra, Trigonometry, Exponential — pick the one that comes first in this order as u. For example, for ∫x·eˣ dx, choose u=x (Algebra) and dv=eˣ dx.
How do you find the inverse of a 2×2 matrix?
For A = [[a,b],[c,d]], |A| = ad−bc. adj(A) = [[d,−b],[−c,a]]. Then A⁻¹ = adj(A)/|A|. If |A|=0, the matrix is singular and has no inverse.
What is Bayes' theorem in simple terms?
Bayes' theorem updates the probability of a cause given that an effect is observed: P(Bᵢ|A) = P(Bᵢ)·P(A|Bᵢ) / ΣP(Bⱼ)·P(A|Bⱼ). It appears as a 5–6 mark problem in CBSE boards using conditional probability trees.
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