In short: Class 12 Maths has 13 chapters with over 200 formulas covering Calculus, Algebra, Vectors, 3D Geometry, and Probability. This complete formula sheet organises every important equation chapter-wise for CBSE 2027 board exam revision. Integration formulas alone fill a full page - and they are all here.
Why This Formula Sheet Matters for Your Board Score
Class 12 Maths is worth 80 marks in CBSE boards. The paper is roughly split into Calculus (~44 marks), Algebra (~10 marks), Vectors and 3D Geometry (~14 marks), Linear Programming (~5 marks), and Probability (~8 marks).
Calculus dominates. Within Calculus, Integration carries the most weight. If you know your integration formulas cold, you can secure 20+ marks from that topic alone.
This sheet follows the CBSE 2026-27 syllabus. Every formula is listed with the chapter it belongs to. Let us dive in.
Chapter 1 - Relations and Functions
| Formula | Expression |
|---|---|
| Reflexive relation | (a, a) in R for all a in A |
| Symmetric relation | (a, b) in R implies (b, a) in R |
| Transitive relation | (a, b) in R and (b, c) in R implies (a, c) in R |
| Equivalence relation | Reflexive + Symmetric + Transitive |
| One-one (injective) | f(a) = f(b) implies a = b |
| Onto (surjective) | For every b in codomain, there exists a in domain with f(a) = b |
| Bijective | One-one + Onto |
| Number of functions from A to B | n(B)^n(A) |
| Number of one-one functions | n(B)! / (n(B)-n(A))! if n(B) >= n(A), else 0 |
Chapter 2 - Inverse Trigonometric Functions
| Formula | Expression | Domain | Range |
|---|---|---|---|
| sin^-1(x) | y such that sin(y) = x | [-1, 1] | [-pi/2, pi/2] |
| cos^-1(x) | y such that cos(y) = x | [-1, 1] | [0, pi] |
| tan^-1(x) | y such that tan(y) = x | R | (-pi/2, pi/2) |
| sin^-1(x) + cos^-1(x) | = pi/2 | - | - |
| tan^-1(x) + cot^-1(x) | = pi/2 | - | - |
| sec^-1(x) + cosec^-1(x) | = pi/2 | - | - |
| tan^-1(x) + tan^-1(y) | = tan^-1((x+y)/(1-xy)), xy < 1 | - | - |
| tan^-1(x) - tan^-1(y) | = tan^-1((x-y)/(1+xy)), xy > -1 | - | - |
| 2 tan^-1(x) | = sin^-1(2x/(1+x^2)) = cos^-1((1-x^2)/(1+x^2)) = tan^-1(2x/(1-x^2)) | - | - |
| sin^-1(-x) | = -sin^-1(x) | - | - |
| cos^-1(-x) | = pi - cos^-1(x) | - | - |
| tan^-1(-x) | = -tan^-1(x) | - | - |
Chapter 3 - Matrices
| Formula | Expression |
|---|---|
| Matrix addition | A + B = a(ij) + b(ij) |
| Scalar multiplication | kA = k a(ij) |
| Matrix multiplication | AB = Sum over r: a(ir) b(rj) |
| Transpose | (A')ij = a(ji) |
| (AB)' | = B'A' |
| (A')' | = A |
| Symmetric matrix | A' = A |
| Skew-symmetric matrix | A' = -A |
| Any square matrix | A = (1/2)(A + A') + (1/2)(A - A') (symmetric + skew-symmetric) |
| Invertible matrix | A^-1 = adj(A) / |
| A x A^-1 | = I (identity matrix) |
Chapter 4 - Determinants
| Formula | Expression |
|---|---|
| 2x2 determinant | |
| 3x3 determinant | Expand along any row/column using cofactors |
| Cofactor | C(ij) = (-1)^(i+j) M(ij), where M(ij) is the minor |
| Adjoint | adj(A) = transpose of cofactor matrix |
| Inverse | A^-1 = adj(A) / |
| Area of triangle | (1/2) |
| Cramer's rule (2 variables) | x = D1/D, y = D2/D where D is not 0 |
| Property | |
| Property | |
| Property |
Chapter 5 - Continuity and Differentiability
