NCERT Solutions for Class 12 Maths - Free Latest PDF (2024-25)

NCERT Class 12 Maths Chapter 1 Relations and Functions

Chapter 1 Relations and Functions of NCERT Class 12 Maths Syllabus deals with types of relations and functions. Throughout this chapter, students will develop a solid understanding of relations and functions, which are essential for understanding various mathematical concepts. They will also learn to analyse and interpret graphical representations of functions, which are used extensively in calculus, algebra, and other branches of mathematics. This chapter serves as a foundation for the more advanced topics covered in later chapters of Class 12 Maths.

Topics Covered in Class 12 Maths Chapter 1 Relations and Functions.

1. Types of Relations

• Symmetric

• Reflexive

• Transitive

• Equivalence.

2. Types of Functions

• One to one functions

• Onto functions

• Composite functions

• Invertible functions.

3. Binary Operations.

NCERT Class 12 Maths Solutions for Chapter 1 Relations and Functions Exercises are given in the table below:

Students can find below the supplementary study materials for Relations and Functions on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Class 12 Maths Chapter Chapter 2 Inverse Trigonometric Functions

In CBSE Class 12 Inverse Trigonometric Functions, there are only two exercises where you will study the restrictions on domains and ranges of trigonometric functions which enables the visibility of their inverses and observe their behaviour through graphical representations.

The inverse trigonometric functions hold great value in calculus, for they serve to define many integrals. The concepts of inverse trigonometric functions are applied vastly in science and engineering.

Here is the list of topics covered in Class 12 Maths Chapter 2 inverse trigonometric function:

• Definition, range, domain, principal value branches. 

• Graphs of inverse trigonometric functions.

• Elementary properties of inverse trigonometric functions.

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Function Exercises are given in the table below:

Students can find below the supplementary study materials for Real Numbers on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Class 12 Maths Chapter 3 Matrices

It covers basic operations like addition, subtraction, multiplication, and scalar multiplication of matrices, which are fundamental in various fields such as computer graphics, quantum mechanics, and economics. Matrices provide a systematic way to solve systems of linear equations.

Chapter 3 of NCERT Class 12 Maths Covers the Following Topics:

• What are Matrices?

• Types of Matrices

• Properties of Matrix Addition

• Properties of Scalar Multiplication

• Multiplication of Matrices

• Properties of Multiplication of Matrices

• Symmetric and Skew-symmetric Matrices

• Elementary Operation or Transformation of a Matrix

• Inverse of a Matrix by Elementary Operations

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Function Exercises are given in the table below:

Students can find below the supplementary study materials for Matrices on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Class 12 Maths Chapter 4 Determinants

There are six exercises which are included in the NCERT Class 12 Maths Chapter 4 Determinants. By solving these exercises, students will be familiar with the concepts of determination, cofactors and the application of determinants in finding the area of a triangle, minors, adjoint and inverse of a square matrix.

Topics Covered in Class 12 Maths Chapter 4 Determinants:

• Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors and 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 of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

The free PDFs of the NCERT Solutions for Class 12 Maths Chapter 4 exercises can be downloaded from the links provided below:

Students can find below the supplementary study materials for Determinants on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Class 12 Maths Chapter 5 Continuity and Differentiability

Understanding continuity and differentiability helps in analysing the behaviour of functions. It provides insights into where functions are smooth and where they have sharp corners or discontinuities. These are the fundamental concepts in calculus. They form the basis for studying limits, derivatives, and integrals, which are extensively used in physics, engineering, economics, and other sciences.

The important concepts that students will get to learn from Class 12 Maths NCERT Solutions are as follows:

• Algebra of continuous function

• Derivatives of composite functions, Implicit functions, Inverse trigonometric functions, Exponential and logarithmic functions

• Logarithmic differentiation

• Derivatives of functions in parametric forms

• Second order derivatives

• Rolle’s and Lagrange’s Mean Value Theorems

The free PDFs of the Class 12 Maths NCERT Solutions exercises can be downloaded from the links provided below:

Here are the links for some important study materials that are available on our website for CBSE Class 12 Maths Chapter 5.

NCERT Class 12 Maths Chapter 6 Application of Derivatives

In this chapter, you will study the applications of the derivative in various disciplines such as engineering, science, and many other fields. For instance, you 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. You will also learn the use of derivatives to find intervals of increasing or decreasing functions, maxima and minima, along with a few simple problems based on all these concepts.

Topics Covered in Class 12 Maths Chapter 6 Application of Derivatives:

• Rate of Change

• Increasing and Decreasing Functions

• Tangents and Normals

• Approximations

• Maxima and Minima

• Optimization Problems

• Rate of Change of Quantities

Students can find below the supplementary study materials for Application of Derivatives on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Class 12 Maths Chapter 7 Integrals

Integrals are used to find the area under curves. This is particularly useful in various fields like physics, where it helps in calculating quantities such as displacement, velocity, and acceleration. Definite integrals are used for precise calculation of quantities such as total distance travelled, total work done, and total profit earned. It is extensively used in economics, engineering, and physics.

Topics in this chapter include an introduction to indefinite integrals, integration as inverse to differentiation, its geometrical interpretation, and properties. Other topics that are discussed in this chapter are the following:

• Comparison of integration and differentiation

• Integration methods including substitution, partial fraction, by parts, and trigonometric identities. 

