**CBSE NCERT Solutions for Class 12 Maths PDF** – Students who are in search of CBSE NCERT Solutions for Class 12 Maths can refer to this article. Going through the CBSE Maths NCERT Solutions is a crucial part of your preparation for Class 12 board exams. Working on CBSE NCERT Solutions for Class 12 Maths will clear your doubts in regards to any question and improve your application skills as well. Working on** **NCERT Solutions for Class 12 Maths PDF will not help only students to secure good marks in their board exams, but also helps to clear the toughest engineering entrance exams such as JEE Main, JEE Advanced, BITSAT etc.

All the NCERT Solutions for Class 12 Maths provided in this page are solved by the top Maths teachers of Embibe. Each question of CBSE NCERT Solutions for Class 12 Maths comes with detailed step-wise solutions. Maths Class 12 NCERT Solutions can also be downloaded for free and access them in offline mode as well. In this article, we will provide you with CBSE NCERT Solutions for Class 12 Maths for all the chapters. Read on to find more about Class 12 Maths NCERT Solutions.

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**CBSE NCERT Solutions for Class 12 Maths PDF Download**

Students can download all the CBSE NCERT Solutions for Class 12 Maths provided in this page for free without any cost. Working on NCERT Solutions for Class 12 Maths will not only help to clear your basics but also help in improving your problem-solving abilities. Just click on the Class 12 Maths NCERT Chapter wise links listed below to download.

**Class 12 Maths NCERT Chapter Wise Solutions**

- Class 12 Maths Chapter 1 – Relations and Functions NCERT Solutions
- Class 12 Maths Chapter 2 – Inverse Trigonometric Functions NCERT Solutions
- Class 12 Maths Chapter 3 – Matrices NCERT Solutions
- Class 12 Maths Chapter 4 – Determinants NCERT Solutions
- Class 12 Maths Chapter 5 – Continuity and Differentiability NCERT Solutions
- Class 12 Maths Chapter 6 – Application of Derivatives NCERT Solutions
- Class 12 Maths Chapter 7 – Integrals NCERT Solutions
- Class 12 Maths Chapter 8 – Application of Integrals NCERT Solutions
- Class 12 Maths Chapter 9 – Differential Equations NCERT Solutions
- Class 12 Maths Chapter 10 – Vector Algebra NCERT Solutions
- Class 12 Maths Chapter 11 – Three Dimensional Geometry NCERT Solutions
- Class 12 Maths Chapter 12 – Linear Programming NCERT Solutions
- Class 12 Maths Chapter 13 – Probability NCERT Solutions

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The advantages of solving CBSE NCERT Solutions for Class 12 Maths from Embibe are listed below:

- All the CBSE NCERT Solutions for Class 12 Maths are provided in the PDF form which can be downloaded for free by anyone, anywhere and make use of them in offline mode.
- Referring to the Class 12 Maths NCERT Solutions will help students understand all basic and fundamental concepts.
- As most of the engineering competitive exam syllabus such JEE Main, JEE Advanced, BITSAT is almost the same as CBSE Class 11 & 12 Maths Syllabus. So having good command over the NCERT Solutions for Class 12 Maths will definitely help to clear the competitive exams easily.
- All the CBSE NCERT Solutions comes with detailed step by step solution which will further help students to solve their homework and assignments on time without any difficulty.

### CBSE NCERT Solutions For Class 12 Maths – Chapter Descriptions

**NCERT Solutions For Class 12 Maths Chapter 1 – Relations and Functions**

In Class 11, you have learnt about the notion of relations and functions, domain, co-domain and range along with different types of specific real-valued functions and their graphs. The concept of the term ‘**relation**’ in mathematics has been drawn from the meaning of relation in English language, according to which two objects or quantities are related if there is a recognisable connection or link between the two objects or quantities. Let **A** be the set of students of Class XII of a school and** B** be the set of students of Class XI of the same school. Then some of the examples of relations from **A to B** are

- {(a, b) ∈ A × B: a is brother of b},
- {(a, b) ∈ A × B: a is sister of b},
- {(a, b) ∈ A × B: age of a is greater than age of b},
- {(a, b) ∈ A × B: total marks obtained by a in the final examination is less than the total marks obtained by b in the final examination},
- {(a, b) ∈ A × B: a lives in the same locality as b}.

