Class 12 Mathematics NCERT Solutions: All Chapters, Questions wth Answer, Preparation

The students can click the links in the below table for the NCERT Class 12 Maths Solutions chapter wise.

Sl. No	Name of Chapter
1	Chapter 1 Relations and Functions
2	Chapter 2 Inverse Trigonometric Functions
3	Chapter 3 Matrices
4	Chapter 4 Determinants
5	Chapter 5 Continuity and Differentiability
6	Chapter 6 Application of derivatives
7	Chapter 7 Integrals
8	Chapter 8 Application of Integrals
9	Chapter 9 Differential equation
10	Chapter 10 Vector algebra
11	Chapter 11 3D Geometry
12	Chapter 12 Linear programming
13	Chapter 13 Probability

Class 12 Mathematics NCERT Solutions Chapter 1 - Relations and Functions

In chapter relations and functions, students will learn different types of relations and equivalence relation, composition of functions, invertible functions and binary operations. A relation R from a set A to a set B is a subset of the Cartesian Product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B.  A function is defined as a relation between a set of inputs having one output each There are two exercises and one miscellaneous exercise in this chapter. Students can expect questions from the NCERT Class 12 Maths Chapter 1 Relations and Functions for board exams as well as competitive exams. The unit 1 Relations and Functions has a weightage of 8 marks in the CBSE 12th board Maths exam. 

Chapter 1 Relations and Functions: Topics Covered

• Types of relations: reflexive, symmetric, transitive and equivalence relations
• One to one and onto functions.

Class 12 Mathematics NCERT Solutions Chapter 2 - Inverse Trigonometric Functions

Inverse trigonometric functions is a chapter of great importance for the Class 12 board exam. Inverse trigonometric functions are the inverse functions of the basic trigonometric functions. Students will learn about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverses and observe their behaviour through graphical representations. Solutions of Chapter 2 Class 12 Maths Inverse Trigonometric Functions helps students to understand the concepts in a faster way. We have also provided shortcuts to find the solutions. 

Chapter 2 Inverse Trigonometric Functions: Topics Covered

• Definition, range, domain
• principal value branch
• Graphs of inverse trigonometric functions

Class 12 Mathematics NCERT Solutions Chapter 3 - Matrices

The chapter Matrices come under Unit II Algebra. A matrix can be defined as an ordered rectangular array of numbers or functions. The weightage of Unit II is 10 marks. Students have to practice the solutions of this chapter thoroughly so that they can boost their problem solving skills. They can expect questions from matrices for competitive exams like JEE, and JEE Advanced. The solutions of Class 12 Maths Chapter 3 Matrices are provided with detailed explanation of steps so that students can easily grasp the concept used. 

Chapter 3 Matrices: Topics Covered

• Concept, notation, order, equality, types of matrices,
• Zero and identity matrix
• Transpose of a matrix
• Symmetric and skew symmetric matrices
• Operations on matrices: Addition and multiplication and multiplication with a scalar
• 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)
• Invertible matrices and proof of the uniqueness of inverse, if it exists

Class 12 Mathematics NCERT Solutions Chapter 4 - Determinants

A determinant is a scalar value which can be calculated from the elements of a square matrix. Students can use the solutions of NCERT Maths Class 12 Chapter 4 Determinants so as to understand the exercises thoroughly. They are advised to practice the NCERT Solutions Maths Class 12 so as to improve their speed of solving the problems. We have provided the accurate solutions of all the exercises in this chapter.

Chapter 4 Determinants: Topics Covered

• Determinant of a square matrix (up to 3 x 3 matrices)
• 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

Class 12 Maths NCERT solutions Chapter 5 - Continuity and Differentiability

In this chapter students will learn the important concepts of continuity, differentiability, relations between them, differentiation of inverse trigonometric functions, exponential and logarithmic functions. The NCERT Solutions of Class 12 Maths Chapter 5 Continuity and Differentiability have been prepared as per the latest update on CBSE Syllabus 2022-23. Students can use these solutions to strengthen their basics.

Chapter 5 Continuity and Differentiability: Topics Covered

• Continuity and differentiability, 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

Class 12 Maths NCERT solutions Chapter 6 - Application of Derivatives

Application of derivatives explains how the derivative can be used to find rate of change of quantities, finding the equations of tangent and normal to a curve at a point, to find turning points on the graph of a function which in turn will help us to locate points at which largest or smallest value (locally) of a function occurs. This chapter also discusses how to find intervals on which a function is increasing or decreasing and how to determine the approximate value of certain quantities. We have provided accurate solutions of the exercises of the chapter Application of derivatives.

