# CBSE Class 12 Maths Revision Notes

## Introduction to CBSE Exam

Central Board of Secondary Education (CBSE) is a national level board of education in India for public and private schools monitored by the Ministry of Education. Approximately 20,299 schools are affiliated with CBSE in India, apart from 220 schools in 28 different foreign countries. It provides a child-centred and holistic education programme to prepare students for college and subsequent careers.

## CBSE Class 12 Mathematics Revision Notes

Mathematics has always been considered a complex subject. The revision Notes are made by keeping the last ten years’ question papers checked and updated according to the latest syllabus. Mathematics is a skill, and the subject demands a lot of practise to grasp concepts. By referring to these Notes, students get to understand Mathematics clearly. It’s crucial to study  all the essential 13 topics of Class 12 Mathematics thoroughly to perform well in exams. Revision Notes are prepared under the guidance of expert faculty where all the necessary and complex topics are explained in more straightforward ways. Extramarks provides the best study material in terms of quality and content. CBSE students get a complete overview of all the chapters with relevant examples, Formulas and derivations.

## Chapter Wise Notes for Class 12 Mathematics

Class 12 Mathematics contains a total of 13 significant chapters. It is essential to solve all the exercises to get a good score. CBSE Class 12 Mathematics Notes and units include Relations and Functions, Algebra, Calculus, Vectors, Three-Dimensional Geometry, Linear Programming and Probability.

A panel of experienced teachers at Extramarks have diligently prepared the chapter wise Notes according to the NCERT syllabus to ensure a smooth and clear understanding of the concepts to score well in boards and competitive exams.

 Chapter Topic Chapter 1 Relations and Functions. Chapter 2 Inverse Trigonometric Functions. Chapter 3 Matrices. Chapter 4 Determinant. Chapter 5 Continuity and Differentiability. Chapter 6 Application of Derivatives. Chapter 7 Integral. Chapter 8 Application of the Integral. Chapter 9 Differential Equations Chapter 10 Vector Algebra Chapter 11 Three Dimensional Geometry Chapter 12 Linear Programming Chapter 13 Probability

### Chapter 1. Relations and Functions

CBSE Class 12 Mathematics Notes start with the first chapter as ‘Relations and Functions’. Students will learn in detail about different types of relations and functions, the composition of functions etc. The concept of relation is used in relating two quantities with each other. A function is a relationship that states that there should be only one output for each input. The topics covered in the relations and functions chapter are :

• Introduction
• Types of relations
• Types of functions
• Binary Operations
• Composition of functions and invertible functions.

### Chapter 2. Inverse Trigonometric Functions.

Inverse Trigonometric Functions are the functions that deal with the inverse of trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant. They are also termed Arcus, Anti Trigonometric or Cyclometric Functions. In geometry, Trigonometry is the relationship between the angles and sides of a right-angled triangle. The inverse trigonometry functions have significant applications in Physics, Geometry, Engineering, and Navigation. CBSE Class 12 Mathematics Notes explain the formulae/functions, which are classified as below.

• Sum and difference of functions
• Reciprocal and Complementary functions
• Double and triple of a function
• Arbitrary Values

### Chapter 3. Matrices.

Matrix, also known as Matrices, is a rectangular array of symbols, numbers, or expressions organised in rows and columns. Matrices are used to do statistics, plot graphs, and research and scientific studies in varied subjects. Matrices can also represent real-world data like the infant mortality rate and population of people, etc. CBSE Class 12 Mathematics Notes explain the meaning of Matrices, Types of Matrices, Important Formulae, etc. All Contents In Matrices are listed below:

• Introduction to Matrices
• Matrix Operations
• Types of Matrices
• The Rank of a Matrix and Special Matrices
• Solving Linear Equations using Matrix
• Adjoint and Inverse of a Matrix

### Chapter 4. Determinant.

A Determinant is the numerical value of the square Matrix. You can define the determinant of a matrix as a Scalar value that can be calculated from the elements of a square Matrix. The determinant of a matrix is always positive. Determinants are built to determine the uniqueness of solutions associated with matrices, which are used to express Algebraic Equations. Determinants are applied in Economics, Engineering and Science. The topics covered in determinants in CBSE Class 12 Mathematics Notes are listed below:

• Introduction
• Area of a Triangle
• Determinant Definition
• Properties
• Adjoint and the inverse of a Matrix
• Determinants and Matrix Applications
• Minors and Cofactors

### Chapter 5. Continuity and Differentiability.

In Continuity and Differentiability chapter Class 12, you will learn essential concepts of differentiability, continuity, and their relationship. CBSE Class 12 Mathematics Notes also concentrates on other important concepts like differentiating inverse trigonometric functions and new functions like logarithmic and exponential functions. Differentiability means that the function has a derivative at a point. Continuity means that the limit from both sides of a value is equal to the function’s value. The list of topics covered under this chapter are:

• Introduction
• Continuity
• Differentiability
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second-Order Derivative
• Mean Value Theorem

### Chapter 6. Application of Derivatives.

In this chapter, you will learn how to determine the rate of change of quantity, find the equations of tangents, find turning points on the graphs for various functions, maxima and minima. CBSE Class 12 Mathematics Notes guide you on how to relate different applications of derivatives in multiple fields like Engineering, Science and varied amounts of calculations for Businesses to develop profit-loss strategies for future investments. Below are the topics covered in the application of derivatives.

• Introduction
• Increasing and Decreasing Functions
• Rate of Change of Quantities
• Approximations
• Tangents and Normals
• Maximum and Minima
• Maxima and Minima

### Chapter 7. Integrals.

The Integrals chapter deals with the study of definite and indefinite integrals and their elementary properties. The development of integral calculus arises to solve the problems of the following types:

1. The problem of finding the area bounded by the graph function under certain conditions.
2. The problem of finding the function whenever the derivatives are given.

