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Class 12 - Maths

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Get NCERT solutions for Class 12 Maths free with videos. Solutions of all exercise questions, examples, miscellaneous exercise, supplementary exercise are given in an easy to understand way

The chapters and the topics in them are

• Chapter 1 Relation and Functions – Types of Relation - Reflexive, Symmetric, Transitive & Equivalence relation, Type of functions - One-one (injective), Onto (surjective) & Bijective, Composition of functions and Inverse of a function, Binary Operation - Commutative, Associative, Identity and Inverse.
• Chapter 2 Inverse Trigonometric Functions –  Finding principle value, solving identities using properties
• Chapter 3 Matrices – Order of matrix, Types of matrices - Column matrix, Row matrix, Square matrix, Diagnol matrix, Scalar Matrix, Identity matrix, Equal matrices, Addition, Subtraction and Multiplication of matrices, Transpose of a Matrix, Symmetric, Skew Symmetric Matrices, Finding inverse of matrix by elementary operations
• Chapter 4 Determinants – Evaluating 2 x 2, 3 x 3 Determinant, Properties of Determinant, Area of triangle, Minors and Cofactors, Adjoint and inverse of matrix, Solution of system of linear equations using matrix method
• Chapter 5 Continuity and Differentiability – Finding whether function is continuous, composition of continuous functions is continuous, Finding Derivative using chain rules, logarithmic and exponential differentiation, Mean value of theorem, Rolle's Theorem.
• Chapter 6 Application of Derivatives – Finding rate of change, Increasing & decreasing functions, Point of Inflexion, Finding slope of tangent  and normal, Finding approximate of functions and numbers, Finding maxima & minima using first derivative & second derivative test and Finding absolute maxima and minima
• Chapter 7 Integrals – Finding integration by substitution, trigonometric identities, Partial Fractions & using special formulas. Finding definite integrals as Limit as a Sum, Finding definite integral using indefinite integral and finding Definite Integral using properties
• Chapter 8 Application of Integrals –  Finding area of region of simple curve, curve and line. Finding area between two curves.
• Chapter 9 Differential Equations – Degree and order of differential equations, General & Particular Solutions of Differential Equation, Forming differential equation when general solution is equation, Solving by variable separating, solving homogeneous differential equations, Linear Differential Equation
• Chapter 10 Vector Algebra – What is vector, Difference between vector and scalar, Types of vector - Unit vector, Equal vector, Addition of vectors, Components of vector, Scalar(Dot) product of vectors, Projection of vector on line, Vector (Cross) product of vectors, Finding area of parallelogram
• Chapter 11 Three Dimensional Geometry – Direction Ratio & Direction cosines of line, Finding equation of line - given point & parallel vector, though two points, Angle between two lines, Distance between two skew lines, Distance between two parallel lines, Finding equation of plane - normal form, perpendicular to given vector & passing through point, through 3 non-collinear points, intercept form, through two planes. Coplanarity of two planes, Angle between two planes, Distance between point and plane, Angle between Line & Plane
• Chapter 12 Linear Programming  – Finding maximum or minimum value of Z using bounded or unbounded region of graphical inequality.  Solving Diet, Manufacturing & Transport problems.
• Chapter 13 Probability – Conditional Probability, Multiplication Theorem of Probability, Independent events, Bayes Theorem, Random Variable & its Probability Distributions, Mean & Variance of Random Variable, Bernoulli Trials & Binomial Distribution

Each chapter is divided into two parts - Serial Order Wise and Concept Wise.

• In Serial  Order Wise, the chapter is taught according to the NCERT Book with exercises, examples and miscellaneous. You should follow this when you are searching for a particular question
• In Concept Wise, we have divided the Chapter into topics. Concept is explained first, and then questions are ordered from easy to difficult. You should always follow this method to do the chapter. This is the Teachoo (टीचू) way of doing a chapter.

In addition to that, Important Questions are marked for your reference.

And, we have also solved Past Year Papers, and Sample Papers with step by step marking as well

Click on a Chapter below to begin

Chapter 1 Class 12 Relation and Functions

Chapter 2 class 12 inverse trigonometric functions, chapter 3 class 12 matrices, chapter 4 class 12 determinants, chapter 5 class 12 continuity and differentiability, chapter 6 class 12 application of derivatives, chapter 7 class 12 integrals, chapter 8 class 12 application of integrals, chapter 9 class 12 differential equations, chapter 10 class 12 vector algebra, chapter 11 class 12 three dimensional geometry, chapter 12 class 12 linear programming, chapter 13 class 12 probability, important questions for exams class 12, solutions of sample papers and past year papers - for class 12 boards.

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NCERT Solutions for Class 12 Maths

NCERT Solutions for Class 12 Maths in Hindi and English Medium for 2024-25 board Exams Tiwari Academy NCERT Solutions for Class 12 Maths emerge as a simple tool for students preparing for their board exams in the 2024-25 session. These NCERT National Council of Educational Research and Training | Official Website: https://ncert.nic.in/ textbook solutions, available in both Hindi and English, are carefully revised to stay in sync with the most recent rationalized NCERT textbooks. They are tailored to meet the needs of students aiming for excellence in their board examinations, ensuring a comprehensive understanding of the latest curriculum.

NCERT Solutions of Class 12 Maths

Each solution is rooted in the National Council of Educational Research and Training (NCERT) curriculum, empowering students to confidently tackle and understand the intricacies of every Maths challenge. Class 12 Chapter Wise Maths NCERT Solutions are as follows:

CBSE Syllabus Class 12 Maths for 2024-25 Chapter 1: Relations and Functions Chapter 2: Inverse Trigonometric Functions Chapter 3: Matrices Chapter 4: Determinants Chapter 5: Continuity and Differentiability Chapter 6: Application of Derivatives Chapter 7: Integrals Chapter 8: Application of Integrals Chapter 9: Differential Equations Chapter 10: Vector Algebra Chapter 11: Three Dimensional Geometry Chapter 12: Linear Programming Chapter 13: Probability

Class 12 Maths NCERT solutions in Bharat serve several important purposes for students and educators. Class 12 mathematics is a critical year for students in Bharat as they prepare for their board exams, such as the Central Board of Secondary Education (CBSE) exams and various state board exams. Please provide your Review and Feedback for further improvement. NCERT solutions provide comprehensive and well-structured solutions to the problems and exercises in the NCERT mathematics textbook for class 12. This helps students prepare effectively for their board exams, as these exams often follow the NCERT curriculum 2024-25 .

Class 12 Solutions Apps for iOS and Android

NCERT solutions for Class 12 Maths delve deep into mathematical techniques and principles. These comprehensive guides bolster a solid Maths foundation, ensuring students excel in board exams, competitive tests, and further academic endeavors.

It also includes examples that will help students better understand the concepts. Oswaal CBSE Question Banks Class 12 2024-25 for 2025 Board Exam | Trusted by 3 Lakh Teachers, 50000 Schools, and 13M Parents . The Tiwari Academy provides NCERT Solutions of Class 12 Maths which can be of great help to students.

NCERT solution for class 12 Maths provide a wide range of solved exercises and problems. Students can use these solutions for practice and revision during the board exams, helping them become proficient in solving mathematical problems of varying complexity. Regular practice is essential to improve problem-solving skills and gain confidence in 12th mathematics particularly Differentiation and Integration.

The Essence of Mathematics Mathematics is a vast field that encompasses the study of numbers, formulas, structures, shapes, spaces, and the changes in quantities. It’s divided into major subdisciplines like number theory, algebra, geometry, and analysis. Interestingly, there’s no universally accepted definition of mathematics among mathematicians, highlighting its diverse and evolving nature. When you encounter problems in the Class 12 NCERT Maths textbook, refer to the NCERT Solutions provided by Tiwari Academy. Follow these steps: 1. Read the solution carefully to understand the approach applying formulae. 2. Attempt to solve the problem on your own first. 3. Compare your solution with the provided solution at Tiwari Academy. 4. Analyze any differences and learn from them. Carry on the process to mater the subject.

NCERT National Council of Educational Research and Training | Official Website: https://ncert.nic.in/ Solutions for Class 12 Maths are indeed valuable resources for students preparing for their Class 12 Mathematics exams. Here’s why these solutions can be helpful for CBSE exams. NCERT Solutions are specifically designed to align with the curriculum prescribed by educational boards in Bharat, including CBSE. This means they cover the topics and concepts that are essential for the Class 12 Mathematics exam.

Preparing for Class 12 Maths board exam becomes more effective with a structured approach, especially when incorporating NCERT Solutions for Class 12 Maths. Initially, it’s vital to build a solid foundation in basic concepts. Make sure you have a clear understanding of the principles from previous grades, as Class 12 Maths often expands on these foundations. Regular review and practice of these earlier concepts can greatly enhance your comprehension. Then, acquaint yourself with the Class 12 Maths syllabus. Identify the topics to be covered, focusing first on those that extend from your existing knowledge. This method eases the transition into more advanced topics. Use NCERT textbooks and other educational materials at Tiwari Academy, to gain a thorough understanding of each subject area. Moreover, solving a variety of problems is crucial for better grasping the concepts and preparing for potential exam questions.

Many competitive exams in Bharat, such as engineering and medical entrance exams (e.g., JEE Main, NEET), require a strong foundation in mathematics. Class 12 mathematics NCERT solutions can be valuable for students preparing for these exams, as they cover essential topics and concepts commonly tested in such competitive exams.

Class 12 Maths NCERT Solutions at Tiwari Academy are known for their clarity and accuracy. They provide step-by-step explanations for solving problems, making it easier for students to understand and follow the solutions. These solutions offer a wide range of practice problems and exercises. Practice is crucial for math exams, as it helps students become more comfortable with the concepts and improves problem-solving skills.

Teachers often use NCERT solutions as teaching aids to explain concepts and solve problems in the classroom. These solutions provide structured explanations and solutions that can assist teachers in delivering effective math lessons. NCERT National Council of Educational Research and Training | Official Website: https://ncert.nic.in/ textbooks and solutions are widely recognized and accepted across Bharat. They ensure a standardized curriculum and help maintain consistency in mathematics education across different states and regions.

Key Points of 12th Maths Solutions

Class 12 Maths NCERT solutions cover advanced mathematical concepts and topics suitable for students in their final year of secondary education. Oswaal CBSE Sample Papers Class 12 2024-25 for 2025 Board Exam | Trusted by 3 Lakh Teachers, 50000 Schools, and 13M Parents .

Here are some key points to consider about Class 12 Maths NCERT solutions:

1. The 12th Maths solutions cover a wide range of topics such as calculus, linear algebra, differential equations, Inverse Trigonometry, probability, and vectors. These concepts are important for building a strong foundation in mathematics and preparing for higher studies.

2. The solutions provide step-by-step explanations for various problems, helping students understand the logic and reasoning behind each step. This approach promotes better comprehension and problem-solving skills.

3. The Maths solutions in grade 12th include a variety of problems, ranging from basic to complex. This allows students to gradually build their skills and tackle more challenging problems. These often provide real-world applications of mathematical concepts, helping students see the practical relevance of what they are learning.

4. Class 12 Maths involves more abstract and analytical thinking. The solutions emphasize mathematical reasoning and logic, helping students develop a deeper understanding of the subject. These are designed to align with the CBSE board exam pattern and guidelines. Practicing with these solutions can help students prepare effectively for their exams.

5. Instead of just memorizing formulas, the solutions encourage students to understand the underlying concepts. This approach is crucial for mastering mathematics. Class 12 Maths often involves graphical representations of functions and equations mainly in chapter 8 and 12. The solutions help students interpret and draw various types of graphs accurately.

6. The solutions of 12th Mathematics include a wide range of practice questions at the end of each chapter, allowing students to reinforce their learning and improve their problem-solving skills. These can be used as a valuable resource by teachers to guide their teaching and help them explain concepts effectively to their students. The official NCERT website and various educational platforms like Tiwari Academy offers Class 12 Maths NCERT solutions online , making them easily accessible for students.

Remember that Class 12 Maths can be challenging, but with consistent practice and a strong grasp of the concepts, you can excel in the subject. It’s a good idea to use the NCERT solutions in conjunction with additional study materials and possibly seek guidance from teachers or tutors if needed.

Class 12th Maths Solutions focus on developing a deep understanding of the underlying mathematical concepts. This is essential for scoring well in exams, as questions may be designed to test your understanding rather than just memorization. Since NCERT textbooks and solutions are created to cater to the syllabus of the respective boards, they are highly relevant to the exams. Questions in the exams are often drawn from NCERT textbooks and similar resources.

Class 12 mathematics NCERT solutions are a valuable for all students and educators in Bharat. They aid in exam preparation, concept clarity, and skill development, making them an essential tool for students aiming to excel in mathematics and related fields.

Class 12 Solutions App Download These solutions are prepared by expert mathematicians and can help students understand the concepts better and score good marks in the exams. NCERT Solutions for Class 12 Maths in English and Hindi Medium updated for CBSE session. It provides a detailed explanation to promote a better conceptual understanding in students for all the topics included in CBSE Class 12 Maths Syllabus.

Digital apps for iOS and Android can easily be updated to reflect the latest changes in NCERT syllabi or correct errors in solutions. This ensures that students always have access to the most accurate and up-to-date information. Using Tiwari Academy apps for NCERT solutions reduces the need for physical textbooks, contributing to a more Eco-friendly learning environment.

While some educational apps may have a cost associated with them, they are often more cost-effective in the long run compared to purchasing physical textbooks. Additionally, many apps offer free versions or trials.

Class 12 Maths NCERT Solutions often include solutions to previous years CBSE board question, which can be extremely helpful for practice and understanding the exam pattern. In addition to the basic solutions, there are often additional exercises and problems provided in the NCERT textbooks and solutions. These can be used to challenge yourself and further strengthen your skills.

NCERT Solutions Class 12, designed by our Expert Maths Teacher to facilitate smooth and clear understanding of concepts and logic. These solution covers all the formulae and theorems with step by step method so that student can easily understand. You can get all chapter wise all exercise solutions for the Class 12 Maths NCERT Books . Students who are practising with these class 12th Maths solution for the academic year 2024-25 can pass the 12th class board exam.