| Formula | Expression |
|---|---|
| Continuity at x = a | lim(x->a-) f(x) = lim(x->a+) f(x) = f(a) |
| d/dx (x^n) | = n x^(n-1) |
| d/dx (sin x) | = cos x |
| d/dx (cos x) | = -sin x |
| d/dx (tan x) | = sec^2 x |
| d/dx (cot x) | = -cosec^2 x |
| d/dx (sec x) | = sec x tan x |
| d/dx (cosec x) | = -cosec x cot x |
| d/dx (e^x) | = e^x |
| d/dx (a^x) | = a^x ln(a) |
| d/dx (ln x) | = 1/x |
| d/dx (log_a x) | = 1/(x ln a) |
| d/dx (sin^-1 x) | = 1/sqrt(1-x^2) |
| d/dx (cos^-1 x) | = -1/sqrt(1-x^2) |
| d/dx (tan^-1 x) | = 1/(1+x^2) |
| Chain rule | d/dx [f(g(x))] = f'(g(x)) g'(x) |
| Product rule | d/dx (uv) = u dv/dx + v du/dx |
| Quotient rule | d/dx (u/v) = (v du/dx - u dv/dx) / v^2 |
| Logarithmic differentiation | For y = f(x)^g(x), take ln both sides |
| Rolle's theorem | If f(a) = f(b), then f'(c) = 0 for some c in (a,b) |
| LMVT | f'(c) = (f(b) - f(a)) / (b - a) for some c in (a,b) |
Chapter 6 - Applications of Derivatives
| Formula | Expression |
|---|---|
| Rate of change | dy/dx at a point |
| Tangent slope | m = dy/dx at (x1, y1) |
| Tangent equation | y - y1 = m(x - x1) |
| Normal equation | y - y1 = (-1/m)(x - x1) |
| Increasing function | f'(x) > 0 |
| Decreasing function | f'(x) < 0 |
| Local maximum | f'(x) = 0 and f''(x) < 0 |
| Local minimum | f'(x) = 0 and f''(x) > 0 |
| Point of inflection | f''(x) = 0 and sign of f'' changes |
| Approximation | f(x + Delta x) approximately equal to f(x) + f'(x) Delta x |
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Chapter 7 - Integrals
This is the longest and most important formula section.
Basic integration formulas:
| Formula | Expression |
|---|---|
| Integral x^n dx | = x^(n+1)/(n+1) + C (n is not -1) |
| Integral 1/x dx | = ln |
| Integral e^x dx | = e^x + C |
| Integral a^x dx | = a^x / ln(a) + C |
| Integral sin x dx | = -cos x + C |
| Integral cos x dx | = sin x + C |
| Integral sec^2 x dx | = tan x + C |
| Integral cosec^2 x dx | = -cot x + C |
| Integral sec x tan x dx | = sec x + C |
| Integral cosec x cot x dx | = -cosec x + C |
| Integral tan x dx | = ln |
| Integral cot x dx | = ln |
| Integral sec x dx | = ln |
| Integral cosec x dx | = ln |
Important standard integrals:
| Formula | Expression |
|---|---|
| Integral dx / (x^2 + a^2) | = (1/a) tan^-1(x/a) + C |
| Integral dx / (x^2 - a^2) | = (1/2a) ln |
| Integral dx / (a^2 - x^2) | = (1/2a) ln |
| Integral dx / sqrt(a^2 - x^2) | = sin^-1(x/a) + C |
| Integral dx / sqrt(x^2 + a^2) | = ln |
| Integral dx / sqrt(x^2 - a^2) | = ln |
| Integral sqrt(a^2 - x^2) dx | = (x/2)sqrt(a^2 - x^2) + (a^2/2) sin^-1(x/a) + C |
| Integral sqrt(x^2 + a^2) dx | = (x/2)sqrt(x^2 + a^2) + (a^2/2) ln |
| Integral sqrt(x^2 - a^2) dx | = (x/2)sqrt(x^2 - a^2) - (a^2/2) ln |
Integration techniques:
| Method | When to Use |
|---|---|
| Substitution | When integrand has a function and its derivative |
| By parts | Integral u dv = uv - Integral v du (ILATE rule for choosing u) |
| Partial fractions | For rational functions P(x)/Q(x) where degree P < degree Q |
| ILATE priority | Inverse trig > Log > Algebraic > Trig > Exponential |
Definite integral properties:
| Property | Expression |
|---|---|
| Integral(a to b) f(x) dx | = F(b) - F(a) |
