NCERT Solutions for Class 12 Mathematics provide detailed, step-by-step answers to all textbook questions. The book is divided into two parts, where Part 1 includes 6 chapters and Part 2 contains 7 chapters. Part 1 covers topics such as trigonometry, matrices, determinants, continuity, and differentiability, while Part 2 includes integrals, differential equations, vector algebra, linear programming, and probability. Since Class 12 Maths requires consistent practice, these solutions serve as a valuable tool to improve speed and accuracy. They are useful for strengthening conceptual understanding as well as preparing for entrance exams.
NCERT Solutions for Class 12 Maths Latest Chapter wise 2026 27
NCERT Solutions for Class 12 Mathematics Part - 1
| S.No. | Chapter Name & Topic |
| 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 |
NCERT Solutions for Class 12 Mathematics Part - 2
| S.No. | Chapter Name & Topic |
| 1 | Chapter 7 - Integrals |
| 2 | Chapter 8 - Application of Integrals |
| 3 | Chapter 9 - Differential Equations |
| 4 | Chapter 10 - Vector Algebra |
| 5 | Chapter 11 - Three Dimensional Geometry |
| 6 | Chapter 12 - Linear Programming |
| 7 | Chapter 13 - Probability |
NCERT Solutions for Class 12 Mathematics Overview Chapter-wise
Mathematics Part I
Chapter 1 - Relations and Functions
The first chapter further expands on relations and functions introduced in Class 11. Students learn how a relation is defined as a subset of the Cartesian product of two sets and how functions are special relations with exactly one output for every input. The chapter discusses important types of functions such as injective, surjective and bijective functions, and explains how these determine the nature of mapping between sets. It also explains the concept of inverse functions, showing that only bijective functions are invertible. It includes composition of functions, where two functions are combined to form a new function.
Chapter 2 - Inverse Trigonometric Functions
Chapter 2 discusses inverse trigonometric functions as the inverse operations of standard trigonometric functions, used to determine angles when their trigonometric ratios are known. Students learn why inverse functions require trigonometric functions to be restricted to specific domains so that they become one-one and onto, allowing proper inverses to exist. The chapter explains the six inverse trigonometric functions, and their domains and principal value ranges. It also covers their graphical representations and basic properties. Students study important identities and relationships between these functions.
Chapter 3 - Matrices
Matrices discusses rectangular arrays of numbers arranged in rows and columns, used to represent and simplify mathematical data and systems of equations. Students learn the concept of order of a matrix, elements, and different types of matrices such as row, column, square, zero, diagonal, scalar, and identity matrices. The chapter explains matrix equality and basic operations including addition, subtraction, scalar multiplication, and matrix multiplication. It includes transpose of a matrix and discusses special types like symmetric and skew-symmetric matrices. Students also study basic operations and the concept of inverse of a matrix, which is used to solve systems of linear equations.
Chapter 4 - Determinants
This chapter discusses determinants as a scalar value associated with a square matrix, used to analyse matrix properties and solve systems of linear equations. Students learn how to evaluate determinants of order 1, 2, and 3 using cofactor expansion, and systematic methods for simplification. The chapter explains important properties such as the effect of row and column operations, interchange of rows or columns, proportional or identical rows leading to zero determinant, and linearity properties. It also teaches minors and cofactors, which are used to expand determinants and find the adjoint of a matrix. It explains the application of determinants in solving systems of linear equations using Cramer’s Rule, and determining whether a system has a unique solution based on whether the determinant is non-zero.
Chapter 5 - Continuity and Differentiability
Continuity and Differrentiability teaches how functions behave smoothly and how rates of change are defined using calculus. Students learn the concept of continuity at a point and over an interval using limits, including the condition that the left-hand limit, right-hand limit, and function value must be equal. The chapter then discusses differentiability and explains that if a function is differentiable at a point, it must be continuous there, though the reverse is not always true. It expands on differentiation from first principles and then builds standard rules of differentiation such as the sum rule, product rule, quotient rule, and chain rule. The chapter also includes derivatives of algebraic, trigonometric, exponential, and logarithmic functions.
