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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.
Experts in Class 12 Mathematics at Embibe have collated all the NCERT Solutions for Class 12 Maths in this article. Each question of CBSE NCERT Solutions provides detailed stepwise answers. This article will provide you with the NCERT class 12 syllabus and a downloadable PDF of NCERT Solutions for Class 12 Mathematics. Read ahead to know more and get access to all the required NCERT Class 12 Solutions.
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 problemsolving abilities. Click on Class 12 Maths NCERT Chapterwise links for the 12th Maths Solution Book PDF.
To download the NCERT Solutions of Class 12 Maths as a PDF, you can download the Class 12 Maths NCERT syllabus PDF from there for free. The syllabus is given as follows:
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=y710
Let, gy=y710, where g : R→R.
Step 2: To prove, gof=IR
gofx=gfx=g10x+7=10x+7710=x=IR
Step 3: To prove, fog=IR
fogy=fgy=fy710=10y710+7=y=IR
Since, gof=fog=IR, f is invertible and inverse of f is x=gy=y710.
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 preimage.
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 F1 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 oneone.
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.
∴ F1:T→S
We can write inverse of the function as: F1=3,a,2,b,1,c.
The NCERT Solutions for each chapter is as follows:
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 chapterbychapter 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.
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 stepwise 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 chapterwise 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.
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 subtopics along with the answer pdf are listed below:
Exercise 1.1  16 Questions  Download Answers PDF 
Exercise 1.2  12 Questions  
Exercise 1.3  14 Questions  
Exercise 1.4  13 Questions 
Chapter 1: Relations and Functions – Download NCERT Solutions PDF
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 subtopics in this chapter are listed below:
Exercise 2.1  14 Questions  Download Answers PDF 
Exercise 2.2  21 Questions 
Chapter 2: Inverse Trigonometric Functions – Download NCERT Solutions PDF
In this chapter from NCERT 12 Maths, you will learn about the fundamentals of matrix and matrix algebra. The subtopics in this chapter are listed below:
Exercise 3.1 
10 Questions 

Exercise 3.2 
22 Questions 

Exercise 3.3 
12 Questions 

Exercise 3.4 
18 Questions 
Chapter 3: Matrices – Download NCERT 12 Maths Solutions PDF
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 subtopics in this chapter are listed below:
Exercise 4.1 
8 Questions 

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 
Chapter 4: Determinants – Download NCERT 12 Maths Solutions PDF
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 subtopics in this chapter are listed below:
Exercise 5.1 
34 Questions 

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 
Chapter 5: Continuity and Differentiability – Download NCERT 12 MathsSolutions PDF
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 subtopics in this chapter are listed below:
Exercise 6.1 
18 Questions 

Exercise 6.2 
19 Questions 

Exercise 6.3 
27 Questions 

Exercise 6.4 
9 Questions 

Exercise 6.5 
29 Questions 
Chapter 6: Application of Derivatives – Download NCERT Solutions PDF
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 subtopics in this chapter are listed below:
Exercise 7.1 
22 Questions 

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 
Chapter 7: Integrals – Download NCERT Solutions PDF
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 abovesaid curves. The subtopics in this chapter are listed below:
Exercise 8.1 
13 Questions 

Exercise 8.2 
7 Questions 
Chapter 8: Application of Integrals – Download NCERT Solutions PDF
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 firstorder – firstdegree differential equation, and some applications of differential equations in different areas. The subtopics in this chapter are listed below:
Exercise 9.1 
12 Questions 

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 
Chapter 9: Differential Equations – Download NCERT Solutions PDF
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 subtopics in this chapter are listed below:
Exercise 10.1 
5 Questions 

Exercise 10.2 
19 Questions 

Exercise 10.3 
18 Questions 

Exercise 10.4 
12 Questions 
Chapter 10: Vector Algebra – Download NCERT Solutions PDF
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 subtopics in this chapter are listed below:
Exercise 11.1 
5 Questions 

Exercise 11.2 
17 Questions 

Exercise 11.3 
14 Questions 
Chapter 11: Three Dimensional Geometry – Download NCERT Solutions PDF
In this chapter, we shall apply the systems of linear inequalities/equations to solve some reallife 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 subtopics in this chapter are listed below:
Exercise 12.1 
10 Questions 

Exercise 12.2 
11 Questions 
Chapter 12: Linear Programming – Download NCERT Solutions PDF
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 subtopics in this chapter are listed below:
Chapter 13: Probability – Download NCERT Solutions PDF
Below we have provided some important questions that will aid you in understanding the chapters as well as help you in scoring better:
1. Let f : N → Y be a function defined as f (x) = 5x + 2, where, Y = {y ∈ N: y = 5x + 2 for some x ∈ N}. Show that f is invertible. Find the inverse. 
2. Find the value of cos^{1} (1/2) + 2 sin^{1} (1/2) 
3. Verify the mean value theorem for the following function f (x) = (x – 4) (x – 8) (x – 12) in [4, 6] 
4. Using matrix method, solve the system of equations 2x + 3y – 2z = 3, x + 3y + 2z = 6 and 3x – 2y + z = 2. 
5. A circular disc of radius 4 cm is being heated. Due to expansion, its radius increases at a rate of 0.05 cm per second. Find the rate at which its area is increasing if the radius is 5.2 cm. 
6. Find the value of ∫tan^{8}x sec^{4} x dx 
7. Determine the area of the region bounded by y^{2} = 8x, x = 3, x = 2 and the xaxis in the first quadrant. 
8. Find the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constants. 
9. Find the vector joining the points P(3, 4, 0) and Q(– 2, – 4, – 5) directed from P to Q. 
10. If a line makes an angle of 80°, 120°, 65° with the x, y, and zaxes, respectively, determine its direction cosines. 
11. There are 5 cards numbered 1 to 5 with one number on each card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X. 
12. The probability of solving a specific problem independently by persons A and B are 1/2 and 1/3, respectively. If both of them try to solve the problem independently, then calculate the probability that the problem is solved. 
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 problemsolving abilities. Students can follow the steps listed below to take a Class 12 mock test or practice questions:
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 stepwise 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 stepwise problem solving approach of the questions and improve their exam performance.