| Integral(a to b) f(x) dx | = -Integral(b to a) f(x) dx |
| Integral(a to b) f(x) dx | = Integral(a to c) f(x) dx + Integral(c to b) f(x) dx |
| Integral(0 to a) f(x) dx | = Integral(0 to a) f(a-x) dx |
| Integral(-a to a) f(x) dx | = 2 Integral(0 to a) f(x) dx if f is even, 0 if f is odd |
| Integral(0 to 2a) f(x) dx | = 2 Integral(0 to a) f(x) dx if f(2a-x) = f(x), 0 if f(2a-x) = -f(x) |
Chapter 8 - Applications of Integrals
| Formula | Expression |
|---|---|
| Area under curve | A = Integral(a to b) f(x) dx |
| Area between two curves | A = Integral(a to b) |
| Area (y-axis) | A = Integral(c to d) g(y) dy |
Chapter 9 - Differential Equations
| Formula | Expression |
|---|---|
| Order | Highest derivative in the equation |
| Degree | Power of the highest order derivative (when polynomial) |
| Variable separable | dy/dx = f(x)g(y), separate and integrate |
| Homogeneous | dy/dx = f(y/x), substitute y = vx |
| Linear (first order) | dy/dx + P(x)y = Q(x), IF = e^(Integral P dx) |
| Solution | y x IF = Integral (Q x IF) dx + C |
Chapter 10 - Vector Algebra
| Formula | Expression |
|---|---|
| Magnitude | |
| Unit vector | a-hat = a/ |
| Dot product | a . b = |
| Cross product magnitude | |
| Cross product | a x b = |
| Projection of a on b | (a . b) / |
| Perpendicular vectors | a . b = 0 |
| Parallel vectors | a x b = 0 |
| Area of parallelogram | |
| Area of triangle | (1/2) |
| Scalar triple product | [a b c] = a . (b x c) |
| Volume of parallelepiped | |
| Coplanar condition | [a b c] = 0 |
Chapter 11 - Three-Dimensional Geometry
| Formula | Expression |
|---|---|
| Direction cosines | l = a/r, m = b/r, n = c/r where r = sqrt(a^2+b^2+c^2) |
| l^2 + m^2 + n^2 | = 1 |
| Line (vector form) | r = a + lambda b |
| Line (Cartesian) | (x-x1)/a = (y-y1)/b = (z-z1)/c |
| Angle between lines | cos(theta) = |
| Parallel lines | a1/a2 = b1/b2 = c1/c2 |
| Perpendicular lines | a1a2 + b1b2 + c1c2 = 0 |
| Plane (vector form) | r . n = d |
| Plane (Cartesian) | ax + by + cz = d |
| Distance from point to plane | d = |
| Angle between line and plane | sin(theta) = (b . n) / ( |
| Angle between two planes | cos(theta) = (n1 . n2) / ( |
| Shortest distance (skew lines) | d = |
Chapter 12 - Linear Programming
| Concept | Formula / Rule |
|---|---|
| Objective function | Z = ax + by (maximise or minimise) |
| Feasible region | Set of all points satisfying all constraints |
| Corner point method | Evaluate Z at each vertex of feasible region |
| Optimal solution | Z is maximum/minimum at a corner point |
Chapter 13 - Probability
| Formula | Expression |
|---|---|
| Conditional probability | P(A |
| Multiplication rule | P(A ∩ B) = P(A) P(B |
| Independent events | P(A ∩ B) = P(A) P(B) |
| Total probability | P(A) = Sum P(B_i) P(A |
| Bayes' theorem | P(B_i |
| Binomial distribution | P(X=r) = nCr p^r q^(n-r), where q = 1-p |
| Mean (binomial) | E(X) = np |
| Variance (binomial) | Var(X) = npq |
| Standard deviation | SD = sqrt(npq) |
How to Memorise 200+ Maths Formulas
1. Master integration formulas first. Integration has the most formulas and carries the most marks. Write out all standard integrals on a single sheet. Practise 3 integrals a day from different categories. Within 2 weeks, they become reflexive.
