# NCERT Solutions for Class 12 Maths

“Why do children dread mathematics? Because of the wrong approach. Because it is looked at as a subject.”

**— Shakuntala Devi**

This quote by the famous Mathematician Shakuntala Devi perfectly recapitulates the fear of Maths that many students live with. Many educationists and policymakers also understand the inhibition Maths, as a subject, enthuses.

To overpower the fear surrounding numbers, **NCERT has reduced and revised the NCERT Syllabus 2023-24. **The class 12 Maths NCERT solutions are an important tool in supporting students in preparing for their class 12 exam.

**These Class 12 Maths NCERT Solutions are more than the answer key of Maths Class 12.** CBSE 12 Maths is the most important milestone in any student’s life. For those who are seeking admission in courses that require them to study from the class 12 Maths book, solutions for class 12 Maths also help prepare for competitive entrance exams.

The class 12 Maths NCERT Solutions covers all class 12 Maths chapters from the NCERT textbook 2023-24. Having these NCERT Solutions for class 12 will let you practice answers of Maths NCERT class 12 as per the correct methodology and pattern of NCERT.

## Class 12 Maths NCERT Solutions

Of course, Maths is there in our everyday life. It is also the foundation for many inventions, discoveries, and research that help humanity. Class 12 Maths students mostly continue their higher studies with Maths as a core subject.

With Maths, you can explore professional opportunities in fields like Engineering, Physics, Biological Sciences, Commerce, Computer Application, and more. Maths can be daunting, but NCERT has made efforts to design the Maths curriculum while relating it to real life situations for better understanding.

**Maths class 12 chapters are based on the National Curriculum Framework 2005 as per the guidelines given in Focus Group on Teaching Mathematics 2005.** NCERT Part 1 and NCERT Maths Part 2 class 12 solutions are designed according to the NCERT objective for the class 12 Maths NCERT book.

Some very sharp minds design Maths plus Two solutions to offer accurate learning for class 12 students preparing for CBSE boards. Mathematics becomes easy when a student focuses more on understanding the concept than scoring more. Once the concept is clear, scoring a 100 in

Maths won’t be impossible. Class 12 Maths NCERT solutions are just a start in, vesting skills like;

- Logical reasoning
- Problem-solving
- Analytical thinking
- Quantitative Reasoning
- Time Management
- Critical Thinking

## Class 12 Maths NCERT All Chapters

- Relations and Functions
- Inverse Trigonometric Functions
- Matrices
- Determinants
- Continuity and Differentiability
- Application of Derivatives
- Integrals
- Application of Integrals
- Differential Equations
- Vector Algebra
- Three-Dimensional Geometry
- Linear Programming
- Probability

## NCERT Solutions for Class 12 Maths Chapter Brief

The class 12 Maths book is divided into two parts. Part one has six chapters, while Part 2 has seven chapters. Class 12 students would have to prepare for their exam with all 13 chapters in the Maths chapter list class 12. The 12th CBSE Maths solutions are designed to provide a better understanding of Mathematical concepts in the class 12 Maths NCERT book.

Easy language or a reduced syllabus do not make Maths easy; it is all about precision, accuracy, and numbers. The subject matters that have created class 12 Mathematics solutions understand this paradox. **The class 12 Maths NCERT solutions are strictly based on the Maths class 12 chapters.** The chapter brief for class 12 Maths NCERT book all chapters are given below:

### Part 1

**Chapter 1: Relations and Functions**

The first chapter of plus two Maths chapters focuses on different types of relations and functions, the composition of functions, invertible functions, and binary operations. The concept has been well-explained with many examples. The highlights of this chapter are;

- Ordered pairs
- Cartesian product of sets
- Number of elements in the Cartesian product of two finite sets
- Cartesian product of the set of reals with itself (upto RxRxR)
- Definition of relation, pictorial diagrams, domain, co-domain, and range of a relation
- Function as a special type of relation
- Pictorial representation of a function, domain, co-domain, and range of a function
- Real valued functions, domain, and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic, and greatest integer functions, with their graphs
- Sum, difference, product, and quotients of functions

Solutions for class 12 Maths finely define the concepts for students to have a better understanding.

