NCERT Solutions For class 12 Maths

Download a Chapter-wise Free PDF of NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Maths – Chapter-wise Overview

Chapter 1: Relations and Functions

This chapter builds on the concepts of relations and functions introduced in Class 11 and takes them to an advanced level. The subtopics include types of relations such as reflexive, symmetric, transitive, and equivalence relations, which are important in understanding the properties of sets and mappings. Students also study functions like one-to-one, onto, and composite functions, along with the concept of invertible functions. Questions in this chapter generally involve proving the properties of given relations, verifying whether a relation is an equivalence relation, and solving problems based on the composition of functions. Higher-order questions may ask students to apply these concepts to real-life examples, such as functions in mathematics and computer science. The NCERT Solutions guide students step by step in handling proofs and logic-based questions, ensuring clarity of concepts. By practicing these solutions, learners build a strong foundation that is later used in topics like inverse trigonometric functions, calculus, and vector algebra.

Chapter 2: Inverse Trigonometric Functions

This chapter introduces inverse trigonometric functions, which are essential for calculus and higher mathematics. The subtopics include principal values of inverse trigonometric functions, domain and range restrictions, and basic properties of inverse trigonometric functions. Students also learn how to express trigonometric identities in terms of inverse functions. Questions in this chapter are usually based on proving identities, simplifying expressions, and evaluating exact values using inverse trigonometric properties. Some problems also require students to apply concepts of restrictions and ranges while solving. The NCERT Solutions provide detailed steps to simplify tricky identities and proofs, helping students avoid common mistakes. Graphical representation of inverse trigonometric functions is another important area where students learn to visualize the functions and their restricted domains. This chapter plays a key role in integral calculus, as inverse trigonometric functions frequently appear in integration formulas. Practicing NCERT Solutions ensures accuracy, logical reasoning, and a strong command of this topic.

Chapter 3: Matrices

The chapter on matrices introduces one of the most powerful tools in mathematics with applications in engineering, computer science, and economics. The subtopics include types of matrices, equality of matrices, addition, subtraction, scalar multiplication, and multiplication of matrices. Special focus is given to properties of matrix operations, which often appear in exam questions. Students also learn about the transpose of a matrix, symmetric and skew-symmetric matrices, and elementary operations used for solving equations. Higher-level problems include verifying algebraic properties and solving questions involving matrix transformations. The NCERT Solutions break down every operation into simple steps, making it easier for students to master the calculation process. Word problems and application-based questions test logical application of matrix properties in real-life contexts. This chapter also serves as a foundation for determinants, where matrices are used to solve systems of linear equations. Regular practice with NCERT Solutions helps students strengthen both conceptual understanding and computational accuracy.

Chapter 4: Determinants

Determinants are an extension of matrices and are crucial in solving systems of linear equations, geometry, and higher mathematics. This chapter covers the definition and properties of determinants, minors, cofactors, and the process of finding the adjoint and inverse of a matrix using determinants. Subtopics also include applications of determinants in solving linear equations through Cramer’s rule. Students practice questions involving expansion of determinants, the verification of properties like row and column operations, and proving identities involving determinants. Higher-order questions may include applying determinant properties to simplify algebraic expressions or solve geometric problems. The NCERT Solutions provide step-by-step explanations of these operations, making it easier for students to apply them in exams. Another important part of this chapter is the use of determinants in calculating the area of triangles and testing the collinearity of points, which often appear in board exams. Mastering determinants prepares students for topics like vector algebra and three-dimensional geometry, where such concepts are applied frequently.

Chapter 5: Continuity and Differentiability

This chapter introduces the advanced concepts of calculus, beginning with the continuity of functions and moving to differentiability and derivatives of functions. Subtopics include algebra of continuous functions, exponential and logarithmic functions, derivatives of composite, implicit, and inverse trigonometric functions. The concept of differentiability at a point and the relationship between continuity and differentiability are also covered in detail. Students also learn the chain rule, derivatives of parametric functions, and second-order derivatives. Questions in this chapter range from proving continuity at a specific point to solving complex differentiation problems. NCERT Solutions guide students through logical steps, helping them understand tricky problems like limits involving trigonometric functions or derivatives using first principles. Advanced questions prepare students for applications in optimization, integration, and curve sketching. This chapter builds the groundwork for the calculus portion of competitive exams, where accuracy and speed are essential.

Chapter 6: Applications of Derivatives

Applications of derivatives take the abstract concept of differentiation and apply it to practical situations. The subtopics include rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, and maxima-minima problems. Students solve a wide variety of questions ranging from proving monotonicity to finding points of tangency and normal equations for curves. Word problems include real-life applications, such as maximizing profit, minimizing cost, and optimizing dimensions of geometrical figures. The NCERT Solutions provide detailed steps to help students approach problems systematically, especially optimization questions that require proper use of the first and second derivative tests. Exam questions usually test problem-solving ability with graphical interpretation and logical reasoning. This chapter not only plays a major role in Class 12 board exams but also has high importance in entrance exams like JEE, where calculus forms a major portion of the question paper. With practice, students gain both conceptual clarity and practical application skills.