These two problems lead to the development of integral calculus (definite integral and indefinite integral). The elaborate explanations of each topic covered under integrals are found in CBSE Class 12 Mathematics Notes.

• Introduction
• Integration as an inverse process of differentiation
• Some properties of indefinite integrals
• Geometrical interpretation of indefinite integral
• Integration by substitution
• Comparison between differentiation and integration
• Methods of integration
• Integration by parts
• Integration using trigonometric identities
• Integrals for some particular functions
• Integration by partial fractions
• Integral of the type
• Integrals for some more types
• Definite integral
• Definite integral as a limit of a sum
• The fundamental theorem of calculus
• Area function
• The first fundamental theorem of integral calculus
• The second fundamental theorem of integral calculus
• Evaluation 0f definite integrals by substitution
• Some properties of definite integrals

### Chapter 8. Application of Integrals

Application of Integrals is practised in various fields like Science, Mathematics Engineering etc. The major integral formulae are used for the calculation of areas. CBSE Class 12 Mathematics Notes gives a clear introduction on integrals based on the Mathematics subject to find areas under simple curves, areas bounded by a curve, a line and area between two curves, and the application of integrals in the Mathematical disciplines with solved problems.

There are two types of Integrals:

1. The definite integrals
2. Indefinite integrals.

Some of the applications of integrals in Mathematics are used to find:

• The centre of mass(Centroid) of an area with curved sides.
• The area between two curves
• The area under a curve
• The average value of a curve

### Chapter 9. Differential Equations

The Differential Equations have a variety of applications in Chemistry, Physics, Biology, Anthropology, Geology, Economics etc. By studying from CBSE Class 12 Mathematics Notes, you will learn the basic concepts related to differential equations in detail.

A Differential Equation is a Mathematical equation that involves variables like x or y and the rate at which those variables change. The differential equations are unique because the solution of a differential equation is itself a function instead of a number. The differential equations are classified into the following:

• Ordinary Differential Equations.
• Partial Differential Equations.
• Linear Differential Equations.
• Nonlinear Differential Equations.
• Homogeneous Differential Equations.
• Non-homogeneous Differential Equations.

The Ordinary differential equations and the Partial differential equations are considered the most essential and standard differential equations.

### Chapter 10. Vector Algebra

Vector is the study of vector and scalar quantities. There are various quantities, which involve magnitude as well as direction. Quantities with a magnitude as well as direction are known as vector quantities.

Examples:

• Displacement
• Acceleration
• Velocity
• Weight
• Force
• Momentum
• Electric intensity

Important concepts like ‘Types and Operations of Vector Algebra’ are included in CBSE Class 12 Mathematics Notes. The list of topics covered are:

• Introduction
• Some Basic Concepts
• Types of Vectors
• Multiplication of a Vector by a Scalar
• Product of Two Vectors

### Chapter 11. Three Dimensional Geometry

Three-Dimensional Geometry has been introduced in Mathematics to understand the different types of shapes and figures. Prime topics covered are direction cosine and direction ratios of a line joining two points. The important topics covered are the distance between two points, section formula, coordinates of a point in space etc. Geometry guides you in deciding what materials to use to design houses and buildings in different geometric shapes in the real world. The CBSE Class 12 Mathematics Notes apply many real-life examples to provide quality deliverance to students to understand the subject better.

The list of topics covered are:

• Introduction
• Coordinate Axes in Three-Dimensional Space
• Coordinate Planes in Three-Dimensional Space
• Coordinates of a Point in Space
• Distance Between Two Points
• Section Formula

### Chapter 12. Linear Programming

Linear programming, also known as ‘Linear Optimization’, is a method used to obtain an optimal solution for a problem with given constraints. In CBSE Class 12 Mathematics Notes for linear programming, students learn to formulate actual problems into a Mathematical model. It involves an objective function and linear inequalities subject to constraints. Linear programming is currently utilised in company management, such as Production, Planning, Technology, Transportation etc.

The list of topics covered are:

• Introduction
• Linear Programming problems.
• Linear Programming Mathematical formulation.
• A Mathematical formulation of the problems.
• Graphical method of solving Linear Programming problems.
• Different types of Linear Programming problems.

### Chapter 13. Probability

Probability is a branch of Mathematics that deals with the occurrence of a random event. The Mathematics presented probability to foresee how likely events are to happen. This is the basic Probability theory, which is also applied in the Probability distribution, where you will learn the possibilities of the outcomes for a random observation. The CBSE Class 12 Mathematics Notes help students judge the accuracy of an event by analysing seemingly persuasive data. This gives students a solid introduction to Probability Mathematics while studying both advantages and limitations of applying Probability theory to different situations under varied conditions.

The list of topics covered are:

• Introduction
• Conditional Probability
• Multiplication Theorem on Probability
• Independent Events
• Bayes’ Theorem
• Random Variables and its Probability Distributions
• Bernoulli’s Trials and Binomial Distribution

Extramarks experts design the Class 12 Mathematics Notes to help students complete their studies in time while giving an in-depth understanding of the subject.

 CBSE Class 12 Mathematics Revision Notes Important Questions for Class 12 Mathematics

The CBSE Class 12 Mathematics Notes are designed to make concepts clear to the students with practical solutions, real-life examples and detailed chapters.

## CBSE Class 12 Mathematics – Distribution of Marks

1. The theory paper constitutes total – 70 marks
2. Practicals constitute total – 30 marks

## How Extramarks Can Help Students for Class 12 Mathematics?

Class 12 Mathematics is the most abstract subject that students find hard to understand. The Notes are prepared by a group of experienced teachers, who designed the curriculum in a self-explanatory way. Regular practice with CBSE Revision Notes will help you get the best out of your exams.