# Ncert 12th mathematics book

National Council of Educational Research and Training (NCERT) Book for Class XII Subject: Maths. NCERT Class XII Maths Book Part I and Part II are given. GanitI. NCERT/CBSE class 12 Mathematics book GanitI · GanitII. NCERT/CBSE class 12 Mathematics book GanitII · MathematicsPartI. NCERT/CBSE class Free NCERT Solutions for Class 12 Maths (chapter-wise) in PDF format to Download online, solved by subject expert teachers from latest edition books and as.

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The solutions are concise, and also discuss alternative methods to solve some problems. We make it a point to provide the students the best help they can get, and the solutions reflect just that. Your performance in the board exams will play a role in determining your admission into a good college or even your dream college! That will undoubtedly induce a lot of pressure upon you and you may tend to get demotivated or stressed out. To help you overcome such situations, LearnCBSE is a platform all the way helps you to crack your target. The students explore several real valued functions, and their graphs. The chapter comprises of four exercises that consolidate the concepts explained in the chapter. This chapter explores the inverse trigonometric through two exercises that teach the students the behavior of the functions, their graphical representations, and the elementary properties. These prove to be of more value in integration later.

Benefits of accessing class 12 maths NCERT solutions are not only confined to your success in Maths, but it also lays a foundation for other super important subjects. Getting a good grip over exhaustive details of some chapters of Maths like Calculus, differential equations etc can be of great help while understanding the derivations and concepts behind the numericals of Science.

The solutions for each exercise have been curated and reviewed by some of the best teachers across India. The chapters are designed in a sequential and logical manner, which allows you to understand the concepts easily.

The first chapter of CBSE Class 12 deals with the notion of relations and functions, domain, co-domain and range which have already been introduced in class 11 syllabus along with various types of specific real valued functions and their respective graphs.

There are many functions which are not one-one, onto or even both and hence we cannot talk of their inverses. In Class 11 studied that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist. In CBSE Class 12 Inverse Trigonometric Functions, there are only two exercises where you will study about the restrictions on domains and ranges of trigonometric functions which enables the visibility of their inverses and observe their behaviour through graphical representations.

Besides, some elementary properties will also be discussed. The inverse trigonometric functions hold a great value in calculus for they serve to define many integrals. You need to know about Matrices as it is instrumental in various branches of mathematics. Matrices are considered as one of the most powerful tools in mathematics. It simplifies our work to a great extent when compared with other methods.

There are total 62 questions in 4 exercises of this chapter. In this chapter, you will be delving deeper into the fundamentals of matrix and matrix algebra. In chapter 3, matrices and algebra of matrices will be explained. Whereas, in this chapter, you will be studying about determinants up to order three only with real entries. Also, the six exercises are distributed in a way that you will study 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 inverse of a matrix in these exercises.

This chapter is the extension of differentiation of functions which you have studies in Class XI. You must have learnt to differentiate certain functions like polynomial functions and trigonometric functions.

This chapter explains the very important concepts of continuity, differentiability and relations between them. You will be learning about differentiation of inverse trigonometric functions. Furthermore, you will be getting acquainted with a new class of functions called exponential and logarithmic functions.

There is a total of eight exercises in this chapter so you will have to dedicate some extra time and effort to get well-versed with this chapter. In Chapter 5, you would learn how to find derivative of composite functions, implicit functions, exponential functions, inverse trigonometric functions, and logarithmic functions. In this chapter, you will study the applications of the derivative in various disciplines such as engineering, science, and many other fields. For instance, we will learn how the derivative can be used to determine rate of change of quantities and to find the equations of tangent and normal to a curve at a point and many more usage of derivatives.

We will also use derivative to find intervals of increasing or decreasing functions. There are five exercises in total where you will get the detailed insight into the Application of Derivatives. Finally, the derivative is used to find approximate value of certain quantities. Differential Calculus is based on the idea of derivative. Derivative came into existence for the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines.

## NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.2

Integral Calculus eases the problem of defining and calculating the area of the region bounded by the graph of the functions. With a total of eleven exercises in total, you will have to learn each and every topic and their related questions with sheer concentration. Here, in this chapter, you will study some specific application of integrals to find the area given under the simple curves, also, area between lines and arcs of circles, parabolas and ellipses standard forms only.

There are two exercises in this chapter in which you will also deal with finding the area bounded by the above said curves. In Differential Equations, you will be studying some basic concepts related to differential equation, general and particular solutions of a differential equation, formation of differential equations, number of methods to solve a question based on first order - first degree differential equation and some applications of differential equations in different areas.

There are total six exercises in this chapter for the students. Differential equations are implemented in a plethora of applications in all the other subjects and areas. Hence, if you study the Differential equations chapter in a detailed manner, it will help you to gain a detailed insight into all modern scientific investigations. Did you know that quantities that involve only one value magnitude , which is a real number are call scalars?

Whereas, quantities that involves magnitude and direction are called Vectors. In the chapter Vector Algebra, you will be studying about some basic concepts of vectors, various operations on vectors, and their algebraic as well as geometric properties.