• Definite integrals

• Fundamental theorem of calculus

• How to evaluate definite integrals by substitution

Students can find below the supplementary study materials for Integrals on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Class 12 Maths Chapter 8 Application of Integrals

Here, in this chapter, you will study some specific applications 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.

Topics Covered in Chapter 8 Application of Integrals:

• Area Under Simple Curves

• Area Between Two Curves

• Area Enclosed Between Two Curves

• Application in Finding Lengths

• Application in Finding Volumes

• Volume by Cross-Sectional Areas

• Volume of Solids with Known Cross-Sections

• Application in Physics and Engineering

Students can find below the supplementary study materials for Application of Integrals on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Solutions for Class 12 Maths 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 equations and some applications of differential equations in different areas.

Topics Covered in Class 12 Maths Chapter 9 Differential Equations

• Introduction to Differential Equations

• Basic Concepts and Terminology

• Formation of Differential Equations

• Methods of Solving First Order, First Degree Differential Equations

• Linear Differential Equations of Second and Higher Order

• Applications of Differential Equations

Students can find below the supplementary study materials for Differential Equations on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

In Class 12 Maths Chapter 10  Vector Algebra, you will be studying some basic concepts of vectors, various operations on vectors, and the algebraic as well as geometric properties. You will also learn about the types of vectors such as zero, unit, equal and collinear vectors.

Other Topics Discussed in this Chapter Include:

• Negative of a vector

• Operations of vectors such as addition

• Properties of vector addition

• Components of vectors

• Multiplication of a vector by a scalar

• Section formulae

• Vector joining two points

• Dot product of two vectors

• Product of two vectors

• Cross product of two vectors

• Projection of a vector on a line

Students can find below the supplementary study materials for Vector Algebra on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

CBSE Class 12 Maths Chapter 11 Three Dimensional Geometry deals with 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 the distance of a point from a plane.

Topics Covered in Class 12 Maths Chapter 11 Three Dimensional Geometry

• Equation of a Line in Space

• Angle between Two Lines

• Shortest Distance between Two Lines

• Cartesian and Vector Equation of a Plane

• Angle between Two Planes

• Distance of a Point from a Plane

• Coplanarity of Two Lines

Students can find below the supplementary study materials for Dimensional Geometry on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Solutions Class 12 Maths 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. Other concepts in this chapter are the mathematical formulation of linear programming problems, different types of linear programming problems, terms such as objective function, optimization and constraints, as well as feasible and infeasible regions and solutions.

Topics Covered in Class 12 Maths Chapter 12 Linear Programming

• Introduction to Linear Programming Problems

• Formulation of L.P. Problems

• Graphical Method of Solution

• Non-negative Constraints and Feasible Region

• Solution of L.P. Problems by Graphical Method

• Corner Points and Optimum Solution

• L.P. Problems with More than Two Variables

Students can find below the supplementary study materials for Linear Programming on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

NCERT Solutions for Class 12 Maths Chapter 13 Probability

Class 12 Maths Chapter 13 Probability will teach you 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. You will also learn the concept of a random variable and its probability distribution, mean and variance of a probability distribution, total probability theorem, Bernoulli trials and binomial distribution.

Topics Covered in Class 12 Maths Chapter 13 Probability

• Introduction to Probability

• Conditional Probability

• Conditional Probability

• Properties of Conditional Probability

• Multiplication Theorem on Probability

• Independent Events

• Bayes'Theorem

• Random Variables and Probability Distribution

Students can find below the supplementary study materials for Dimensional Geometry on Vedantu. These resources are downloadable, providing extra assistance for your Maths preparation.

CBSE Class 12 Maths Weightage (2024-25)

Given below is the unit-wise mark distribution for CBSE Class 12 Maths exam for session 2024-25.

Internal Assessment Class 12 Marks Distribution

CBSE Class 12 Maths Exam Pattern

The CBSE Class 12 Maths Exam Pattern for 2024 follows a competency-based approach, focusing on application and concept understanding. Here's a breakdown:

Total Marks: 100 (80 Theory + 20 Internal Assessment)

Theory Paper (80 Marks):

• Exam Duration: 3 Hours

• Question Paper Format: Five sections (A, B, C, D, and E) - all compulsory with internal choices in some sections.

Internal Assessment (20 Marks):

• Conducted by the school throughout the year

• May include projects, assignments, practical work, or quizzes

The Class 12 Maths NCERT Solutions provided on Vedantu 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 Maths NCERT Solutions Class 12 from Vedantu

• You can easily clear your doubts by accessing this PDF anytime and also be able to learn some important tips to answer some complex questions in a detailed manner. 

• You no longer have to turn in several pages for revision before the examination. The NCERT Maths Class 12 PDF Download is available on our website for FREE, which you can access anytime, anywhere.

• Also, all the NCERT Class 12 Maths 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 the 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 topic-wise detailed solutions will help you to take your preparation to the next level. Vedantu’s Class 12 Maths NCERT Solutions will definitely help you in improving your mathematical skills as well.

Go through these Chapter-wise Exercise Solutions that are provided for CBSE Class 12 Maths and begin your preparation for the Class 12 Board exam confidently. For any more information and study materials, visit Vedantu’s website.

Related Links for NCERT Class 12 Maths in Hindi

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