However, it’s important to note that while NCERT Solutions of 12th Maths are a valuable resource, they may not cover every possible type of question or provide extensive practice for highly competitive exams. Depending on your goals and the level of your preparation, you may also want to explore other textbooks, reference books, online resources, and practice papers.

NCERT Solutions for Class 12 Maths are an excellent starting point for your exam preparation, but it’s advisable to supplement your study with a variety of resources to ensure comprehensive preparation for your Class 12 Mathematics exam.

You can download NCERT Class 12 Maths Solutions App for all exercises by just click these links here. Class 12 Maths solutions play a vital role in supporting students’ learning and understanding of complex mathematical concepts. It covers advanced topics like calculus, verctor-3D, and probability, which can be challenging to grasp. Solutions provide clear explanations, breaking down complex concepts into digestible steps, helping students understand the underlying principles.

Board exams are a significant milestone. Class 12 Maths solutions align with the CBSE curriculum 2024-25 , providing practice questions similar to those that might appear in the exams. This helps students prepare effectively and boosts their performance.

In the next phase, concentrate on problem-solving and practical application, utilizing NCERT Solutions for Class 12 Maths. Engage in consistent practice with different problem types, including exercises from textbooks, previous exam papers, and additional questions from various math workbooks or online platforms. If you face challenges, seek assistance from teachers, tutors, or educational resources like videos and online forums. It’s also important to understand the theories underlying mathematical concepts, as this can make problem-solving more intuitive. Furthermore, managing your time and stress effectively is crucial. Develop a study plan that includes regular mathematics practice, using NCERT Solutions for Class 12 Maths, while also balancing time for other subjects and activities. Remember, consistent and gradual study is more beneficial than last-minute cramming for exams.

Class 12 Maths Syllabus Revised for 2024-25

• CBSE Syllabus for Class 12 Maths

NCERT Class 12 Maths all Chapters

Class 12 Maths covers advanced topics in mathematics such as calculus, matrix and determinants, differential equations, probability, and vectors. The solutions for these topics aim to provide a clear understanding of the concepts and guide students through solving various types of mathematical problems.

It’s important to note that Class 12 Maths solutions are not only meant for providing answers but also for enhancing understanding and building problem-solving skills. They serve as a valuable resource for students, helping them master the concepts and excel in their studies.

NCERT Solutions for Class 12 Maths Chapter 1 – Relations and Functions

Class 12 Maths, Relations and Functions is the first chapter of the Class 12 Maths NCERT Book. There are 3 exercises in this NCERT Solutions for class 12 Maths Chapter 1 .

Class 12 Maths Exercise 1.1 Class 12 Maths Exercise 1.2 Class 12 Maths Miscellaneous 1

Class 12 Maths Exercise 1.1 in Hindi Class 12 Maths Exercise 1.2 in Hindi Class 12 Maths Miscellaneous 1 in Hindi

These exercises cover types of relations and functions, the composition of functions, domain, co-domain, and composite functions, the inverse of a function and invertible function etc. NCERT Solutions for class 12 Maths Chapter 1 Relations and Functions is an important chapter which provides detail knowledge of relation and functions. All 12th class Maths book solution given here are well solved each problem sums by the experienced Maths teacher. The solution of class 12 Maths chapter 1 are in dual language – Hindi and English.

NCERT Solutions for Class 12 Maths Chapter 2- Inverse Trigonometric Functions

NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions covers various topics such as concepts of inverse trigonometric functions, inverse trigonometric functions properties, and miscellaneous examples.

Class 12 Maths Exercise 2.1 Class 12 Maths Exercise 2.2 Class 12 Maths Miscellaneous 2

Class 12 Maths Exercise 2.1 in Hindi Class 12 Maths Exercise 2.2 in Hindi Class 12 Maths Miscellaneous 2 in Hindi

There are two exercises which explain ranges of trigonometric functions and observe the behaviour in the graphical representation. Study Material for Class 12 Maths Chapter 2 for all exercises are prepared by Mathematics teacher and solutions are well-structured so that it would be easier to learn and understand.

NCERT Solutions for Class 12 Maths Chapter 3 – Matrices

NCERT Solutions for Class 12 Maths Chapter 3 Matrices you can get the definition of Matrices, various types of matrices, equality of matrices, operations on matrices such as the addition of matrices and multiplication of matrices by a scalar.

Class 12 Maths Exercise 3.1 Class 12 Maths Exercise 3.2 Class 12 Maths Exercise 3.3 Class 12 Maths Exercise 3.4 Class 12 Maths Miscellaneous 3

Class 12 Maths Exercise 3.1 in Hindi Class 12 Maths Exercise 3.2 in Hindi Class 12 Maths Exercise 3.3 in Hindi Class 12 Maths Exercise 3.4 in Hindi Class 12 Maths Miscellaneous 3 in Hindi

You will learn here about the properties of matrices addition, properties of scalar multiplication, multiplication of matrices, properties of multiplication of matrices, transpose of matrices, properties of the transpose of the matrices, symmetric and elementary operation or transformation of a matrix, skew symmetric matrices, inverse of a matrix with by elementary operations, and at last, it covers miscellaneous examples. Matrices exercise answers of 12th Maths are given here in Hindi and English medium text and videos format.

NCERT Solutions for Class 12 Maths Chapter 4 – Determinants

NCERT Solutions for Class 12 Maths Chapter 4 Determinants covers all topic of determinants. You can learn and study here about the definition and meaning of determinants.

Class 12 Maths Exercise 4.1 Class 12 Maths Exercise 4.2 Class 12 Maths Exercise 4.3 Class 12 Maths Exercise 4.4 Class 12 Maths Exercise 4.5 Class 12 Maths Miscellaneous 4

Class 12 Maths Exercise 4.1 in Hindi Class 12 Maths Exercise 4.2 in Hindi Class 12 Maths Exercise 4.3 in Hindi Class 12 Maths Exercise 4.4 in Hindi Class 12 Maths Exercise 4.5 in Hindi Class 12 Maths Miscellaneous 4 in Hindi

There are six exercises in this chapter which explains remarks based on the order of determinants, properties of determinants, finding the area of a triangle using determinants, minors, adjoint, and cofactors of determinants, systems of linear equation in two as well as three variables using the inverse of matrices.

Class 12 Maths Chapter 5 – Continuity and Differentiability

Class 12 Maths Chapter 5 Continuity and Differentiability textbook has eight exercises which are based on 12th Class Maths Book Solution, in the 1st exercise you just learn about the definition of continuity.

Class 12 Maths Exercise 5.1 Class 12 Maths Exercise 5.2 Class 12 Maths Exercise 5.3 Class 12 Maths Exercise 5.4 Class 12 Maths Exercise 5.5 Class 12 Maths Exercise 5.6 Class 12 Maths Exercise 5.7 Class 12 Maths Miscellaneous 5

Class 12 Maths Exercise 5.1 in Hindi Class 12 Maths Exercise 5.2 in Hindi Class 12 Maths Exercise 5.3 in Hindi Class 12 Maths Exercise 5.4 in Hindi Class 12 Maths Exercise 5.5 in Hindi Class 12 Maths Exercise 5.6 in Hindi Class 12 Maths Exercise 5.7 in Hindi Class 12 Maths Miscellaneous 5 in Hindi

Then you proceed with continuity, algebra of continuous function, definition and meaning of differentiability, derivative of implicit functions, derivatives of composite functions, derivative of inverse trigonometric functions, logarithmic differentiation, Concept of exponential and logarithmic functions, derivatives of functions in parametric forms, exponential and logarithmic functions, second order derivatives, mean value theorem with different examples for practice in board preparation.

NCERT Solutions for Class 12 Maths Chapter 6 – Applications of Derivatives

NCERT Solutions for Class 12 Maths Chapter 6 Applications of Derivatives has five exercises which cover the definition of derivatives, tangents and normal, and rate of change of quantities, increasing and decreasing functions.

Class 12 Maths Exercise 6.1 Class 12 Maths Exercise 6.2 Class 12 Maths Exercise 6.3 Class 12 Maths Miscellaneous 6

Class 12 Maths Exercise 6.1 in Hindi Class 12 Maths Exercise 6.2 in Hindi Class 12 Maths Exercise 6.3 in Hindi Class 12 Maths Miscellaneous 6 in Hindi

It describes the use of derivatives in approximation maxima and minima, approximations, first derivative test, function in a closed interval and miscellaneous examples. Here in this NCERT Solutions for 12th Mathematics Chapter 6 you can get a detailed knowledge of Applications of Derivatives. Student can learn Chapter 6 in Hindi and English Medium as per their requirement.

NCERT Solutions for Class 12 Maths Chapter 7 – Integrals

Class 12 Maths Chapter 7 Integration has twelve exercises which cover the definition of integration, definite and indefinite integrations and limit of sums based questions.

Class 12 Maths Exercise 7.1 Class 12 Maths Exercise 7.2 Class 12 Maths Exercise 7.3 Class 12 Maths Exercise 7.4 Class 12 Maths Exercise 7.5 Class 12 Maths Exercise 7.6 Class 12 Maths Exercise 7.7 Class 12 Maths Exercise 7.8 Class 12 Maths Exercise 7.9 Class 12 Maths Exercise 7.10 Class 12 Maths Miscellaneous 7

In mathematics, integration is the process of finding the integral of a function. Integrals are used to calculate areas, volumes, and other quantities. The fundamental theorem of calculus states that integration and differentiation are inverse operations. This means that the integral of the derivative of a function is equal to the original function, plus a constant.

Class 12 Maths Chapter 7 in Hindi Medium Class 12 Maths Exercise 7.1 in Hindi Class 12 Maths Exercise 7.2 in Hindi Class 12 Maths Exercise 7.3 in Hindi Class 12 Maths Exercise 7.4 in Hindi Class 12 Maths Exercise 7.5 in Hindi Class 12 Maths Exercise 7.6 in Hindi Class 12 Maths Exercise 7.7 in Hindi Class 12 Maths Exercise 7.8 in Hindi Class 12 Maths Exercise 7.9 in Hindi Class 12 Maths Exercise 7.10 in Hindi Class 12 Maths Miscellaneous 7 in Hindi

NCERT Solutions for Class 12 Maths Chapter 8 – Applications of Integrals

In NCERT Solutions for Class 12 Maths Chapter 8 Applications of Integrals, students can get an overall knowledge about the Integrals and specific application of integrals.

Class 12 Maths Exercise 8.1 Class 12 Maths Miscellaneous 8

Class 12 Maths Exercise 8.1 in Hindi Class 12 Maths Miscellaneous 8 in Hindi

It covers the definition of Integrals, especially lines, circles/ parabolas/ellipses, area under simple curves, area between two curves and at last miscellaneous examples. There are two exercises in NCERT Solutions for Class 12 Maths Chapter 8. We have prepared the solutions based on 12th Class Maths NCERT Book part 2. Get Applications of Integrals Class 12 Maths Solutions in Hindi medium.

NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations

Students can find concentrate on the definition of Differential Equations in the NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations.

Class 12 Maths Exercise 9.1 Class 12 Maths Exercise 9.2 Class 12 Maths Exercise 9.3 Class 12 Maths Exercise 9.4 Class 12 Maths Exercise 9.5 Class 12 Maths Miscellaneous 9

Class 12 Maths Exercise 9.1 in Hindi Class 12 Maths Exercise 9.2 in Hindi Class 12 Maths Exercise 9.3 in Hindi Class 12 Maths Exercise 9.4 in Hindi Class 12 Maths Exercise 9.5 in Hindi Class 12 Maths Miscellaneous 9 in Hindi

It clears basic concepts related to differential equations and many more with step by step solutions so that students can easily understand. It covers all the solutions based on 12th Class Maths Book Solution including the degree of a differential equation, order of a differential equation, general and particular solutions of a differential equation, homogeneous differential equations, linear differential equations, procedure involved formation of a differential equation etc. Student can use Differential Equations of 12th standard in Hindi and English medium.

NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra, you can understand the introduction of vector algebra. And you can know how to find position vector, some basic concepts related to vector algebra etc.

Class 12 Maths Exercise 10.1 Class 12 Maths Exercise 10.2 Class 12 Maths Exercise 10.3 Class 12 Maths Exercise 10.4 Class 12 Maths Miscellaneous 10

Class 12 Maths Exercise 10.1 in Hindi Class 12 Maths Exercise 10.2 in Hindi Class 12 Maths Exercise 10.3 in Hindi Class 12 Maths Exercise 10.4 in Hindi Class 12 Maths Miscellaneous 10 in Hindi

In this chapter, you can also learn about algebraic as well as geometric properties. And multiplication of a vector by a scalar, r, vector joining two points, components of a vector, section formula, product of two vector etc. This chapter 10 has four exercises excluding miscellaneous. All solutions are based on 12th Class Maths Book Solution. Get the complete solutions with step by step description in Hindi and English medium.

NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry, you can get know about Three Dimensional Geometry. The student can constitute topics related to three dimensional geometry, direction cosines and direction ratios.

Class 12 Maths Exercise 11.1 Class 12 Maths Exercise 11.2 Class 12 Maths Miscellaneous 11

Class 12 Maths Exercise 11.1 in Hindi Class 12 Maths Exercise 11.2 in Hindi Class 12 Maths Miscellaneous 11 in Hindi

Learn the properties of line, equation of a line through a given point and parallel direction cosines of a line passing through two points, relation between the direction cosines of a line, equation of a line in space, plane passing through the intersection of two given planes, angle between two planes, co-planarity of two lines and many more. And last it contains miscellaneous examples. There are three exercises and a miscellaneous exercise in chapter 11 of 12th Mathematics NCERT.

NCERT Solutions for Class 12 Maths Chapter 12 – Linear Programming

You can get introduced with to NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming. Here this chapter covered linear programming problem and its mathematical formulation.

