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• NCERT Solutions
• NCERT Class 12
• NCERT 12 Maths

NCERT Solutions For Class 12 Maths

Ncert solutions for class 12 maths – free 2023-24 cbse pdf download.

NCERT Solutions for Class 12 Maths provides a solid conceptual base for all the topics included in the CBSE Class 12 Maths Syllabus. It covers all the important theorems and formulae with detailed explanations to ensure better conceptual understanding in students. NCERT Solutions for Class 12 Maths will help students in preparation for undergraduate-level Engineering entrance exams like JEE Mains, BITSAT, VITEEE, etc., to be conducted for the year 2023-2024.

BYJU’S brings you NCERT Solutions for Class 12 Maths, designed by our subject experts to facilitate a smooth and precise understanding of concepts. These Class 12 Solutions of NCERT have detailed step-by-step explanations of problems given in the textbooks. Apart from solutions for each chapter of CBSE Class 12 Maths, BYJU’S also provides Class 12 Maths notes, solved exemplar problems, and previous year question papers, along with worksheets as study materials for students who are preparing for their Class 12 board exams.

NCERT Solutions for Class 12 Maths Chapter-Wise PDF

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 applications of integrals, chapter 9 differential equations, chapter 10 vector algebra.

• Chapter 11 Three dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 probability, class 12 maths ncert solutions exercises.

Students can download NCERT Solution of Maths of Class 12 , chapter-wise from the links given below.

Chapter 1 of NCERT textbook deals with relations and functions , types of relation, types of functions, the composition of functions and invertible function, binary operations and miscellaneous examples. Here, you can find the exercise solution links for the topics covered in this chapter.

Topics Covered in Class 12 Maths Chapter 1 Relations and Functions: Types of relations: Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

Also, access the following resources for Class 12 Chapter 1 Relations and Functions at BYJU’S:

• Relations And Functions Class 12 Maths Notes Chapter 1
• Important Questions For Class 12 Maths Chapter 1 – Relations and Functions
• Maths Revision Notes for Class 12 Chapter 1 Relations and Functions
• NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions

Chapter 2, Inverse Trigonometric Functions of NCERT Class 12 Maths, gives an account of various topics such as the remarks based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions and miscellaneous examples. These concepts are explained very well through examples. Solutions to the exercises from the chapter can be accessed from here.

Topics Covered in Class 12 Maths Chapter 2 Inverse Trigonometric Functions: Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Also access the following resources for Class 12 Chapter 2 Inverse Trigonometric Functions at BYJU’S:

• Inverse Trigonometric Functions Class 12 Notes – Chapter 2
• Important Questions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions
• Maths Revision Notes for Class 12 Chapter 2 Inverse Trigonometric Functions
• NCERT Exemplar Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

In Chapter 3 of NCERT textbook, we shall see the definition of a matrix, types of matrices, equality of matrices, operations on matrices such as the addition of matrices and multiplication of a matrix by a scalar, properties of matrix addition, properties of scalar multiplication, multiplication of matrices, properties of multiplication of matrices, transpose of a matrix, properties of the transpose of the matrix, symmetric and skew-symmetric matrices, elementary operation or transformation of a matrix, the inverse of a matrix by elementary operations and miscellaneous examples. Here, you can find the exercise solution links for the topics covered in this chapter.

Topics Covered in Class 12 Maths Chapter 3 Matrices: Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

Also access the following resources for Class 12 Chapter 3 Matrices at BYJU’S:

• CBSE Class 12 Maths Notes Chapter 3 Matrices
• Important Questions for Class 12 Maths Chapter 3 – Matrices
• Maths Revision Notes for Class 12 Chapter 3 Matrices
• NCERT Exemplar Solutions for Class 12 Maths Chapter 3 Matrices

Chapter 4 Determinants –

Chapter 4 of 12 Class NCERT Maths Solution discusses the topic of determinants. Students will get to learn about the definition and meaning of determinants, remarks based on order of determinants, properties of determinants, finding area of triangle using determinants, minors and cofactors of determinants, adjoint of a matrix, inverse of matrix , applications of determinants and matrices, and miscellaneous examples. Below, we have links provided to each exercise solution covered in this chapter.