Chapter 6 - Application of Derivatives
This chapter discusses how derivatives are used to study the behaviour of functions. Students learn how to determine whether a function is increasing or decreasing by analyzing the sign of its derivative. The chapter talks about important applications such as finding tangents and normals to curves and interpreting the derivative as the slope of the tangent at a point. It also explains how derivatives are used to find maxima and minima, helping identify the maximum or minimum values of functions. Students apply these ideas to rate of change problems, where one quantity depends on another and changes over time.
Mathematics Part II
Chapter 7 - Integrals
This chapter teaches integration as the inverse process of differentiation and a key part of calculus. Students learn the concept of antiderivatives and indefinite integrals, where a function is found whose derivative is given. The chapter discusses standard methods of integration such as substitution, partial fractions, and integration by parts for solving different types of functions. It also explains basic properties of indefinite integrals and builds the connection between integration and differentiation through the Fundamental Theorem of Calculus. Students are taught definite integrals and their interpretation as the area under a curve.
Chapter 8 - Application of Integrals
Application of Integrals discusses the practical use of definite integrals to find areas of regions bounded by curves. Students learn how integration is applied to calculate the area under a curve, the area between a curve and the x-axis, and the area enclosed between two curves. The chapter explains how to set up integrals for different geometric regions in the Cartesian plane and evaluate them to obtain exact areas. It also includes cases involving standard curves such as lines, circles, parabolas, and ellipses.
Chapter 9 - Differential Equations
This chapter teaches equations that involve an unknown function and its derivatives, used to describe how quantities change. Students learn the meaning of order and degree of a differential equation and how to identify them. The chapter includes first-order, first-degree differential equations, and methods of solving them such as variables separable form, homogeneous differential equations, and linear differential equations. It explains how to form a differential equation from a given function or family of curves, and the concept of a general solution and a particular solution, where the general solution contains arbitrary constants equal to the order of the equation.
Chapter 10 - Vector Algebra
Chapter 10 discusses vectors as quantities that have both magnitude and direction, used to represent physical quantities such as displacement, velocity, and force. Students learn different types of vectors including zero vector, unit vector, position vector, collinear vectors, and coplanar vectors. The chapter explains how vectors are represented both geometrically and in component form using unit vectors along the coordinate axes. It covers basic vector operations such as addition, subtraction, and scalar multiplication, along with their geometric interpretation using the triangle law and parallelogram law. Students also study important products of vectors such as the dot product and cross product, which are used to find angles, projections, and areas.
Chapter 11 - Three Dimensional Geometry
Three Dimensional Geometry explores coordinate geometry into a three-dimensional space, where a point is represented using three coordinates (x, y, z). Students learn how to describe the position of points using the Cartesian coordinate system and understand concepts such as direction ratios and direction cosines, which help define the direction of a line in space. The chapter explains the equations of lines in three-dimensional space and how to find the shortest distance between two lines, as well as the distance between a point and a line. It also includes the study of angles between lines.
Chapter 12 - Linear Programming
Linear Programming teaches the mathematical method used to find the optimal value (maximum or minimum) of a linear function subject to a set of linear constraints. Students learn mathematical models using decision variables, objective functions, and constraints. The chapter explains how inequalities define a feasible region in the Cartesian plane, which contains all possible solutions satisfying the given conditions. Students learn the graphical method to solve linear programming problems by identifying the vertices of the feasible region and evaluating the objective function at these points to determine the optimal solution.
Chapter 13 - Probability
The last chapter discusses probability and teaches more advanced tools for analyzing uncertainty in random experiments. Students learn conditional probability, which measures the likelihood of an event given that another event has already occurred. The chapter also covers the multiplication theorem of probability, which helps find the probability of the simultaneous occurrence of events. It covers independent events and explains how the occurrence of one event does not affect another in such cases. The chapter explains the Bayes’ theorem, which is used to update probabilities based on new information. Students also study random variables and their probability distributions, including measures like mean and variance.
