# CBSE Class 12 Maths – Syllabus, Question Papers, Notes, NCERT Solutions

CBSE Class 12 Maths: Are you preparing for CBSE 12th Maths Exam? Have you selected the right study materials to prepare? Do you have proper preparation to score better marks?

Don’t worry! In this article, we are covering everything related to CBSE Class 12 Maths Exam. Here you will get brief information on, Class 12

• Basic Introduction,
• Syllabus,
• Books
• Question Papers
• Solutions
• Worksheet
• Revision Notes
• Formulas

## Basic Introduction

Maths plays an important role in CBSE Class 12 Exam. It has a broader prospect if you have taken Science Stream in 12th. Apart from that, you can easily crack various competitive and entrance exams with a better concept on the subject. For those who are want to go for the research field, you should have depth concept on the subject.

By studying Class 12 Maths, you will improve your mathematical abilities. The CBSE experts have considered various points while creating the book for CBSE Class 12 Maths.

A few points are given below.

• In the Introduction, they have highlighted the importance of the topic, connection with earlier studied topics, a brief mention of the new concepts to be discussed in the chapter.
• They have organized the chapter into sections comprising one or more concepts or sub-concepts.
• They have used clear and simple language.
• Multiple examples are provided to build the student’s better base.
• Varieties of solutions are provided so students can understand the multiple ways to solve a problem.
• In Proofs and solutions, a learner can develop a clear and logical way of expressing the arguments.
• All answers to the exercises and solutions or hints are given where required. So a student can become more proficient in the subject.

## CBSE Class 12 Maths

Maths is the essential subject for class 12. It plays an important role in Student’s life. It is neither an easy subject nor a tougher one. With a clear concept on every chapter and the daily practice of exercises, you can score a better mark in the 12th Board Exam.

## CBSE Class 12 Maths Books

To score the better Marks in the 12th Maths Exam, you should have the right books. You cannot take any random book while selecting the book. Make sure that it has covered the latest CBSE Maths Syllabus.

For that, you must have NCERT Class 12 Maths book. Along with that, you can follow the books as given below as a reference,

• RD Sharma 12th Maths Book

You will find the following chapters in 12th Maths,

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives / Derivative of Functions
• Chapter 7 Integral Functions
• Chapter 8 Application of Integrals
• 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

Relations and Functions is the first chapter of NCERT Class 12 Maths. Here you will learn more advanced topics on the concepts of relations, functions, domain, and codomain introduced in Class 11.

You will also know the several real-valued functions and their graphs. The chapter has four exercises-based concepts explained in the chapter.

In NCERT Class 12 Maths Chapter 1, students study different types of relations and equivalence relations, the composition of functions, invertible functions, and binary operations.

The main features and topics of the CBSE Class 12 Maths chapter are as follows:

• Class of functions
• Types of Functions / Variety of functions
• Composite functions
• Implicit functions
• Exponential functions
• Inverse functions
• Approximate of functions
• Empty relation
• Universal relation
• Reflexive relation
• Symmetric relation
• Transitive relation
• Equivalence relation
• A function f : X → Y is one-one (or injective) if f (x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈
• A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
• A function f: X → Y is one-one and onto (or bijective), if f is both one-one and onto.
• A function f: X → Y is invertible if and only if f is one-one and onto.

### Chapter 2 Inverse Trigonometric Functions

The second chapter of NCERT 12th Maths comprises Inverse Trigonometric Functions. Here, you will get knowledge on the trigonometric functions such as sine, cosine, tangent, cot, cosec, sec.

Also, the concept of an inverse trigonometric function, finding the principal value of the inverse trigonometric function, domain, and range of the inverse trigonometric function. You will also find the discussion on the graphs of inverse trigonometric functions.

You can explore more about inverse trigonometric through two exercises.

You will learn to:

• Find principal value of inverse trigonometry functions like sin-1, cos-1, tan-1, cot-1, cosec-1, sec-1
• Solve inverse trigonometry questions using formulas
• Solve by changing trigonometric variables like sin-1 to cos-1 or sec-1 to tan-1 and then applying formulas

### Chapter 3 Matrix

In CBSE 12th Chapter 3, you will learn about the Matrix method. The students learn the fundamentals of Matrix, operations on matrices, equality of matrices, and other Matrix Algebra. Here, students will learn how matrices are associated with different fields.

