CBSE Class 12 Maths Revision Notes

CBSE Class 12 Mathematics covers functions, algebra, calculus, vectors, geometry, linear programming and probability through 13 NCERT chapters. These chapter-wise notes organise the main concepts, formulas and problem-solving methods required for CBSE preparation in India.

Class 12 Mathematics develops ideas introduced in earlier classes and applies them to more advanced problems. Its chapters cover algebraic operations, changing quantities, spatial relationships, optimisation and uncertainty.

Use these CBSE Class 12 Maths Revision Notes for the 2026–27 academic session to revise important definitions, formulas and methods. The notes follow the current NCERT chapter sequence and help students identify the main focus of every chapter.

Key Takeaways

• 13 chapters: The current NCERT Mathematics syllabus is divided between Part I and Part II.
• Five broad areas: The chapters cover functions, algebra, calculus, vectors and geometry, optimisation and probability.
• Calculus sequence: Continuity, derivatives, integrals and differential equations form one connected group.
• Formula recall: Most chapters require students to understand when and how a formula applies.

Access CBSE Class 12 Maths Revision Notes in 30 Minutes

Revise the complete syllabus in three parts:

• First 10 minutes: Relations and functions, inverse trigonometric functions, matrices and determinants
• Next 10 minutes: Continuity, differentiation, applications of derivatives, integrals and differential equations
• Final 10 minutes: Vector algebra, three-dimensional geometry, linear programming and probability

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Complete Chapter List for CBSE Class 12 Maths Revision Notes

The Class 12 Maths revision notes chapter wise sequence begins with relations and ends with probability. Each chapter introduces a separate concept while also connecting with earlier topics.

Algebra and Functions in Class 12 Maths Notes

The opening chapters establish the language of functions and matrix-based algebra. They provide essential methods used in later mathematical applications.

Relations and Functions

A relation from set A to set B is a subset of A × B. It connects selected elements of one set with elements of another set.

The chapter covers reflexive, symmetric, transitive and equivalence relations. A relation becomes an equivalence relation when it satisfies all three required properties.

A function assigns exactly one element of the codomain to each element of its domain. Important types include one-one and onto functions.

For two functions f and g:

(g o f)(x) = g(f(x))

A function is invertible when it is both one-one and onto. Its inverse reverses the original mapping.

Inverse Trigonometric Functions

Inverse trigonometric functions return angles for given trigonometric values. Their domains are restricted so that each inverse gives only one principal value.

For example:

sin⁻¹x has domain [-1, 1] and principal value range [-π/2, π/2].

cos⁻¹x has domain [-1, 1] and principal value range [0, π].

An important relation is:

sin⁻¹x + cos⁻¹x = π/2

Students need to distinguish between sin⁻¹x and 1/sin x. The expression sin⁻¹x represents an inverse function, while cosec x represents the reciprocal.

Matrices

A matrix is a rectangular arrangement of numbers or functions in rows and columns. Its order is written as m × n.

Two matrices can be added only when they have the same order. Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second.

In general:

AB ≠ BA

The transpose of matrix A is represented by Aᵀ. It is obtained by interchanging rows and columns.

Important transpose properties include:

(Aᵀ)ᵀ = A

(A + B)ᵀ = Aᵀ + Bᵀ

(AB)ᵀ = BᵀAᵀ

A square matrix A is symmetric if Aᵀ = A. It is skew-symmetric if Aᵀ = -A.

Determinants

A determinant is a scalar value associated with a square matrix. For a 2 × 2 matrix:

|a b|
|c d|

The determinant is:

ad - bc

Interchanging two rows or columns changes the sign of a determinant. If two rows or columns are identical or proportional, the determinant is zero.

The minor Mᵢⱼ is obtained after deleting the ith row and jth column. The corresponding cofactor is:

Aᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ

For a square matrix A:

A⁻¹ = adj A / |A|, provided |A| ≠ 0

A matrix with a non-zero determinant is invertible.

Calculus Concepts in Maths Class 12 Notes

Calculus studies changing quantities and accumulation. The five calculus chapters move from continuity and derivatives to integration and differential equations.

Continuity and Differentiability

A function f is continuous at x = a when:

lim x→a f(x) = f(a)

This condition requires the left-hand limit, right-hand limit and function value to be equal.

Differentiability refers to the existence of a derivative. Every differentiable function is continuous, but every continuous function need not be differentiable.

Important derivative rules include:

d/dx (u + v) = du/dx + dv/dx

d/dx (uv) = u(dv/dx) + v(du/dx)

d/dx (u/v) = [v(du/dx) - u(dv/dx)]/v²

For a composite function:

dy/dx = dy/du × du/dx

This is the chain rule.

The chapter also covers derivatives of exponential and logarithmic functions, logarithmic differentiation, parametric forms and second-order derivatives.

Applications of Derivatives

Derivatives measure the rate at which one quantity changes with another. If y = f(x), then dy/dx gives the instantaneous rate of change.

A function is increasing in an interval when:

f′(x) > 0

It is decreasing when:

f′(x) < 0

At a possible local maximum or minimum:

f′(x) = 0

The sign of f′(x) around the critical point helps identify the nature of the point. The second derivative may also be used where applicable.