**Chapter 2: Inverse Trigonometric Functions**

The second chapter is a step ahead of what students learned in class 11, Trigonometry. Inverse Trigonometric Functions throw light on the restrictions on domains and ranges of trigonometric functions, ensuring the existence of their inverse and observing their behaviour through graphical representations. Trigonometry is one of the most important topics for students, as it is also used in Science and Engineering. Students who want to pursue these fields in the future must have proper knowledge of Trigonometry. Our subject matter experts know this topic's importance and have formulated the 12th Maths solution book that acts as the perfect study material for you. The chapter covers the following major areas;

- Definition
- Range
- Domain
- Principal value branch
- Graph of inverse trigonometric functions

**Chapter 3: Matrices**

Maths experts design Class XII Maths NCERT solutions as a tool to help class 12 students practice the Maths problems. The third chapter for class 12 Maths would include the following;

- Concept
- Notation
- Order
- Equality
- Types of matrices
- Zero and identity matrix
- Transpose of a matrix
- Symmetric and skew-symmetric matrices
- Operations on matrices: Addition and multiplication and multiplication with a scalar
- Simple properties of addition, multiplication, and scalar multiplication
- Non-commutativity of multiplication of matrices and the existence of non-zero matrices whose product is the zero matrix
- Invertible matrices and proof of the uniqueness of inverse, if it exists

**Chapter 4: Determinants**

Chapter 4 of the class 12 Maths NCERT book focuses on determinants up to order three only with real entries. The chapter will focus on various properties of determinants like Minors, Co-factors, Applications of determinants in finding the area of a triangle, adjoint, and inverse of a square matrix, consistency and inconsistency of system of linear equations, and solutions of linear equations in two or three variable using inverse of a matrix. The areas that are covered are;

- Determinant of a square (up to 3 X 3), minors, co-factors, and applications of determinants in finding the area of a triangle
- Adjoint and inverse of a square matrix
- Consistency, inconsistency, and number of solutions of a system of linear equations by examples, solving system of linear equations in two or three variables using the inverse of a matrix

Class 12 Maths NCERT book solution offers answers to all the topics under this chapter.

**Chapter 5: Continuity and Differentiability**

Students will be introduced to concepts of continuity, differentiability, and relations between them, and a new class of functions called exponential and logarithmic functions in the fifth chapter of NCERT 12 Maths. The chapter also covers geometrically obvious conditions through differential calculus. The content is divided under the following headings;

- Continuity and differentiability
- Chain rule
- Derivative of inverse trigonometric functions
- Derivative of implicit functions
- Concept of exponential and logarithmic functions
- Derivatives of logarithmic and exponential functions
- Logarithmic differentiation, derivative of functions expressed in parametric forms
- Second order derivatives

**Chapter 6: Application of Derivatives**

Chapter 6 of class 12 Maths further explains the concept of Derivatives. Students will learn the application of derivatives in various disciplines like Engineering, Science, Social Science, and other fields. The chapter covers topics like;

- Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima, and minima
- Simple problems

**Chapter 7: Integrals**

Students who have downloaded the class 12 Maths NCERT pdf from the official website of NCERT will have had a look at the class 12 Maths all chapters. Chapter 7 is one of the important topics in the NCERT class 12 Maths textbook. Students will be studying Integration as the inverse process of differentiation. Apart from that, the focus would be on integrating a variety of functions by substitution, partial fractions and parts, and Evaluation of simple integrals.

**Chapter 8: Application of Integrals**

**Mathematics is not just about learning to do sums and equations. More important is to learn to apply the concepts. Chapter 8 of NCERT Maths, Application of Integrals, focuses on studying a specific application of integrals to find the area under simple curves, the area between lines, and arcs of circles, parabolas, and ellipses, and more. **

**Chapter 9: Differential Equations**

Mathematics class 12 cannot be overlooked. Each chapter in the NCERT Maths textbook also deals with topics covered in higher studies. In Differential Equations, students would be studying;

- Definition, order and degree, general and particular solutions of a differential equation
- Solution of differential equations by the method of separation of variables, solutions of homogenous differential equations of the first order and first degree
- Solutions of linear differential equations

**Chapter 10: Vector Algebra**

Chapter 10 of 12th Mathematics covers basic concepts about vectors, various operations on vectors, and their algebraic and geometric properties. Other areas under this topic are;

