 # NCERT Solutions For Class 12 Maths

## NCERT Solutions of Class 12 Maths – Free CBSE PDF Download

NCERT Solutions Class 12 Maths provides a solid conceptual base for all the topics included in CBSE Class 12 Maths Syllabus. It covers all the important theorems and formulae with detailed explanations to promote a better conceptual understanding in students. NCERT Maths Solutions for Class 12 will help students in the preparation of undergraduate level Engineering entrance exams like JEE Mains, BITSAT, VITEEE, etc, to be conducted for the year, 2020-2021. BYJU’S brings you NCERT Solutions Class 12, designed by our subject experts to facilitate 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 questions 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 ## Class 12 Maths NCERT Solutions Exercises

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

### Chapter 1 Relations and Functions

Chapter 1 of NCERT textbook deals with relations and functions, types of relation, types of functions, 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:
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function.

Maths Class 12 NCERT Solutions Chapter 1 Exercises
Exercise 1.1
Exercise 1.2
Exercise 1.3
Exercise 1.4
Miscellaneous Exercise

### Chapter 2 Inverse Trigonometric 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:
Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions Elementary properties of inverse trigonometric functions.

Maths Class 12 NCERT Solutions Chapter 2 Exercises
Exercise 2.1
Exercise 2.2
Miscellaneous Exercise

### Chapter 3 Matrices

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:
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 and existence of non-zero matrices whose product is the zero matrix (restricted 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).

Maths Class 12 NCERT Solutions Chapter 3 Exercises
Exercise 3.1
Exercise 3.2
Exercise 3.3
Exercise 3.4
Miscellaneous-Exercise

### Chapter 4 Determinants

Chapter 4 of NCERT Solutions Maths 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:
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. 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.

### Chapter 5 Continuity and Differentiability

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:
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. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

### Chapter 6 Applications of Derivatives

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:
Applications of derivatives: rate of change of bodies, 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).

Maths Class 12 NCERT Solutions Chapter 6 Exercises
Exercise 6.1
Exercise 6.2
Exercise 6.3
Exercise 6.4
Exercise 6.5
Miscellaneous Exercise

### Chapter 7 Integrals

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:
Integration as an 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. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof).Basic properties of definite integrals and evaluation of definite integrals.

### Chapter 8 Applications of 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:
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).

Maths Class 12 NCERT Solutions Chapter 8 Exercises
Exercise 8.1
Exercise 8.2
Miscellaneous Exercise

### Chapter 9 Differential Equations

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:
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, solutions of 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 or constants.
d𝑥 /d𝑦 + px = q, where p and q are functions of y or constants.

### Chapter 10 Vector Algebra

In this chapter, students are introduced to the vector algebra. The concepts in this chapter cover 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:
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, scalar triple product of vectors.

Maths Class 12 NCERT Solutions Chapter 10 Exercises
Exercise 10.1
Exercise 10.2
Exercise 10.3
Exercise 10.4
Miscellaneous Exercise

### 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:
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. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.

Maths Class 12 NCERT Solutions Chapter 11 Exercises
Exercise 11.1
Exercise 11.2
Exercise 11.3
Miscellaneous Exercise

### Chapter 12 Linear Programming

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:
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 (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Maths Class 12 NCERT Solutions Chapter 12 Exercises
Exercise 12.1
Exercise 12.2
Miscellaneous Exercise

### Chapter 13 Probability

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, Baye’s 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, which have explained the concept thoroughly.
Topics Covered:
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean and variance of random variable. Binomial probability distribution.

Maths Class 12 NCERT Solutions Chapter 13 Exercises
Exercise 13.1
Exercise 13.2
Exercise 13.3
Exercise 13.4
Exercise 13.5
Miscellaneous Exercise

## CBSE Class 12 Maths Exam Pattern 2020-2021

CBSE Class 12 Maths Exam Pattern 2020-2021 is based on the revised guidelines of the Central Board of Secondary Education. CBSE Class 12 marking scheme is the guide that is used for allocating the marks to the students. In Class 12 Maths, the board exam will be conducted for 100 marks, with three sections that are very short answer type, short answer type and long answer type of questions. Students who are appearing for the exam must go through the Marking Scheme 2020-2021 after solving the CBSE Class 12 CBSE Sample Papers and previous year question paper. This also helps to attempt the competitive exams as it gives more importance to very short answer type questions. Here you can find the CBSE Class 12 Maths Exam pattern.

 CBSE Class 12 Maths (Basic and Standard Paper) Marking Scheme Sections No. of Questions Marks per Question Total Marks Section A 10 Qs 1 Mark 10 Section B 12 Qs 4 Marks 48 Section C 7 Qs 6 Marks 42 Total 40 Qs 100

### CBSE Class 12 Chapter-wise marks weightage

The marking scheme for the exam 2020-2021 is released by the CBSE on the official site. Here you can find the CBSE Class 12 Chapter-wise marks weightage according to the previous year question papers.

