NCERT Solutions for Class 12 Maths

Maths is among the highest-scoring subjects in the Class 12th board exam 2024-25. The syllabus of the CBSE Maths exam is based on the latest NCERT Maths syllabus. So, GeeksforGeeks has curated the NCERT Class 12 Maths Solution for you to prepare.

Students can also download the NCERT Solution PDF of Math for reference. This Math solution can help students build a deeper understanding of concepts like Matrices & Determinants, Integrals, Differentials, trigonometric functions, inverse trigonometric functions etc.

These NCERT solutions can be used as a reference by students if they get stuck answering issuesby students as a reference or as a source to clarify topics.

Chapter 1 - Relations and Functions

This section covers Class 12 Maths NCERT Solutions Chapter 1 Relations and Functions and discusses the introduction of relations and functions, types of relations, types of functions, the composition of functions and invertible functions, binary operations, and miscellaneous examples in Class 12 Maths.

 Topics Covered in Chapter 1 – Relations and Functions:

• Types of Relations: Reflexive, symmetric, transitive, and equivalence relations
• Types of Functions: One-to-one (injective), onto (surjective), and bijective functions
• Composite Functions
• Inverse of a Function
• Binary Operations

Chapter 2 - Inverse Trigonometric Functions

Chapter 2 of the NCERT Class 12 Mathematics textbook deals with Inverse Trigonometric Functions. It covers essential concepts such as definitions, domain, range, and principal value branches of inverse trigonometric functions. 

Topics Covered in Chapter 2 – Inverse Trigonometric Functions:

• Definition, Domain, and Range of Inverse Trigonometric Functions
• Principal Value Branches
• Graphs of inverse trigonometric functions
• Elementary Properties of Inverse Trigonometric Functions
• Miscellaneous Examples for conceptual clarity and application

Chapter 3 - Matrices

Topics Covered in Chapter 3 – Matrices:

• Types and order of matrices
• Matrix operations and their properties
• Transpose, symmetric, and skew-symmetric matrices
• Elementary row/column operations
• Inverse of a matrix and its uniqueness

Chapter 4 - Determinants

This chapter introduces determinants of square matrices (up to 3×3) and explores their properties and applications, especially in solving systems of linear equations.

Topics Covered in Chapter 4 – Determinants:

• Definition and evaluation (2×2 and 3×3)
• Properties of Determinants
• Minors and Cofactors
• Area of a Triangle
• Adjoint and Inverse of a Matrix
• Solving Linear Equations
• Consistency of Systems

Chapter 5 - Continuity and Differentiability

Chapter 5 introduces the fundamental concepts of continuity and differentiability, forming the basis for advanced calculus. It explores the behavior of functions under various operations and provides techniques to differentiate complex functions, including composite, implicit, inverse trigonometric, exponential, and logarithmic functions. The chapter also covers logarithmic differentiation, parametric forms, second-order derivatives, and Rolle’s and Lagrange’s Mean Value Theorems.

Topics Covered in Chapter 5 – Continuity and Differentiability:

• Continuity and Differentiability
• Introduction
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Derivatives of Functions in Parametric Forms
• Second-Order Derivative
• Mean Value Theorem

Chapter 6 - Applications of Derivatives

Chapter 6 explores the practical use of derivatives in various mathematical and real-life contexts. It covers the rate of change of quantities, behavior of functions (increasing or decreasing), and the equations of tangents and normals.

Topics Covered in Chapter 6 – Application of Derivatives:

• Rate of change of quantities
• Increasing and decreasing functions
• Tangents and normals
• Approximations
• Maxima and minima (first and second derivative tests)
• Application-based and real-life problems

Chapter 7 - Integrals

Chapter 7 introduces the concept of integration as the inverse process of differentiation. It covers the definition and properties of indefinite integrals, along with their geometric interpretation. Various integration methods are discussed, including substitution, partial fractions, by parts, and using trigonometric identities. The chapter also explains definite integrals, their interpretation as limits of sums, and the Fundamental Theorem of Calculus, followed by techniques for evaluating definite integrals and their properties.

Topics Covered in Chapter 7 – Integrals:

• Integrals
• Integration by Substitution
• Integration by Partial Fractions
• Integration by Parts
• Integration of Trigonometric Functions
• Functions Defined by Integrals
• Definite Integral
• Computing Definite Integrals

Chapter 8 - Applications of Integrals

Chapter 8 extends the concept of integrals to practical applications, primarily in finding areas. It includes methods to calculate the area under simple curves, the area bounded by a curve and a line, and the area between two curves. The chapter focuses on standard forms and clearly defined regions, supported by illustrative examples and exercises.