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.

Also access the following resources for Class 12 Chapter 4 Determinants at BYJU’S:

• Determinants Class 12 Notes – Chapter 4
• Important Question for Class 12 Maths Chapter 4 – Determinants
• Maths Revision Notes for Class 12 Chapter 4 Determinants
• NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants

Chapter 5 of NCERT textbook starts with the definition of continuity. Students go on to learn about continuity, algebra of continuous function, definition and meaning of differentiability, derivatives of composite functions, derivative of implicit functions, derivative of inverse trigonometric functions, exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric forms, second order derivatives , mean value theorem via miscellaneous examples. Here, students can find the exercises explaining these concepts properly with solutions.

Topics Covered in Class 12 Maths Chapter 5 Continuity and Differentiability: Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. 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 (without proof) and their geometric interpretations.

Also access the following resources for Class 12 Chapter 5 Continuity and Differentiability at BYJU’S:

• Continuity And Differentiability Class 12 Notes Chapter 5
• Important Questions for Class 12 Maths Chapter 5 – Continuity and Differentiability
• Maths Revision Notes for Class 12 Chapter 5 Continuity and Differentiability
• NCERT Exemplar Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

Chapter 6 of NCERT textbook provides definition of derivatives, rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, maxima and minima , first derivative test, maximum and minimum values of a function in a closed interval, and miscellaneous examples. Here, students can find the exercises explaining these concepts properly.

Topics Covered in Class 12 Maths Chapter 6 Applications of Derivatives: Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normals, 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).

Also access the following resources for Class 12 Chapter 6 Applications of Derivatives at BYJU’S:

• Application of Derivatives Class 12 Notes Chapter 6
• Important Questions for Class 12 Maths Chapter 6 – Applications of Derivatives
• Maths Revision Notes for Class 12 Chapter 6 Application of Derivatives
• NCERT Exemplar Solutions for Class 12 Maths Chapter 6 Application of Derivatives

In Chapter 7, Integrals, we shall see the 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, integration by parts , integration using trigonometric identities, integration of some integral functions, definition and meaning of definite integrals, definite integral as the limit of a sum, fundamental theorem of calculus, evaluation of definite integrals by substitution, some properties of definite integrals and miscellaneous examples. Here, we have given exercises with solutions based on these topics from the chapter.

Topics Covered in Class 12 Maths Chapter 7 Integrals: Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type

Also access the following resources for Class 12 Chapter 7 Integrals at BYJU’S:

• Integrals Class 12 Notes Chapter 7
• Important Questions for Class 12 Maths Chapter 7 – Integrals
• Maths Revision Notes for Class 12 Chapter 7 Integrals
• NCERT Exemplar Solutions for Class 12 Maths Chapter 7 Integrals

In Chapter 8 of NCERT textbook, we continue the discussions about integrals starting with definition, area under simple curves, area of the region bounded by a curve and a line , area between two curves and miscellaneous examples. From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

Topics Covered in Class 12 Maths Chapter 8 Applications of Integrals: Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ ellipses (in standard form only), area between the two above-said curves (the region should be clearly identifiable).

Also, access the following resources for Class 12 Chapter 8 Applications of Integrals at BYJU’S:

• Application of Integrals Class 12 Notes Chapter 8
• Important Questions Class 12 Maths Chapter 8-Applications of Integrals
• Maths Revision Notes for Class 12 Chapter 8 Application of Integrals
• NCERT Exemplar Solutions for Class 12 Maths Chapter 8 Application of Integrals

In this chapter, students concentrate on the definition of differential equations, basic concepts related to differential equations, degree of a differential equation , order of a differential equation, general and particular solutions of a differential equation, formation of a differential equation whose general solution is given, procedure to form a differential equation that will represent a given family of curves, methods of solving first order, first degree differential equations, differential equations with variable separable, homogeneous differential equations, linear differential equations, procedure involved to solve first order linear differential equations, and miscellaneous examples. Solutions to the exercises from the chapter can be accessed from here.