There are a total of 62 questions in the 4 exercises of this chapter. The main features of this chapter are as follows:

• An ordered rectangular array of numbers or functions is called a matrix.
• A matrix that has m rows and n columns is called a matrix of order m × n.
• Matrices – Column matrix: [aij]m×1 is a column matrix.
• [aij]1×n is a row matrix.
• An m × n matrix is a square matrix if m=n.
• A = [aij]m×m is a diagonal matrix if aij=0, when i≠j.
• A = [aij]n×n is a scalar matrix if aij=0, when i≠j, aij=k (k is some constant), when i=j.
• A = [aij]n×n is an identity matrix if aij=1, when i=j, aij=0, when i≠j.
• A zero matrix has all its elements as zero.
• A non-zero matrices has elements other than zero.
• A = [aij] = [bij] = B if (i)A and B of the same order, (ii) aij = bij for all possible values of i and j.
• 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 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 constant.

Also, check out few topics which are included in this chapters like:

• Multiplication of matrices
• Types of matrices
• Inverse of matrix
• Operation on matrices
• Properties of matrix
• Symmetric and skew-symmetric matrices, and so on.

### Chapter 4 Determinants

Chapter 4 covers an important part of Matrices, i.e. Determinants. Here you will learn about the multiple topic of determinants, the meaning of determinants, cofactors of determinants, exercises with solutions of a determinant of orders up to 3, their cofactors, and inverses.

CBSE Class 12 Maths also talks about various properties of determinants, cofactors, and applications of determinants in finding the area of a triangle, minors, adjoint and inverse of a square matrix, consistency, and inconsistency of system of linear equations, and solution of linear equations in two or three variables using the inverse of a matrix in these exercises.

Chapter 4 cover the following points and formulas.

• If any two rows or any two columns are identical or proportional, then the value of the determinant is zero
• A square matrix A has an inverse if and only if A is non-singular
• Unique solution of equation AX = B is given by X = A–1 B, where A ≠ 0
• If a system of equation is consistent, then it has a solution
• If a system of equations is inconsistent, then there is no solution
• For a square matrix A in matrix equation AX = B:
• If | A| ≠ 0, then there exists a unique solution
• If | A| = 0 and (adj A) B ≠ 0, then there exists no solution
• If | A| = 0 and (adj A) B = 0, then system may or may not be consistent

### Chapter 5 Continuity and Differentiability

Class 12 Maths Chapter 5 – Continuity: It is the advanced version of the Differentiation of Functions introduced in Class 11. Here you can learn the definition of continuity, important concepts of Continuity, Differentiability and establishes the relationship between them.

You can also learn about the continuity and differentiability of Inverse Trigonometric Functions. It also introduces exponential and logarithmic functions, step explanations for chapter-wise solutions, and so on.

The main features of this chapter are as follows:

• Sum, difference, product, and quotient of continuous functions are continuous.
• Every differentiable function is continuous, but the converse is not true.
• Rolle’s Theorem: If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b) such that 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 6 Application of Derivatives

In 12th Maths Chapter 6, you will learn the differentiability of functions, and introduction to derivatives. You will also find the derivatives of specific functions, and their applications, namely the tangents and normal to curves, simple curves, the family of curves, and the rate of change in some quantities. It also talks about increasing and decreasing functions.

This chapter also includes miscellaneous examples, Maths Class 12 NCERT Solutions, and different method of solution to the derivatives questions.

The main features of this chapter are as follows:

• A function f is said to be
• Increasing on an interval (a, b) if x1 < x2 in (a, b) ⇒ f(x1) < f(x2) for all x1, x2 ∈ (a, b)
• Decreasing on (a, b) if x1 < x 2 in (a, b) ⇒ f(x1 ) > f(x2 ) for all x1 , x2 ∈ (a, b).
• Constant in (a, b), if f (x) = c for all x ∈ (a, b), where c is a constant.
• First Derivative Test
• Second Derivative Test

### Chapter 7 Integrals

You should know that chapter 7 is important and time-consuming in the whole Class 12 Mathematics syllabus. The chapter has the integration of functions and their applications in terms of calculation of areas. You will also learn the introduction to Integral Calculus. There are a total of 11 exercises in this chapter, making it a very practice-driven part of the syllabus.

The main features of this chapter are as follows:

• Integration as anti-derivative
• Integration using Trigonometry Formulas
• Integration by substitution
• Integration by parts
• By parts integration of ex
• Integration by partial fractions
• Integration by special formulas
• Integration as limit as a sum
• Definite Integration

### Chapter 8 Application of Integrals

Chapter Eighth deals with the applications of integrals in calculating areas under the simple curve, lines, parabolas, and ellipses. The two exercises provide enough questions for practicing the area calculations.

The main features of this chapter are as follows: ### Chapter 9 Differential Equations

In Chapter 9 Differential Equations you will find the concept of Differential equations, the general and particular solutions, linear differential equation, first-order first-degree differential equations, and up to the high order and degree of an equation.

It also talks about the applications of differential equations in the six exercises. It also provides CBSE NCERT Solutions For Class 12 differential equations exercises.

It also covers:

• Process of differentiation
• Logarithmic differentiation
• Comparison between differentiation

exponential differentiation, and so on.