If f′′(a) < 0, f(a) may be a local maximum.

If f′′(a) > 0, f(a) may be a local minimum.

Integrals

Integration is the inverse process of differentiation. An indefinite integral represents a family of antiderivatives:

∫f(x) dx = F(x) + C

Here, F′(x) = f(x), and C is the constant of integration.

Common results include:

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ -1

∫1/x dx = log |x| + C

∫eˣ dx = eˣ + C

∫sin x dx = -cos x + C

∫cos x dx = sin x + C

Major integration methods include substitution, partial fractions and integration by parts.

Integration by parts is:

∫u dv = uv - ∫v du

For a definite integral:

∫ₐᵇ f(x) dx = F(b) - F(a)

The Fundamental Theorem of Calculus connects definite integration with antiderivatives.

Applications of Integrals

Definite integrals can calculate the area bounded by curves and coordinate axes.

When a curve y = f(x) lies above the x-axis from x = a to x = b:

Area = ∫ₐᵇ f(x) dx

For two curves y = f(x) and y = g(x), where f(x) ≥ g(x):

Area = ∫ₐᵇ [f(x) - g(x)] dx

A sketch helps identify the upper curve, lower curve and correct limits. The interval may need to be divided when the curves cross.

Differential Equations

A differential equation contains derivatives of a dependent variable with respect to an independent variable.

The order is the order of the highest derivative present. The degree is the power of the highest-order derivative after removing radicals and fractions involving derivatives.

A general solution contains arbitrary constants. A particular solution is obtained after applying the given conditions.

For a variable-separable equation:

dy/dx = g(x)h(y)

Separate the variables:

dy/h(y) = g(x) dx

Then integrate both sides.

A homogeneous differential equation can often be solved using the substitution:

y = vx

Therefore:

dy/dx = v + x(dv/dx)

Vector Algebra and Three-Dimensional Geometry Notes

Vectors represent quantities with magnitude and direction. Three-dimensional geometry uses vector ideas to describe lines and their positions in space.

Vector Algebra

A vector is commonly written as:

a = a₁i + a₂j + a₃k

Its magnitude is:

|a| = √(a₁² + a₂² + a₃²)

A unit vector in the direction of a is:

â = a/|a|

The dot product of vectors a and b is:

a · b = |a||b| cos θ

It produces a scalar quantity.

The cross product is:

a × b = |a||b| sin θ n̂

It produces a vector perpendicular to both original vectors.

The dot product helps calculate angles and projections. The cross product helps calculate areas and perpendicular directions.

Three-Dimensional Geometry

Direction cosines describe the angles made by a line with the coordinate axes.

If l, m and n are the direction cosines, then:

l² + m² + n² = 1

A vector equation of a line passing through point a and parallel to vector b is:

r = a + λb

The Cartesian form is:

(x - x₁)/a = (y - y₁)/b = (z - z₁)/c

The angle between two lines is found using their direction ratios.

For direction vectors b₁ and b₂:

cos θ = (b₁ · b₂)/(|b₁||b₂|)

The chapter also examines the shortest distance between two lines.

Linear Programming and Probability Revision Notes

These chapters deal with decision-making under restrictions and the mathematical study of uncertain events.

Linear Programming

A linear programming problem aims to maximise or minimise a linear objective function.

Its main parts are:

• Decision variables: Unknown quantities to be determined
• Objective function: Expression to maximise or minimise
• Constraints: Linear inequalities restricting the variables
• Feasible region: Common region satisfying all constraints
• Optimal solution: Feasible point giving the required maximum or minimum

For two variables, constraints are represented graphically. The objective function is checked at the corner points of the feasible region.

An optimal solution may occur at one corner point, more than one point or may not exist when the feasible region is unbounded in the required direction.

Probability

Conditional probability measures the probability of an event when another event has already occurred.

P(A|B) = P(A ∩ B)/P(B), where P(B) ≠ 0

The multiplication theorem is:

P(A ∩ B) = P(A)P(B|A)

Two events A and B are independent when:

P(A ∩ B) = P(A)P(B)

Bayes’ theorem reverses conditional probabilities:

P(Aᵢ|B) = [P(Aᵢ)P(B|Aᵢ)] / ΣP(Aⱼ)P(B|Aⱼ)

The events A₁, A₂, … must form a partition of the sample space.

Class 12 Maths Important Formulas for Quick Revision

These Class 12 Maths important formulas summarise frequently used results. Their conditions and meanings remain as important as memorising them.

Connections Between Class 12 Maths Chapters

Several chapters become easier when their connections are understood.

How to Use These Class 12 Maths Short Notes

Begin each chapter by reviewing its definitions and standard formulas. Then solve NCERT examples and exercises related to the same concept.

Maintain a separate formula sheet for calculus, vectors and probability. Write the condition beside each formula because many formulas apply only under specific conditions.

While solving questions, identify the chapter concept before selecting a method. For example, a maximum-value problem usually requires a derivative sign test rather than direct substitution.

Revise connected chapters together. Matrices can be paired with determinants, while vector algebra can be revised with three-dimensional geometry.

Use quick revision notes for Class 12 Maths to recall methods, but practise full solutions regularly. Mathematics improves through written steps, correct notation and repeated application.