Chapter 6 Application of Derivatives: Topics Covered

• Applications of derivatives: rate of change of quantities
• Increasing/decreasing functions
• Maxima and minima 
• Simple problems 

Class 12 Maths NCERT solutions Chapter 7 - Integrals

Chapter 7 Integrals explains some indefinite and definite integrals and their elementary properties including some techniques of integration. Integration is the calculation of an integral. There are so many formulas in this chapter which students have to learn thoroughly. Integration is a process which is the inverse of differentiation. Students should understand the basics of integrals and learn them thoroughly. Class 12 Integration NCERT solutions will definitely help students to understand the chapter. 

Chapter 7 Integrals: Topics Covered

• Integration as 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
• Fundamental Theorem of Calculus (without proof)
• Basic properties of definite integrals and evaluation of definite integrals

Class 12 Maths NCERT solutions Chapter 8 - Application of Integrals

Application of Integrals explains how to determine the area under simple curves, area between lines and arcs of circles, parabolas and ellipses (standard forms), and the area bounded by the curves. This is a chapter of great importance for the Class 12 board exam. Students can expect 2-3 questions from this chapter for entrance exams too. We have provided detailed solutions of this chapter for free.

Chapter 8 Application of Integrals: Topics Covered

• Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

Class 12 Maths NCERT solutions Chapter 9 - Differential Equations

An equation having derivatives of the dependent variable with respect to independent variables (variables) is known as a differential equation. In the chapter Differential Equations, students will learn some basic concepts related to differential equations, formation of differential equations, general and particular solutions of a differential equation, methods to solve a first order - first degree differential equation and some applications of differential equations in different areas. There are five exercises and one miscellaneous exercise in this chapter.

Chapter 9 Differential Equations: Topics Covered

• Definition, order and degree, general and particular solutions of a differential equation
• 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) + pxy = q, where p and q are functions of y or constants

Class 12 Maths NCERT solutions Chapter 10 - Vectors Algebra

In this chapter, students will learn some of the basic concepts about vectors, various operations on vectors, and their algebraic and geometric properties. Vector algebra and 3D geometry has a weightage of 14 marks in the Class 12 board exam. This chapter has four exercises and one miscellaneous exercise. Students have to learn this chapter thoroughly to score high marks in board exams and entrance exams.

Chapter 10 Vector Algebra: 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.

Class 12 Mathematics NCERT solutions Chapter 11 - Three Dimensional Geometry

In 3D Geometry, students will learn the direction cosines and direction ratios of a line joining two points, equations of lines and planes in space under different conditions, angle between two lines, two planes, a line and a plane, shortest distance between two skew lines and distance of a point from a plane. There are two exercises and one miscellaneous exercise in this chapter. We provide complete solutions of Three Dimensional Geometry which are prepared by experts.

Chapter 10 Three Dimensional Geometry: Topics Covered

• Direction cosines and direction ratios of a line joining two points
• Cartesian equation and vector equation of a line, skew lines
• Shortest distance between two lines
• Angle between two lines

Class 12 Mathematics NCERT solutions Chapter 12 - Linear Programming

In this chapter, students will learn some linear programming problems and their solutions by graphical method only. Linear programming has a weightage of 5 marks for the Class 12 board exam. We provide accurate solutions of this chapter in a detailed manner which are prepared by expert teachers. 

Chapter 12 Linear Programming: Topics Covered

• Introduction, related terminology such as constraints, objective function, optimization
• 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). 

Class 12 Mathematics NCERT solutions Chapter 13 - Probability

Probability can be defined as measuring the likelihood of an event to occur. Probability is an important chapter which has a weightage of 8 marks for the Class 12 board exam. In chapter Probability, students will learn the important concept of conditional probability of an event given that another event has occurred, which will be helpful in understanding the Bayes' theorem, multiplication rule of probability and independence of events. This chapter also covers the concept of random variable and its probability distribution, mean and variance of a probability distribution and Binomial distribution. Probability class 12 NCERT solutions covers the following topics.

Chapter 13 Probability: Topics Covered

• Conditional probability
• Multiplication theorem on probability
• Independent events, total probability
• Bayes’ theorem
• Random variable and its probability distribution, mean of random variable

Check the weightage of units below. 

No.	Units	Marks
I	Relations and Functions	8
II	Algebra	10
III	Calculus	35
IV	Vectors and Three – Dimensional Geometry	14
V	Linear Programming	5
VI	Probability	8
	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