NCERT Solutions for Class 12 Maths – Free PDF Download
Class 12 is an important milestone in your life as you take some serious decisions about your future based on your performance. Indeed, Mathematics forms a vital component of Class 12 as well as every competitive exams’ syllabus. Whether you choose to be an engineer or a doctor, you will have to deal with Mathematics in every area. Making the concepts of Class 12 concrete can immensely help you in securing good marks in the CBSE Class 12 board exam as well as in various competitive exams. CBSE NCERT Solutions for Class 12 Maths plays a vital role in your exam preparation as it has detailed chapter-wise solutions for all exercises. First and foremost, you should imprint this in your mind that NCERT books are the biggest tool that you can have to get well-versed with the fundamentals. This will help you to not leave out any important question in your exam preparation.
Benefits of accessing Class 12 CBSE 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 numerical of Science. You can easily download FREE CBSE Class 12th NCERT Maths Solutions PDF of every Chapter of Mathematics. The solutions for each exercise have been curated and reviewed by some of the best teachers across India. It covers all the answers to the questions from 13 important chapters from the NCERT Mathematics textbook. Class 12 Maths NCERT Solutions covers crucial and intricate chapters such as Differentiation, Integration, Algebra, and others. The chapters are designed sequentially and logically, which allows you to understand the concepts easily.
Let us look into the Class 12 Maths Chapters in detail:-
Chapter 1: Relations and Functions
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. In this chapter, you will be solving a total of four exercises’ questions entailing all the essential topics in them.
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. 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 great value in calculus for they serve to define many integrals. Inverse trigonometric functions’ concepts are applied vastly in science and engineering
Chapter 3: Matrix
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 a total of 62 questions in 4 exercises of this chapter. In this chapter, you will be delving deeper into the fundamentals of matrix and matrix algebra.
Chapter 4: Determinants
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.
Chapter 5: Continuity and Differentiability
This chapter is the extension of differentiation of functions which you have studied 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 the differentiation of inverse trigonometric functions. Furthermore, you will be getting acquainted with a new class of functions called exponential and logarithmic functions. There are 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.
Chapter 6: Application of Derivatives
In Chapter 5, you will learn how to find the 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 the 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 the derivative to find intervals of increasing or decreasing functions. There are five exercises in total where you will get a detailed insight into the Application of Derivatives. Finally, the derivative is used to find an approximate value of certain quantities.
Chapter 7: Integrals
Differential Calculus is based on the idea of a derivative. Derivative came into existence for the problem of defining tangent lines to the graphs of functions and calculating the slope of such lines. Integral Calculus eases the problem of defining and calculating the area of the region bounded by the graphs of the function. With a total of eleven exercises, you will have to learn every topic and their related questions with sheer concentration.
Chapter 8: Application of Integrals
Here, in this chapter, you will study some specific application of integrals to find the area given under the simple curves, also, the 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.
Chapter 9: Differential Equations
In Differential Equations, you will be studying some basic concepts related to the 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 a total of 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.
Chapter 10: Vector Algebra
Did you know those quantities that involve only one value (magnitude), which is a real number is called scalars? Whereas, quantities that involve magnitude and direction are called Vectors. In chapter Vector Algebra, you will be studying about some basic concepts of vectors, various operations on vectors, and they’re algebraic as well as geometric properties.
Chapter 11: Three Dimensional Geometry
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 the angle between two lines, a line and a plane, two planes, the shortest distance between two skew lines and distance of a point from a plane.
Chapter 12: Linear Programming
This chapter has only two exercises in which you will study some linear programming problems and their solutions by graphical method only, though there are many other methods also available to solve such problems.
Chapter 13: Probability
The probability will take you through the important concepts of the 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 a random variable and its probability distribution and the mean and variance of a probability distribution.
The Class 12 Maths NCERT Solutions provided on CoolGyan have been written to help you understand all the chapter-wise problems in the latest and updated textbooks prescribed by NCERT. Being the most preferred study material, NCERT Solutions will help the students in various ways. Let us look into this:-
Key Benefits of CBSE Class 12 Maths NCERT Solutions from CoolGyan:
- You can easily grasp some crucial tips to answer some particular complex questions in a detailed manner.
- You no longer have to turn several pages for revision before the examination. The Maths Class 12th NCERT Solutions PDF Download is available on our website for FREE which you can access anytime, anywhere.
- Also, all the chapters’ solutions are arranged in a sequential manner which enables smooth learning. Maths is a sequential subject and it is important that it is studied in a right manner.
- The solutions are explained in a detailed manner. Step by step solutions provided for all the exercise questions helps you to know the subject in a detailed manner.
- The shortcuts help you to take your preparation into next level. CoolGyan NCERT Solutions will help you in improving your mathematical skills as well.
Mathematics is a subject which has been relevant to humanity since ancient times. 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. It is a great guide that can be used to prepare for the CBSE Board exams, as well as engineering entrance examinations like JEE. 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. It helps the students to get good marks in their Math board exams 2019-20. One of the best advantages of NCERT Solutions is it acts as an expert tool which can be used as a gateway to the mind of the examiner.