NCERT Solutions for Class 12 Maths: Free PDF - Embibe
• Written By Varsha
• Written By Varsha

# NCERT Solutions for Class 12 Maths Class 12 Maths NCERT Solutions: CBSE has published the revised NCERT Solutions for Class 12 Maths for students. To build a strong conceptual foundation for all the chapters of Class 12, NCERT Solutions can be of great help. The NCERT solutions included in PDF format can help in studying offline for the Class 12 board exam, and also in searching for a particular term across the article.

Students can download all the NCERT Solutions of Class 12 Maths provided on this page for free. Working on NCERT Math 12th Class solutions will not only help clear your basics but will also help in improving your problem-solving abilities. Click on Class 12 Maths NCERT Chapter-wise links for the 12th Maths Solution Book PDF.

Q.1. Let f:R→R be defined as fx=10x+7. Find the function g:R→R such that gof=fog=IR.

Solution: Given that f(x)=10x+7
Step 1: Find the inverse of f
Let, fx=y ⇒10x+7=y ⇒x=y-710
Let, gy=y-710, where g : R→R.
Step 2: To prove, gof=IR
gofx=gfx=g10x+7=10x+7-710=x=IR
Step 3: To prove, fog=IR
fogy=fgy=fy-710=10y-710+7=y=IR
Since, gof=fog=IR, f is invertible and inverse of f is x=gy=y-710.

Q.2. Find the number of all onto functions from the set {1,2,3,…,n} to itself

Solution: Let A={1,2,3,…,n}

A function f : X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under
f, i.e., for every y ∈ Y, there exists an element x in X such that f(x) = y.
Since, function is onto on set A, So, all elements in 1,2,3,…..,n have unique pre-image.
Number of onto functions from A→A follows the permutation logic of n people sitting in n chairs.
∴ Number of onto function from A to A=n !

Q.3. Let S=a,b,c and T=1,2,3. Find F-1 of the given function F from S to T, if it exists.
F=a,3,b,2,c,1

Solution: Given, S={a,b,c},T={1,2,3}
F:S→T is given as F=a,3,b,2,c,1 ⇒Fa=3, Fb=2, Fc=1
Since each element has unique image.
∴F is one-one.
And, since for every image, there is a corresponding element,
∴F is onto.
And, if a function is one – one and onto then we can find the inverse of a function.
∴ F-1:T→S
We can write inverse of the function as: F-1=3,a,2,b,1,c.

The NCERT Solutions for each chapter is as follows:

### Why Refer to Class 12 Maths NCERT Solutions from Embibe?

• The advantages of using NCERT Solutions Class 12 Maths by Embibe are listed below:
• All CBSE NCERT Solutions for Class 12 Maths are provided in PDF form, which can be downloaded for free by anyone, anywhere.
• Referring to the Class 12 Maths NCERT Solutions will help students understand all basic concepts.
• As most of the engineering competitive exam syllabus such as JEE Main, JEE Advanced, BITSAT is almost the same as CBSE Class 11 and 12 Maths Syllabus. So, having a good command over the NCERT Solutions for Class 12 Maths will help to clear the competitive exams easily.
• The CBSE NCERT Solutions will help students to solve their homework and assignments on time without any difficulty.

### NCERT 12 Maths Solutions (Chapter-wise)

Mathematics is an important part of the Class 12 curriculum and the syllabus of all competitive exams. Whether you want to be an engineer or a doctor, you will have to deal with math in every aspect of your life. Making Class 12 concepts tangible will greatly assist you in achieving high results in the CBSE Class 12 board exam and other competitive examinations. CBSE NCERT Solutions for Class 12 Maths are essential for exam preparation since they include thorough chapter-by-chapter solutions to all problems.

First and foremost, you should remember that NCERT books are the most powerful tool you can use to learn the foundations. This will ensure that you do not overlook any important questions throughout your exam preparation.

The advantages of using NCERT 12 Maths Solutions 2021 – 22 are not limited to math performance; they also build the groundwork for other crucial courses. Additionally, every chapter of the CBSE Class 12th NCERT Maths Solutions PDF is available for free download.

### NCERT Solutions for Class 12 Maths: Chapter Descriptions

All the NCERT 12 Maths PDF solutions we have provided on this page are solved by the top Maths teachers of Embibe. Each Class 12 Maths NCERT Solution comes with a detailed step-wise explanation so that students get clarity on the topic. Download Maths Class 12 NCERT Solutions PDF for free and access them in offline mode as well.