Class 12 Maths Exercise 12.1

Class 12 Maths Exercise 12.1 in Hindi

You will learn about the mathematical formulation of the problem, different types of all linear programming problems graphical feasible and infeasible method solution. NCERT Solutions for Class 12 Maths Chapter 12 covers two exercises in all. Take the Linear Programming of 12th Mathematics in Hindi and English medium here.

NCERT Solutions for Class 12 Maths Chapter 13 – Probability

In this last NCERT Solutions for Class 12 Maths Chapter 13 Probability, you can find deals with probability and Multiplication rule of Probability.

Class 12 Maths Exercise 13.1 Class 12 Maths Exercise 13.2 Class 12 Maths Exercise 13.3 Class 12 Maths Miscellaneous 13

Class 12 Maths Exercise 13.1 in Hindi Class 12 Maths Exercise 13.2 in Hindi Class 12 Maths Exercise 13.3 in Hindi Class 12 Maths Miscellaneous 13 in Hindi

You can understand here the concept of random variables and its probability in this chapter. Other than this chapter also covers mean of random variable, variance of random variable, and binomial distribution and Bernoulli trials with miscellaneous examples. NCERT Solutions for Class 12 Maths Chapter 13 describes about the five topics with well-defined solutions by the expert math teacher.

As Class 12 Maths is an important subject that plays a vital role in next classes and future career. Class 12 Maths NCERT Solutions acts as the best tool for students in their exam Class 12 boards and undergraduate competitive examination in future. These above NCERT Class 12 Maths solutions for all chapter exercises can be your best education support for board exam preparations.

Mathematics is a subject you cannot get good marks in the exam by reading and memorizing the formulae. So you need to practice well to succeed. Learning from Tiwari Academy’s NCERT solutions will help students of Class 12 to clear all the logic of every problem quite effectively.

Students can get many benefits if they learn and practice with Maths Class 12 NCERT Solutions from Tiwari Academy. Class 12th Maths plays an important role for the preparation of many undergraduate competition exams. So well described Maths solution at Tiwari Academy help you to clear all doubts and concepts of all chapters. It is the best study material to score more marks in your class 12 board exam and competitive exams in future.

There are total of 13 chapters which are given by the expert math teachers with extensive reviewed. By learning properly and practice this NCERT solution for Class 12 Maths , you will be top on 12th class board exam. Provides Authentic Information to the Students All Maths Class 12 NCERT Solutions are provided by the experienced Math after extensive research on each and every chapter to provide correct and authentic information to the students.

Follows the CBSE Curriculum

This is another most prominent reason that why you learn NCERT Solutions from us. These solutions for every chapter strictly adhere to the CBSE curriculum 2024-25 .

Clears all Fundamental Logic & Concepts

These NCERT solutions help the students to make their concepts crystal clear once you clear the concept then you no need to read the whole stuff again and again.

Helpful For Revisions during Examinations

These all solutions are designed in such a way that Students can revise quickly during their Examinations.

Save Your Preparation Time

The well-defined solution saves your revision time by providing all the necessary logic and formulae in a proper way.

How are CBSE 12 Maths Solutions of NCERT Helpful for Board Exams?

CBSE 12 Maths Solutions help you secure marks in the 12 class Board exam as efficiently as they clear your conceptual doubts. The class 12 Maths NCERT solutions help students solve all the exercises given in the textbooks prescribed by the NCERT, which are easy to understand the concepts and get good marks in their board exam.

If the students do practice and revise CBSE 12 Maths Solutions of NCERT in a proper way then you may surely passed your Class 12 Board Exams with above 90% marks in the subject Maths. It is seen that most of the student achieved a high score in math by learning CBSE 12 Maths Solutions of NCERT.

12th Maths Solutions are an invaluable resource for self-study. They allow students to revisit topics at their own pace, helping them reinforce their understanding and address any doubts. It serve as a reference for checking answers while doing homework or practice exercises. This helps students identify errors and rectify mistakes, contributing to accurate learning. With grade 12 Math solutions, students gain insights into the logic behind mathematical techniques rather than memorizing formulas. This clarity enhances their comprehension and critical thinking.

Many solutions include real-world applications of mathematical concepts, demonstrating their practical significance and fostering an appreciation for mathematics in daily life. Solutions aid teachers in explaining complex topics effectively, ensuring a consistent understanding among students.

Teachers can also use these solutions as teaching aids to supplement their lessons. A strong understanding of Class 12 Maths concepts is essential for pursuing higher education in fields like engineering, mathematics, economics, and sciences.

How to get class 12 Maths NCERT Solutions?

Get the revised and modified Class 12 Maths NCERT Solutions to prepare for board exams. As you know the syllabus is reduced for academic year 2024-25. So, follow the instruction to get appropriate NCERT textbook solution for final exams. If you still have doubts or questions, don’t hesitate to ask your teacher or a peer for clarification. Clearing up any confusion is essential to build a strong foundation in mathematics. Follow the steps given here to get Class 12 Maths solutions in Hindi and English Medium.

Step 1: Search the website for class 12 Maths NCERT Solutions.

Step 2: visit to class 12 maths solutions based educational websites like tiwari academy., step 3: use class 12 maths solutions app to get hindi and english medium content., step 4: prepare class 12 maths for board exams using ncert solutions., step 5: crosscheck the answers and solution using ncert solutions of class 12 maths..

Class 12 Maths textbook part 1 and part 2 solutions are available in various formats, including books, online platforms, and mobile apps. This accessibility ensures that students can choose the format that suits their learning preferences. Class 12 Maths solutions provide clarity, guidance, and practice for students facing intricate mathematical challenges.

Is NCERT Enough for Class 12 Maths Board Exams?

Yes, NCERT Class 12 Maths Books are enough to score maximum marks in the Class 12 Board exams. Just do practice all the exercise and learn the theorems. Also, do not skip any topic of CBSE 12th Maths syllabus and do revision 2-3 times before your class 12 board exam. The NCERT books for Class 12 are prescribed by the CBSE itself. So these NCERT books are known for their clear explanation of each topic and a good level of practice questions. It the best study material for Class 12 Board’s Maths. These NCERT textbooks are enough for you to score well in your CBSE Class 12 Maths board exam.

Which Book is Best for Class 12 Maths Board Exam?

NCERT Book is the best and most trusted book for Class 12 board exam. Students can clear their concepts and logics through NCERT Class 12 Maths book by solving the examples and all exercise problems. Students are recommended to solve the NCERT book as this book follows the CBSE syllabus and pattern.

Is Class 12 Maths Difficult?

No, Class 12 Maths is not difficult than class 11 in CBSE board. It is easier than the syllabus of class 11 Maths. By the practicing in perfectly you can score more marks in Class 12 Maths. The student who have cleared all concept of the early class 11 Maths syllabus, then no need to worry about the Class 12 Maths topic, they can easily understand all topics. If the student faces any doubt about the syllabus then they can learn and get solutions from Class 12 Maths NCERT solutions. It can help you to clear all doubts and concept of all topics.

Which website is best for Class 12 maths NCERT solutions?

You can find many websites online which are providing Maths NCERT solutions for class 12 . But tiwariacademy.com provides a solid conceptual base for all the topics included in CBSE Class 12 Maths Syllabus. All solutions cover all the important theorems and formulae with detailed explanations so that student can easily clear the concept. A detailed step-by-step explanations of problems given in the NCERT textbooks on tiwariacademy.com website. Apart from solutions for each chapter of CBSE Class 12 Maths are the best study materials for students who are preparing for their Class 12 board exams. From this website, you can get FREE Class 12 maths NCERT solutions with unlimited solution PDFs.

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Providing help in science for class 6 to 10. Adviser in Tiwari Academy for the science related subjects subject as well as videos contents. Ample teaching experience in schools. I am not only working for Tiwari Academy but also provide new ideas for the website and apps.

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NCERT Solutions Class 12 Maths

NCERT solutions for class 12 maths comprise all the answers to questions that are available in the NCERT class 12th textbook. These solutions can be used by kids as a reference if they get stuck while solving questions or can act as a source to clarify concepts. These NCERT solutions for class 12 maths are prepared with immense care by math experts to ensure that a student of any intelligence quota can easily understand the explanations.

Class 12 maths NCERT solutions are extremely helpful for students preparing for the 12th board exams as they give kids an idea of the type of questions that can appear in the examination. They are also very useful for competitive exams such as IIT-JEE, BITS Pilani, etc. CBSE NCERT solutions for class 12 maths help kids build an ironclad mathematical foundation that is also applicable to college-level questions.

NCERT Solutions Class 12 Maths PDF Download

The grades a student gets in the class 12 board exam act as a deciding factor for a child’s future and hence, following well-structured solutions is very important. The precise and brief explanation of class 12th NCERT solutions is available for free pdf download. The links are arranged according to chapters and are given below.

Class 12 Maths NCERT Solutions Chapter 1 to 13

By regularly practicing the solutions linked above a student can not only get an excellent score in the home and board exams but will also develop an in-depth knowledge of the subject that will be useful for him at the college level. Download the NCERT class 12 maths book given below and effectively prepare for any examination.

☛ Download Class 12 Maths NCERT Book

NCERT Solutions for Class 12 Maths - PART 1

Ncert solutions for class 12 maths chapter 1 - relations and functions.

NCERT solutions for class 12 maths relations and numbers give a detailed and simplified explanation of types of relations and functions, binary functions , and certain operations that can be performed on the elements of a function.

NCERT Solutions Class 12 Maths Chapter 1 Study Tips

NCERT solutions class 12 chapter 1 can prove to be very confusing thus, kids can follow the tips to help with the learning process.

• Try to visualize mapping questions.
• Solve numerous problems till you do not get confident with them.

Class 12 Maths Chapter 1 Relations and Functions

Topics Covered: Class 12 Maths NCERT Solutions talks about empty, universal, reflexive, transitive relations along with injective, bijective, and surjective mapping of functions. Apart from this, questions based on binary operations, invertible elements, and commutative operations are also included in this chapter.

Total Questions: Class 12 maths chapter 1 relations and functions have a total of 74 exemplar problems out of which 45 are challenging sums, 20 are easy and 9 are medium.

NCERT Solutions Class 12 Maths Chapter 2 - Inverse Trigonometric Functions

NCERT solutions for class 12 maths inverse trigonometric functions introduce students to the domain and graphical representation of inverse trigonometric functions . Kids have the opportunity to learn about the properties of these functions and solve questions using the given concepts.

Important Formulas for NCERT Solutions Class 12 Maths Chapter 2

• sin –1 (–x) = – sin –1 x
• tan –1 x + cot –1 x = π/2
• sin –1 x + cos –1 x = π/2
• cos –1 (–x) = π – cos –1 x
• cot –1 (–x) = π – cot –1 x

Class 12 Maths Chapter 2 Inverse Trigonometric Functions

Topics Covered: Identifying the domain and finding the value of inverse trigonometric functions at a particular point, as well as simplifying trigonometric expressions using the properties of inverse trigonometry are very important topics under Class 12 Maths NCERT Solutions

Total Questions: Class 12 maths chapter 2 inverse trigonometric functions has a total of 52 questions out of which 20 are simple formula-based sums, 18 are moderate and 14 are tough problems.

NCERT Solutions Class 12 Maths Chapter 3 - Matrices

NCERT solutions for class 12 maths matrices help students amass the tools required to solve problems based on the elements of matrices . This chapter includes the various categories of matrices, how to add , subtract and multiply matrices , along with applying elementary operations to get the inverse of a matrix.

Important Formulas for NCERT Solutions Class 12 Maths Chapter 1

• – A = (–1)A
• A – B = A + (–1) B
• A + B = B + A
• (A + B) + C = A + (B + C)
• k(A + B) = kA + kB

where A, B and C are matrices.

Class 12 Maths Chapter 3 Matrices

Topics Covered: Class 12 Maths NCERT Solutions covers topics such as types of matrices, finding unknown quantities using equivalent matrices, performing arithmetic operations on matrices, as well as how to transpose matrices and subsequently find the inverse using different methods. 

Total Questions: There are 78 questions in Class 12 maths chapter 3 matrices. 36 are complex sums, 14 are easy and 28 are intermediate problems. 

NCERT Solutions Class 12 Maths Chapter 4 - Determinants

NCERT solutions for class 12 maths determinants clearly explain the concept of the determinant of a square matrix and how to find it. Properties of determinants, minors, cofactors, the inverse of a matrix , and linear equations are crucial sub-topics that kids need to have a good understanding of.

Class 12 Maths Chapter 4 Determinants

Topics Covered: Evaluating the determinants, proving expressions using the properties of determinants, finding the inverse of a matrix using minors, cofactors , and the adjoint of that matrix as well as finding the solution of a linear system of equations using matrices are topics covered under Class 12 Maths NCERT Solutions that are important for board exams.

Total Questions: Class 12 maths chapter 4 determinants have a pool of 87 fantastic questions. 17 are simple property-based questions, 30 are moderately difficult, and 40 conceptual sums meant to challenge a student’s thinking.

NCERT Solutions Class 12 Maths Chapter 4 Summary

The NCERT solutions class 12 maths chapter 4 builds on the previous chapter of matrices to include concepts of determinants, the inverse of a matrix, and other elements. The lesson then teaches children how to apply these properties learned to real-life examples.

NCERT Solutions Class 12 Maths Chapter 5 - Continuity and Differentiability

NCERT solutions for class 12 maths continuity and differentiability consist of well-structured explanations pertaining to establishing the continuity or differentiability of a particular function at a point. This topic is very important as it is required to improve a child’s understanding of calculus .

Important Formulas for NCERT Solutions Class 12 Maths Chapter 5

• Chain Rule: df/dv = (dt/dx) * (dt/dx)
• d/dx (sin -1 x) = 1/√(1 - x 2 )
• d/dx (cos -1 x) = - 1/√(1 - x 2 )
• d/dx (tan -1 x) = 1/(1 + x 2 )
• d/dx (cot -1 x) = 1/(1 + x 2 )
• d/dx (sec -1 x) = 1/x √(1 - x 2 )

Class 12 Maths Chapter 5 Continuity and Differentiability

Topics Covered: Proving the continuity of a function, and performing algebraic operations on continuous functions, are topics covered under Class 12 Maths NCERT Solutions. Other subject matter included that is essential for exams are finding the derivatives of composite, implicit, trigonometric, exponential, and logarithmic functions , differentiation of functions in parametric form, second-order derivatives, mean value theorem as well as the chain rule for differentiation.