Topics Covered in Chapter 8 – Applications of Integrals:

• Area under Simple Curves
• Area Between Two Curves - Calculus
• Area between Polar Curves
• Area as Definite Integral

Chapter 9 - Differential Equations

Class 12 Maths Chapter 9 – Differential Equations introduces the basic concepts of differential equations, including their definition, order, and degree. It explains the difference between general and particular solutions and how to form differential equations from given functions or curves. The chapter also focuses on various methods for solving first-order, first-degree differential equations, including the method of separation of variables, solving homogeneous differential equations, and linear differential equations of the form:

Topics Covered in Chapter 9 – Differential Equations:

• Differential Equations
• Particular Solutions to Differential Equations
• Homogeneous Differential Equations
• Separable Differential Equations
• Exact Equations and Integrating Factors
• Implicit Differentiation
• Implicit Differentiation - Advanced Examples
• Advanced Differentiation
• Disguised Derivatives - Advanced differentiation | Class 12 Maths
• Derivative of Inverse Trig Functions
• Logarithmic Differentiation

Chapter 10 - Vector Algebra

Class 12 Maths Chapter 10: Vector Algebra introduces students to the foundational concepts of vectors and their operations. The chapter begins by distinguishing between vectors and scalars, followed by an exploration of the magnitude and direction of a vector. Students learn how to determine the position vector of a point and understand direction cosines and direction ratios.

Topics Covered in Chapter 10 – Vector Algebra:

• Vector Algebra
• Dot and Cross Products on Vectors
• How to Find the Angle Between Two Vectors?
• Section Formula - Vector Algebra

Chapter 11 - Three-Dimensional Geometry

Class 12 Maths Chapter 11: Three-Dimensional Geometry introduces students to concepts in 3D space, including direction cosines and direction ratios of a line, equations of lines and planes in vector and Cartesian form, and the relationships between them. Key topics include the angle between two lines, the shortest distance between lines (skew and parallel), coplanarity, and various forms of plane equations—normal, intercept, and point-normal. The chapter also covers angles between planes, angle between a line and a plane, and the distance of a point from a plane. Exercise solutions are provided below for practice and clarity.

Topics Covered in Chapter 11 – Three-Dimensional Geometry :

• Direction Cosines and Direction Ratios
• Equation of a Line in 3D
• Angles Between two Lines in 3D Space
• Shortest Distance Between Two Lines in 3D Space | Class 12 Maths
• Points, Lines, and Planes

Chapter 12 - Linear Programming

Class 12 Maths Chapter 12: Linear Programming introduces students to the basics of linear optimization. It covers the mathematical formulation of linear programming (LP) problems, terminology such as constraints, objective function, feasible region, and optimization. Students learn to solve L.P. problems involving two variables using the graphical method. The chapter also explores different types of L.P. problems and discusses feasible, infeasible, and optimal solutions. Exercise solutions are provided to help reinforce understanding through practice.

Topics Covered in Chapter 12 – Linear Programming:

• Linear Programming
• Graphical Solution of Linear Programming Problems

Chapter 13 - Probability

Class 12 Maths Chapter 13: Probability revisits foundational concepts from Class 11 and builds on them with advanced topics. Key concepts include conditional probability, properties of independent events, the multiplication theorem of probability, total probability, and Bayes’ theorem. The chapter also introduces random variables, their probability distributions, and how to calculate their mean and variance. It concludes with Bernoulli trials and the binomial distribution. Solutions to the exercises are provided for thorough understanding and practice.

Topics Covered in Chapter 13 – Probability:

• Conditional Probability and Independence - Probability | Class 12 Maths
• Multiplication Theorem
• Dependent and Independent Events
• Bayes' Theorem
• Probability Distribution
• Binomial Distribution in Probability

NCERT Class 12 Maths Topic-Wise Weightage 2024-25

The table given shows the unit-wise marks weightage. The Maths theory paper is divided into two parts , one for the internal assessment of the 20 marks and the main paper of 80 marks,Three-dimensional bringing the total marks to 100.

Also Read

• CBSE Class 12 Maths Notes
• CBSE Class 12 Maths Formulas
• CBSE Class 12 Maths Syllabus
• RD Sharma Solutions Maths Class 12
• Class 12 Maths NCERT Solution in Hindi

NCERT Solution for Class 12 Maths Download PDF

Click on the link below to download the NCERT Class 12 Math Solution Download PDF.