Students can go through the description of each chapter to get an overview of the contents of the chapters. The chapter-wise description has been included in the following sections. To download the NCERT Solutions PDF for each chapter, click on the link given at the end of the following sections.

#### NCERT Solutions for Class 12 Maths Chapter 1: Relations and Functions

The term ‘relation’ in Mathematics has been drawn from the meaning of relationships in the English language, according to which two objects or quantities are related if there is a recognizable connection or link between the two objects or quantities.

In Class 12 Chapter 1, you will learn about different types of relations and functions, the composition of functions, invertible functions, and binary operations. The sub-topics along with the answer pdf are listed below:

• 1.1 Introduction
• 1.2 Types of Relations
• 1.3 Types of Functions
• 1.4 Composition of Functions and Invertible Function
• 1.5 Binary Operations

#### NCERT Solutions for Class 12 Maths Chapter 2: Inverse Trigonometric Functions

This chapter from NCERT 12 Maths will delve into the restrictions on domains and ranges of trigonometric functions that ensure the existence of their inverses and observe their behaviour through graphical representations. Besides, some elementary properties will also be discussed. The inverse trigonometric functions play an important role in calculus, for they serve to define many integrals. The concepts of inverse trigonometric functions are used in science and engineering. The sub-topics in this chapter are listed below:

• 2.1 Introduction
• 2.2 Basic Concepts
• 2.3 Properties of Inverse Trigonometric Functions

#### NCERT Solutions for Class 12 Maths Chapter 3: Matrices

In this chapter from NCERT 12 Maths, you will learn about the fundamentals of matrix and matrix algebra. The sub-topics in this chapter are listed below:

• 3.1 Introduction
• 3.2 Matrix
• 3.3 Types of Matrices
• 3.4 Operation on Matrices
• 3.5 Transpose of a Matrix
• 3.6 Symmetric and Skew Symmetric Matrices
• 3.7 Elementary Operation (Transformation) of a Matrix
• 3.8 Invertible Matrices
 Exercise 3.1 10 Questions Download Answers PDF Exercise 3.2 22 Questions Exercise 3.3 12 Questions Exercise 3.4 18 Questions

#### NCERT Solutions for Class 12 Maths Chapter 4: Determinants

The determinant is a scalar value that can be measured from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. In the case of a 2 × 2 matrix, the determinant may be defined as:

Similarly, for a 3 × 3 matrix A, its determinant is:

So in this chapter, you will study determinants up to order three only with real entries. The sub-topics in this chapter are listed below:

• 4.1 Introduction
• 4.2 Determinant
• 4.3 Properties of Determinants
• 4.4 Area of a Triangle
• 4.5 Minors and Cofactors
• 4.6 Adjoint and Inverse of a Matrix
• 4.7 Applications of Determinants and Matrices
 Exercise 4.1 8 Questions Download Answers PDF Exercise 4.2 16 Questions Exercise 4.3 5 Questions Exercise 4.4 5 Questions Exercise 4.5 18 Questions Exercise 4.6 16 Questions

#### NCERT Solutions for Class 12 Maths Chapter 5: Continuity And Differentiability

In this chapter, we will learn everything about the Continuity and Differentiability of a function. In this chapter, students will learn about the very important concepts of continuity, differentiability, and relations between them.

You will also learn the differentiation of inverse trigonometric functions. Further, this chapter introduces a new class of functions called exponential and logarithmic functions. The sub-topics in this chapter are listed below:

• 5.1 Introduction to Continuity and Differentiability
• 5.2 Exponential and Logarithmic Functions
• 5.3 Logarithmic Differentiation
• 5.4 Derivatives of Functions in Parametric Forms
• 5.5 Second Order Derivative
• 5.6 Mean Value Theorem
 Exercise 5.1 34 Questions Download Answers PDF Exercise 5.2 10 Questions Exercise 5.3 15 Questions Exercise 5.4 10 Questions Exercise 5.5 18 Questions Exercise 5.6 11 Questions Exercise 5.7 17 Questions Exercise 5.8 6 Questions

#### NCERT Solutions for Class 12 Maths Chapter 6: Application of Derivatives

In the chapter Continuity and Differentiability, you have learned how to find a derivative of composite functions, inverse trigonometric functions, implicit functions, exponential functions, and logarithmic functions. In this chapter, you will study applications of the derivative in various disciplines, e.g., in engineering, science, social science, and many other fields.