Total Questions: Class 12 maths chapter 5 continuity and differentiability has a total of 144 questions. 80 are easy formula-based sums, 35 are slightly more complicated, and 29 are difficult problems that require great use of a student’s acumen.

NCERT Solutions Class 12 Maths Chapter 6 - Application of Derivatives

NCERT solutions for class 12 maths applications of derivatives is a chapter that introduces the real-life utilization of derivatives in judging the rate of change of quantities, verifying the nature of a function, finding tangents to a function, and various other associated concepts.

Class 12 Maths Chapter 6 Application of Derivatives

Topics Covered: Class 12 Maths NCERT Solutions encapsulates topics such as finding the rate of change of a particular quantity, determining if a function is increasing or decreasing, detecting the slope of a tangent, and using differentiation to approximate a number. Additionally, this lesson also includes problems on finding the maxima and minima of a function along with establishing the results of the first and second derivative test.

Total Questions: There are 126 well-researched questions in the Class 12 maths chapter 6 application of derivatives. Out of these 40 are short answer type simple problems, 75 are of moderate complexity and 11 are tough sums.

NCERT Solutions for Class 12 Maths - PART 2

Ncert solutions class 12 maths chapter 7 - integrals.

NCERT solutions for class 12 maths integrals introduce students to the concept of the anti-derivative also known as an integral. Kids have the opportunity to learn about the geometric representation of integrals, and how to use various techniques with formulas to perform the integration of functions. In addition to this, children also study definite integrals.

Class 12 Maths Chapter 7 Integrals

Topics Covered: Class 12 Maths NCERT Solutions comprise topics such as finding the integral of a function using integration by substitution, partial fractions , parts, and by using trigonometric identities. Definite integrals, their limits, related properties, and the fundamental theorem of calculus are also part of this lesson.

Total Questions: Class 12 maths chapter 7 integrals has a pool of 271 questions. 115 are formula-based sums of a moderate level, 40 are easy and 116 are tough problems based on the mathematical skills of a student.

NCERT Solutions Class 12 Maths Chapter 7 Summary

The NCERT solutions class 12 maths chapter 7 gives a holistic view of integrals to students. There are several concepts such as integration by different methods, using integral formulas , and definite integrals that are very important from a board examination point of view. Thus, kids are advised to go through all concepts for a thorough understanding of integrals .

NCERT Solutions Class 12 Maths Chapter 8 - Application of Integrals

NCERT solutions for class 12 maths application of integrals aids students in understanding the practical use of integrals such as finding the area under a curve, or the area between a curve and a line segment. This chapter gives students a precise and comprehensive cognizance of calculus.

Class 12 Maths Chapter 8 Application of Integrals

Topics Covered: Class 12 Maths NCERT Solutions explains in detail topics such as area bound by the curve of a function, area between two curves , and other associated miscellaneous examples.

Total Questions: Class 12 maths chapter 8 application of integrals has 39 problems. 15 are easy, 8 are of intermediate level and 16 are tough sums.

NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

NCERT solutions for class 12 maths differential equations help kids build crystal clear concepts of how to work with differential equations. Students learn about the order, degree , and solutions of a differential equation , as well as the procedure of forming such an equation.

NCERT Solutions Class 12 Maths Chapter 9 Study Tips

NCERT solutions class 12 maths chapter 9 can help kids combat confusion and strive to clear concepts of calculus. Students can follow the study tips mentioned below to streamline their learning process.

• Ensure that you are well versed with the previous chapters before studying differential equations.
• Learn formulas and make notes.
• Practice regularly.

Class 12 Maths Chapter 9 Differential Equations

Topics Covered: Determining the order and degree of an equation, verifying the solution of a differential equation, and forming such an equation from the general solution, are covered under the Class 12 Maths NCERT Solutions. Solving a homogeneous and linear differential equation are other topics in this lesson that are equally important to learn. 

Total Questions: There are 113 sums in Class 12 maths chapter 9 differential equations of a wide-ranging variety. 72 are tough problems, 15 are simple and 26 are of a medium level.

NCERT Solutions Class 12 Maths Chapter 10 - Vector Algebra

NCERT solutions for class 12 maths vector algebra have a step-wise elucidation of problems based on vectors and their algebra . Students study how to perform dot and cross products, arithmetic vector operations, along with other vital topics that are important not only for Mathematics but also Physics.

Class 12 Maths Chapter 10 Vector Algebra

Topics Covered: Addition, multiplication, and laws of vector operations, section formula , computing the magnitude of vectors , vector projections, as well as dot and cross product, are topics that form the building blocks of Class 12 Maths NCERT Solutions.

Total Questions: Class 12 maths chapter 10 vector algebra has 63 sums divided into 20 straightforward questions, 37 intermediate, and 6 complicated problems.

NCERT Solutions Class 12 Maths Chapter 11 - Three Dimensional Geometry

NCERT solutions for class 12 maths three dimensional geometry builds on previous lessons for introducing kids to direction cosines, equations and other geometrical aspects related to a line. This chapter also helps to clarify concepts of a plane

Important Formulas for NCERT Solutions Class 12 Maths Chapter 11

• Direction cosines of a line joining two points A and B:

(x 2 - x 1 )/AB,(y 2 - y 1 )/AB,(z 2 - z 1 )/AB

where AB = √[(x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 + (z 2 - z 1 ) 2 ]

• cosθ = |[a 1 a 2 + b 1 b 2 + c 1 c 2 ] / √(a 1 + b 1 + c 1 ) 2 √(a 2 + b 2 + c 2 ) 2 ]|

where a 1 , b 1 , c 1 and a 2 , b 2 , c 2 are direction ratios and θ is the acute angle between the two lines.

Class 12 Maths Chapter 11 Three Dimensional Geometry

Topics Covered: Class 12 Maths NCERT Solutions includes topics pertaining to finding the direction cosines of a line , determining the distance and angle between two lines , as well as writing the equation of a plane when different constraints are given. Moreover, the distance and angle between two planes together with the coplanarity of two lines are also covered in this chapter.

Total Questions: Class 12 maths chapter 11 three dimensional geometry has 59 fantastic sums that can be split into 34 easy formula-based problems, 12 complex, and 13 moderate level questions.

NCERT Solutions Class 12 Maths Chapter 12 - Linear Programming

NCERT solutions for class 12 maths linear programming is one of the most essential chapters that is widely used in industrial applications. Students learn how to mathematically formulate a linear programming problem, and its real-life applications.

Class 12 Maths Chapter 12 Linear Programming

Topics Covered: Solving linear programming problems graphically, maximizing and minimizing a given equation, as well as using these concepts for various applications such as diet and manufacturing questions fall under the scope of Class 12 Maths NCERT Solutions.

Total Questions: There are 31 questions in Class 12 maths chapter 12 that are necessary for board examinations. Out of these 14 are simple, 10 are slightly tougher and 7 are higher-order thinking questions.

NCERT Solutions Class 12 Maths Chapter 12 Summary

NCERT solutions class 12 maths chapter 12 give various methods to solve and analyze linear equations. Problems related to maximizing or minimizing an equation and then marking the required area on a graph helps students to understand the practical implications of that question. Children also grasp the various techniques that are used to solve a linear equation.

NCERT Solutions Class 12 Maths Chapter 13 - Probability

NCERT solutions for class 12 maths probability enhances a student’s skills to solve probability sums by giving a well-detailed explanation of conditional probability , probability distribution, and many other concepts that are required by kids to successfully attempt their test paper.

Class 12 Maths Chapter 13 Probability

Topics Covered: Class 12 Maths NCERT Solutions encompass topics like conditional probability and its properties, applying the multiplication theorem on probability to sums, independent events, Bayes’ theorem , probability distribution, mean and variance of a random variable, Bernoulli trials, as well as binomial distribution . 

Total Questions: Class 12 maths chapter 13 probability has 100 well-curated questions. 23 are simple sums based on formulas , 37 are moderate and 40 are tricky problems.

Important Formulas in NCERT Solutions Class 12 Maths

Formulas are a generalized representation of the relationship between two or more given quantities. It includes various theorems and expressions used to deliver information in a concise manner. They can be seen as vital keys that are required to solve questions. Some of the important formulas covered in NCERT solutions for class 12 maths that students must have knowledge of are given below.

Inverse Trigonometry

• sin -1 (1/x) = cosec -1 x
• cos -1 (1/x) = sec -1 x
• sin -1 (-x) = -sin -1 x
• cos -1 (-x) = π - cos -1 x
• sin -1 x + cos -1 x = π/2

Derivatives

• d/dx (e x ) = e x

Importance of NCERT Solutions Class 12 Maths

Class 12 Maths NCERT Solutions are extremely helpful for CBSE students because they provide a reliable way for kids to cross-check their answers and clear doubts. The following are the advantages of the NCERT solutions.

• Board exam preparation - While appearing for board exams it becomes essential for students to write their papers in a format that can be easily comprehended by the people who correct them. The NCERT solutions have a clear and step-wise explanation that gives kids an insight into how to solve any sum. As a certain set of marks are allotted to steps, hence if children follow the pattern of these NCERT solutions they are sure to get the best grade possible.
• Conceptual Understanding - The NCERT Class 12 Maths Solutions help students to clear their conceptual doubts. The topics in class 12 are not only important for passing exams but also lay the foundation for studies in college. Therefore, it is imperative that kids refer to these NCERT solutions when they hit roadblocks so as to develop the correct understanding of these topics.
• Revision - During the examination days, children have a very cramped up studying schedule and cannot solve all coursebook questions. In such a situation, the NCERT Class 12 Maths Solutions helps to give the students a quick overview of how to solve different kinds of problems and can help in saving a lot of time.

FAQs on NCERT Solutions for Class 12 Maths

Why are ncert solutions for class 12 maths important.

Getting a good score in class 12th is crucial as this can be the deciding factor for a child’s career. Hence, it is important to consult the NCERT solutions class 12 maths as they give the most reliable, and to-the-point explanation of a question ensuring that kids do not have any doubts regarding a topic.

How CBSE Students can utilize Class 12 Maths NCERT Solutions effectively?

Before attempting any problem kids must be well versed with the theory and the formulas. They can then use the class 12 maths NCERT solutions to strengthen their understanding of that topic. Students must also structure their lesson plans in such a way that they can revise the entire syllabus at least twice so as to get the maximum benefit from these solutions.

Do I Need to Make Notes while Referring to NCERT Solutions Class 12 Maths?

When students solve a particular question they might come up with certain pointers detailed in notes that can supplement their learnings. Thus, it is always advised to make well-written notes for NCERT solutions class 12 maths to aid kids in recalling the concepts given in that specific sum. 

Do I Need to Practice all Questions Provided in NCERT Solutions Class 12 Maths?

Constant practice and due diligence ensure success. Thus, kids must make it a point to practice all questions in the NCERT solutions class 12 maths. This not only helps them to build a robust mathematical foundation but also enables them to gain confidence and attempt questions with ease.

Where Can I Get Chapter-wise NCERT Solutions for Class 12 Maths?

The links provided above give students chapter-wise access to NCERT solutions for class 12 maths. Moreover, kids can also find the solutions to all the exercises that have been given by experts with certain tips and tricks to score the highest marks possible.

Why Should I Practice Class 12 Maths NCERT Solutions Regularly?

In order to master any subject, practice is key. Regularly revising NCERT solutions class 12 maths helps students to get a clearer perspective of a topic enabling them to apply concepts proficiently. As students will be well-versed with all the chapters they can maximize the probability of getting an excellent score in their board examination.

What are the Important Topics Covered in NCERT Solutions Class 12 Maths?

All topics such as probability, linear programming, application of derivatives, etc. given in the NCERT solutions class 12 maths are equally important for board examinations. The syllabus of these exams is based on the NCERT curriculum. Thus, it is vital for students to study each and every topic with immense care so as to successfully solve all problems given in the test papers.

Are NCERT Solutions Class 12 Maths Sufficient for CBSE Boards Exams?

NCERT solutions class 12 maths are sufficient for board exams as the paper pattern follows the structure set by these textbooks. However, it is best to consult more subject material based on the same topics to have a broader scope of learning. It is always better to be over-prepared as opposed to being under-prepared.

Why Choose Cuemath for Class 12 Maths NCERT Solutions?

Cuemath is the forerunner for class 12 maths NCERT solutions. These solutions are prepared by math geniuses who are IIT graduates. The methods used to solve questions are brief, accurate, and simple to understand, therefore, helping students to combat any doubts or confusions.

What are the Benefits of Class 12 Maths NCERT Solutions for CBSE Students?

The class 12 maths NCERT solutions give students an insight into how to give a step-by-step detailed solution while attempting a problem. This ensures that kids have written their examination in a clear manner thus, improving their chances of getting better marks.

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NCERT Solutions for Class 12 Maths Exercises

• NCERT Solutions

NCERT Solutions for Class 12 Maths Exercises - PDF Solutions

Class 12 Maths syllabus is covering basic and advanced mathematical concepts and fundamental principles. To study this CBSE syllabus , you will need the assistance of the complete study material with all the exercise solutions. The experts of Vedantu have designed the NCERT Class 12 Maths All Exercises Solutions to download and refer to while studying the respective chapters. Follow how the experts have solved the questions and develop your mathematical skills accordingly.

NCERT Solutions for Class 12 Maths Chapter 1

NCERT Solutions for Class 12 Maths Chapter 2

Ncert solutions for class 12 maths chapter 3, ncert solutions for class 12 maths chapter 4, ncert solutions for class 12 maths chapter 5, ncert solutions for class 12 maths chapter 6, ncert solutions for class 12 maths chapter 7, ncert solutions for class 12 maths chapter 8, ncert solutions for class 12 maths chapter 9, ncert solutions for class 12 maths chapter 10, ncert solutions for class 12 maths chapter 11, ncert solutions for class 12 maths chapter 12, ncert solutions for class 12 maths chapter 13.