However, abstracting from this, we define mathematically a relation R from A to B as an arbitrary subset of A × B. If (a, b) ∈ R, we say that a is related to b under the relation R and we write as **a** R **b**. So as you learnt in Class 11, the functions are special kind of relations whereas in this Class 12 Chapter 1, you will learn about different types of relations and functions, the composition of functions, invertible functions and binary operations.

**NCERT Solutions For Class 12 Maths Chapter 2 – Inverse Trigonometric Functions **

In Chapter 1, you have studied that the inverse of a function f, denoted by f^{–1}, exists if f is one-one and onto. There are many functions which are not one-one, onto or both and hence we can not talk of their inverses. In Class XI, you have learnt that trigonometric functions are not one-one and onto over their natural domains and ranges and hence their inverses do not exist.

In this chapter, you will study the restrictions on domains and ranges of trigonometric functions which 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. Also, the concepts of inverse trigonometric functions are also used in science and engineering.

**NCERT Solutions For Class 12 Maths Chapter 3 – Matrices **

The knowledge of matrices is necessary for various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrices are not only used as a representation of the coefficients in system of linear equations, but utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet programs for personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analysing the results of an experiment etc. Also, many physical operations such as magnification, rotation and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology and industrial management.

In this chapter, you will learn about the fundamentals of matrix and matrix algebra.

**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*|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the *n*-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of *n*-space.

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. Also, you will study various properties of determinants, minors, cofactors and 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 solution of linear equations in two or three variables using the inverse of a matrix.

**NCERT Solutions For Class 12 Maths Chapter 5 – Continuity and Differentiability**

Continuity of a function is the characteristic of a function by virtue of which, the graphical form of that function is a continuous wave. A differentiable function is a function whose derivative exists at each point in its domain. In this chapter, we will learn everything about Continuity and Differentiability of a function. In this chapter, students will learn about the very important concepts of continuity, differentiability and relations between them. We will also learn differentiation of inverse trigonometric functions. Further, this chapter introduces a new class of functions called exponential and logarithmic functions. These functions lead to powerful techniques of differentiation. Also, this chapter illustrates certain geometrically obvious conditions through differential calculus.

**NCERT Solutions For Class 12 Maths Chapter 6 – Application of Derivatives NCERT Solutions**

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

For instance, you will learn about how the derivative can be used

(i) to determine the rate of change of quantities

(ii) to find the equations of tangent and normal to a curve at a point

(iii) to find turning points on the graph of a function which in turn will help us to locate points at which the largest or smallest value (locally) of a function occurs.

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.

**NCERT Solutions For Class 12 Maths Chapter 7 – Integrals NCERT Solutions**

Differential Calculus is centred on the concept of the derivative. The original motivation for the derivative was the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines. Integral Calculus is motivated by the problem of defining and calculating the area of the region bounded by the graph of the functions.

The functions that could possibly have given function as a derivative are called anti derivatives (or primitive) of the function. Further, the formula that gives all these anti derivatives is called the indefinite integral of the function and such process of finding anti derivatives is called integration. Such type of problems arise in many practical situations. For instance, if we know the instantaneous velocity of an object at any instant, then there arises a natural question, i.e., can we determine the position of the object at any instant? There are several such practical and theoretical situations where the process of integration is involved. The development of integral calculus arises out of the efforts of solving the problems of the following types:

(a) the problem of finding a function whenever its derivative is given

(b) the problem of finding the area bounded by the graph of a function under certain conditions. These two problems lead to the two forms of the integrals, e.g., indefinite and definite integrals, which together constitute the Integral Calculus.

**NCERT Solutions For Class 12 Maths Chapter 8 – Application of Integrals NCERT Solutions**

In geometry, we have learnt formulae to calculate areas of various geometrical figures including triangles, rectangles, trapezias and circles. Such formulae are fundamental in the applications of mathematics to many real-life problems. The formulae of elementary geometry allow us to calculate areas of many simple figures. However, they are inadequate for calculating the areas enclosed by curves.

For that we shall need some concepts of Integral Calculus. In the previous chapter, we have studied to find the area bounded by the curve y = f (x), the ordinates x = a, x = b and x-axis, while calculating definite integral as the limit of a sum. Here, in this chapter, we shall study a specific application of integrals to find the area under simple curves, area between lines and arcs of circles, parabolas and ellipses (standard forms only). We shall also deal with finding the area bounded by the above said curves.

