CBSE Class 12 Maths All Formulas List 2027 - Chapter Wise Complete Reference

In short: This is the complete chapter-wise formula reference for CBSE Class 12 Maths 2027. Covers all key formulas for Calculus (differentiation, integration), Algebra (matrices, determinants), Vectors and 3D Geometry, Probability, Relations and Functions, and Linear Programming. Bookmark this page and use it for daily revision during the last month of preparation.

How to Use This List

1. Calculus formulas are the backbone of the paper (35+ marks). Master differentiation and integration first.
2. Algebra formulas cover matrices and determinants (10-13 marks). Know the properties and shortcuts.
3. Vectors and 3D Geometry carry 14 marks. Memorize the distance, section, and angle formulas.
4. Probability carries 8-10 marks. Bayes' theorem and distribution formulas are critical.

Relations and Functions (Chapter 1)

Types of Relations:

• Reflexive: (a, a) ∈ R for all a
• Symmetric: (a, b) ∈ R ⟹ (b, a) ∈ R
• Transitive: (a, b) ∈ R and (b, c) ∈ R ⟹ (a, c) ∈ R
• Equivalence: Reflexive + Symmetric + Transitive

Types of Functions:

• One-one (injective): f(a₁) = f(a₂) ⟹ a₁ = a₂
• Onto (surjective): For every b in codomain, there exists a in domain such that f(a) = b
• Bijective: One-one and Onto

Inverse Trigonometric Functions (Chapter 2)

Principal Value Ranges:

• sin⁻¹x: [-π/2, π/2], domain [-1, 1]
• cos⁻¹x: [0, π], domain [-1, 1]
• tan⁻¹x: (-π/2, π/2), domain R

Key Properties:

• sin⁻¹x + cos⁻¹x = π/2

• tan⁻¹x + cot⁻¹x = π/2

• sec⁻¹x + cosec⁻¹x = π/2

• sin⁻¹(-x) = -sin⁻¹x

• cos⁻¹(-x) = π - cos⁻¹x

• tan⁻¹(-x) = -tan⁻¹x

• tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1-xy)), if xy < 1

• 2tan⁻¹x = sin⁻¹(2x/(1+x²)) = cos⁻¹((1-x²)/(1+x²)) = tan⁻¹(2x/(1-x²))

Matrices and Determinants (Chapters 3-4)

Matrix Operations:

• (A + B)' = A' + B'
• (AB)' = B'A'
• (kA)' = kA'

Properties of Determinants:

• |AB| = |A| × |B|
• |A'| = |A|
• |kA| = kⁿ|A| (for n×n matrix)
• |A⁻¹| = 1/|A|

Inverse of a Matrix: A⁻¹ = (1/|A|) × adj(A)

For 2×2 matrix: If A = [a b; c d], then A⁻¹ = (1/(ad-bc)) × [d -b; -c a]

Cramer's Rule (for solving equations): x = D₁/D, y = D₂/D, z = D₃/D

Area of triangle using determinants: Area = ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

Continuity and Differentiability (Chapter 5)

Standard Derivatives:

• d/dx(xⁿ) = nxⁿ⁻¹
• d/dx(eˣ) = eˣ
• d/dx(aˣ) = aˣ ln a
• 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
• d/dx(sec x) = sec x tan x
• d/dx(cosec x) = -cosec x cot x

Inverse Trigonometric Derivatives:

• d/dx(sin⁻¹x) = 1/√(1-x²)
• d/dx(cos⁻¹x) = -1/√(1-x²)
• d/dx(tan⁻¹x) = 1/(1+x²)

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))/v²

Logarithmic Differentiation: For y = f(x)^g(x), take log both sides first.

Applications of Derivatives (Chapter 6)

Rate of change: dy/dx at x = a gives rate of change of y w.r.t. x

Tangent at (x₁, y₁): y - y₁ = m(x - x₁), where m = dy/dx at (x₁, y₁)

Normal at (x₁, y₁): y - y₁ = (-1/m)(x - x₁)

Increasing function: f'(x) > 0 in interval Decreasing function: f'(x) < 0 in interval

Maxima/Minima:

• First derivative test: f'(x) changes from +ve to -ve → maxima; -ve to +ve → minima
• Second derivative test: f''(x) < 0 → maxima; f''(x) > 0 → minima

Integrals (Chapter 7)

Standard Integrals:

• ∫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
• ∫sec²x dx = tan x + C
• ∫cosec²x dx = -cot x + C
• ∫sec x tan x dx = sec x + C
• ∫cosec x cot x dx = -cosec x + C

Important Integrals:

• ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a) + C
• ∫dx/√(a²-x²) = sin⁻¹(x/a) + C
• ∫dx/√(x²+a²) = ln|x + √(x²+a²)| + C
• ∫dx/√(x²-a²) = ln|x + √(x²-a²)| + C
• ∫dx/(x²-a²) = (1/2a)ln|(x-a)/(x+a)| + C
• ∫dx/(a²-x²) = (1/2a)ln|(a+x)/(a-x)| + C