You will also use the derivative to find intervals on which a function is increasing or decreasing. Finally, here you will use the derivative to find the approximate value of certain quantities. The sub-topics in this chapter are listed below:

• 6.1 Introduction
• 6.2 Rate of Change of Quantities
• 6.3 Increasing and Decreasing Functions
• 6.4 Tangents and Normals
• 6.5 Approximations
• 6.6 Maxima and Minima
 Exercise 6.1 18 Questions Download Answers PDF Exercise 6.2 19 Questions Exercise 6.3 27 Questions Exercise 6.4 9 Questions Exercise 6.5 29 Questions

#### NCERT Solutions for Class 12 Maths Chapter 7: Integrals

Differential Calculus is centred on the concept of the derivative. Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions. The sub-topics in this chapter are listed below:

• 7.1 Introduction to Integral Calculus
• 7.2 Integration as an Inverse Process of Differentiation
• 7.3 Methods of Integration
• 7.4 Integrals of Some Particular Functions
• 7.5 Integration by Partial Fractions
• 7.6 Integration by Parts
• 7.7 Definite Integral
• 7.8 Fundamental Theorem of Calculus
• 7.9 Evolution of Definite Integrals by Substitution
• 7.10 Some Properties of Definite Integrals
 Exercise 7.1 22 Questions Download Answers PDF Exercise 7.2 39 Questions Exercise 7.3 24 Questions Exercise 7.4 25 Questions Exercise 7.5 23 Questions Exercise 7.6 24 Questions Exercise 7.7 11 Questions Exercise 7.8 6 Questions Exercise 7.9 22 Questions Exercise 7.10 10 Questions Exercise 7.11 21 Questions

#### NCERT Solutions for Class 12 Maths Chapter 8: Application Of Integrals

In this chapter, you will study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabola, and ellipses (standard forms only). We shall also deal with finding the area bounded by the above-said curves. The sub-topics in this chapter are listed below:

1. 8.1 Introduction
2. 8.2 Area under Simple Curves
3. 8.3 Area Between Two Curves

#### NCERT Solutions for Class 12 Maths Chapter 9: Differential Equations

In this chapter, we will study some basic concepts related to the differential equation, general and particular solutions of a differential equation, formation of differential equations, some methods to solve a first-order – first-degree differential equation, and some applications of differential equations in different areas. The sub-topics in this chapter are listed below:

• 9.1 Introduction to Differential Equations
• 9.2 Basic Concepts
• 9.3 General and Particular Solutions of a Differential Equation
• 9.4 Formation of a Differential Equation whose General Solution is given
• 9.5 Methods of Solving First Order, First Degree Differential Equations
 Exercise 9.1 12 Questions Download Answers PDF Exercise 9.2 12 Questions Exercise 9.3 12 Questions Exercise 9.4 23 Questions Exercise 9.5 17 Questions Exercise 9.6 19 Questions

#### NCERT Solutions for Class 12 Maths Chapter 10: Vector Algebra

In this chapter, we will study some of the basic concepts about vectors, various operations on vectors, and their algebraic and geometric properties. When considered together, these two types of properties fully realise the concept of vectors and lead to their vital applicability in various areas. The sub-topics in this chapter are listed below:

• 10.1 Introduction to Vectors
• 10.2 Some Basic Concepts
• 10.3 Types of Vectors
• 10.5 Multiplication of a Vector by a Scalar
• 10.6 Product of Two Vectors
 Exercise 10.1 5 Questions Download Answers PDF Exercise 10.2 19 Questions Exercise 10.3 18 Questions Exercise 10.4 12 Questions

#### NCERT Solutions for Class 12 Maths Chapter 11: Three Dimensional Geometry

In this chapter, we shall study the direction cosines and direction ratios of a line joining two points and also discuss the equations of lines and planes in space under different conditions, the angle between two lines, two planes, a line and a plane, the shortest distance between two skew lines and distance of a point from a plane. Most of the above results are obtained in vector form. The sub-topics in this chapter are listed below:

• 11.1 Introduction
• 11.2 Direction Cosines and Direction Ratios of a Line
• 11.3 Equation of a Line in Space
• 11.4 Angle between Two Lines
• 11.5 Short Distance between Two Lines
• 11.6 Plane
• 11.7 Coplanarity of Two Lines
• 11.8 Angle between Two Planes
• 11.9 Distance of a Point from a Plane
• 11.10 Angle between a Line and a Plane

