# NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Maths: The performance of students in the board exams plays a significant role to determine their future so that they can take admission into a good college. In CBSE class 12 maths there are 13 chapters that cover important theorems and formulas. In this article, you will find chapter-wise solutions of NCERT for class 12 maths which will help you to perform well in the board exams. Practicing NCERT questions is very important in mathematics, to understand the concepts. If you are not able to solve, you can take help of these CBSE NCERT solutions for class 12 maths provided in this article by clicking on the chapter you wish to study. The entire syllabus of NCERT Class 12 maths has six major parts that cover relation and function, algebra, calculus, Vector and 3-D, Linear programming and Probability. In the first part "Relation and Function" you will learn types of relations, reflective, transactive and equivalence relation, types of functions, range, domain and codomain of function, inverse function and binary operations. "Algebra" consists of Matrix and determination, and it's operation. Calculus has three main parts continuity, differentiability, and integrals. Here you will find all NCERT solutions at a single place which will be helpful when you are not able to solve the NCERT practice questions on your own. Also, you can use these NCERT solutions as a quick revision of concepts before the exam to score more marks.

Before solving the NCERT exercise question, understand the concept and solve the solved examples of the NCERT 12 maths textbook. You should try to solve every question including examples on your own, if you are not able to do, you can take help with these NCERT solutions for class 12 maths in which detailed explanation using step-by-step approach is there to facilitate an easy and smooth understanding of concepts. It will give you a new insight to solving CBSE maths problems. These solutions of NCERT for class 12 maths will help you to learn CBSE class 12 maths as well as the crack JEE Main and other competitive exams like BITSAT, VITEEE, etc. With the help of these NCERT solutions for class 12 maths, you can grasp the concepts based on their application very easily.

## CBSE Class 12 Weightage Distribution Of Mathematics-

 Units Marks Relations & Functions 10 Algebra 13 Calculus 44 Vector & 3-Dimensional Geometry 17 Linear Programming 6 Probability 10 Total 100

## NCERT Solutions for Class 12 Maths Chapter Wise:-

If two sets of elements are related in some ways, we say these two sets have a relation. Father, mother, sister, brother are real-life examples of relations. Relation becomes function when for every input there is only only one output. For different inputs, the output can be the same. Every function is a relation, but every relation doesn't represent a function. In this chapter, you will learn about types of relations, reflective, transactive and equivalence relation, functions, range, domain and codomain of function, which have already been introduced in maths of NCERT class 11. In this chapter, there are 4 exercises. Click on the above link to get CBSE NCERT solutions for class 12 maths of this chapter.

In class 11 maths you have already learnt trigonometric function. The inverse trigonometric function is the inverse of a trigonometric function. If you are good at trigonometric function, it won't take much effort to get a command on the inverse trigonometric functions. In solutions of NCERT class 12 maths of this chapter, you will learn concepts related to the definition of inverse trigonometric functions,  domain, range, and principal value, graph, and properties of inverse trigonometric functions. In class 11 you have already learnt that trigonometric functions are not one to one and onto in their natural ranges. We restrict domains and ranges of trigonometric functions to enable their inverse one-one and onto. This chapter contains 2 exercises with 35 questions.

The array of numbers or mathematical objects called "Matrix", for that operation like addition, multiplications are defined. It was introduced to solve simultaneous linear equations. Matrix has a lot of applications in engineering, scientific studies, computer graphics (image or video that you see are matrices of color intensities), storage of big data, etc. In this chapter, you will learn about matrices and its properties. Also, you are going to study the types, operation transpose and the inverse of the matrix, and properties of matrices which is one of the most powerful tools of mathematics. This tool simplifies our work when compared with other straight forward methods. In Chapter 3 Matrices there are 4 exercises with 62 questions.

In the previous chapter you have already learnt matrices and its properties. For every square matrix, you can compute a number called "Determinant". It shows the factor by which times area or volume of linear transformation has changed. In CBSE NCERT solutions for class 12 maths of this chapter, you will learn about the determinants of a square matrix, and properties of determinant, consistency, inconsistency, the number of solutions of linear equations and solving system of linear equations which is an important part of matrices. So, we are dealing with determinants of order up to three only. In this chapter, there are 6 exercises.

In the class 11 NCERT you have learnt about the differentiation of certain functions. In NCERT solutions for class 12 maths of this chapter, you will learn about differentiability, continuity and how differentiability is related to continuity. The important concepts that you will study are continuity, differentiability, the derivative of a composite function, a derivative of a trigonometric function, the derivative of the inverse trigonometric functions, derivative of the logarithm, the derivative of the exponential function, chain rule, role's theorem, and Lagrange mean value theorem, etc. In this chapter, there are a total of eight exercises.