What could be better if you get the simpler and readymade solutions of your Class 12 NCERT textbook exercises? Moreover, all these are created by an experienced subject matter expert. Can you imagine how beneficial these could be? Sounds amazing, right! You must opt for these NCERT solutions for Class 12 Maths right away. You can download NCERT solutions Class 12 Maths from Vedantu official website (vedantu.com) by registering yourself with your phone number/ email address. You can also access the same study materials from Vedantu app as well. In that case, you need to download the Vedantu app first from Google play store.

You will get Class 12 Maths NCERT solutions written in a simple and lucid manner. Solve all the problems/ questions given in the exercises and evaluate your performance with these 100% accurate solutions so that you can learn how to write answers to the questions from any perspective during the final exam. Each of the answers in the Maths NCERT solutions Class 12 is student-friendly and customized by the experts. 

These NCERT solutions Class 12 Maths contain step-by-step answers to all the questions from the exercise which are considered as an invaluable aid to the students when it comes to regular study, exam preparation or doing homework. Class 12 Maths NCERT solutions will clear all your doubts in very less time. Hence your confidence will be boosted. By practising the questions from these books, you will be able to improve your time management skills. That will help you in solving all the questions in the stipulated time frame during the final exam. 

NCERT Solutions for Class 12 Maths are designed as per the latest CBSE curriculum and guidelines. Vedantu’s study materials are updated regularly as per the latest changes. Hence students of Class 12 who are referring to these materials will be able to score the best marks in the final exam. Once you start using these materials, the preparation for Class 12 exam and scoring the highest percentile amongst your peers will seem like a cakewalk at the end of the day.

Class 12 Maths Chapter 1 Relations and Functions

Chapter 1 Relations and Functions of NCERT Class 12 Maths Syllabus deals with types of relations and functions along with a few miscellaneous examples.

NCERT Solutions for Class 12 Maths Chapter 1 Exercises are given in the table below:

Class 12 Maths Chapter 2 Inverse Trigonometric Functions

In CBSE Class 12 Inverse Trigonometric Functions , there are only two exercises where you will study the restrictions on domains and ranges of trigonometric functions which enables the visibility of their inverses and observe their behaviour through graphical representations.

CBSE Class 12 Maths Chapter 3 Matrices

There are 4 exercises included in this chapter. The links to these exercises are provided in the table below.

CBSE Class 12 Maths Chapter 4 Determinants

There are six exercises which are included in the NCERT Class 12 Maths Chapter 4 Determinants. By solving these exercises, students will be familiar with the concepts of determination, cofactors and the application of determinants in finding the area of a triangle, minors, adjoint and inverse of a square matrix. 

The free PDFs of the NCERT Solutions for Class 12 Maths Chapter 4 exercises can be downloaded from the links provided below:

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

Class 12 Maths Chapter 5 Continuity and Differentiability explains the definition of continuity and meaning of differentiability.

The free PDFs of the NCERT Solutions for Class 12 Maths Chapter 5 exercises can be downloaded from the links provided below:

NCERT Solutions for Class 12 Maths Chapter 6 Application of Derivatives

In this chapter, you will study the applications of the derivative in various disciplines such as engineering, science, and many other fields. For instance, you will learn how the derivative can be used to determine the rate of change of quantities and to find the equations of tangent and normal to a curve at a point. You will also learn the use of derivatives to find intervals of increasing or decreasing functions, maxima and minima, along with a few simple problems based on all these concepts. 

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

Topics in this chapter include an introduction to indefinite integrals, integration as inverse to differentiation, its geometrical interpretation, and properties. 

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

Here, in this chapter, you will study some specific applications of integrals to find the area given under the simple curves, also, the area between lines and arcs of circles, parabolas and ellipses (standard forms only). There are two exercises in this chapter in which you will also deal with finding the area bounded by the above-said curves.

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations

In Differential Equations, you will be studying some basic concepts related to the differential equation, general and particular solutions of a differential equation, formation of differential equations, number of methods to solve a question based on first order - first-degree differential equations and some applications of differential equations in different areas.

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

In Class 12 Maths Chapter 10  Vector Algebra, you will be studying some basic concepts of vectors, various operations on vectors, and the algebraic as well as geometric properties. You will also learn about the types of vectors such as zero, unit, equal and collinear vectors.

Other topics discussed in this chapter include the negative of a vector, operations of vectors such as addition, its properties, components and how to multiply a vector by a scalar. You will also learn about section formulae, vector joining two points, dot product of two vectors, product of two vectors, cross product of two vectors, and projection of a vector on a line.

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

CBSE Class 12 Maths Chapter 11 Three Dimensional Geometry deals with the direction cosines and direction ratios of a line joining two points and also about the equations of lines and planes in space under different conditions. Along with that, you will also get to know about the angle between two lines, a line and a plane, two planes, the shortest distance between two skew lines and the distance of a point from a plane.

You will also learn about cartesian equations and vector equations of a line, skew and coplanar lines, and the cartesian and vector equations of a plane.

NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming

This chapter has only two exercises in which you will study some linear programming problems and their solutions by graphical method. Other concepts in this chapter are the mathematical formulation of linear programming problems, different types of linear programming problems, terms such as objective function, optimization and constraints, as well as feasible and infeasible regions and solutions.

NCERT Solutions for Class 12 Maths Chapter 13 Probability

Class 12 Maths Chapter 13 Probability will teach you the important concepts of the conditional probability of an event given that another event has occurred, which will, in turn, clarify the Bayes' theorem, multiplication rule of probability and independence of events. You will also learn the concept of a random variable and its probability distribution, mean and variance of a probability distribution, total probability theorem, Bernoulli trials and binomial distribution.

CBSE Class 12 Maths Weightage (2022-23)

Given below is the unit-wise mark distribution for CBSE Class 12 Maths exam for session 2022-23.

Importance of Chapter Wise List of Class 12 Maths All Exercises

The vivid exercises of a chapter in Class 12 Maths have questions that check your preparation level. This is where you will need the support of exercise solutions to learn the exact ways of answering fundamental questions.

The experts have framed the answers by following the fundamental principles explained in the chapters and the latest CBSE guidelines . The 12th Maths All Exercises Book Answers will enable students to focus on their preparation and proceed with the exam syllabus faster.

Benefits of NCERT Maths All Exercises Book Class 12 Solutions PDF

Resolve doubts on your own with these solutions faster.

Learn the ideal answering formats to follow and solve problems accurately.

Prepare chapters in no time by using the solutions as revision material.

Understand the concepts and principles of Class 12 Maths chapters from the simpler solutions easily.

Here is the list of exercise solutions for all chapters in the CBSE Class 12 Maths syllabus .

Download NCERT Maths All Exercises Book Class 12 Solutions PDF

Perform the NCERT Maths All Exercises Book Class 12 Solutions PDF free download according to your syllabus. Focus on how the experts have solved the exercise problems. Cover a syllabus faster and develop your problem-solving skills in a better way to score more in the exams.

Important Related Links for CBSE Class 12 Maths

NCERT Solutions for Class 12 Math in Hindi Chapter wise free pdf 

Important Questions for CBSE Class 12 Math

RS Aggarwal Class 12 Solutions for Math book

FAQs on NCERT Solutions for Class 12 Maths Exercises

1. How can I prepare a Class 12 Maths chapter?

Complete studying the theoretical part of a chapter and solve the exercises. Refer to the solutions to compare your answers and find out where you need to change your techniques.

2. What is the advantage of using Class 12 Maths solutions?

You will the precise answers compiled following the latest CBSE guidelines to follow and develop your answering skills.

3. Can I practise using the CBSE Class 12 Maths solutions?

The solutions are to the point and comply with the NCERT guidelines. You can surely practise solving problems using the solutions.

4. How can I resolve doubts about Class 12 Maths exercises?

Refer to the exercise solutions framed by the experts to resolve all your doubts in no time.

5. What is the best way to develop my concepts for the Class 12 Maths chapters?

Apart from the notes, you can use the simpler solutions developed by the experts to learn how to use the concepts and principles of an exercise in a Class 12 Maths chapter. This is how you can develop your concepts.

NCERT Solutions for Class 12 Maths

Ncert solutions for class 12 maths – free pdf solutions.

Maths NCERT Solutions for Class 12 acts as a vital tool when students are preparing for their Board examinations and even while solving the home work or any assignments. It also helps students to crack any competitive exam such as IIT JEE, AIIMS, NDA, UPSC, NEET, BITSAT, VITEEE or any government exam. In short, any undergraduate engineering entrance exam or government exam requires NCERT solutions since they are your best ally and foot forward towards your exam preparations. Scoring in Maths can be very difficult process as it is not plain wrote-learning but practical application of the conceptual knowledge you have learned. This can be only done by mere practice and hardwork.

Class 12 NCERT Maths Solutions prepare a strong foundation of conceptual learning of all the chapters and topics covered in 12 th standard portion as prescribed by CBSE. The solution PDF compromises of important formulas, theorems and equations. Each point is further explained with detailed explanations so that students can grasp it easily and understand the concepts efficiently. Mathongo provides detailed step by step explanation of the problems. They are free download and in PDF format. They are designed by the best professors and teachers to ensure students get good marks in their examination.

The maths solutions of CBSE NCERT Class 12 gives in-depth clarity answers to all the questions asked in the NCERT textbook. All the topics of NCERT are equally important and must be studied thoroughly to attain good marks. Students should study chapters like probability, calculus, coordinate geometry, trigonometry with consistence as they can become very easy to solve with proper practice and then these units will help in fetching high marks.

NCERT Class 12 Maths Chapterwise Solutions

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Application of Derivatives
• Chapter 7 Integrals
• Chapter 8 Applications of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three-Dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability

Why Go for Mathongo’s NCERT Maths Solutions for Class 12

Mathongo provides NCERT class 12 maths solution at free of cost online and it’s easily accessible in the form of PDF. Each PDF constitutes of the chapter in which exercise wise solutions are given. It is very easy for the students to locate any answer from the exercise.

All the solutions are made with the help of the best teachers and academicians. The solutions are plain easy and can be understood by anyone. The approaches and the methods used to solve the questions are also given in step by step method so that students score good in paper. For additional conceptual clarity, diagrams and formulas are also used.

Mathongo has provided the best solutions for class 12 maths Ncert textbooks. Students can refer these solutions which will help them in solving the previous year question papers. These NCERT maths class 12 solution will help the students to study more effectively and also present the answers in the most effective way to get the highest scores.

The back-end exercise questions given in NCERT books are very important for the class 12 finals. These exercises often test the conceptual clarity and the subject knowledge of the student. NCERT Solutions will help you to understand these concepts better. On the other hand, the students who have problem in solving maths and require assistance and guidance can also refer class 12 Ncert Maths Solutions to clarify the complex questions concepts.

Maths is all about practice. The more you practice, the better you will be. Ncert maths class 12 pdf download can help students to solve all their problems and also save their time as everything is at one place, one click away.

Which book is best for class 12 Maths?

The best book for not only class 12 Mathematics but for any class is the NCERT textbook which is prescribed by CBSE. The conceptual base required to solve HOTs (High Order Thinking Problems) is built by practicing NCERT maths book examples and exercise questions. Once the students have studied thoroughly the NCERT textbook, the students can through books like RD Sharma and RS Aggarwal for further practice. However, students must not underestimate the power of solving NCERT since majority of the questions in the boards come either directly from NCERT or are NCERT based.

Is NCERT enough for Class 12 board Maths?

NCERT textbook is well enough for class 12 board examinations for Maths. Solving questions from NCERT textbook helps student to grasp concepts easily. If they wish to practice more as Maths is all about practice, they can use other reference books or even solve question banks and past year question papers.

Is class 12 NCERT Maths difficult?

The idea of the subject being difficult or easy lies in the mind of the beholder. Some students since childhood have Maths phobia engraved in their minds. But I would like to clear that. NCERT Maths of class 12 is not difficult. Once you get the conceptual clarity, you can start solving solved examples and problems followed by exercise questions. The only way you can master Maths is by practicing variety of questions.

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NCERT Solutions for Class 12 Maths

Table of Contents

NCERT Solutions for Class 12 Maths offer a comprehensive grasp of the core concepts covered in the CBSE Class 12 Maths Syllabus. Encompassing vital theorems and formulas, these solutions provide thorough elucidations, fostering a deeper comprehension among students. For those seeking Class 12 Maths NCERT Solutions , this resource proves to be the ultimate destination. The provided solutions are concise, accurate, and meticulously error-free. Aligned with the esteemed reputation of NCERT textbooks , these solutions maintain the same level of excellence.

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Delivered by Infinity Learn, the NCERT Solutions for Class 12 Maths are meticulously crafted by subject experts, catering to a seamless and accurate comprehension of subjects. These solutions entail meticulous, step-by-step breakdowns of problems sourced from the textbooks. Beyond chapter-wise solutions for CBSE Class 12 Maths , Infinity Learn also extends a trove of resources including comprehensive Class 12 Maths notes, meticulously solved exemplar problems, prior year question papers, and supplementary worksheets . These resources collectively serve as indispensable study aids, catering to students gearing up for their imminent Class 12 board examinations.

Also Check: CBSE Class 12 Maths Syllabus 2023-24

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Applications of Derivatives
• Chapter 7 Integrals
• Chapter 8 Applications of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three-dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability

It may be difficult for pupils to understand what is taught in class. Students should look up information online to clear their doubts about dealing with this issue. Regular practice of concepts in which they are weak will enhance their confidence in terms of the exam. Students should study the Chapter from the NCERT textbook and answer the exercise problems to gain a sound conceptual understanding. Students’ problem-solving and analytical thinking skills will be improved if they use the NCERT Solutions available at Infinity Learn to react to the challenges.

