# NCERT Solutions for CBSE Class 12 Maths

### CBSE Class 12 Maths NCERT Solution All Chapters

NCERT Solutions for Class 12 Maths has been written to help students like you who are appearing for different board exams as well as the IIT JEE, AIEEE and Other Engineering Entrance Exams.. It provides sound knowledge and understanding of all important concepts covered in each chapter in the Class 12 Maths NCERT textbook. There are 13 important chapters from the NCERT textbook. Both problems and their detailed solutions are provided. The solutions have been devised by India’s best teachers to help students grasp basic concepts better and faster.

Class 12 Maths is key in the preparation for boards as well as JEE Main and Advance, and our solutions are made keeping in mind the very same thing. The solutions are concise, and also discuss alternative methods to solve some problems.

### NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions

The chapter builds up on the concepts of relations, functions, domain and codomain introduced in Class 11. The students explore several real valued functions, and their graphs. The chapter comprises of four exercises that consolidate the concepts explained in the chapter.

### NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

This chapter plays in the restricted range of trigonometric functions that enables them to be one on one, and hence makes inverse of the functions visible. 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.

### NCERT Solutions for Class 12 Maths Chapter 3 Matrix

This chapter takes you through the properties of one of the most important tools of Mathematics, namely Matrices. The students learn the basics of Matrices and Matrix Algebra..

### NCERT Solutions for Class 12 Maths Chapter 4 Determinants

Correctly preceded by Matrix, this chapter deals with an important part of Matrices, namely Determinants. The chapter deals with determinants of orders upto 3, and their cofactors and inverses.This chapter also talks about the applications of determinants, like calculation of area of triangles, adjoint of matrices, consistency of a set of linear equations.

### NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

This chapter picks up after the Differentiation of Functions introduced in Class 11. The chapter explores the important concepts of Continuity, Differentiability and establishes the relationship between them. It also deals with the continuity and differentiability of Inverse Trigonometric Functions. It also introduces exponential and logarithmic functions.

### NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives

With Continuity and Differentiability marking the start of Calculus, this chapter deals with the differentiability of functions, and introduction to derivatives. It delves into the derivatives of specific functions, and their applications, namely the tangents and normal to curves, and the rate of change in some quantities. It also talks about increasing and decreasing functions.

### NCERT Solutions for Class 12 Maths Chapter 7 Integrals

This is one of the most important and time consuming chapters in the whole Class 12 Mathematics. This chapter deals with integration of functions, and its applications in terms of calculation of areas. This marks the introduction to Integral Calculus. There are a total of 11 exercises in this chapter, making it a very practice driven part of the syllabus.

### NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

This chapter deals with the applications of integrals in calculating areas under simple curve, lines, parabolas and ellipses. The two exercises provide enough questions for practicing the area calculations.

### NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations

This chapter introduces the concept of Differential equations, the general and particular solutions, and the order and degree of an equation. This also talks about the applications of differential equations in the six exercises.

### NCERT Solutions for Class 12 Maths Chapter 10 Vector Equations

This chapter deals with the vector quantities, that is, the quantities with both magnitude and direction. The Vector algebra summarises the rules of addition of vector quantities, and other important properties. Vector Algebra plays a big role in Physics.

### Class 12 Maths Chapter 11 Three Dimensional Geometry

This chapter utilises vectors to calculate the distance between lines, a point and a line, and a line and a plane, along with the angles between them.It also introduces the concept of direction cosines

### Class 12 Maths Chapter 12 Linear Programming

This chapter explores the concept of linear programming problems, and their solutions using graphical method only.

### Class 12 Maths Chapter 13 Probability

This chapter deals with the concept of Conditional Probability and the encounters of the same in our daily lives. This explores Bayes’ Theorem, the multiplication rule, and the independence of events. This chapter is best understood by practicing as many questions as possible.

## CBSE Class 12 Maths Syllabus 2020-21(Revised)

 No. Units No. of Periods Marks I. Relations and Functions 17 08 II. Algebra 35 10 III. Calculus 57 35 IV. Vectors and  Three  – Dimensional Geometry 26 14 V. Linear Programming 13 05 VI. Probability 20 08 Total 168 80 Internal Assessment 20

Unit-I: Relations and Functions

Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

Inverse Trigonometric Functions

Definition, range, domain, principal value branch.

Unit-II: Algebra

Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Non- commutativity of multiplication of matrices Invertible matrices

Determinants

Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

Applications of Derivatives

Applications of derivatives: increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

Integrals

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

Applications of the Integrals

Applications in finding the area under simple curves, especially lines, parabolas; area of circles /ellipses  (in standard form only) (the region should be clearly identifiable

Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of  differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree of the type: dy/dx = 𝑓(y/x). Solutions of linear differential equation  of the type:

Solutions of linear differential equation of the type:

(dy/dx) + py = q, where p and q are functions of x or constants.

Unit-IV: Vectors and Three-Dimensional Geometry

Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a  vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar,  position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation,  properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

Three – dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation  of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation  of a plane. Distance of a point from a plane.

Unit-V: Linear Programming

Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, different types of  linear programming (L.P.) problems. graphical method of solution for problems in two variables, feasible  and infeasible regions (bounded), feasible and infeasible solutions, optimal feasible solutions (up to  three non-trivial constraints).

Unit-VI: Probability

Probability

Conditional probability, multiplication theorem on probability, independent events, total probability,  Bayes’ theorem, Random variable and its probability distribution.

### Reference Books for Class 12 Math

• CBSE Chapter wise Solutions – Mathematics (Class 12) by Hemant Malhotra
• CBSE Examination Mathematics : 15 Sample Question Papers (Class 12) by Prem Kumar
• Mathematics (Class – XII) by NCERT
• Problems Plus In IIT Mathematics by A Das Gupta
• Senior Secondary School Math 12 by R. S. Aggarwal
• Solutions of Objective Mathematics: Solutions of Exercises and Chapter Tests by Dhanpat Rai
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