## UP Board Solutions for Class 12 Maths गणित

The chapter Three Dimensional Geometry will take you through the study of the direction cosines and direction ratios of a line joining two points and also about the equations of lines and planes in space under different conditions. Along with that, you will also get to know about angle between two lines, a line and a plane, two planes,shortest distance between two skew lines and distance of a point from a plane. This chapter has only two exercises in which you will study about some linear programming problems and their solutions by graphical method only, though there are many other methods also available to solve such problems. Probability will take you through the important concepts of conditional probability of an event given that another event has occurred, which will in turn clarify the Bayes' theorem, multiplication rule of probability and independence of events.

With a total of five exercises, you will also learn an important concept of random variable and its probability distribution and the mean and variance of a probability distribution.

Mathematics is a subject which has been relevant to humanity since ancient times. As a matter of fact, archaeologists have discovered that a numbering system was even used by the Indus Valley civilization. They used this mainly for trade and commerce purposes with different parts of the world.

Today, Math holds a very important place in almost all aspects of our daily life. For instance, at the supermarket billing counter, we inevitably use the computerized numerical system to calculate our total bill.

The NCERT solutions for class 12 maths has been curated and reviewed by some of the best Maths teachers in the country to help students grasp all the fundamental concepts. The Class 12 Maths NCERT solutions are very helpful during individual or group studies, teaching students how to best approach a problem.

The Maths NCERT solutions class 12 is an aid that should complement the students' school textbook and help them understand the problems in a much better manner. One of the best advantages NCERT Solutions is it acts as an expert tool which can be used as a gateway to the mind of the examiner.

Students who need additional guidance while solving Class 12 Maths Questions can avail the online tutoring option that is available at Vedantu. The 12th board exam can be the stepping stone that can ensure admission into a good college. The professional help extended by the Vedantu faculty will help the students become more confident as any doubt they have while solving questions with the help of NCERT Solutions for class 12 Maths can be immediately addressed.

## NCERT Books for Class 12 All Subjects in PDf form (Download)

Chapter 2: Inverse Trigonometric Functions There are many functions which are not one-one, onto or even both and hence we cannot talk of their inverses. Matrix You need to know about Matrices as it is instrumental in various branches of mathematics. Chapter 4: This chapter teaches students How to calculate determinants, the different properties of determinants, the adjoint and inverse of a matrix, the area of a triangle, the various applications of determinants and matrix, the minors and cofactors, and the determinants and matrices as equation solver.

These questions and solutions are also helpful for students preparing for other exams like JEE and engineering exams. Chapter 5 — Continuity and Differentiability The 5th chapter of the book is dedicated to the theories of continuity and differentiability.

Some of the topics covered in this chapter are the Algebra of continuous functions, Derivatives of composite functions and implicit functions, Derivatives of inverse trigonometric functions, Derivatives of Functions in Parametric Forms, Exponential and Logarithmic Functions and their differentiation, Mean Value Theorem, and Second Order Derivative.

The chapter also helps the students examine certain functions of continuity and generate conclusions. There are four exercises that will teach the students to find the points of discontinuity for a certain variable when it is defined by a function, etc. Chapter 6 — Application of Derivatives This chapter teaches students the application of derivatives.

A few topics covered in this chapter are approximations, increasing and decreasing functions, maxima, and minima, the rate of change of quantities, tangents and normals, etc. Each chapter is well-explained in a step by step manner for the students to understand the concept well. There are about four exercises that students can practice to get a better grasp of the concept. Students can also refer to the explanations and other guidelines in the chapter while preparing for exams.

Chapter 7 — Integrals The 7th chapter of the book discusses the theory of Integral calculus. This is an important chapter in mathematics and students need to pay careful attention. The topics covered in this chapter are inverse Process of Differentiation, Geometrical interpretation of indefinite integral, Indefinite integrals and their properties, the difference Between Differentiation and Integration, the Methods of Integration, the usage of trigonometric identities in Integration, the Formulas and Types of Integrals, the Integrals Techniques, the Properties of Definite Integrals, the Calculus Fundamental Theorems, the First and Second fundamental theorems of integral calculus, the evaluating Definite Integrals by Substitution, among many others. Chapter 8 — Application of Integrals After discussing all about integrals in the last chapter, the 8th chapter of the book discusses the applications of integrals in various cases. In this chapter, the students are taught several important concepts of application like how to find the area under simple curves, the area between two curves, and the area of the region bounded by a curve and a line.

All these three concepts of applications are explained with simple diagrams and various examples so that the students can understand the concept of integrals and its various applications in a better way. Chapter 9 — Differential Equation This is an important chapter not only in Mathematics but is also used in solving many Physics derivations and is used in some of the chapters in the field of engineering too.

Some of the topics covered in this chapter are the degree of a differential equation, the First-degree differential equations, the general and particular solutions of a differential equation, the Methods of Solving First Order, among the others.

There are six exercises at the end of the chapter too that have been solved for the students to understand and grasp the subject in a better way. Chapter 10 — Vector Algebra The 10th chapter of the NCERT Books for Class 12 Maths covers a few important topics like applying vectors to figures, how to differentiate vectors and scalars, functions on vectors, among the many others.

To have a clear and better understanding of the topics, there are also four solved exercises at the end of the chapter. All the exercises are designed according to the CBSE syllabus and follow the requisite guidelines. Students can practice the same to master the concepts as well as perform well in the board examination and other competitive exams.

Chapter 11 — Three Dimensional Geometry After studying vector algebra in the previous chapter, students are introduced to the concept of 3D geometry in the 11th chapter of the book. Students are taught all about the concept of 3D geometry and various topics related to the subject are discussed in details for the students to understand.