NCERT Solutions for Class 12 Maths - PDF Download

For students like you who are taking the JEE as well as several board exams, NCERT Solutions for Class 12 Math has been published by eSaral. It offers thorough knowledge and comprehension of all important concepts presented in each chapter of the NCERT textbook for class 12 Math. The NCERT textbook of class 12 math contains 13 major chapters. For a clear and precise understanding of the topics, eSaral presents NCERT Solutions for Class 12 Math. These solutions are created by our subject experts. In addition to the Class 12 Maths solutions for each chapter, There are problems and detailed solutions of these problems have been provided. The answers are developed by the top educators of eSaral to help students in better and quicker understanding of the concepts.The problems in these NCERT Class 12 Solutions are thoroughly explained step-by-step.

NCERT Solutions for Class 12 Math gives a strong conceptual foundation for each topic. For students to have a better conceptual understanding of the material, it thoroughly covers all the key theorems and formulas. Students can prepare for undergraduate engineering admission examinations like JEE Mains, BITSAT, VITEEE, and others with the help of NCERT Solutions for Class 12 Maths.

There are 13 chapter in class 12 Maths NCERT book which are listed below:

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

Chapter 1: Relations and Functions

Chapter 1 of class 12 NCERT Maths textbook deals with Relations and Functions. Here, you will find exercise solution link for all the topics in this chapter.

Topics covered in chapter 1 are listed below:

• Types of Relations: Reflexive, Symmetric, transitive and equivalence relations.

• One to one and onto textbook

• Composite functions

• Inverse of a function

• Binary operations  

Chapter 2: Inverse Trigonometric Functions

Chapter 2 of class 12 NCERT Maths textbook deals with Inverse Trigonometric Functions. Here, you will find exercise solution link for all the topics in this chapter.

Topics covered in chapter 2 are listed below:

• Definition 

• Range 

• Domain 

• Principle value branches 

• Graphs of inverse trigonometric functions

Chapter 3: Matrices

Chapter 3 of class 12 NCERT Maths textbook deals with matrices. Here, you will find exercise solution link for all the topics in this chapter.

Topics covered in chapter 3 are listed below:

• Concept 

• Notation, order, equality

• Types of matrices

• Zero matrix

• Transpose of a matrix

• Symmetric and skew symmetric matrices

• Addition, multiplication and scaler multiplication of matrices

• Simple properties of addition, multiplication and scalar multiplication 

• Non Commutativity of multiplication of matrices 

• Concept of elementary row and column operations

Chapter 4: Determinants 

Chapter 4 of class 12 NCERT Maths textbook deals with determinants. Here, you will find exercise solution link for all the topics in this chapter

Topics covered in chapter 4 are listed below:

• Determinant of a square matrix

• Properties of determinants 

• Minors and cofactors

• Applications of determinants in finding the area of a triangle

• Adjoint and inverse of a square matrix

• Consistency, inconsistency and number of solutions of system in linear equations 

• Solving a system of linear equations in two or three variables using inverse of matrix.

Chapter 5: Continuity and Differentiability 

Chapter 5 of class 12 NCERT Maths textbook deals with continuity and Differentiability. Here, you will find exercise solution link for all the topics in this chapter

Topics covered in chapter 5 are listed below:

• Definition of continuity and differentiability 

• Derivative of composite functions

• Chain rule

• Derivatives of inverse trigonometric functions

• Derivative of implicit functions

• Concept of exponential and logarithmic functions and their derivatives 

• Logarithmic differentiation

• Derivative of functions expressed in parametric forms

• Second order derivatives

• Rolle’s and lagrange’s mean value theorems

Chapter 6: Applications of Derivatives 

Chapter 6 of class 12 NCERT Maths textbook deals with Application of derivatives.

Topics covered in chapter 6 are listed below:

• Rate of change

• Increasing and decreasing functions

• Tangents and normals

• Approximation 

• Maxima and minima

Chapter 7: Integrals

Chapter 7 of class 12 NCERT Maths textbook deals with Integrals.

Topics covered in chapter 7 of class 12 are listed below:

• Definition of indefinite integral

• Integration as an inverse process of differentiation

• Geometrical interpretation of indefinite integral

• Properties of indefinite integral

• Comparison between differentiation and integration

• Methods of integration such as integration by substitution

• Integration using partial fractions

• Integrations by parts

• Integrations using trigonometric identities

• Definite and meaning of definite integrals 

• Definite integrals as the limit of a sum

• Fundamental theorem of calculus

• Evaluation of definite integral by substitution 

Chapter 8: Application of Integrals

Chapter 8 of class 12 NCERT Maths textbook deals with Application of derivatives.

Topics covered in chapter 8 are listed below:

• Applications in finding the area under simple curves

• Arcs of circles/parabolas/ellipses (in standard form only)

• Area of the region bounded by a curve and a line

Chapter 9: Differential Equations

Chapter 9 of class 12 NCERT Maths textbook deals with Differential Equations.

Topics covered in chapter 9 are listed below: 

• Degree of a differential equation

• Order of a differential equation

• General and particular solutions of a differential equation.

