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# NCERT Solutions For Class 12 Maths

## NCERT Solutions of Class 12 Maths – Free 2022-23 CBSE PDF Download

NCERT Solutions for Class 12 Maths provide 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 Solutions for Class 12 Maths will help students in the preparation of undergraduate level Engineering entrance exams like JEE Mains, BITSAT, VITEEE, etc, to be conducted for the year, 2022-2023.

BYJU’S brings you NCERT Solutions for Class 12 Maths, 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.

## Class 12 Maths NCERT Solutions Exercises

### Chapter 1 Relations and Functions

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.

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

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

### 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 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.

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

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

### 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 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).

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

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

### 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:

### 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 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:

### 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 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).

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

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

### 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 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

$\int \frac{dx}{x^{2}\pm a^{2}}, \int \frac{dx}{\sqrt{x^{2}\pm a^{2}}}, \int \frac{dx}{\sqrt{a^{2}- x^{2}}}, \int \frac{dx}{ax^{2}+ bx+c}, \int \frac{dx}{ax^{2}+bx+c}$

$\int \frac{px+q}{ax^{2}+bx+c}dx, \int \frac{px+q}{\sqrt{ax^{2}+bx+c}}dx, \int \sqrt{a^{2}\pm x^{2}}dx, \int \sqrt{x^{2}-a^{2}}dx$ Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof).
Basic properties of definite integrals and evaluation of definite integrals.

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

### 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 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).

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

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

### 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 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:

### 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 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.

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

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

### 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.

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

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

### 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 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).

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

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

### 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, 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, which have explained 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.

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

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

### CBSE Class 12 Maths 2022-23 Unit wise weightage

 No. Units Marks I Relations and Functions 08 II Algebra 10 III Calculus 17 IV Vectors and Three – Dimensional Geometry 05 V Linear Programming 40 VI Probability 10 Total 80 Internal Assessment 20

#### Internal Assessment Class 12

 Internal Assessment Marks Periodic Tests (Best 2 out of 3 tests conducted) 10 Marks Mathematics Activities 10 Marks Total Marks 20 Marks

### 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. These solutions PDF of Class 12 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 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, 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.
• The CBSE Class 12th Maths NCERT Solutions PDF Download option enables logical approach among students 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

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

The NCERT textbook of Class 12 Maths has 2 parts. Part 1 contains chapter 1 to 6 whereas part 2 contains chapter 7 to 13. So the chapters are – Matrices, Inverse Trigonometric Functions, Relations and Functions, Determinants, Applications of Derivatives, Continuity and Differentiability, Applications of Integrals, Vector Algebra, Differential Equations, Three Dimensional Geometry, Probability and Linear Programming.

### Can I get answers for miscellaneous exercise in the NCERT Solutions for Class 12 Maths?

The solutions for the miscellaneous exercise are present in the BYJU’S website. Students can download the solutions PDF and access them while solving the problems from the NCERT textbook. The solutions are designed in a systematic way based on the marks weightage for the concepts in accordance to the CBSE board. The level of these exercises are high to help the students prepare for the board exams with much confidence.

### Is NCERT Solutions for Class 12 Maths the best source of reference materials for the students?

NCERT books are considered to be one of the best sources of reference materials for the students of Class 12. Mugging up of concepts a few days before the board exams will not help students to score well. So conceptual understanding is very much important in order to ace the exam without fear. Ample time should be spent in solving the textbook problems on a daily basis. This will help students to realise the type of problems that would arise in the board exams and work on them for a better academic score.

### Where can I download the NCERT Solutions for Class 12 Maths?

Class 12 is a major step in a students life as the academic score will help them to achieve their career goals. Students can refer to the solutions of Class 12 in online mode or download and refer to them while solving the textbook problems to score good marks in the exams. The chapter wise Maths Class 12 NCERT Solutions PDF download option provided at BYJU’S helps students understand the concepts in a better way.

#### 1 Comment

1. Manvi Pathak

Very good