# CBSE Class 12 Maths Notes

• Difficulty Level :Medium
• Last Updated :25 Jul, 2022

The last destination of the school – Class 12. If you wish to get into your dream college or university, this is the class where you’ve to score the best to obtain a ride to your future career destination. Especially when you don’t find notes for subjects like Maths and Science which terrorize students to the core when it comes to preparing for boards exam. This is why, to make your search easier, GeeksforGeeks curated CBSE Class 12 Maths Notes, guided by experts.

Our CBSE Class 12 Mathematics Notes are written in simple language and cover nearly all of the chapters in the CBSE class 12 Maths Syllabus. Preparing from these CBSE Class 12 Maths Revision 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.

CBSE Class 12 Maths Syllabus covers all the important chapters given in the revised NCERT textbooks that include some Important topics of Class 10 Maths like Composite functions, Matrices and its Types, Equation of Tangents and Normals, etc.

To improve the basic concepts of students, GeeksforGeeks also covered 1500+ Most asked Questions of Mathematics, Chapterwise Important Formulas, and many more based on the new Class 12th CBSE syllabus. These notes and other study materials provide a helping hand for students to prepare for their Board Examinations.

## Chapter 1: Relations and Functions

The term ‘relation’ in mathematics is derived from the English language’s definition of relationship, which states that two objects or quantities are linked if there is an observable connection or relation between them. This Class 12 Chapter 1 might be very confusing, therefore students can even use the strategies to improve in their learning.

Let’s look at Class 12 Maths Chapter 1 Notes. Chapter 1 – Relation and functions discuss the introduction of relations and functions, types of relations, types of functions, the composition of functions and invertible functions, and binary operations.

Important formulae covered in CBSE Class 12 Chapter 1- Relations and Functions,

• RelationAn Empty relation R in X, can be defined relation as: R = φ ⊂ X × X
• An Equivalence relation R in X is defined as a relation that can represent all the three types of relations: Reflexive, Symmetric, and Transitive relations.
• Symmetric relation R in X: (a, b) ∈ R ⇒ (b, a) ∈ R.
• Reflexive relation R in X: (a, a) ∈ R, ∀ a ∈ X.
• Transitive relation R in X: (a, b) ∈ R and (b, c) ∈ R, ⇒ (a, c) ∈ R.
• While, the Universal relation R in X: R = X × X.
• Function-Depending on the conclusion obtained functions  f: X → Y can be of different types like,

Chapter 1 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 1

## Chapter 2: Inverse Trigonometric Functions

The NCERT Class 12 Maths Chapter 2, Inverse Trigonometric Functions, covers a variety of subjects, including notes based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions, and miscellaneous examples. These principles are well-explained with examples. In calculus, inverse trigonometric functions are essential because they are used to define various integrals. Inverse trigonometric functions have applications in science and engineering.

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.

Here is the list of some important formulas covered in CBSE Class 12 Chapter 2- Inverse Trigonometric Functions,

• y = sin−1x ⇒ x = sin y
• x = sin y ⇒ y = sin−1x
• sin−1(1/x) = cosec−1x
• cos−1(1/x) = sec−1x
• tan−1(1/x) = cot−1x
• cos−1(−x) = π−cos−1x
• cot−1(−x) = π−cot−1x
• sec−1(−x) = π−sec−1x
• sin−1(−x) = −sin−1x
• tan−1(−x) = −tan−1x
• cosec−1(−x) = −cosec−1x
• tan−1x + cot−1x = π/2
• sin−1x + cos−1x = π/2
• cosec−1x + sec−1x = π/2
• tan−1x + tan−1y = tan−1{x + y / (1−xy)}
• 2tan−1x = sin−1{2x / 1+x2} = cos−1{1−x2}/{1+x2}
• 2tan−1x = tan−12x / {1−x2}
• tan−1x + tan−1y = π + tan−1(x+y / 1−xy); xy > 1; x, y > 0

Chapter 2 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 2

## Chapter 3: Matrices

A Matrix is said to have an ordered rectangular array of functions or numbers. A matrix of order m × n consists of m rows and n columns. This chapter provides crucial knowledge of matrices that have applications in different areas such as business, sales, cost estimation, etc.

Class 12 Maths Notes for Chapter 3 Matrics covers topics such as types of matrices, finding unknown quantities using equivalent matrices, and performing arithmetic operations on matrices. Also, how to transpose matrices and followed by the finding the inverse using different methods are covered in this chapter.

Basic Operations of matrices that are introduced in CBSE Class 12 Chapter 3- Matrices,

• kA = k[aij]m × n = [k(aij)]m × n
• – A = (– 1)A
• A – B = A + (– 1)B
• A + B = B + A
• (A + B) + C = A + (B + C); where A, B and C all are of the same order
• k(A + B) = kA + kB; where A and B are of the same order; k is constant
• (k + l)A = kA + lA; where k and l are the constant

If A = [aij]m × n and B = [bjk]n × p, then

• AB = C = m × p ; where cik = ∑nj=1aijbjk
• A.(BC) = (AB).C
• A(B + C) = AB + AC
• (A + B)C = AC + BC

If A= [aij]m × n, then A’ or AT = [aji]n × m also,

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

Chapter 3 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 3

## Chapter 4: Determinants

Class notes for Chapter 4 Determinants clearly demonstrate the image of the determinant of a square matrix and the way to find it. Characteristics of determinants, minors and cofactors, and linear equations are important sub-topics that are explained in this chapter thoroughly.

