# CBSE Class 12 Maths Notes

CBSE Class 12 Mathematics Notes, which are written in simple language, cover nearly all of the chapters of class 12th Mathematics. Preparing from these notes will assist students in achieving high grades in their 12th grade as well as successful tests such as JEE Mains and JEE Advanced. These notes provided by GeeksforGeeks will help students understand each concept and revise thoroughly before the tests. These notes were written by content experts, and have a major advantage in that students would be well prepared to answer some kind of question that could be asked in the exams.

### Chapter 1: Relations and Functions

The Chapter Relation and functions discuss the introduction of relations and functions, types of relation, types of functions, the composition of functions and invertible function, binary operations.

- Types of Functions
- Composite functions
- Invertible Functions
- Composition of Functions
- Inverse Functions
- Verifying Inverse Functions by Composition

### Chapter 2: Inverse Trigonometric Functions

Inverse Trigonometric Functions gives an account of various topics such as the graphs of inverse trigonometric functions, different properties of inverse trigonometric functions, along with their domain, range, and other important attributes.

- Basic Concepts
- Graphs of Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
- Inverse Trigonometric Identities

### Chapter 3: Matrices

This chapter provides crucial knowledge of matrices that have applications in different areas such as business, sales, cost estimation, etc. In this chapter, the study of various properties of matrices, elementary row and column operations in matrices, invertible matrices, transpose of a matrix, and identity matrix are discussed.

- Matrices and its Types
- Mathematical Operations on Matrices
- Properties of Matrix Addition and ScalarMultiplication
- How to Multiply Matrices
- Transpose of a matrix
- Symmetric and Skew Symmetric Matrices
- Elementary Operations on Matrices
- Inverse of a Matrix by Elementary Operations
- Invertible Matrices

### Chapter 4: Determinants

This chapter is a continuation of the previous chapter of Matrices. This chapter helps to learn about the determinants, their properties, how determinants can be used to calculate the area of a triangle, and in solving a system of linear equations.

- Determinants
- Properties of Determinants
- Area of a Triangle using Determinants
- Minors and Cofactors
- Adjoint of a Matrix
- Application of Determinants and Matrices

### Chapter 5: Continuity and Differentiability

This chapter is a continuation of the concepts learned in the previous class, Differentiation of Polynomial and Trigonometric Functions. Now here in this class, the difference is that the level of the problems and the introduction of the two theorems given by Rolle and Lagrange.

- Continuity and Discontinuity in Calculus
- Differentiability of a Function
- Derivatives of Implicit Functions
- Derivatives of Composite Functions
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Proofs for the derivatives of eˣ and ln(x) – Advanced differentiation
- Rolle’s and Lagrange’s Mean Value Theorem
- Derivative of functions in parametric forms
- Second-Order Derivatives in Continuity and Differentiability
- Mean value theorem
- Algebra of Continuous Functions

### Chapter 6: Applications of Derivatives

This chapter deals with applications of derivatives which helps to learn how to determine the rate of change of quantities, find the minimum and maximum values of a function, and equations of tangents and normals to a curve.

- Critical Points
- Derivatives as Rate of Change
- Increasing and Decreasing Functions
- Increasing and Decreasing Intervals
- Tangents and Normals
- Relative Minima and Maxima
- Absolute Minima and Maxima
- Concave Function
- Inflection Points
- Curve Sketching
- Approximations & Maxima and Minima – Application of Derivatives

### Chapter 7: Integrals

In this chapter, the methods to determine the function when its derivative is given and the area under a graph of a function are discussed. Basic properties of integrals and the fundamental theorem of calculus are also included in this chapter.

- Introduction to Integrals
- Integration by Substitution
- Integration by Partial Fractions
- Integration by Parts
- Integration using Trigonometric Identities
- Functions defined by Integrals
- Definite integrals
- Computing Definite Integrals
- Fundamental Theorem of Calculus
- Finding Derivative with Fundamental Theorem of Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Definite integrals of piecewise functions
- Improper integrals
- Riemann sums
- Riemann sums with sigma notation
- Trapezoidal rule
- Definite integral as the limit of a Riemann sum
- Antiderivatives
- Indefinite integrals
- Particular Solutions to Differential Equations
- Integration by U-substitution
- Reverse chain rule
- Partial fraction expansion
- Trigonometric substitution

### Chapter 8: Applications of Integrals

This chapter included topics like how to find the area of different geometrical figures such as circles, parabolas, and ellipses.

- Areas under Simple Curves
- Area Between Two curves
- Area defined by Polar Curves
- Area as Definite Integral

### Chapter 9: Differential Equations

In this chapter, the concepts of differential equations and how to find solutions to a differential equation are discussed. This topic holds various applications in Physics, Economics, Chemistry, and Biology.

- Basic Concepts of differential equations
- Particular Solutions to Differential Equations
- Homogeneous Differential Equations
- Separable Differential Equations
- Exact equations and integrating factors
- Implicit Differentiation
- Implicit differentiation – Advanced Examples
- Disguised Derivatives – Advanced differentiation
- Differentiation of Inverse Trigonometric Functions
- Logarithmic Differentiation

### Chapter 10: Vector Algebra

In this chapter, the concepts of vector quantities, how to find the position vector of a point, geometrical interpretation of vectors, and scalar and cross product of vectors are discussed. These concepts have great importance in higher education (engineering and technology).

### Chapter 11: Three-dimensional Geometry

Based on the vector algebra discussed in the previous chapter, here are the concepts like, how it can be applied to three-dimensional geometry. Also, the introduction to topics like direction cosines and direction ratios, cartesian and vector equations of a line, and how to find the shortest distance between two lines using these concepts are discussed in this part.

- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in 3D
- Angle between two lines
- Shortest Distance Between Two Lines in 3D Space
- Points, Lines and Planes

### Chapter 12: Linear Programming

This chapter is a continuation of the concepts of linear inequalities and the system of linear equations in two variables studied in the previous class. This chapter helps to learn how these concepts can be applied to solve real-world problems and how to optimize the problems of linear programming so that one can maximize resource utilization, minimize profits, etc.

### Chapter 13: Probability

This chapter deals with probability, the concept of probability are also studied in earlier classes. This chapter in the present class helps to learn about conditional probability. Further, the topics like Bayes’ theorem, independence of events, the probability distribution of random variables, mean and variance of a probability distribution, and Binomial distribution are discussed in this chapter.

- Conditional Probability and Independence
- Multiplication Theorem
- Dependent and Independent Events
- Bayes’ Theorem
- Probability Distribution
- Binomial Random Variables and Binomial Distribution
- Binomial Mean and Standard Deviation
- Bernoulli Trials and Binomial Distribution
- Discrete Random Variables
- Expected Value