Chapter 7: Integrals

This is one of the most important chapters in Class 12 Maths, as it introduces integration, the reverse process of differentiation. The subtopics include integration as an inverse process of differentiation, methods of integration such as substitution, partial fractions, and integration by parts. Students also learn to integrate trigonometric functions, exponential functions, and rational functions. Another major focus is on definite integrals, their properties, and their application in evaluating limits. Questions in this chapter range from simple integrals to complex multi-step problems requiring the use of multiple techniques. NCERT Solutions explain each step clearly, ensuring students understand how to choose the right method for solving an integral. Definite integrals also involve questions on applying properties to simplify calculations. This chapter is highly important for board exams and entrance tests like JEE, where integration carries significant weightage. Mastery of integrals sets the stage for applications in areas like probability, physics, and engineering.

Chapter 8: Applications of Integrals

After learning the techniques of integration, students move to applications in real-world contexts. This chapter deals mainly with finding areas under simple curves, areas between two curves, and areas bounded by a curve and the coordinate axes. Subtopics include the use of definite integrals in calculating these areas. Typical questions involve evaluating definite integrals to find the area enclosed by standard curves like parabolas, circles, ellipses, and straight lines. Word problems are also included, where students apply integration to geometric and real-life situations. NCERT Solutions guide students through diagrams and detailed steps, making visualization easier while solving. This chapter connects calculus with geometry, helping learners see the practical use of integration beyond just symbolic manipulation. Board exam questions often come directly from solved examples and exercises, while competitive exams may present slightly twisted versions of the same concepts. Practicing these solutions ensures speed and accuracy while strengthening conceptual clarity.

Chapter 9: Differential Equations

Differential equations form the foundation for many applications in physics, engineering, and economics. This chapter covers the definition of a differential equation, order and degree, and general and particular solutions. Subtopics also include methods of solving differential equations of the first order and first degree, such as variable separable, homogeneous equations, linear equations, and equations reducible to linear form. Higher-level problems involve forming differential equations based on given conditions and solving them to find particular solutions. Students also practice application-based questions like modeling real-life situations using differential equations. The NCERT Solutions systematically explain the solving process, helping students avoid confusion while applying different methods. This chapter requires strong algebraic manipulation and logical reasoning, as problems often involve multiple steps. Board exam questions typically ask for general and particular solutions, while competitive exams focus on applications and shortcuts. Mastery of this topic gives students an edge in both school and entrance exams.

Chapter 10: Vector Algebra

Vector Algebra introduces students to one of the most powerful mathematical tools used in physics, engineering, and computer science. The chapter begins with the basics of vectors, including types of vectors such as zero, unit, collinear, and coplanar vectors. Subtopics include operations on vectors like addition, subtraction, and multiplication by scalars, along with properties of these operations. Students also learn about the scalar (dot) product and vector (cross) product of two vectors, which are essential for solving problems in geometry and physics. Questions often involve proving properties of vectors, calculating magnitudes, and finding angles between vectors using dot and cross product formulas. Application-based problems include verifying collinearity, computing work done, and determining areas of parallelograms and triangles using vector methods. NCERT Solutions provide step-by-step explanations for each operation, making complex vector manipulations simpler. Board exam questions frequently test the application of dot and cross products, while entrance exams focus on problem-solving involving multiple vector concepts. This chapter prepares students for 3D geometry and strengthens their logical reasoning with algebraic-geometric connections.

Chapter 11: Three-Dimensional Geometry

This chapter extends the study of coordinate geometry into three dimensions, where students learn about lines, planes, and the distances between geometric elements in space. Subtopics include direction cosines, direction ratios of a line, equation of a line in 3D, angle between two lines, shortest distance between skew lines, and equations of planes. Students also study angles between a line and a plane, distance of a point from a plane, and the intersection of lines and planes. Typical questions require finding equations, verifying perpendicularity or parallelism, and solving problems involving distances and angles. NCERT Solutions simplify these calculations by breaking them into manageable steps. Diagrams are often used to improve visualization, helping students understand the geometry of three-dimensional space. This chapter has high importance in board exams, where most questions are application-based. For competitive exams, 3D geometry combines with vector algebra to test problem-solving skills in physics-related contexts. Mastering this chapter enhances both spatial reasoning and algebraic manipulation.