2. Link derivative and integral formulas. Every derivative formula has a corresponding integral. d/dx(sin x) = cos x means integral cos x dx = sin x + C. Learning them in pairs cuts your memorisation effort in half.
3. Use the ILATE mnemonic. For integration by parts, ILATE tells you which function to pick as u: Inverse trig > Log > Algebraic > Trig > Exponential. This one mnemonic solves dozens of by-parts problems.
4. Practise vector formulas with geometry. Dot product, cross product, and scalar triple product become intuitive when you visualise them as angles, areas, and volumes. Draw the vectors, then write the formula.
5. Solve Bayes' theorem problems step by step. Probability formulas are best learned through tree diagrams. Draw the probability tree, label each branch, and the formula applies itself.
Common Mistakes in Class 12 Maths
- Forgetting the constant of integration. Every indefinite integral must end with + C. Losing this mark on every integration question adds up quickly.
- Wrong ILATE choice. In integration by parts, choosing the wrong u makes the integral harder. Follow ILATE strictly unless you have a clear reason to deviate.
- Confusing dot and cross product. Dot product gives a scalar (number). Cross product gives a vector. Using the wrong one changes your answer type entirely.
- Ignoring domain restrictions in inverse trig. sin^-1(x) exists only for x in [-1, 1]. tan^-1(x + y) formula requires xy < 1. Violating these conditions gives wrong answers.
- Not checking feasible region in LPP. Plot all constraints, shade the feasible region, identify corner points, then evaluate. Skipping the graph leads to wrong vertices.
Exam-Day Quick Revision Strategy
Minutes 1-15: Integration formulas - all basic integrals, standard forms (1/(x^2+a^2), 1/sqrt(a^2-x^2)), definite integral properties. This is your highest-value revision.
Minutes 16-25: Differentiation formulas, inverse trig derivatives, and applications (tangent/normal equations, maxima/minima conditions).
Minutes 26-35: Vector and 3D geometry - dot product, cross product, line and plane equations, shortest distance formula.
Minutes 36-40: Probability - Bayes' theorem, binomial distribution formulas (mean = np, variance = npq). Matrices inverse formula.
FAQ
How many integration formulas should I know?
About 30-35 integration formulas (basic + standard forms). Beyond that, you need to know integration techniques (substitution, by parts, partial fractions) rather than more formulas.
Which chapter carries the most marks?
Integrals (Chapter 7) and Applications of Integrals (Chapter 8) together carry about 18-20 marks. Continuity and Differentiability adds another 8-10 marks. Calculus as a whole is about 44 marks out of 80.
Is Bayes' theorem difficult?
The formula looks intimidating but is straightforward once you use a probability tree. Draw the tree, multiply along branches, divide by total. Practise 5 problems and it becomes mechanical.
Should I memorise all inverse trig identities?
Yes, especially sin^-1(x) + cos^-1(x) = pi/2, the tan^-1 addition formula, and 2tan^-1(x) expansions. These appear as 2-3 mark questions regularly.
Are Class 12 Maths formulas enough for JEE?
The formulas are identical. JEE tests them in more complex multi-step problems. You need strong problem-solving skills on top of formula knowledge. Padhaao can help you practise at JEE difficulty level.
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