• Formation of differential solution whose general solution is given

• Solution of differential equations by method of separation of variables

• Homogeneous differential equations of first order and first degree 

Chapter 10: Vector Algebra 

Chapter 10 of class 12 NCERT Maths textbook deals with Vector Algebra.

Topics covered in chapter 10 of class 12 maths are listed below: 

• Vectors and scalars

• Magnitude and direction of a vector

• Direction cosines/ratios of vectors

• 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

• Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors.  

Chapter 11: Three Dimensional Geometry

Chapter 11 of class 12 NCERT Maths textbook deals with Vector Algebra.

Topics covered in chapter 10 of class 12 are listed below: 

• direction cosines and direction ratios of a line

• relation between the direction cosines of a line

• direction cosines of a line passing through two points

• equation of a line in space

• equation of a line through a given point and parallel to a given vector

• derivation of cartesian form from vector form

• equation of a line passing through two given points

• angle between two lines, shortest distance between two lines, distance between two skew lines

• distance between two parallel lines, plane in cartesian form, equation of a plane in a normal form

• equation of a plane perpendicular to a given vector and passing through a given point, equation of a plane passing through three non collinear points

• intercept form of the equation of a plane, plane passing through the intersection of two given planes,

• coplanarity of two lines

• angle between two planes, distance of a point from a plane, angle between a line and a plane

Chapter 12: Linear Programming

Chapter 12 of class 12 NCERT Maths textbook deals with Linear Programming. It covers topics such as linear programming problems and their mathematical formulae, the mathematical formulae of the problem to be solved, the graphical approach to solving the linear programming problem, and various types of the linear programming problem with various examples. Furthermore, students can find exercises that adequately explain these concepts with solutions.

Topics covered in chapter 12 are listed below: 

• Introduction

• related terminology such as constraints, objective function, optimization

• different types of linear programming (L.P.) problems

• mathematical formulation of L.P. problems

• graphical method of solution for problems in two variables

• feasible and infeasible regions

• optimal feasible solutions (up to three non-trivial constrains).

Chapter 13: Probability

This chapter is all about probability. We'll go over the basics and formulas we looked at in class 11, and we'll do the same in this chapter with the new ideas we've learned.

Topics covered in chapter 13 are listed below: 

• Multiplication theorem on probability

• Conditional probability

• independent events

• total probability

• Baye’s theorem

• Random variable and its probability distribution

• mean and variance of haphazard variable

• Repeated independent (Bernoulli) trials

• Binomial distribution

Key Benefits of CBSE Class 12 Maths NCERT Solutions from eSaral

CBSE Class 12 Mathematics NCERT solutions from eSaral provide several advantages for students preparing for board and competitive exams. Some of the main advantages of NCERT solutions are:

Complete Coverage: All topics and chapters prescribed by CBSE Class 12 Syllabus are covered in NCERT solutions. Students will have access to a complete set of study material to help them prepare for their exams effectively.

Accurate & Verified Answers: NCERT solutions are provided by subject matter experts to ensure accuracy and reliability of answers. Students can depend on these solutions for getting the correct approach to solve problems.

Understanding Concepts: NCERT solutions provide step by step explanations for each problem to help students understand the basic concepts and improve their problem solving skills.

Save time and effort: Having a ready-made solution saves students’ time since they don’t have to guess the answers themselves. They can use the extra time for revision and practicing.

Practice & revision: Using NCERT solutions helps students practice and revise. Students can try a variety of problems to strengthen their understanding of different concepts.

Exam preparation: CBSE board examinations are based on NCERT textbooks. Practicing with solutions helps students get familiar with the question format and marking system.

Homework & Assignments. NCERT solutions are useful for completing homework & assignments on time. Students can refer to NCERT solutions whenever they encounter challenging questions.

Higher Performance: With regular practice, students improve their problem solving skills, which helps them perform better in exams & other competitive tests

Frequently Asked Questions

Question 1: Why is a Class 12 maths book important for engineering exams?

Answer 1: The importance of a Class 12 Maths book for engineering exams can be summarised as follows:

• Save Time & Effort: Students can save time and effort by having ready-made solutions as they don’t have to work on their own. They can use the extra time to revise and practice.

• Practice & revision: Using NCERT solutions helps students practice and revise. Students can try a variety of problems to strengthen their understanding of different concepts. 

• Exam preparation: CBSE board examinations are based on NCERT textbooks. Practicing with solutions helps students get familiar with the question format and marking system. Homework & Assignments NCERT solutions are useful for completing homework & assignments on time. Students can refer to NCERT solutions whenever they encounter challenging questions. 

• Clarification: Students may face doubts and misconceptions while studying independently. By using NCERT solutions, they can get clarity on their doubts. Higher Performance: With regular practice, students improve their problem solving skills, which helps them perform better in exams & other competitive tests

Question 2: How can students use NCERT solution effectively?

Answer: In order to effectively utilize NCERT Solutions, students must first solve the exercises on their own and then compare their answers to the solutions provided. Students should also analyse solution steps to understand the problem solving methods.