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

Here is the list of some important formulas used to understand the concepts in CBSE Class 12 Chapter 4- Determinants,

• Definition of Determinant: For a given matrix, A = [a11]1 × 1 its determinant is defined as det a11 or |a11| = a11
• For a 2 × 2 matrix, X =   the determinant is defined as,
• For a 3 × 3 matrix, A =   the determinant is defined as |A| =
• Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by

• Minor: If the matrix given is:

The Minor of a12 will be the determinant:

• Cofactor: Cofactors are related to minors by a small formula, for an element aij, the cofactor of this element is Cij and the minor is Mij then, cofactor can be written as:

Cij = (-1)i+jMij

• Sum Property of determinants:: If some or all elements of a row or column can be expressed as the sum of two or more terms, then the determinant can also be expressed as the sum of two or more determinants.

Chapter 4 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 4

## Chapter 5: Continuity and Differentiability

The Chapter Continuity and Differentiabilityis the extension of the Differentiation of functions studied in Class 11. Now, in this class, you will understand functions, such as polynomial and trigonometric functions. This chapter focuses on the ideas of continuity, differentiability, and their interrelations.

The topics covered in Chapter 5 Continuity and Differentiability are how to differentiate inverse trigonometric functions. In addition, you’ll learn about a new class of functions known as exponential and logarithmic functions. The derivatives of exponential and logarithmic functions, as well as logarithmic differentiation, will be covered. This chapter also covers the ideas of function derivatives in terms of parametric forms and second-order derivatives and introduction to the two theorems given by Rolle and Lagrange

Major Important formulas for CBSE Class 12 Chapter 5- Continuity and Differentiability,

df/dx = dv/dt . dt/dx

• Rolle’s Theorem: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) where as f(a) = f(b), then there exists some c in (a, b) such that f ′(c) = 0.
• Mean Value Theorem:If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that

f′(c) = f(b)−f(a) / b−a

Chapter 5 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 5

## Chapter 6: Applications of Derivatives

Applications of Derivatives in Class 12 deals with the basic introduction of derivatives, 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

This is not enough students we have also covered increasing and decreasing functions and intervals, Equation of Tangents, and Normals. Important topics for the exam point of view are covered in a very easy-to-learn way. Such topics are relative and absolute Minima and Maxima, critical points, Curve Sketching, and Approximations.

Important formulas of Derivatives studied in CBSE Class 12 Chapter 6- Applications of Derivatives,

(y – f(a))/(x – a) = f'(a)

(y – f(a))/(x – a) = -1/f'(a)

Chapter 6 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 6

## Chapter 7: Integrals

The anti-derivative, also known as an integral, is introduced to students in CBSE Notes Class 12 maths Integrals. Students are taught about the geometric representation of integrals as well as how to perform function integration using numerous methods and formulas. In addition, students are taught about definite 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.

The most crucial part of this chapter is covered well versed in the below links. Such topics are various methods used to determine the integration of a function such as integration by substitution, integration using partial fractions, integration by parts, integration using trigonometric identities, integration of some integral functions, and definition and concept of definite integrals. Along with Riemann sums with sigma notation, Trapezoidal rule, Definite integral as the limit of a Riemann sum, Indefinite integrals, and some methods to determine definite integrals like Integration by U-substitution, Reverse chain rule are discussed in these notes for chapter 7 integrals.

Standard formulas of Integrals studied in CBSE Class 12 Chapter 7- Integrals,

• ∫xndx = xn+1/n+1+C, where n≠−1.
• ∫cos x dx = sin x + C
• ∫sin x dx = −cos x + C
• ∫sec 2x 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
• ∫dx / √1−x2 = sin−1x + C
• ∫dx / √1−x2 = -cos−1x + C
• ∫dx / 1+x2 = tan−1 x + C
• ∫dx / 1+x2 = −cot−1x + C
• ∫ex dx = ex + C
• ∫ax dx = axlog a + C
• ∫dx / x√x2−1 = sec−1x + C
• ∫dx / x√x2−1 = −cosec−1x + C
• ∫1 / x dx = log |x| + C

Chapter 7 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 7

• Class 12 NCERT Solutions Maths Chapter 7
• Class 12 RD Sharma Solutions Integrals Chapter 1 and Chapter 2
• All important formulas for Class 12 Chapter 7

## Chapter 8: Applications of Integrals

Through this chapter Applications of Integrals, we’ll be continuing to discuss integrals. A different application of Integrals like area under simple curves, area of the region bounded by a curve and a line,area between two curves, and miscellaneous examples. From the below-given links, students can access the chapter-wise notes explaining the concepts from this chapter.