 Unit Marks 1. (a) Relations and Functions (b) Inverse trigonometric Functions 10 2. (a) Matrices (b) Determinants 13 3. (a) Continuity and Differentiability (b) Applications of Derivatives (c) Integrals (d) Applications of Integrals (e) Differential Equations 44 4. (a) Vectors (b) 3 – dimensional Geometry 17 5. Linear Programming 06 6. Probability 10 Total Marks 100

MATHEMATICS (Code No. – 041)

Question Paper Design Class – XII (2020 – 2021)

Time – 3 hours Max. Marks – 80

 Sl. No. Typology of Questions Total Marks Percentage weightage (approx.) 1 Remembering: Having memory of previously studied materials by terms, answers, basic concepts and recalling facts. Understanding: Show understanding of ideas by comparing, organizing, interpreting, translating, describing and stating main ideas 44 55 2 Applying: Solve problems to situations by applying the learnt facts, knowledge, rules and techniques in various ways. 20 25 3 Analysing: Examine the information into parts by recognizing the causes. Make inferences and find evidence to support generalizations Evaluating: Present opinions by making judgments about validity of ideas, information or quality of work based on a set of criteria. Creating: Combine information in a different way by grouping elements in a new pattern or creating alternative solutions 16 20 Total 80 100
 Internal Assessment 20 Marks Periodic Tests ( Best 2 out of 3 tests conducted) 10 Marks Mathematics Activities 10 Marks

#### NCERT Solutions Class 12 Maths Chapter 1 to 13

12 Class Maths NCERT Solutions 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. These NCERT Class 12 maths solutions can be your best education support for board exam preparations. It’s difficult to score in Mathematics simply by reading and memorizing the formulae. Instead, practise is the perfect method to succeed. Learning from BYJU’S NCERT solutions will help students of Class 12 to squeeze conceptual meaning out of every problem quite effectively. CBSE Class 12 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, trigonometry are easy and with proper practice, students can easily score good marks 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.
• These CBSE Class 12 Maths solutions contain important tips, useful for answering typical questions accurately.
• 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 Class 12 Maths

### Which book is best for Class 12 Maths?

The most trusted book for Class 12 board exam is NCERT Book. Students must build up their concepts through NCERT Class 12 Maths book by solving the examples and exercise problems. After that, they can practise questions from other reference books like RD Sharma and RS Aggarwal. But firstly they are recommended to solve the NCERT book as this book follows the CBSE syllabus and pattern.

### Is NCERT enough for Class 12 board Maths?

Yes, NCERT books are enough for board exams. Students can easily grasp the concepts from NCERT books by solving the examples and exercise problems. For practising more questions, they can refer to other books also.

### Is Class 12 NCERT Maths difficult?

The class 12 NCERT Maths is not at all difficult. If students have understood the concept and gone through the solved examples then they can easily solve the questions related to it. For this, they have to practise the questions daily. Practise is the only way through which students can master the subject.

### How many chapters are there in NCERT Maths Class 12?

The NCERT Class 12 Maths book has 2 parts. Part 1 contains chapters 1 to 6 and part 2 contains chapters 7 to 13. Therefore, there is a total of 13 chapters in NCERT Maths Class 12. Viz, Relations and Functions, Inverse Trigonometric Functions, Matrices, Determinants, Continuity and Differentiability, Applications of Derivatives, Integrals, Applications of Integrals, Differential Equations, Vector Algebra, Three Dimensional Geometry, Linear Programming and Probability.

### Where can I get NCERT Class 12 Maths Solutions with Miscellaneous Exercise?

The NCERT Class 12 Maths Solutions are provided on the BYJU’S site. These solutions are prepared by our subject experts. The questions are solved in the simplest method so that students can easily understand them. Also, the solutions for miscellaneous exercises are provided.

### Do Miscellaneous Exercise questions come in board exams?

Questions may be asked from the miscellaneous exercise in the board exam. So, to be on the safer side, students must solve the miscellaneous exercise. Also, the level of these exercises is quite high so it will be good practice for students.

The NCERT Solutions for Class 12 Maths can be used both in online and offline mode from the BYJU’S website. The detailed solutions for all the problems available in the NCERT textbook for Class 12 Maths are answered by experts after conducting vast research on the concepts as per the CBSE guidelines.

### How many chapters are present in NCERT Class 12 Maths?

A total of 13 chapters are present in NCERT Class 12 Maths. 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 are the chapters in NCERT Class 12 Maths.

### Which is the best source for Class 12 board exam preparation?

NCERT books are the best source of study material for CBSE board exam preparation. Mugging up the concepts present in NCERT books a few days before the exam will not help the students in any ways. Students should be well versed in the solutions ahead of the final exams for a better academic score. Ample time should be given in practising problems from other reference books like RD Sharma, RS Aggarwal etc. After completing the solutions, students can move on to sample papers and previous year question papers for a better idea of the exam pattern.

### Which is the official website for NCERT?

The official website for NCERT is http://ncert.nic.in/. NCERT provides students and teachers with a wide range of study materials like textbooks, reference books, etc.