Topics Covered in Class 12 Maths Chapter 9 Differential Equations: Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type: dy/dx + Py = Q, where P and Q are functions of x 

Also access the following resources for Class 12 Chapter 9 Differential Equations at BYJU’S:

• Differential Equation Class 12 Notes Chapter 9
• Important Questions Class 12 Maths Chapter 9 – Differential Equations
• Maths Revision Notes for Class 12 Chapter 9 Differential Equations
• NCERT Exemplar Solutions for Class 12 Maths Chapter 9 Differential Equations

In this chapter, students are introduced to vector algebra. The concepts in this chapter include how to find position vector, some basic concepts related to vector algebra, direction cosines, types of vectors such as zero vector, unit vector, collinear vector, equal vector, negative of a vector, addition of vectors, properties of vector addition, multiplication of a vector by a scalar, components of a vector, vector joining two points, section formula, product of two vectors, scalar or dot product of two vectors, properties of scalar product, projection of a vector on a line, vector or cross product of two vectors and miscellaneous examples. From the below-given links, students can access the exercises with solutions explaining the concepts from this chapter.

Topics Covered in Class 12 Maths Chapter 10 Vector Algebra: 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.

Also, access the following resources for Class 12 Chapter 10 Vector Algebra at BYJU’S:

• Vector Algebra Class 12 Notes – Chapter 10
• Important Questions Class 12 Maths Chapter 10 – Vectors
• Maths Revision Notes for Class 12 Chapter 10 Vector Algebra
• NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 11 of NCERT Class 12 Maths constitutes topics related to three dimensional geometry, 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 and miscellaneous examples. Below, we have listed links to the solutions for the exercises from this chapter.

Topics Covered in Class 12 Maths Chapter 11 Three Dimensional Geometry: Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.

Also access the following resources for Class 12 Chapter 11 Three Dimensional Geometry at BYJU’S:

• Three Dimensional Geometry Class 12 Notes Chapter 11
• Important Questions Class 12 Maths Chapter 11-Three Dimensional Geometry
• Maths Revision Notes for Class 12 Chapter 11 Three Dimensional Geometry
• NCERT Exemplar Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

In this chapter, students are introduced to linear programming. The concepts covered in this chapter are linear programming problem and its mathematical formulation, mathematical formulation of the problem, graphical method of solving linear programming problems, different types of linear programming problems with miscellaneous examples. Students can find the exercises explaining these concepts properly with solutions.

Topics Covered in Class 12 Maths Chapter 12 Linear Programming: 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, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).

Also access the following resources for Class 12 Chapter 12 Linear Programming at BYJU’S:

• Linear Programming Class 12 Notes Chapter 12
• Important questions Class 12 Maths Chapter 12 Linear Programming
• Maths Revision Notes for Class 12 Chapter 12 Linear Programming
• NCERT Exemplar Solutions for Class 12 Maths Chapter 12 Linear Programming

This chapter deals with probability. By reviewing the basic facts and formulae studied earlier in Class 11, we shall implement those formulas again in this chapter along with the advance concepts mentioned, i.e., conditional probability, properties of conditional probability , multiplication theorem on probability, multiplication rule of probability for more than two events, independent events, Bayes’ theorem, partition of a sample space, theorem on total probability, random variables and its probability distributions, probability distribution of random variable, mean of random variable, variance of random variable, Bernoulli trials and binomial distribution with miscellaneous examples. We have compiled here the solutions to exercises from this chapter that explain the concept thoroughly.

Topics Covered in Class 12 Maths Chapter 13 Probability: 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 and Binomial distribution.

Also access the following resources for Class 12 Chapter 13 Probability at BYJU’S:

• Probability Class 12 Notes Chapter 13
• Important Questions for Class 12 Maths chapter 13 Probability
• Maths Revision Notes for Class 12 Chapter 13 Probability
• NCERT Exemplar Solutions for Class 12 Maths Chapter 13 Probability

CBSE Class 12 Maths 2023-24 Unit-Wise Weightage

Internal Assessment Class 12

NCERT Solutions Class 12 Maths Chapter 1 to 13

NCERT Solutions PDF Class 12 Maths acts as a vital tool for students in their exam preparation and home assignments. Maths plays a vital role in the success of students in Class 12 boards and competitive exams. Solutions PDF of Class 12 can be your best support for board exam preparations. It’s difficult to score in Mathematics simply by reading and memorizing the formulae. Instead, practice is the perfect method to succeed.