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

Important formulas of Integrals studied in CBSE Class 12 Chapter 8- Applications of Integrals,

Area = ∫bay dx=∫baf(x) dx

Area = ∫dcx dy=∫dcϕ(y) dy

Area = ∫ba[f(x)−g(x)]dx

where, f(x) ≥ g(x) in [a,b].

• If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in , a < c < b, then:

Area = ∫ca[f(x)−g(x)]dx+∫ca[g(x)−f(x)]dx

Chapter 8 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 8

## Chapter 9: Differential Equations

In this Chapter, Differential Equations, students will be introduced to the concept of differential equations, basic concepts related to differential equations, degree of a differential equation, order of a differential equation, and general and particular solutions of a differential equation. The next section of the unit covers the formation of a differential equation, first-degree differentiable equations, and methods of solving first order,

These concepts of differential equations and how to find solutions to a differential equation are very useful in various applications in Physics, and Economics.

Important concepts discussed in CBSE Class 12 Chapter 9- Differential Equations,

• Order of differential equation:In the given differential equation, the greatest order of the derivative existent in the dependent variable with respect to the independent variable.

• General and Particular Solution of a Differential Equation: The general solution of the differential equation is the solution that contains arbitrary constants. A particular solution of the differential equation is one that is free of arbitrary constants and is produced from the general solution by assigning particular values to the arbitrary constants.
• Methods of Solving First Order, First Degree Differential Equations

Chapter 9  of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 9

## Chapter 10: Vector Algebra

In this chapter, the concepts of Vector algebra, 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).

Major topics covered in this chapter cover how to find position vector, some basic concepts related to vector algebra, direction cosines, types of vectors such as zero vector, unit vector, collinear vector, equal vector, negative of a vector, addition of vectors, properties of vector addition. Along with the multiplication of a vector by a scalar, components of a vector, vector joining two points, section formula, a product of two vectors, scalar or dot product of two vectors, properties of scalar product, projection of a vector on a line, vector or cross product of two vectors are discussed in this chapter.

Here are all-important formulas for CBSE Class 12 Chapter 10- Vector Algebra,

• Commutative Law – A + B = B + A
• Associative Law – A + (B + C) = (A + B) + C
• Dot Product – (A • B )= |P| |Q| cos θ
• Cross Product – (A × B )= |P| |Q| sin θ
• k (A + B )= kA + kB
• Additive Identity – A + 0 = 0 + A

Chapter 10 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 10

## 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.

Here are all-important formulas for CBSE Class 12 Chapter 11- Three-dimensional Geometry,

Chapter 11 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 11

• Class 12 NCERT Solutions Maths Chapter 11
• Class 12 RD Sharma Solutions Three-dimensional Geometry Chapter 1, Chapter 2 and Chapter 3
• All important formulas for Class 12 Chapter 11

## 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.

Important concepts studied in CBSE Class 12 Chapter 12- Linear Programming,

• The common region determined by all the constraints including the non-negative constraints x ≥ 0, y ≥ 0 of a linear programming problem is called the feasible region (or solution region) for the problem.
• Points within and on the boundary of the feasible region represent feasible solutions of the constraints. Any point outside the feasible region is an infeasible solution.
• Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.

Chapter 12 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 12

## 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.

Important formulas studied in CBSE Class 12 Chapter 13- Probability,

P(E|F) = E ∩ F / P(F), P(F)≠0

• Total Probability: Let E1 , E2 , …. , En be the partition of a sample space and A be any event; then,

P(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + … + P (En) . P(A|En)

• Bayes Theorem: If E1 , E2 , …. , En are events contituting in a sample space S; then,

P(Ei|A) = P(Ei) P(A|Ei) / ∑nj=1 P(Ej) P(A|Ej)

Chapter 13 of CBSE Class 12 Maths Notes covers the following topics:

More resources for CBSE Class 12 Maths Chapter 13

## Frequently Asked Questions (FAQs)

Question 1: What are the major topics covered in Class 12 Maths Chapter 5 Continuity and Differentiability?

The major topics covered in Chapter 5 Continuity and Differentiability are Continuity of a function, and performing algebraic operations on continuous functions. Other topics included which are essential for exams are finding the derivatives of composite, implicit, trigonometric, exponential, and logarithmic functions, differentiation of functions in parametric form, second-order derivatives, mean value theorem as well as the chain rule for differentiation.

Question 2: What are some important tips to study Chapter 9 Differential Equations in Class 9?

The CBSE Class 12 Maths Notes for Chapter 9 may guide students in overcoming difficulty and trying to understand calculus concepts. Students can use the study tips listed below to help them speed up their learning.

• Before diving into differential equations, make sure you’ve completed the previous chapters.
• Make a list of formulas and take notes.
• Practice on a regular basis.

Question 3: How these CBSE Class 12 Maths Notes are helpful for students in their board exams?