Latest: CBSE Class 12 Maths Answer Key 2024 | CBSE Class 12 Maths Paper Analysis 2024

NCERT Solutions Class 12 Maths – Chapter Wise Detail

It is important to study NCERT Solutions Class 12 Maths for board exams and competitive examinations. According to the CBSE syllabus and test structure, the solutions are presented in simple language to aid students. Every Chapter in the most recent NCERT textbook is covered in the solutions. The solutions explain various fundamental concepts to provide accurate information to the students.

NCERT Solutions for Class 12 Maths Chapter 1

This chapter delves into the concepts of relations and functions, exploring their properties and classifications. It discusses various types of functions, their domains, and ranges. The solutions cover topics like one-to-one and onto functions, composition of functions, and inverse functions.

NCERT Solutions for Class 12 Maths Chapter 2

This chapter focuses on the inverse trigonometric functions and their properties. It covers topics like domain, range, graphs, and properties of these functions. The solutions help students understand how to solve equations involving inverse trigonometric functions.

NCERT Solutions for Class 12 Maths Chapter 3

Matrices are the central theme of this chapter. Students will learn about matrix operations, types of matrices, and their properties. The solutions cover concepts like matrix multiplication, determinants, and the inverse of a matrix.

NCERT Solutions for Class 12 Maths Chapter 4

This chapter delves deeper into the concept of determinants, exploring their properties and uses in solving linear equations. The solutions guide students through calculating determinants, using properties to simplify calculations, and solving problems related to them.

NCERT Solutions for Class 12 Maths Chapter 5

The concepts of continuity and differentiability of functions are explored in this chapter. The solutions cover topics like the definition of continuity, differentiability, and methods to determine whether a function is continuous or differentiable.

NCERT Solutions for Class 12 Maths Chapter 6

This chapter delves into the applications of derivatives in various fields. It covers concepts like rate of change, increasing and decreasing functions, maxima and minima, and related rates problems. The solutions provide step-by-step guidance on solving real-world problems using derivatives.

NCERT Solutions for Class 12 Maths Chapter 7

Integrals and their properties are the focus of this chapter. Students will learn about indefinite and definite integrals, methods of integration, and applications of integrals in finding areas and volumes. The solutions help in understanding integration techniques and their applications.

NCERT Solutions for Class 12 Maths Chapter 8

Building on the previous chapter, this one focuses on the applications of integrals in practical scenarios. It covers topics like area under curves, area between curves, and applications in finding volumes and lengths of curves. The solutions guide students through solving problems related to these applications.

NCERT Solutions for Class 12 Maths Chapter 9

This chapter introduces the basics of differential equations and their classifications. It covers concepts like order and degree of differential equations, solutions of differential equations, and their applications. The solutions help students understand the methods to solve different types of differential equations.

NCERT Solutions for Class 12 Maths Chapter 10

Vector algebra is the main theme of this chapter. It covers concepts like vectors in space, scalar and vector products, and the geometrical interpretation of vectors. The solutions provide clarity on vector operations and their geometric interpretations.

NCERT Solutions for Class 12 Maths Chapter 11

This chapter explores three-dimensional space and geometry. It covers topics like direction cosines, direction ratios, equations of lines and planes in space, and the distance between points and lines. The solutions help students grasp the spatial concepts and solve problems related to them.

NCERT Solutions for Class 12 Maths Chapter 12

This chapter introduces linear programming as a mathematical method to optimize solutions. It covers topics like linear inequalities, graphical and algebraic methods of solving linear programming problems, and applications in real-life scenarios. The solutions guide students through solving linear programming problems.

NCERT Solutions for Class 12 Maths Chapter 13

Probability theory is the focus of this chapter. It covers concepts like probability distribution functions, mean and variance of a random variable, and various probability distributions like binomial, Poisson, and normal distributions. The solutions help students understand probability concepts and solve problems involving probability distributions.

Key Features of NCERT Solutions for Class 12 Maths

• Comprehensive Coverage: NCERT Solutions comprehensively cover all the chapters and topics included in the Class 12 Maths syllabus, ensuring that students have access to solutions for every concept.
• Detailed Explanations: Each solution is accompanied by step-by-step explanations, making it easier for students to understand the problem-solving process. This aids in building a strong conceptual foundation.
• Accuracy and Clarity: The solutions provided are accurate and presented in a clear and concise manner. This clarity helps students to follow the solution methodology without confusion.
• Variety of Exercises: The solutions cover a wide range of exercises, including solved examples, practice problems, and additional questions from each chapter. This variety assists students in practicing different types of questions.
• Problem-Solving Approach: The solutions emphasize problem-solving techniques and strategies, guiding students on how to approach and tackle various types of mathematical problems effectively.
• Conceptual Understanding: The solutions not only provide answers but also focus on explaining the underlying concepts. This helps students develop a deeper understanding of the subject matter.
• Revision Aid: These solutions serve as an excellent revision resource. Students can refer to them while revising for exams to quickly recap concepts and problem-solving methods.

Benefits of NCERT Solutions for Class 12 Maths

• Strong Foundation: The solutions lay the groundwork for a solid understanding of mathematical concepts, preparing students for advanced topics and higher education.
• Exam Preparation: NCERT Solutions are invaluable resources for exam preparation. They help students practice a wide range of questions and gain confidence in solving problems effectively.
• Clarity and Confidence: By providing clear explanations and solutions, these resources enhance students’ clarity and confidence in their problem-solving abilities.
• Time Management: The step-by-step solutions aid in time management during exams. Students learn efficient problem-solving techniques that enable them to solve questions more quickly.
• Conceptual Application: The solutions demonstrate how to apply mathematical concepts in various scenarios, encouraging students to think critically and creatively.
• Consistency with Curriculum: The solutions are aligned with the official curriculum set by educational boards, ensuring that students are studying the right material.
• Accessibility: These solutions are easily accessible online, making it convenient for students to refer to them whenever needed, whether for homework, assignments, or exam preparation.
• Supplementary Materials: In addition to solutions, many resources provide additional study materials like notes, exemplar problems, and previous year question papers, aiding students in a comprehensive preparation strategy.

FAQs on NCERT Solutions for Class 12 Maths

What are ncert solutions for class 12 maths.

NCERT Solutions for Class 12 Maths are comprehensive answers and explanations to the problems and exercises found in the NCERT Class 12 Maths textbook. These solutions assist students in understanding and solving mathematical concepts effectively.

How can NCERT Solutions for Class 12 Maths benefit students?

NCERT Solutions aid students in building a strong conceptual foundation, enhancing problem-solving skills, preparing for exams, and gaining clarity on mathematical concepts. They serve as valuable resources for both learning and revision.

Are NCERT Solutions for Class 12 Maths available for all chapters?

Yes, NCERT Solutions cover all the chapters in the Class 12 Maths textbook. They provide solutions for a variety of exercises, including examples, practice problems, and additional questions from each chapter.

Are NCERT Solutions aligned with the official curriculum?

Yes, NCERT Solutions are designed to be in alignment with the official Class 12 Maths curriculum prescribed by educational boards. This ensures that students are studying relevant and accurate material.

Is NCERT Solutions for Class 12 Maths a recommended reference material for students?

Certainly, NCERT books hold a reputable status as excellent reference materials for Class 12 students. Rote memorization shortly before board exams isn't conducive to high scores. Thus, cultivating a genuine grasp of concepts is paramount for approaching exams with confidence. Dedicate sufficient time daily to solving textbook problems. This practice unveils the spectrum of problems likely to appear in board exams, fostering preparedness and enhancing academic performance.

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NCERT Solutions for Class 12 Maths PDF

For students like you who are taking the JEE as well as several board exams, NCERT Solutions for Class 12 Math has been published by eSaral. It offers thorough knowledge and comprehension of all important concepts presented in each chapter of the NCERT textbook for class 12 Math. The NCERT textbook of class 12 math contains 13 major chapters. For a clear and precise understanding of the topics, eSaral presents NCERT Solutions for Class 12 Math. These solutions are created by our subject experts. In addition to the Class 12 Maths solutions for each chapter, There are problems and detailed solutions of these problems have been provided. The answers are developed by the top educators of eSaral to help students in better and quicker understanding of the concepts.The problems in these NCERT Class 12 Solutions are thoroughly explained step-by-step.

NCERT Solutions for Class 12 Math gives a strong conceptual foundation for each topic. For students to have a better conceptual understanding of the material, it thoroughly covers all the key theorems and formulas. Students can prepare for undergraduate engineering admission examinations like JEE Mains, BITSAT, VITEEE, and others with the help of NCERT Solutions for Class 12 Maths.

There are 13 chapter in class 12 Maths NCERT book which are listed below:

• Chapter 1 : Relations and Functions 
• Chapter 2 : Inverse Trigonometric Functions
• Chapter 3 : Matrices
• Chapter 4 : Determinants
• Chapter 5 : Continuity and Differentiability
• Chapter 6 : Applications of Derivatives
• Chapter 7 : Integrals
• Chapter 8 : Application of Integrals
• Chapter 9 : Differential Equations
• Chapter 10 : Vector Algebra
• Chapter 11 : Three Dimensional Geometry
• Chapter 12 : Linear Programming
• Chapter 13 : Probability

Chapter 1: Relations and Functions

Chapter 1 of class 12 NCERT Maths textbook deals with Relations and Functions. Here, you will find exercise solution link for all the topics in this chapter.

Topics covered in chapter 1 are listed below:

Types of Relations: Reflexive, Symmetric, transitive and equivalence relations.

One to one and onto textbook

Composite functions

Inverse of a function

Binary operations  

Chapter 2: Inverse Trigonometric Functions

Chapter 2 of class 12 NCERT Maths textbook deals with Inverse Trigonometric Functions. Here, you will find exercise solution link for all the topics in this chapter.

Topics covered in chapter 2 are listed below:

Definition 

Principle value branches 

Graphs of inverse trigonometric functions

Chapter 3: Matrices

Chapter 3 of class 12 NCERT Maths textbook deals with matrices. Here, you will find exercise solution link for all the topics in this chapter.

Topics covered in chapter 3 are listed below:

Notation, order, equality

Types of matrices

Zero matrix

Transpose of a matrix

Symmetric and skew symmetric matrices

Addition, multiplication and scaler multiplication of matrices

Simple properties of addition, multiplication and scalar multiplication 

Non Commutativity of multiplication of matrices 

Concept of elementary row and column operations

Chapter 4: Determinants 

Chapter 4 of class 12 NCERT Maths textbook deals with determinants. Here, you will find exercise solution link for all the topics in this chapter

Topics covered in chapter 4 are listed below:

Determinant of a square matrix

Properties of determinants 

Minors and cofactors

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 in linear equations 

Solving a system of linear equations in two or three variables using inverse of matrix.

Chapter 5: Continuity and Differentiability 

Chapter 5 of class 12 NCERT Maths textbook deals with continuity and Differentiability. Here, you will find exercise solution link for all the topics in this chapter

Topics covered in chapter 5 are listed below:

Definition of continuity and differentiability 

Derivative of composite functions

Derivatives of inverse trigonometric functions

Derivative of implicit functions

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

Chapter 6: Applications of Derivatives 

Chapter 6 of class 12 NCERT Maths textbook deals with Application of derivatives.

Topics covered in chapter 6 are listed below:

Rate of change

Increasing and decreasing functions

Tangents and normals

Approximation 

Maxima and minima

Chapter 7: Integrals

Chapter 7 of class 12 NCERT Maths textbook deals with Integrals.

Topics covered in chapter 7 of class 12 are listed below:

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

Integrations by parts

Integrations using trigonometric identities

Definite and meaning of definite integrals 

Definite integrals as the limit of a sum

Fundamental theorem of calculus

Evaluation of definite integral by substitution 

Chapter 8: Application of Integrals

Chapter 8 of class 12 NCERT Maths textbook deals with Application of derivatives.

Topics covered in chapter 8 are listed below:

Applications in finding the area under simple curves

Arcs of circles/parabolas/ellipses (in standard form only)

Area of the region bounded by a curve and a line

Chapter 9: Differential Equations

Chapter 9 of class 12 NCERT Maths textbook deals with Differential Equations.

Topics covered in chapter 9 are listed below: 

Degree of a differential equation

Order of a differential equation

General and particular solutions of a differential equation.

Formation of differential solution whose general solution is given

Solution of differential equations by method of separation of variables

Homogeneous differential equations of first order and first degree 

Chapter 10: Vector Algebra 

Chapter 10 of class 12 NCERT Maths textbook deals with Vector Algebra.

Topics covered in chapter 10 of class 12 maths are listed below: 

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.  

Chapter 11: Three Dimensional Geometry                                                                                      

Chapter 11 of class 12 NCERT Maths textbook deals with Vector Algebra.

Topics covered in chapter 10 of class 12 are listed below: 

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

Chapter 12: Linear Programming

Chapter 12 of class 12 NCERT Maths textbook deals with Linear Programming. It covers topics such as linear programming problems and their mathematical formulae, the mathematical formulae of the problem to be solved, the graphical approach to solving the linear programming problem, and various types of the linear programming problem with various examples. Furthermore, students can find exercises that adequately explain these concepts with solutions.

Topics covered in chapter 12 are listed below: 

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

optimal feasible solutions (up to three non-trivial constrains).

Chapter 13: Probability

This chapter is all about probability. We'll go over the basics and formulas we looked at in class 11, and we'll do the same in this chapter with the new ideas we've learned.

Topics covered in chapter 13 are listed below: 

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

Binomial distribution

Key Benefits of CBSE Class 12 Maths NCERT Solutions from eSaral

CBSE Class 12 Mathematics NCERT solutions from eSaral provide several advantages for students preparing for board and competitive exams. Some of the main advantages of NCERT solutions are:

Complete Coverage: All topics and chapters prescribed by CBSE Class 12 Syllabus are covered in NCERT solutions. Students will have access to a complete set of study material to help them prepare for their exams effectively.

Accurate & Verified Answers: NCERT solutions are provided by subject matter experts to ensure accuracy and reliability of answers. Students can depend on these solutions for getting the correct approach to solve problems.