In chapter 5, you have learnt how to find the derivative of implicit functions, composite functions, exponential functions, inverse trigonometric functions, and logarithmic functions. In this chapter, you are going to study the applications of derivatives in various fields like science, social science, engineering, and many other fields. Solutions of NCERT class 12 maths of this chapter "Application of Derivatives" will cover the rate of change, maxima, minima of functions, equation of tangents and normal, etc. There are five exercises in this chapter and here you will find the solutions to all exercises.

Differential calculus is centered around the concept of differentiation, defining tangent and calculating the slope of the line. Integral calculus is centered around defining and calculating the area of the region bounded by the graph of a function. Integration means 'bringing-together' which is the inverse process of differentiation(Set-apart). The function is integrable only if the function is already differentiated. Remember every function is not integrable means you can't integrate every function. In this chapter, you will learn about the integrals of various functions and also discuss its application like calculation of areas under the graphs. There are 11 exercises in this chapter.

- In the previous chapter you have learnt integration of different complicated functions. Well, it has a lot of applications like calculating the area between curves, calculating distance, velocity, acceleration, finding the average value of a function, probability,  surface area, etc. In this chapter, you will study a specific application of integrals to find the area between the lines and arcs of circles, ellipses and parabolas and area under simple curves. The two exercises of this chapter deal with finding the area bounded by the above-mentioned curves.

You have learnt about differentiation and integration. The differential equation is an equation of a function and with its one or more derivative. In this chapter, you are going to learn some basic concepts related to the differential equation, particular and general solutions of a differential equation, different methods to solve a first-order-first degree differential equation, solutions of linear differential equations, the formation of differential equations and discuss some applications of differential equations in different areas.

- Think about the situation where you are applying force on a box. So it has two quantities, one the magnitude that you are applying and second the direction in which you are applying the force. Such quantities with both magnitude and direction are termed as a vector quantity. In the physics and engineering, you often come across these quantities. In the CBSE NCERT solutions for class 12 maths, you will learn about some basic concepts of vectors, types of vectors, the component of vectors, algebraic properties addition, multiplication by a scalar, dot product, cross product, and their geometric properties.

In the previous class you have learned the 2-D geometry. 3-D geometry is an extension of 2-D geometry with taking consideration of 3 orthogonal axes. Augustus De Morgan quotes "the moving power of mathematical invention is not reasoning but imagination". To visualize in 3-D geometry it requires more of your imagination power. This chapter makes the study of 3-D geometry simple. In the solutions of NCERT class 12 maths of this chapter "Three Dimensional Geometry" you will study the direction ratio and direction cosines of a line joining two points, equations of planes and lines in space under different conditions, the angle between two planes, two lines, a plane, and a line and other important concepts.

NCERT solutions for class 12 maths chapter 12 Linear Programming: You often come across problems where you seek to maximize profit or minimize cost. Linear programming is useful to find the optimal solution for such problems. We formulate these real-life problems into a mathematical model. In solutions of NCERT class 12 maths of this chapter, we deal with some problems on linear programming, different types of linear programming and their solutions by the graphical method, feasible and infeasible region, feasible and infeasible solutions, optimal feasible solutions and also about many other methods to solve such problems. There are two exercises in Chapter 12 Linear programming.

NCERT solutions for class 12 maths chapter 13 Probabily: In previous classes, you have learnt that probability means how likely something is to happen. In this chapter, you will learn the important concept of the conditional probability of an event given that another event has occurred, which will be helpful in understanding the multiplication rule of probability, Bayes' theorem and independence of events. In this chapter, you will also study an important concept of the random variable and its probability distribution and also the mean and variance of a probability distribution.

## NCERT Solutions for different classes:

You may come cross some students who are performing well in the class tests but are not able to perform in the final exam. Why is it so?

• ## Stick to the syllabus that is provided by NCERT and solve it completely including all examples and all exercises.

• First, try to solve every question on your own, if you are not able to do, you can take help of CBSE NCERT solutions for class maths that are provided here. Don't ever think that reading the solutions is enough. You have to solve on your own, even after reading the solutions
• If you have solved all NCERT questions then you can solve previous years paper CBSE board to get familiar with the question paper
• If you have difficulties in remembering the formulas of some chapters, you can use formula sheet of that chapter but if you really want to remember the formulas do just one thing "write all the formulas which you are using in every question when you are solving the problems"
• Formula sheet is always helpful in revision before the exam.

Happy learning!!!