In differential equations,

• we are given an equation like dy/dx = 2x + 3 and we need to find y
• An equation of this form dy/dx = g(x)  is known as a differential equation.

### Chapter 10 Vector Equations

In Chapter 10, you will know about the detailed solutions for vector quantities, i.e. the quantities with both magnitude and direction. You will also learn the rules of addition of vector quantities in Vector algebra a product of vectors, parallel vector, position vector, collinear vectors, and other basic properties or important properties of vector.

Vector Algebra plays a big role in Physics, as it is a common topic for Science Maths. It also provides you detailed explanations on vector equations by examples.

The main features of this chapter are as follows:

• The scalar components of a vector are its direction ratios and represent its projections along the respective axes.
• The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as l=(a/r), m=(b/r) n=(c/r)
• The vector sum of the three sides of a triangle taken in order is 0.
• The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
• The multiplication of a given vector by a scalar λ, changes the magnitude of the vector by the multiple |λ|, and keeps the direction the same (or makes it opposite) according to as the value of λ is positive (or negative).

### Chapter 11 Three Dimensional Geometry

In Chapter 11, you will find the geometric properties, vectors to calculate the distance between lines, a point & a line, and a line & an equation of plane, along with the angles between them. It also introduces the concept of direction cosines

The main features of this chapter are as follows:

• The direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.
• If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1
• Direction ratios of a line are the numbers that are proportional to the direction cosines of a line.
• Skew lines are lines in space that are neither parallel nor intersecting. They lie in different planes.
• The angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines.
• If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then cosθ = |l1l2 + m1m2 + n1n2|

### Chapter 12 Linear Programming

This chapter explores the concept of linear programming problems and their solutions using a graphical method only, which helps you to find out infeasible regions, feasible solutions, and unbounded region.

The main features of this chapter are as follows:

• Finding minimum or maximum value of Z (known as an objective function), using constraints – We use the corner point method and the basics which we learned in the Linear Equality Chapter of Class 11.
• Handing cases where the region is not feasible or unbounded.
• In the previous topic, the equation is already given to us. But, we also need to create the equations and then solve them. We cover different types of problems like Diet, Manufacturing, Transport and Other Problems. In these questions, we first make the equations and then find the minimum or maximum value of Z.

### Chapter 13 Maths Probability

This is the last chapter of CBSE Class 12 Mathematics. Here you will learn the concept of Conditional Probability and the encounters of the same in our daily lives.

This explores Bayes’ Theorem, the multiplication rule, the independence of events, the probability distribution, the rule of probability, the Binomial probability distribution, and another important theorem on probability. This chapter is best understood by practicing as many questions as possible.

The main features of this chapter are as follows:

1. 0 ≤ P (E|F) ≤ 1, P (E′|F) = 1 – P (E|F)P ((E ∪ F)|G) = P (E|G) + P (F|G) – P ((E ∩ F)|G)
2. P (E ∩ F) = P (E) P (F|E), P (E) ≠ 0P (E ∩ F) = P (F) P (E|F), P (F) ≠ 0
3. If E and F are independent, then P (E ∩ F) = P (E) P (F)P (E|F) = P (E), P (F) ≠ 0P (F|E) = P (F), P(E) ≠ 0
4. Theorem of total probability
5. Bayes’ theorem
6. A random variable is a real-valued function whose domain is the sample space of a random experiment.
7. Var (X) = E (X2) – [E(X)]2
8. Trials of a random experiment are called Bernoulli trials if they satisfy the
9. following conditions :
1. There should be a finite number of trials.
2. The trials should be independent.
3. Each trial has exactly two outcomes: success or failure.
4. The probability of success remains the same in each trial.

## CBSE Class 12 Maths Syllabus

Here you will get the detailed CBSE Class 12 Maths Syllabus along with section-wise marks distribution.

You can take a look at the units included in CBSE Mathematics Syllabus along with the marks distribution:

 Unit Topic Marks I. Relations and Functions 10 II. Algebra 13 III. Calculus 44 IV. Vectors and 3-D Geometry 17 V. Linear Programming 06 VI. Probability 10 Total 100

## CBSE Class 12 Maths Solutions

After the textbook, you should have CBSE Class 12 Maths solutions to prepare correctly 12th Exam. You will easily find the exercise-wise explanation, step solutions, textbook solutions, chapter solutions, Maths NCERT Solutions of all the problems.

You will also find various Miscellaneous Exercise questions from every topic.

### CBSE Class 12 Maths NCERT Solutions

Experts have created NCERT Maths Solutions for Class 12 to provide students depth concepts on every chapter. They have covered all questions from all exercises based on the latest NCERT pattern.

You should not worry about the content because the NCERT experts have created the solutions in such a way that any student can easily understand without having a depth concept in the section.