Understanding Concepts: NCERT solutions provide step by step explanations for each problem to help students understand the basic concepts and improve their problem solving skills.

Save time and effort: Having a ready-made solution saves students’ time since they don’t have to guess the answers themselves. They can use the extra time for revision and practicing.

Practice & revision: Using NCERT solutions helps students practice and revise. Students can try a variety of problems to strengthen their understanding of different concepts.

Exam preparation: CBSE board examinations are based on NCERT textbooks. Practicing with solutions helps students get familiar with the question format and marking system.

Homework & Assignments. NCERT solutions are useful for completing homework & assignments on time. Students can refer to NCERT solutions whenever they encounter challenging questions.

Higher Performance: With regular practice, students improve their problem solving skills, which helps them perform better in exams & other competitive tests

Frequently Asked Questions

Question 1: Why is a Class 12 maths book important for engineering exams?

Answer 1: The importance of a Class 12 Maths book for engineering exams can be summarised as follows:

Save Time & Effort: Students can save time and effort by having ready-made solutions as they don’t have to work on their own. They can use the extra time to revise and practice.

Practice & revision: Using NCERT solutions helps students practice and revise. Students can try a variety of problems to strengthen their understanding of different concepts. 

Exam preparation: CBSE board examinations are based on NCERT textbooks. Practicing with solutions helps students get familiar with the question format and marking system. Homework & Assignments NCERT solutions are useful for completing homework & assignments on time. Students can refer to NCERT solutions whenever they encounter challenging questions. 

Clarification: Students may face doubts and misconceptions while studying independently. By using NCERT solutions, they can get clarity on their doubts. Higher Performance: With regular practice, students improve their problem solving skills, which helps them perform better in exams & other competitive tests

Question 2: How can students use NCERT solution effectively?

Answer: In order to effectively utilize NCERT Solutions, students must first solve the exercises on their own and then compare their answers to the solutions provided. Students should also analyse solution steps to understand the problem solving methods.

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Welcome, learners!

Unit 1: relations and functions, unit 2: inverse trigonometric functions, unit 3: matrices, unit 4: determinants, unit 5: continuity and differentiability, unit 6: application of derivatives, unit 7: integrals, unit 8: application of integrals, unit 9: differential equations, unit 10: vector algebra, unit 11: three dimensional geometry, unit 12: probability.

• NCERT solutions for class 12 Maths
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Class 12 th maths is already considered one of the advanced mathematics. The team at Physics Wallah has come forward to provide NCERT Solutions for class 12 maths . The complete study material has been prepared as per the latest CBSE guidelines. This study material in overall assists your progress to attain desired marks. Do solve all questions given in Physics Wallah Important questions for class 12 Maths .

Scoring optimum marks is crucial is this stage of life. Since marks scored in class 12 th maths would forge your future steps. On the other hand, maths holds a significant position and marks in all the competitive examination. Students looking forward to progress in maths area would require maths as hand in hand relation with NCERT Solutions for class 12 maths.

NCERT Solutions for Class 12 Maths Chapters (Updated 2022-23)

Chapter 1 relations and functions, chapter 2 inverse trigonometric functions, chapter 3 matrices, chapter 4 determinants, chapter 5 continuity and differentiability, chapter 6 applications of derivatives, chapter 7 integrals, chapter 8 applications of integrals, chapter 9 differential equations, chapter 10 vector algebra, chapter 11 three dimensional geometry, chapter 12 linear programming, chapter 13 probability, ncert solutions for class 12 maths free pdf download.

NCERT textbook Chapter 1 deals with relations and functions, types of relations, types of functions, the composition of functions and invertible functions, binary operations, and various examples. You can find links to exercise Solutions for the topics covered in this chapter.

Topics Covered in Chapter- 1 Relations and Functions for Class 12 Maths:

Types of relations: symmetric, Reflexive, transitive, and equivalence relations. The inverse of a function, Binary operations, composite functions, and One to one and onto functions.

Also, access the helpful resources for Class 12 Chapter 1 Relations and Functions at Physics Wallah

• Important questions for class 12 Maths Chapter 1 Relations and Functions
• RD Sharma Solution for Class 12 Maths Chapter 1 Relations and Functions

Chapter-2 Inverse Trigonometric Functions of NCERT Class 12 Mathematics, provides an overview of various topics, such as notes based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions, and various examples. These concepts are very well explained with examples. Solutions to exercises from this chapter can be accessed here.

Topics Covered in Chapter- 2 Inverse Trigonometric Functions of Class 12 Maths:

Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Also, access the helpful resources for Class 12 Chapter 2 Inverse Trigonometric Functions at Physics Wallah

• Important questions for class 12 Maths Chapter 2 Inverse Trigonometric Functions
• RD Sharma Solution for Class 12 Maths Chapter 2 Inverse Trigonometric Functions

In chapter 3 of the NCERT textbook, we will see the definition of the matrix, types of matrices, equality of matrices, matrix operations like the addition of matrices and multiplication of a matrix by a scalar, properties of addition of matrices, properties of scalar. Multiplication, matrix multiplication, matrix multiplication properties, matrix transpose, matrix transpose properties, symmetric and skew-symmetric matrices, elementary operations or matrix transformations, matrix inversions by elementary operations, and various examples. You can find links to exercise Solutions for the topics covered in this chapter.

Topics Covered in Chapter- 3 Matrices of Class 12 Maths:

Notation, order, equality, Concept, types of matrices, transpose of a matrix, zero matrix, symmetric and skew-symmetric matrices. Scalar multiplication of matrices and addition multiplication, multiplication, and scalar multiplication, simple properties of addition Existence of non-zero matrices, and Noncommutativity of multiplication of matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Proof of the uniqueness of inverse and Invertible matrices, if it exists; (Here, all matrices will have real entries).

Also, access the helpful resources for Class 12 Chapter 3 Matrices at Physics Wallah

• Important questions for class 12 Maths Chapter 3 Matrices
• RD Sharma Solution for Class 12 Maths Chapter 3 Matrices

Chapter 4 Class 12 NCERT Solutions for Maths deals with the topic of Determinants. Students will be introduced to the definition and meaning of determinants, notations based on the order of determinants, properties of determinants, finding the area of ​​a triangle using determinants, minor and cofactors of determinants, adjoint of a matrix, the inverse of matrix, determinants and matrix applications, and various examples. Below are links to the individual Solutions to the exercises covered in this chapter.

Topics Covered in Chapter- 4 Determinants of Class 12 Maths:

Determinants of a square matrix (3 × 3 matrices), minors, properties of determinants, cofactors, and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Inconsistency, consistency, and number of Solutions of a system of linear equations, for example, a system of linear equations in two or three variables (having a unique Solutions) using the inverse of a matrix.

Also, access the helpful resources for Class 12 Chapter 4 Determinants at Physics Wallah

• Important questions for class 12 Maths Chapter 4 Determinants
• RD Sharma Solution for Class 12 Maths Chapter 4 Determinants

NCERT textbook of Chapter 5 begins with the definition of continuity. Students will also learn about continuity, the algebra of a continuous function, the definition and meaning of differentiability, derivatives of complex functions, derivatives of implicit functions, derivatives of inverse trigonometric functions, exponential and logarithmic functions, logarithmic derivatives, derivatives of functions in parametric forms, derivatives of the second order, the mean theorem value through various examples. Here students will find exercises that correctly explain these concepts with Solutions.

Topics Covered in Chapter- 5 Continuity and Differentiability of Class 12 Maths:

Continuity and differentiability, chain rule, a derivative of composite functions, derivatives of the inverse derivative of an implicit function, trigonometric functions. 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 helpful resources for Class 12 Chapter 5 Continuity and Differentiability at Physics Wallah

• Important questions for class 12 Maths Chapter 5 Continuity and Differentiability
• RD Sharma Solution for Class 12 Maths Chapter 5 Continuity and Differentiability

NCERT textbook of Chapter 6 provides a definition of derivatives, increasing and decreasing functions, rate of change of quantities, first derivative test, tangents and normals, approximations, maximum and minimum values ​​of a function in a closed interval, maxima and minima, and various examples. Here, students will find exercises that properly explain these concepts.

Topics Covered in Chapter- 6 Applications of Derivatives of Class 12 Maths:

Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normals, approximation, maxima, and minima. Simple problems (that illustrate the basic principles and understanding of the subject and real-life situations).

Also, access the helpful resources for Class 12 Chapter 6 Application of Derivatives at Physics Wallah

• Important questions for class 12 Maths Chapter 6 Application of Derivatives
• RD Sharma Solution for Class 12 Maths Chapter 6 Application of Derivatives

In Chapter 7, Integrals, we will see the definition of the indefinite integral, integration as the inverse process of differentiation, the geometric interpretation of the indefinite integral, properties of the indefinite integral, comparison of differentiation and integration, methods of integration such as integration by substitution, integration by partial fractions, integration by parts, integration by of trigonometric identities, integration of some integral functions, definition and meaning of definite integrals, definite integral as limit of sum, basic theorem of calculus, calculation of definite integrals by substitution, some properties of definite integrals and various examples. Here we have provided exercises with Solutions based on these topics from the chapter.

Topics Covered in Chapter- 7 Integrals of Class 12 Maths:

Integration is the inverse process of differentiation. Integration of a variety of functions by substitution, partial fractions, and by parts, only simple integrals of the type

Basic properties of evaluation of definite integrals and definite integrals. Fundamental Theorem of Calculus (without proof).

Also, access the helpful resources for Class 12 Chapter 7 Integrals at Physics Wallah

• Important questions for class 12 Maths Chapter 7 Integrals
• RD Sharma Solution for Class 12 Maths Chapter 7 Integrals

NCERT textbook of In chapter 8, we continue the discussions on integrals, area under simple curves, starting with definition, area of the region bounded by a curve and a straight line, area between two curves, and various examples. From the links below, students can access exercises with Solutions explaining the concepts from this chapter.

Topics Covered in Chapter- 8 Applications of Integrals of Class 12:

Applications in finding the area under simple curves, lines, arcs of parabolas/circles/ellipses, area between the two above curves (the region should be clearly identifiable).

Also, access the helpful resources for Class 12 Chapter 8 Application of Integrals at Physics Wallah

• Important questions for class 12 Maths Chapter 8 Application of Integrals
• RD Sharma Solution for Class 12 Maths Chapter 8 Application of Integrals

In this chapter 9, students focus on the definition of differential equations, degree of a differential equation, basic concepts related to differential equations, general and particular Solutions of a differential equation, order of a differential equation, formation of a differential equation whose general Solutions is given, the procedure for formation of a differential equation that will represent a family of curves, first-order Solutions methods, first-degree differential equations, differential equations with a separable variable, homogeneous differential equations, linear differential equations, the procedure for solving a first-order linear differential equation, and various examples. Solutions to exercises from this chapter can be accessed here.

Topics Covered in Chapter- 9 Differential Equations of Class 12 Maths:

Definition, general and particular Solutions of a differential equation, order, and degree. Formation of differential equation whose available Solutions is mentioned. Solutions of differential equations by separating variables, first degree and homogeneous differential equations of the first order. Solutions of linear differential equation: dy/dx + Py = Q, where P and Q are the functions of x.

Also, access the helpful resources for Class 12 Chapter 9 Differential Equations at Physics Wallah

• Important questions for class 12 Maths Chapter 9 Differential Equations
• RD Sharma Solution for Class 12 Maths Chapter 9 Differential Equations

In this chapter- 10, students will be introduced to vector algebra. The concepts in this chapter cover how to find the 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, vector addition, properties of vector addition, multiplication of a vector by a scalar, components of a vector, joining vectors of two points, cut formula, scalar or dot product of two vectors, scalar product properties, product of two vectors, projection of a vector onto a line vector or cross product of two vectors and various examples. From the links below, students can access exercises with Solutions explaining the concepts from this chapter.

Topics Covered in Chapter- 10 Vector Algebra of Class 12 Maths:

Vectors and Scalars, magnitude and direction of a vector. Ratios of vectors/Direction cosines. Types of vectors (unit, equal, zero, parallel, and collinear vectors), position vector of a point, negative of a vector, components of a vector, multiplication of a vector by a scalar, the addition of vectors, 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.

Also, access the helpful resources for Chapter 10 Vector Algebra at Physics Wallah

• Important questions for class 12 Maths Chapter 10 Vector Algebra
• RD Sharma Solution for Class 12 Maths Chapter 10 Vector Algebra

NCERT Class 12 Mathematics Chapter 11 contains topics related to three dimensional geometry, direction cosines and direction ratios of a line, relationship between direction cosines of a line, direction cosines of a line passing through two points, equation of lines in space, equation of a line passing through a given point and parallel to a given vector, derivation Cartesian form from vector form, the equation of a line passing through the two given points, angle between two lines, shortest distance between two lines, distance between two oblique lines, distance between two parallel lines, the equation of a plane in normal form, plane in Cartesian form, equation of a perpendicular plane to a given vector and passing through the given point, equation of a plane passing through three non-collinear points, intersection 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 various examples. Below are links to Solutions to exercises from this chapter.

Topics Covered in Chapter- 11 Three Dimensional Geometry of Class 12 Maths:

Direction cosines/ratios of a line joining two points. The shortest distance between the two lines, Cartesian and vector equation of a line, coplanar and skew lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii)a line and a plane, (iii) two planes. Distance of a point from a plane.

Also, access the helpful resources for Class 12 Chapter 11 Three Dimensional Geometry at Physics Wallah

• Important questions for class 12 Maths Chapter 11 Three Dimensional Geometry
• RD Sharma Solution for Class 12 Maths Chapter 11 Three Dimensional Geometry

In this chapter, students will discuss about the linear programming. Concepts discussed in this chapter are linear programming problem and its mathematical formulation, graphical method of solving linear programming problems, mathematical formulation of the problem, different types of linear programming problems with different examples. students can find exercises that correctly explain these concepts with Solutions.

