NCERT Solutions For class 12 Maths

Class 12 Maths Solutions

Get detailed NCERT solutions for class 12 maths here. Are you the one preparing for your board examinations for class 12? Are you worried about getting maximum marks in Maths board examinations? If you are a student, you sincerely need to work hard for your board examinations as your marks help you get into a good college in your future.

If you are searching for the complete solutions to your Maths NCERT Book, your search ends here! We provide the best NCERT solutions to help you understand the underlying concept in each question. NCERT Solutions for Class 12 Maths is a complete resource that enables students to properly understand the Class 12 NCERT questions. Here on our website, we provide the Best online tutors with experts who help students understand the concepts of each topic, and they will also help you solve your queries and doubts. 

While you solve your NCERT Book on your own, you may face some difficulties while solving questions, and you must be curious about the accuracy of your answers.

Here comes the role of the platforms, which provide you with the questions with the best explanation and accuracy so that you don't have to face any difficulty while dealing with numerous questions in your NCERT Book. 

We have included solutions to every question for the NCERT textbook Class 12 for Maths. Practising the questions will give you a better understanding of all the concepts. You can even download these Class 12 Maths solutions in PDF format for free.

Here are chapter Wise NCERT Solutions for Class 12 Maths:

Here, we are providing the complete chapter-wise NCERT Solutions for Class 12 Maths. Keeping in mind the language barrier, we have provided these solutions in two languages: English and Hindi. 

1.    Relation & Function

    This chapter deals with linking a pair of elements from two sets and then introduce relations between the two elements in the pair. The relation is good for understanding how variables behave together. 

    Finally, we will learn about the function which is a special type of relationship where there is an angled output for any combination of given inputs hence, we can say that for a relationship to be a function, each input must have one and only one output. 

    Before solving the linear and non-linear equations, one must develop a proper understanding of the difference between relation and function. 

          In this chapter, we will study different types of relations and functions, the composition of functions, invertible functions and binary operations. 

   2.    Inverse Trigonometric Function

    In the chapter on relation and function, we have studied that the inverse of a function exists only if that function is one-one and onto both.

     Trigonometric functions are not one-one and onto over their natural domain and range hence their inverse does not exist. In this chapter, we will study the restrictions on domain and range of trigonometric functions so that they become one-one and onto which ensure the existence of their inverse and observe their behaviours through graphical representation.

    The concept of an inverse trigonometric function is very important for the understanding of calculus, and it is also used in science and engineering.

3.    Matrices

    A matrix is an ordered rectangular array of numbers or functions. The number of functions is called the elements or entries of the matrix. The concept of the matrix is mainly developed to obtain a compact and simple method of solving a system of linear equations.

    Matrices are one of the most powerful tools in mathematics. Matrices are not only used as a representation of the coefficients in a system of linear equations, but their utility far exceeds that use.

    Matrix notation and operations are used in electronic spreadsheet programs for personal computers, which in turn are used in different areas of business and science like budgeting, sales projection, cost estimation, analyzing the results of an experiment etc.

    In this chapter, we will study matrices and the algebra of matrices.

4.    Determinants

    To every square matrix, we can associate a number (real or complex) called the determinant of the square matrix. In this chapter, we will study determinants up to order three only with real entries. Also, we 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 equation in two or three variables using the inverse of a matrix.

5.    Continuity and Differentiability

    As we have already studied the limits and derivatives in class XI and this chapter is a continuation of the study of that. In this chapter, we will learn the very important concepts of continuity, differentiability, and relation between them.

    Continuity of a function at any point means there is no break in the graph of the function at that point. If the function is not continuous at a point, it is called a discontinuous function.

    In this chapter, we will also learn the differentiation ion of inverse trigonometric function apart from polynomial function, trigonometric function, logarithmic function, exponential function etc.

 6.             Application of Derivatives

    This chapter deals with the application of derivatives in various disciplines such as engineering, science, social science and many other fields.

    In this chapter, we will learn mainly, how the derivative can be used to determine the rate of change, the question of tangent and normal, and points of maxima and minima at which the graph of the function turns. We will also learn the use of derivatives in increasing and decreasing functions and to find approximate values of certain quantities. 

7.    Integrals

    In simple language, we can say that the integral is the reverse of the derivative. If a derivative of a function exists at each point of an interval, then the function that could have given a function as a derivative is called antiderivative or primitive of the function.

    The formula that gives all these antiderivatives is called Indefinite Integral of the function and such process of finding antiderivatives is called Integration.

    The problem of finding the area bounded by the graph of a function under certain conditions is also dealt with by Integral Calculus.

8.    Application of Integrals

    In this chapter, we use the concept of integral calculus for determining the area enclosed by curves.

    Here we will learn a specific application of integrals to find the area under simple curves, the area between lines and arc of circles, parabola, and ellipses etc. we can also find the area enclosed between the circles, parabola, and ellipse.

    In this chapter basically, we think of the area under the curve is composed of many very thin vertical strips. The area of these vertical strips is called an elementary area.

9.    Differential Equations

    A differential equation is an equation containing differential coefficients, variables (x, y) and constants.

    In this chapter, we will study some basic concepts related to differential equations, general and solutions of a differential equation, the formation of differential equations, some methods to solve the first order and first-degree differential equations and some applications of differential equations in different types of situations.

    The concept of a differential equation is very important as it is used in Physics, Chemistry, Biology, Geology and Economics.

 10.    Vector Algebra

    A quantity that involves the magnitude only is called a scalar whereas the quantities that involve magnitude and direction both are called vectors.

    In this chapter, we will learn the concepts of vectors, their operations and their algebraic and geometric properties.

    In mathematics and Physics, we frequently use the vector quantities like displacement, velocity, acceleration, force, weight, momentum etc. and scalar quantities like mass, time, distance, length, speed, area, volume, temperature etc.

11.    Three-Dimensional Geometry

     This chapter is a continuation of what we have already studied in class XI and we will use vector algebra for 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. Nevertheless, we shall also translate these results in the Cartesian form which, at times, presents a clearer geometric and analytic picture of the situation.     

12.    Linear Programming

    A special but very important class of optimization problems is linear programming problems.

    Such type of problems which seek maximum/minimum profit is called optimization problems. Thus an optimization problem may involve finding maximum profit, minimum cost etc.

    Many applications in mathematics involve systems of inequalities or equations. In this chapter, we will apply the system of linear inequalities/equations to solve some real-life problems.

    In this chapter, we will learn some linear programming problems and their solution by graphical method only. 

13.    Probability

    In general, probability is the chance of occurrence of an event. Probability is defined as a quantitative measure of uncertainty.  The value of probability always lies between 0 and 1 including 0 and 1.

    There is a lot of usage of probability in real life such as in weather forecasting, clinical estimate, making business plans, sports etc.   

          In this chapter, we will learn the important concept of the conditional probability of an event given that another event has occurred. Further with the help of conditional probability, we will learn Bayes’ theorem, multiplication rule and independence of events. We will also learn random variables, probability distribution, mean and variance of a probability distribution, Binomial distribution etc. 

Benefits of NCERT solutions:

Do you want to excel in Mathematics in class 12? Do you want to get overall good marks in your final board examinations? If your answer is Yes, you need to work hard on your goals, and we are here to help you with the same! 

We provide you with the NCERT Book solutions for class 12 Maths. It will help you save your time searching for the answer to any particular question on multiple platforms. We provide the best-detailed solutions to help you thoroughly prepare for your exams.

You can score well in your Class 12th Board examination if you prepare well from the detailed NCERT Solutions provided by our expert team!

CBSE Class 12 Maths Academic Resources 2023-24