**NCERT Solutions For Class 12 Maths Chapter 9 – Differential Equations NCERT Solutions **

An equation of the form (1) is known as a differential equation. A formal definition will be given later. These equations arise in a variety of applications, may it be in Physics, Chemistry, Biology, Anthropology, Geology, Economics etc. Hence, an in-depth study of differential equations has assumed prime importance in all modern scientific investigations. 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.

**NCERT Solutions For Class 12 Maths Chapter 10 – Vector Algebra NCERT Solutions**

In our day to day life, we come across many queries such as – What is your height? How should a football player hit the ball to give a pass to another player of his team? Observe that a possible answer to the first query may be 1.6 meters, a quantity that involves only one value (magnitude) which is a real number. Such quantities are called scalars. However, an answer to the second query is a quantity (called force) which involves muscular strength (magnitude) and direction (in which another player is positioned). Such quantities are called vectors. In mathematics, physics and engineering, we frequently come across with both types of quantities, namely, scalar quantities such as length, mass, time, distance, speed, area, volume, temperature, work, money, voltage, density, resistance etc. and vector quantities like displacement, velocity, acceleration, force, weight, momentum, electric field intensity etc. In this chapter, we will study some of the basic concepts about vectors, various operations on vectors, and their algebraic and geometric properties. These two type of properties, when considered together give a full realisation to the concept of vectors, and lead to their vital applicability in various areas as mentioned above.

**NCERT Solutions For Class 12 Maths Chapter 11 – Three Dimensional Geometry NCERT Solutions**

In Class XI, while studying Analytical Geometry in two dimensions, and the introduction to three-dimensional geometry, we confined to the Cartesian methods only. In the previous chapter of this book, we have studied some basic concepts of vectors. We will now use vector algebra to three-dimensional geometry. The purpose of this approach to 3-dimensional geometry is that it makes the study simple and elegant.

In this chapter, we shall study the direction cosines and direction ratios of a line joining two points and also discuss about the equations of lines and planes in space under different conditions, angle between two lines, two planes, a line and a plane, shortest distance between two skew lines and distance of a point from a plane. Most of the above results are obtained in vector form. Nevertheless, we shall also translate these results in the Cartesian form which, at times, presents a more clear geometric and analytic picture of the situation.

**NCERT Solutions For Class 12 Maths Chapter 12 – Linear Programming NCERT Solutions **

In earlier classes, we have discussed systems of linear equations and their applications in day to day problems. In Class XI, we have studied linear inequalities and systems of linear inequalities in two variables and their solutions by graphical method. Many applications in mathematics involve systems of inequalities/equations. In this chapter, we shall apply the systems of linear inequalities/equations to solve some real life problems of the type as given below: A furniture dealer deals in only two items–tables and chairs. He has Rs 50,000 to invest and has storage space of at most 60 pieces. A table costs Rs 2500 and a chair Rs 500. He estimates that from the sale of one table, he can make a profit of Rs 250 and that from the sale of one chair a profit of Rs 75. He wants to know how many tables and chairs he should buy from the available money so as to maximise his total profit, assuming that he can sell all the items which he buys. Such type of problems which seek to maximise (or, minimise) profit (or, cost) form a general class of problems called optimisation problems. Thus, an optimisation problem may involve finding maximum profit, minimum cost, or minimum use of resources etc. A special but a very important class of optimisation problems is linear programming problem. The above stated optimisation problem is an example of linear programming problem. Linear programming problems are of much interest because of their wide applicability in industry, commerce, management science etc. In this chapter, we shall study some linear programming problems and their solutions by graphical method only, though there are many other methods also to solve such problems.

**NCERT Solutions For Class 12 Maths Chapter 13 – Probability NCERT Solutions **

In earlier Classes, we have studied the probability as a measure of uncertainty of events in a random experiment. We discussed the axiomatic approach formulated by Russian Mathematician, A.N. Kolmogorov (1903-1987) and treated probability as a function of outcomes of the experiment. We have also established equivalence between the axiomatic theory and the classical theory of probability in case of equally likely outcomes. On the basis of this relationship, we obtained probabilities of events associated with discrete sample spaces. We have also studied the addition rule of probability. In this chapter, we shall discuss the important concept of conditional probability of an event given that another event has occurred, which will be helpful in understanding the Bayes’ theorem, multiplication rule of probability and independence of events. We shall also learn an important concept of random variable and its probability distribution and also the mean and variance of a probability distribution. In the last section of the chapter, we shall study an important discrete probability distribution called Binomial distribution. Throughout this chapter, we shall take up the experiments having equally likely outcomes, unless stated otherwise.

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