#### NCERT Solutions for Class 12 Maths Chapter 12: Linear Programming

In this chapter, we shall apply the systems of linear inequalities/equations to solve some real-life problems. Also, we shall study some linear programming problems and their solutions by graphical method only. However, there are many other methods to solve such problems. The sub-topics in this chapter are listed below:

• 12.1 Introduction to Linear Programming
• 12.2 Linear Programming Problem and its Mathematical Formulation
• 12.3 Different Types of Linear Programming Problems

#### NCERT Solutions for Class 12 Maths Chapter 13: Probability

In this chapter, we shall discuss the important concept of the conditional probability of an event given that another event has occurred, which will help us understand the Bayes’ theorem, multiplication rule of probability, and independence of events.

We shall also learn an important concept of a random variable, its probability distribution, and the mean and variance of a probability distribution. The sub-topics in this chapter are listed below:

• 13.1 Probability Introduction
• 13.2 Conditional Probability
• 13.3 Multiplication Theorem on Probability
• 13.4 Independent Events
• 13.5 Bayes’ Theorem
• 13.6 Random Variables and its Probability Distributions
• 13.7 Bernoulli Trials and Binomial Distribution

### How to Get Good Marks in Class 12 Maths?

• Students can follow the tips listed below to get better at the subject and improve their scores:
• First and most important is the regular practice of problems and exercises. You cannot get good marks in Maths if you do not practice regularly.
• You should be thorough with the syllabus of CBSE Class 12 Maths. You should be familiar with all the concepts and topics in every chapter.
• Start your preparation with the NCERT Class 12 Maths textbook. Once you are done with the NCERT textbook, you can pick up a reference book of your choice for practice.
• Solve all the exercises at the end of each chapter. Refer to the NCERT solutions for Class 12 Maths if you are stuck on a problem.
• Give special attention to the chapters/concepts that you find difficult. Do not skip any chapter or topic.
• Take timed mock tests to improve your speed and accuracy.
• Solve previous year’s question papers to understand what questions are asked in the final exam.

### Some Important Questions for 12th Class Maths

Below we have provided some important questions that will aid you in understanding the chapters as well as help you in scoring better:

#### Learn & Practice Lessons Online With Embibe

At Embibe, you can solve free Class 12 Practice Questions for Science and Maths or take free Class 12 Mock Tests. These tests and practice questions will help you to strengthen your concepts and problem-solving abilities. Students can follow the steps listed below to take a Class 12 mock test or practice questions:

• 1st Step: Visit embibe.com. To have the best experience, log in to your account. If you are visiting the site for the first time, register yourself.
• 2nd Step: At the time of registration, you need to select your goal. Choose your board for example, “CBSE” then “12th CBSE”, and click on “Done”.
• 3rd Step: To practice questions for Class 12 CBSE, click on “Practice” option at the top. If you want to take a mock test, click on “Test” option.
• 4th Step: Select the subject as “Mathematics” as you want to take the test/practice questions for Maths.
• 5th Step: Click on the chapter name for which you want to practice questions. Then click on ‘Practice Now’ to start practising questions.
• 6th Step: If you want to take mock tests for any subject, you have the option to take chapter-wise tests or full subject tests. Click on any test link and then click on “Start Test” to start the test.

### FAQs on Class 12 Maths NCERT Solutions

Q.1: How NCERT Solutions for Class 12 Maths can help me in improving my score?
Ans: The NCERT solutions include all the details in an easy and step-wise manner. Hence, you will easily understand the problems. With the help of these solutions, you can revise the syllabus quickly. They are available both offline and online and thus, you can access them anytime you want.

Q.2: Is NCERT enough for Class 12 boards?
Ans: As per CBSE guidelines, NCERT books and NCERT solutions for Class 12 are enough for the board exam, but students can still practice from other reference books to widen their level of understanding.

Q.3: Is the NCERT Mathematics textbook enough for Class 12 exam preparation?
Ans: According to many teachers and experts, the NCERT Mathematics textbook is sufficient to clear Class 12 Maths exams. However, students can also refer to other textbooks if they are aiming for the top marks.

Q.4: Which book is the best for the Class 12 Maths exam preparation?
Ans: Many experts consider the NCERT Mathematics textbook the best book for the Class 12 exam preparation.

Q.5: What are the benefits of refering to the NCERT solutions of Embibe?
Ans:
By refering to the NCERT solutions from Embibe, students can increase their confidence levels, understand step-wise problem solving approach of the questions and improve their exam performance.

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