Learning from BYJU’S NCERT Solutions will help students of Class 12 to grasp the conceptual meaning of every problem quite effectively. CBSE Class 12th NCERT Maths Solutions provides in-depth answers to all the problems covered in the syllabus. All the above-mentioned chapters are equally important and can be mastered with consistent practice and hard work. The topics like probability, calculus, coordinate geometry, and trigonometry are easy, and with proper practice, students can easily score good marks for questions from these units. We have also provided solved examples at the end of every chapter to make concepts clear to our readers.

Key Features of CBSE Class 12 Maths NCERT Solutions

• Students can easily practise and revise concepts by referring to these solutions.
• It saves your time by providing all the necessary concepts and formulae in a proper sequence.
• The CBSE Class 12th Maths NCERT Solutions   PDF Download option promotes among students the logical approach, which is crucial to ace the exam.
• These solutions are extremely helpful for quick revisions during examinations.

Students are advised to follow the NCERT Class 12 books and solutions on a regular basis to get in-depth knowledge about all the concepts covered in the syllabus.

Frequently Asked Questions on NCERT Solutions for Class 12 Maths

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BSc 1st Year Mathematics Notes: PDF (For Sem I & II)

Are you looking for BSc 1st year mathematics notes in pdf format? Well, if it is so, then you are in the right place.

Today, we are here with free to download mathematics notes for BSc 1st year students. In this post, we have discussed a detailed guide on how you can download these notes without spending a single penny.

In our last few posts, we have also covered detailed guides on how to download physics & chemistry notes for BSc 1st year students.

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Along with that, you can also download free BSc books & much more on our Free Study Material section.

I think it’s enough about free study material available on our website. Let’s go back to our main topic, i.e. BSc 1st year mathematics notes.

Let us first discuss a bit about the syllabus of BSc mathematics!

BSc 1st Year Mathematics Syllabus

The syllabus of BSc 1st year is distributed into two semesters. The first semester comprises books like Elementary Algebra , Matrices , Trigonometry , Differential Calculus & Vector Analysis .

Likewise, the syllabus for the second semester comprises Group Theory , Integral Calculus & Analytical Geometry .

The detailed syllabus of both the semesters in tabular form is given below:

You can also download the complete syllabus of BSc mathematics here .

BSc 1st Year Mathematics Notes: Download Here

The link to download BSc 1st year mathematics notes is given below. In order to download these notes, you have to use any of the below share buttons.

Just after you share this post using the below share buttons, the download links will be visible for each book.

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** Notes for Group Theory are taken from Michigan Technological University (mtu.edu) & that of Integral Calculus from Northwestern University (northwestern.edu)

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• BSc Sem I Important Questions in Chemistry
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Module 5- Math KA - Course Guide

Module overview:.

In this module, students will circle numbers in a set after counting. Students will compare attributes and classify by attributes. Students will use matching and counting strategies.

• Assignments

Module Materials:

Module Objectives:

Module Key Words:

Module Assignments:

Learning Coach Notes:

• Guiding Questions
• Video Questions
• Suggested Books

Module Guiding Questions:

When a student starts a lesson ask them questions to check for prior knowledge and understanding and to review concepts being taught. At the end of the lesson ask the questions again to see if their answer changes.

Module Video Questions:

When a student watches a video take time to ask them questions about what they watched. Suggested questions for the videos in this module are listed here. Suggestion: Have the student watch the entire video first all the way through. Then have them watch the video a second time, as they watch it pause the video and ask the questions.

Module Suggested Read Aloud Books:

• What is the main idea?
• What are three things new you learned?
• How does this book relate to what you are learning about?

Module Outing:

Take some time to apply what your student is learning to the real world. Suggested outings are below.