Chapter 12: Linear Programming

Linear Programming (LPP) introduces optimization techniques widely used in business, economics, and operations research. The chapter focuses on problems involving inequalities that form a feasible region on a coordinate plane. Subtopics include formulation of linear programming problems, graphical methods of solving LPP, and finding optimal solutions using corner point methods. Questions typically require forming inequalities from word problems, plotting feasible regions, and determining maximum or minimum values of an objective function. Application-based problems may involve maximizing profit, minimizing cost, or optimizing resources under given constraints. NCERT Solutions provide clear diagrams and step-by-step explanations, ensuring that students understand both the graphical interpretation and logical reasoning behind the solution. While board exam questions are mostly based on direct applications, competitive exams often test quick interpretation of constraints and logical formulation of inequalities. This chapter helps students connect mathematics with real-world scenarios, making it one of the most practical topics in the Class 12 syllabus.

Chapter 13: Probability

Probability in Class 12 deals with advanced concepts that extend beyond the basics introduced earlier. The chapter covers conditional probability, multiplication theorem of probability, independent events, Bayes’ theorem, and random variables with probability distributions. Students also learn about the mean and variance of random variables, which are crucial in statistics and real-world applications. Questions often involve finding conditional probabilities, solving problems using Bayes’ theorem, and calculating probabilities in multi-step experiments. Higher-order problems focus on applying probability rules in real-life contexts like card games, dice, and selection-based questions. The NCERT Solutions provide detailed explanations for each type of question, ensuring clarity in using formulas and logical reasoning. This chapter is very scoring in board exams, as most questions are direct applications of theorems and properties. In competitive exams like JEE and CUET, probability problems are designed to test both speed and accuracy. By practicing NCERT Solutions, students can handle complex probability models confidently and apply them to advanced fields such as data science, machine learning, and finance.

Features of NCERT Solutions for Class 12 Maths by Home-Tution.com

The NCERT Solutions for Class 12 Maths prepared by the Home-Tution.com team are designed to provide accurate, step-by-step solutions to all textbook problems, ensuring conceptual clarity and effective exam preparation. Here are the key features:

1. Chapter-wise & Step-by-Step Solutions

Each chapter is covered in detail with step-by-step explanations for all exercises, ensuring a strong conceptual understanding of Algebra, Calculus, Probability, and Vector Algebra.

2. Based on the Latest CBSE Syllabus

The solutions strictly follow the latest CBSE curriculum, making them highly relevant for board exams and competitive exams like JEE and NDA.

3. Easy-to-Understand Language

Solutions are explained in simple, student-friendly language, helping students grasp difficult mathematical concepts with ease.

4. Error-Free & Expert-Verified Solutions

All answers are 100% accurate and reviewed by experienced Maths educators, ensuring high quality and reliability.

5. Free PDF Download & Online Access

Students can access the solutions anytime, anywhere, through free downloadable PDFs and an interactive online platform.

These solutions by Home-Tution.com help students strengthen their problem-solving skills, improve accuracy, and boost confidence for Class 12 Maths exams.

The Right Way to Use NCERT Solutions for Class 12 Maths Effectively

NCERT Solutions for Class 12 Maths are a valuable resource for board exam preparation and competitive exams like JEE and NDA. However, to maximize their benefits, students must use them strategically. Here’s the right way to use NCERT Solutions for Class 12 Maths effectively:

1. Read the Chapter First

Before jumping to solutions, read the chapter thoroughly from the NCERT textbook. Understand the concepts, formulas, and theorems explained in each topic.

2. Try Solving Problems on Your Own

Attempt the exercises without looking at the solutions. This helps in identifying your weak areas and improving problem-solving skills.

3. Refer to the NCERT Solutions for Doubts

Use the NCERT Solutions only after making an effort to solve the question. Compare your approach with the given solution and analyze any mistakes.

4. Understand the Step-by-Step Approach

Pay attention to the detailed step-by-step explanations in the solutions. This helps in learning the correct methods and logical reasoning behind each step.

5. Make Notes of Important Concepts

While using the solutions, note down key formulas, shortcuts, and tricky concepts to revise later. This improves retention and speeds up problem-solving during exams.

6. Practice Regularly

Maths requires consistent practice. Solve different types of questions multiple times using NCERT solutions to gain confidence and improve accuracy.

7. Use for Last-Minute Revision

Before exams, revise important questions from NCERT Solutions to brush up on concepts and boost speed in solving problems.

By following these steps, students can use NCERT Solutions for Class 12 Maths effectively to improve their conceptual clarity, problem-solving speed, and overall performance in exams.

Benefits of NCERT solutions for class 12 Maths

Do you want to excel in Mathematics in class 12? Do you want to get overall good marks in your final board examinations? If your answer is Yes, you need to work hard on your goals, and we are here to help you with the same!  We provide you with the NCERT Book solutions for class 12 Maths. It will help you save your time searching for the answer to any particular question on multiple platforms. We provide the most detailed solutions to help you thoroughly prepare for your exams. You can score well in your Class 12th Board examination if you prepare well from the detailed NCERT Solutions provided by our expert team!

CBSE Class 12 Maths Academic Resources