Integration by Parts: ∫u v dx = u∫v dx - ∫(du/dx × ∫v dx) dx

ILATE order: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential

Special integral: ∫eˣ[f(x) + f'(x)] dx = eˣf(x) + C

Properties of Definite Integrals:

• ∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx
• ∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx
• ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx
• ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx
• ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even, = 0 if f is odd

Applications of Integrals (Chapter 8)

Area under curve: A = ∫ₐᵇ y dx (between curve and x-axis)

Area between two curves: A = ∫ₐᵇ |f(x) - g(x)| dx

Differential Equations (Chapter 9)

Order: Highest order derivative present Degree: Highest power of highest order derivative (after removing fractions)

Variable Separable: dy/dx = f(x)g(y) → ∫dy/g(y) = ∫f(x)dx

Homogeneous: dy/dx = F(y/x), substitute y = vx

Linear DE: dy/dx + Py = Q Solution: y × IF = ∫Q × IF dx + C, where IF = e^(∫P dx)

Vectors (Chapter 10)

Position vector: OP = xi + yj + zk

Magnitude: |a| = √(a₁² + a₂² + a₃²)

Unit vector: â = a/|a|

Section formula: Internal: (mb + na)/(m+n) External: (mb - na)/(m-n)

Dot product: a · b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃

Cross product: |a × b| = |a||b|sin θ

a × b = |i j k| |a₁ a₂ a₃| |b₁ b₂ b₃|

Projection of a on b: (a · b)/|b|

Area of parallelogram: |a × b| Area of triangle: ½|a × b|

Three Dimensional Geometry (Chapter 11)

Direction cosines: l² + m² + n² = 1

Distance formula: d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)

Equation of line (vector form): r = a + λb

Equation of line (Cartesian): (x-x₁)/a = (y-y₁)/b = (z-z₁)/c

Angle between two lines: cos θ = |l₁l₂ + m₁m₂ + n₁n₂| or cos θ = |(a·b)/(|a||b|)|

Shortest distance between skew lines: d = |(b₁ × b₂) · (a₂ - a₁)| / |b₁ × b₂|

Equation of plane (vector form): r · n = d

Equation of plane (Cartesian): ax + by + cz = d

Distance of point from plane: d = |ax₁ + by₁ + cz₁ - d| / √(a² + b² + c²)

Angle between two planes: cos θ = |(n₁ · n₂)/(|n₁||n₂|)|

Linear Programming (Chapter 12)

Steps:

1. Formulate objective function: Z = ax + by
2. Identify constraints as linear inequalities.
3. Plot the feasible region.
4. Find corner points.
5. Evaluate Z at each corner point.
6. Maximum/minimum Z is the answer.

Probability (Chapter 13)

Conditional Probability: P(A|B) = P(A∩B)/P(B)

Multiplication Rule: P(A∩B) = P(A) × P(B|A) = P(B) × P(A|B)

Bayes' Theorem: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ)

Independent Events: P(A∩B) = P(A) × P(B)

Binomial Distribution: P(X = r) = ⁿCᵣ × pʳ × qⁿ⁻ʳ

Mean = np, Variance = npq (where q = 1-p)

Quick Revision Tips

1. Integration is the biggest section. Practice at least 5 integrals daily. Cover all types: substitution, partial fractions, by parts.
2. Vectors + 3D Geometry carry 14 marks combined. Learn the formulas and practice 5 problems daily.
3. Write formulas from memory every morning. Check against this list and note what you missed.
4. Determinant properties save time. Learn all 7 properties and use them to simplify calculations.
5. Bayes' theorem is a guaranteed 4-5 mark question. Practice at least 10 different problems.

For chapter-wise marks distribution, see our CBSE Class 12 Maths marks weightage guide.

Frequently Asked Questions

Which chapter has the most formulas?

Integration (Chapter 7) has the most formulas with 20+ standard integrals plus integration techniques. Vectors and 3D Geometry combined come second.

How should I memorize so many formulas?

Group them by chapter. Write all formulas from one chapter on a single page. Practice writing them from memory daily. Start with integration and derivatives since they carry the most marks.

Are all formulas tested in the exam?

The most frequently tested formulas are: integration standard forms, Bayes' theorem, 3D geometry distance formulas, matrix inverse, and differentiation rules. Focus on these first, then cover the rest.

Should I derive formulas or memorize them?

Memorize the standard ones (integration table, derivative table). Understand the derivation of key formulas (section formula, Bayes' theorem, integration by parts) since understanding helps retention.

How many marks can formulas alone help me score?

If you know all the formulas and can apply them correctly, you can potentially score 60+ out of 80. The remaining marks require problem-solving skills and practice.

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