- Vectors and Scalars, magnitude, and direction of the vector
- Direction cosines and direction ratios of a vector
- Types of vectors (equal, unit, zero, parallel, and collinear vectors), position vector of a point, negative of a vector, components of a vector, the addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio
- Definition, geometrical interpretation, properties, and application of scalar (dot) product of vectors, vector (cross) product of vectors

**Chapter 11: Three-Dimensional Geometry**

Chapter 11 from Maths chapter list class 12 students would be focusing on;

- Direction cosines and direction ratios of a line joining two points
- Cartesian equation and vector equation of a line, skew lines, the shortest distance between two lines
- The angle between two lines

**Chapter 12: Linear Programming**

Linear Programming is a concept that is applied daily more often than we know. It deals with issues like maximising or minimising profit (cost) or minimum use of resources and is related to professional industries like Commerce, Management, Science, etc. This chapter from Mathematics Class 12 focuses on Linear Programming problems and their solutions by graphical method only. Chapter 12 of the class 12 NCERT book would cover the following;

- Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded)
- Feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints)

**Chapter 13: Probability**

Chapter 13 of NCERT class 12 textbook Maths deals with the important concept of the conditional probability of an event given that another event has occurred, which will help understand Bayes’ theorem, multiplication rule of probability and independence of events. The chapter will cover many important aspects of the concept that defines many things in our lives. From businesses to sports and commerce to weather, the probability quotient is important in determining the road ahead and formulating pathbreaking policies.

## Chapter-wise Marking Scheme 2023-24

Units | Marks |

Relations and Functions | 8 |

Algebra | 10 |

Calculus | 35 |

Vectors and Three-Dimensional Geometry | 14 |

Linear Programming | 5 |

Probability | 8 |

Total | 80 |

## Internal Assessment

Units | Marks |

Periodic Tests (Best 2 out of 3 tests conducted) | 10 |

Mathematics Activities | 10 |

Total | 20 |

## Key Features of Class 12 Maths NCERT Solutions

Maths is not a one-time thing. You do not read and understand. Maths is about doing and then applying it as well. Maths helps you in analysis, inventions, and research. That is why we see a **high demand for Maths guide for class 12.** Of course, without practicing, Maths becomes all the more complex. **Class 12 Maths NCERT solutions act as a perfect start for class 12 students to kickstart their preparations for the academic year.** Why you should study from these Class 12 Maths NCERT solutions, let’s find out:

**Free Solution for all class 12 Maths chapters:**There are two parts and 13 chapters in NCERT 12 Maths to cover. These solutions for class 12 Maths help you cover all the chapters in one click. The best part is, you don’t have to pay at all.**Class 12 Maths NCERT Solutions in your hands:**These CBSE class 12 Maths NCERT solutions are available online. You do not need to purchase any books. There are no bundles of books to clutter your bookshelf, all you have to do is just visit www.oswalpublishers.com, and you can access the study material from your phone and PC as well.**Chapterwise Class 12 Maths NCERT Solutions:**Highly proficient Maths subject matter experts have designed these chapters CBSE 12 Maths NCERT solutions. Students will get solutions for all the chapters and a detailed understanding of the equations, theorems, and applications.

## FAQs on Maths Class 12 NCERT Solutions

**Ans:** Class 12 Mathematics Solutions helps students prepare for the CBSE board examination and many competitive entrance examinations like NEET, CUET, JEE, and more. Mostly, the questions in these examinations are based on Maths 12 chapters. Therefore, class 12 Mathematics solutions hold a lot of importance for students either in class 12 or preparing for their competitive or entrance examination.

**Ans: **Yes, the grade 12 Maths NCERT solutions are free. Students do not have to pay fees or subscribe to the website to access the 12th Maths CBSE solutions.

**Ans: **Regarding Maths, the market is swarmed with many reference books. However, class 12 Maths NCERT book solutions are the best to begin your academic session. The NCERT 12 Maths solutions are based on the NCERT textbook and NCERT syllabus and help students understand the CBSE curriculum better.

**Ans:** There is no right day or time for class 12 students to start preparing. Class 12 is when you must give you all to shape your life. Instead of stressing later, planning your studies will help you easily sail the boat. For the most effective tips on planning your studies, click here.