### CBSE Class 12 Maths RD Sharma Solutions

After NCERT solutions, RD Sharma Solutions are the best Reference Materials you should follow to prepare class 12 Maths. The solutions are created by the well-known author RD Sharma to provide you with a detailed guide.

### CBSE Class 12 Maths RS Aggarwal Solutions

You can also use the RS Aggarwal Maths Solution to build your concept in much depth. They have elaborated on every topic in detail. They have also covered every exercise like RD Sharma.

## CBSE Class 12 Maths Question Paper

It is an important materials for students who are preparing for the Class 12 Board Exam.

You should solve the class 12 exam question papers after completion of your 12th Maths chapters and exercises for Class board exam preparation, various engineering entrance exams, and other competitive exams.

• Also, you will get an idea about the type of questions asked in the upcoming exam and also identify the repetitive questions.
• By solving more question papers on CBSE Class 12, you will understand the exam pattern more accurately, and also develops proper exam understanding in students.
• You can also develop time management skills and boost your speed to solve a problem by practicing more question papers.

You should have the following materials to find the complete question papers related to CBSE Class 12th Exam,

• Sample Question Papers
• Previous Year Question Papers
• Question Bank

### Class 12 Maths Sample Question Papers

It is an important resource or tool for students who are preparing for the Class 12 Board Exam.

You may know that to provide you with a clear idea about exam questions paper. Central Board of Secondary Education (CBSE) releases every year sample question papers and marking schemes for all subjects of class 12. By solving these sample papers, you will have a preview of the board question paper.

You can understand the current pattern of the question papers after solving sample papers.

### Class 12 Maths Previous year Papers

Previous exam papers are also one of the good study materials to prepare for the class 12 Maths exam. You must solve the Maths previous year papers before the board exams.

It will give you an idea about the previous exams and the changes throughout the period. You will also find the important repetitive questions and sections of class 12 maths. It will also help you to build concepts on various types of questions.

By solving the papers in a certain time as the real exam, you can know your current preparation level and identify the conceptual weaknesses. You can easily cover those before the board exam.

### Class 12 Maths Question Bank

Experts have created a CBSE class 12 maths question bank by analyzing the chapter-wise exercise and the previous exams. You will find multiple questions for every single topic.

You can build depth concepts on every topic after solving multiple questions from chapters and previous exams.

## CBSE Class 12 Maths Worksheets

You can enhance your logical, lingual, analytical, and problem-solving capabilities after solving CBSE Class 12 Maths Worksheet.

We have covered all the Class 12 Maths important questions and answers in the worksheets included in the latest CBSE NCERT Syllabus.

## CBSE Class 12 Maths Revision Notes

You should have CBSE Class 12 Maths Revision Notes. It will help you to revise all chapters in an easier way. The experts have created the notes by analyzing previous years of question papers and syllabus in mind.

## CBSE Class 12 Maths Formulas

You will find multiple formulas and theorems in class 12 Maths. There is a chance to forget formulas from various chapters. To help you to memorize and revise these, we are providing you with the complete set of class 12 Maths formulas in PDF format.

Make sure that you should have CBSE Class 12 Maths handy. You can utilize them and score a better grade in the exam.

We have covered a detailed guide on CBSE Class 12 Maths. Feel Free to ask any questions in the comment section below.

### 1. Is 12th Mathematics Easier Than 11th?

No, both are of the same level. If you are ideological in class 11, then you will have to face the problem in CBSE class 12th.

### 2. Is Class 12 Maths Difficult?

No, class 12 Maths on the CBSE board is not harder than class 11. You can get more marks in class 12 Maths by practicing completely.

### 4. Can I get answers to the various practices of Class 12 Maths in NCERT Solutions?

Solutions for various exercises oF Class 12 Maths are available on Kopykitab’s website. You can download and access PDFs by solving problems from the NCERT textbook.

### 5. Is the NCERT Mathematics textbook enough for Class 12 exam preparation?

According to many teachers and experts, the NCERT Mathematics textbook is sufficient to pass the Class 12 math exams. However, students can also refer to other textbooks if they are aiming for the top marks.

### 6. How many chapters are present in NCERT Solutions for Class 12 Maths?

The NCERT textbook of Class 12 Maths has 2 parts. For complete details, read the above article.

### 7. What are the main topics discussed in NCERT Solutions for Class 12 Maths Chapter 3?

Matrix, types of matrices, operations on matrices, transpose of a matrix, symmetric and skew-symmetric matrices, elementary operation on matrix, and invertible matrices are the main topics discussed in Chapter 3 of Class 12 Maths.

### 8. What are the uses of determinants according to NCERT Solutions for Class 12 Maths Chapter 4?

Determinants are one of the main concepts having a number of applications in algebra. This concept is useful to solve a set of linear equations, and so on.
Go through the above blog for more details.