Topics Covered in Chapter- 12 Linear Programming of Class 12 Maths:

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 Solutions for problems in two variables, feasible and infeasible regions, feasible and infeasible Solutions, optimal feasible Solutions (up to three non-trivial constrains).

Also, access the helpful resources for Class 12 Chapter 12 Linear Programming at Physics Wallah

• Important questions for class 12 Maths Chapter 12 Linear Programming
• RD Sharma Solution for Class 12 Maths Chapter 12 Linear Programming

This chapter discusses probability. Reviewing the basic facts and formulas studied earlier in Class 11, we reintroduce these formulas in this chapter along with the advanced concepts are mentioned, i.e. conditional probability, properties of conditional probability, multiplication theorem of probability, multiplication rule of probability for more than two events, independent events, Bayesian theorem, sample space distribution, total probability theorem, random variables and their probability distribution, mean of a random variable, probability distribution of a random variable, variance of a random variable, Bernoulli trials and binomial distribution with various examples. Here we have compiled the Solutions to the exercises from this chapter that have thoroughly explained the concept.

Topics Covered in Chapter- 13 Probability of Class 12 Maths:

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.

Also, access the helpful resources for Chapter 13 Probability at Physics Wallah

• Important questions for class 12 Maths Chapter 13 Probability
• RD Sharma Solution for Class 12 Maths Chapter 13 Probability

Why NCERT Solutions for Class 12 Maths are important?

The NCERT Solutions for class 12 maths study material provided by the team would help students get top ranks. The complete study material has been divided into various segments. For ease in understanding topics, students can find it divided into chapter wise section.

The very ideology with which the team has been formed is to enhance students' foundations. A stronger foundation of concepts will help students grow in their career. Students can also find important questions in NCERT Solutions for class 12 maths provided by Physics Wallah. NCERT books already known among students for their comprehensive study program. Many a time students cannot complete and solve NCERT textbooks. These NCERT Solutions for class 12 maths are anytime helpful for students at an instant of time.

Maths is among the subject which not only helps students in qualifying their examinations but also helps them in understanding advanced studies in future. The complete study material has been curated by the experts of their field. The duration and efforts put on by this faculties are incomparable. The students learning form NCERT Solutions for class 12 maths attain an extra compatible edge over the rest of the students.

How to Study NCERT Solutions for Class 12 Maths effectively?

Student can enhance their problem-solving skills through, our study material. Many a time students get stuck with difficult numerical. In order to prevent the wastage of time, students can follow NCERT Solutions for class 12 maths prepared byPhysics Wallah.

The entire team of NCERT Solutions for class 12 maths at Physics Wallah have already aced various national level exams. In order to prepare this study material, various previous year question papers have been referred. The complete study material has been prepared flawless and checked various times for any error.

Why Physics Wallah is best for NCERT Solutions for class 12 maths?

Several times students have to travel long distances to get their doubts cleared from expert faculties. Physics Wallah has provided this complete study material right at your home. All the students need to do is study every bit of it and score maximum marks. In order to minimize the efforts students can also find various important questions marked in the study material.

The complete NCERT Solutions for class 12 maths has been provided for free. Students can access the world-class study material with a single click. All that students need to do is, to sign up with us. Study from NCERT Solutions for class 12 maths and climb the ladders of success.

Key Benefit for NCERT Solutions for Class 12 Maths

Physics Wallah academic team consist of expert faculty members across country who prepared NCERT Solutions for class 12 Maths the key highlights of these Solutions are.

• All the questions given in the NCERT exercise are solved.
• All NCERT Solutions for class 12 Maths are errorless and proof-readied several time.
• All answers are explained and written as per the guideline of CBSE to score good marks in class 12 final exam.
• Based on new pattern of CBSE class 12 we have added additional questions which are MCQ based and added several solved examples apart from NCERT Solutions for class 12 Maths.
• Physics Wallah NCERT Solutions for class 12 Maths consist of exercise wise pages to help you find your maths doubts instantly.
• Each question of NCERT Solutions for class 12 Maths is added with all required step which are supposed to write in final exam.
• Step wise Solutions of questions will help you to understand the method of solving the questions and clear your concepts.
• While solving any questions of NCERT text book one must have clarity on the concept for that while reading the theory of maths chapter you must prepare proper notes of the theory and make sure you write and understand all the concept. Write down the important formula use in the chapter in separate notes which helps you in final revision.
• Learning the application of maths and to know how to apply concepts of maths in the numerical required lot of effort to develop such method you must try more than one time and give time to question which are difficult make sure you have tried multiple time to solve the particular question before using NCERT Solutions for class 12 Maths. This way you will develop a skill to attempt the MCQ based question as well as you can handle the new questions.

Frequently Asked Question (FAQs)

Q1. What is Right strategies to score 100 percent marks in CBSE Board of class 12 Maths subject?

Ans. Right approach start with strong planning and it start with preparing right time table and notes. Start your preparation of class 12 Maths subject from the month of April. Do follow NCERT text book for class 12 Maths and read the entire theory mentioned in the text book. While reading and understanding the theory of NCERT text book make sure you have prepared a handout note of the subject.

Try to summarise the entire chapter formula in one page. Solve all questions given in the exercise with the help of Physics Wallah NCERT Solutions for class 12 Maths. Complete your syllabus in the month of October and then start revision with your own notes and do solve previous year papers.

Q2. How to Make balance between Board exam and competitive exam?

Ans. This is most common problem every student face when he was in class 12 and preparing for competitive entrance exam. Right planning is required to make a balance between your competitive entrance exam with board exam. Now a days all most all competitive entrance exam is conducted by NTA and the recommended book of NTA is NCERT.

So start your preparation with NCERT text book. try to solve all question in the exercise and solve questions from NCERT exemplar. Do solve MCQ based question will help you in your entrance exam as well as board exam. Allocate equal time for board as well as competitive exam with effective revision of class 11.

Q3. What are the Important topics of class 12 Maths for CBSE board exam?

Ans. The questions asked in CBSE board exam are from all most all chapters so it is difficult to prepare strategies in important chapters still few topics are important like Differentiability of any functions (This chapter consist of 2 and 4 marks questions)

• Logarithmic Differentiation – (This chapter consist of 4 marks questions)
• Substitution method (In Inverse) – (This chapter consist of 2 and 4 marks questions)
• Higher-order Derivative – (This chapter consist of 4 marks questions)
• Maxima Minima Word Problem – (This chapter consist of 6 marks questions)
• Equation of Tangent and Normal – (This chapter consist of 2 marks questions)

Q4. How sample papers of class 12 Maths can help to score good marks in Board exam?

Ans .If you are preparing for CBSE board exam sample paper for class 12 Maths are must for you. There are few thinks must be keep in mind while solving the sample papers such as solve one sample paper completely at once.

Do solve the sample papers based on time allocated this way you can check your speed of solving the paper and find out your mistake. Physics Wallah uploaded CBSE Sample paper for class 12 Maths with detail Solutions to help you out.

Q5. Should we solve additional MCQ based questions with NCERT text book?

Ans. Yes certainly you must solve MCQ based questions for class 12 Maths. Although NCERT text book consist of objective questions in the exercise but it is highly recommended to solve some addition Maths Questions objective type.

Q6. How to use previous year question papers of CBSE class 12 Board exam?

Ans. To score good marks in any exam need good level of understanding of exam pattern which can be developed through understanding the pattern of entrance exam. You can develop understanding of exam paper by doing deep analysis of previous year question papers. Do solve the questions paper understand the pattern of exam.

Q7. Is NCERT text book and NCERT exercise are enough for board as well as competitive entrance exam?

Ans. Yes NCERT text book is enough for competitive as well as board exam. NCERT text book is well designed and can be completed within the time limit so that you can do perfect revision of entire syllabus which play very important role in entrance as well as board exam.

Related Chapters

• NCERT Solutions for Class 12 Biology
• NCERT solutions for class 12 Chemistry

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• chapter 12-Linear Programming Miscellaneous Exercise

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Grade 12 Math Problems with Solutions and Answers

Grade 12 math problems with detailed solutions are presented. More grade 12 math practice test are included in this website.

Free Practice for SAT, ACT and Compass Math tests

(x + 2) 2 + (y + 1) 2 = 13

x 2 - 3|x - 2| - 4x = - 6

(x - 2) 2 + (y - 2) 2 = 4 (x - 1) 2 + (y - 1) 2 = 4

• Four large and 2 small pumps can fill a swimming pool in 2 hours. Two large and 6 small pumps can also fill the same swimming pool in 2 hours. How long does it take 8 large and 8 small pumps to fill 50% of the swimming pool. (NOTE: all the large pump have same power and all the small pumps have the same power).
• Let R and r the rate of work of the large and the small pumps respectively 4(2R + r) = 1 : 2 large and 1 small work for 4 hours to do 1 job 4(R + 3r) = 1 : 1 large and 3 small work for 4 hours to do 1 job T(4R + 4r) = 1 : Find time T if 4 large and 4 small are to do one job. Solve for R and r the system of first two equations then substitute in the third and solve for T to find the time. T = 5/3 hours = 1 hour 40 minutes.
• x + y + H = 60 : perimeter , x, y and H be the two legs and the hypotenuse of the right triangle (1/2)xy = 150 : area x 2 + y 2 = H 2 : Pythagora's theorem. 3 equations with 3 unknowns. (x + y) 2 - 2xy = H 2 : completing the square in the third equation. x + y = 60 - H : express x + y using the first equation and use the second equation to find xy = 300 and substitute in equation 5. (60 - H) 2 - 600 = H 2 : one equation with one unknown. Solve for H to find H = 25 cm. Substitute and solve for x and y to find x = 15 cm and y = 20 cm.
• √((-6 + 2) 2 + (0 + 3) 2 ) = √((a + 3) 2 + (0 + 2) 2 ) : distances from center to any point on the circle are equal. (25) = (a + 3) 2 + 4 : Simplify and square both sides (a + 3) 2 = 21 : Rewrite the above equation as Solve for a a = -3 + √(21)   ,   a = -3 - √(21) : solve for a and find two solutions.
• (-2 , -1) : center of circle m = (2 - -1) / (0 - -2) = 3 / 2 : slope of line through the center and the point of tangency (0 , 2) The line through the center and the point of tangency (0 , 2) is perpendicular to the tangent. M = -2 / 3 : slope of tangent y = -(2/3)x + 2 : equation of tangent given its slope and point (0 , 2).
• 3 C 2 × 8 C 6 × 1 = 84 : Use of fundamental theorem of counting
• x 2 - 3|x - 2| - 4x = - 6 : given Let Y = x - 2 which gives x = Y + 2 (Y + 2) 2 - 3|Y| - 4(Y + 2) = - 6 : substitute in above equation Y 2 - 3|Y| + 2 = 0 Y 2 = |Y| 2 : note |Y| 2 - 3|Y| + 2 = 0 : rewrite equation as (|Y| - 2)(|Y| - 1) = 0 |Y| = 2 , |Y| = 1 : solve for |Y| Y = 2, -2 , 1 , -1 : solve for Y x = 4 , 0 , 3 , 1 : solve for x using x = Y + 2.
• h = -b / 2a = 2 : x coordinate of the vertex of the parabola k = -(2) 2 + 4(2) + C = 4 + C : y coordinate of vertex x = (2 + √(4 + C)) , x = (2 - √(4 + C)) : the two x intercepts of the parabola. length of BA = k = 4 + C length of AC = 2 + √(4 + C) - 2 = √(4 + C) area = (1/2)BA × AC = (1/2) (4 + C) × √(4 + C) (1/2) (4 + C) × √(4 + C) = 32 : area is equal to 32 C = 12 : solve above for C.
• y = a(x + 2)(x - 3) : equation of the parabola in factored form 10 = a(5 + 2)(5 - 2) : (5 , 10) is a point on the graph of the parabola and therefore satisfies the equation of the parabola. a = 5/7: solve the above equation for a.
• Divide x 3 + 3x 2 -2Ax + 3 by (x 2 + 1) to obtain a remainder = -x(1 + 2A) -x(1 + 2A) = 5x : remainder given -(1 + 2A) = 5 : polynomials are equal if they corresponding coefficient area equal. A = -3 : Solve the above for A.
• P(1) = 1 5 + 2(1 3 ) +A × (1) + B = 2 : remainder theorem P(-3) = (-3) 5 + 2(-3) 3 +A × (-3) + B = -314 A = 4 and B = -5 : solve the above systems of equations.
• x 2 - 4x + 2 + y 2 - 4y + 2 = 4 : expand equation of first circle x 2 - 2x + 1 + y 2 - 2y + 1 = 4 : expand equation of second circle -2x - 2y - 6 = 0 : subtract the left and right terms of the above equations y = 3 - x : solve the above for y. 2x 2 - 6x + 1 = 0 : substitute y by 3 - x in the first equation, expand and group like terms. (3/2 + √(7)/2 , 3/2 - √(7)/2) , (3/2 - √(7)/2 , 3/2 + √(7)/2) : solve the above for x and use y = 3 - x to find y.
• I + 200 = A 2 : 200 added to I (unknown integer) gives a square. I + 276 = B 2 : 276 added to I (unknown integer) gives another square. B 2 = A 2 + 76 : eliminate I from the two equations. add squares A 2 (0, 1, 4, 9, 16, 25,...) to 76 till you obtain another square B 2 . 76 + 18 2 = 400 = 20 2 A 2 = 18 2 and B 2 = 20 2 I = A 2 - 200 = 124
• sum1 = a + a r + a r 2 = 42: the sum of the three terms given, r is the common ratio. sum2 = a 2 + a 2 r 2 + ar 2 r 4 = 1092: the sum of the squares of the three terms given . sum1 = a + ar + ar 2 = a(r 3 - 1) / (r - 1) = 42 : apply formula for a finite sum of geometric series. sum2 = a 2 + a 2 r 2 + ar 2 r 4 = a 2 (r 6 - 1) / (r 2 - 1) = 1092: the sum of squares is a also a sum of geometric series. sum2/sum1 2 = 1092 / 42 2 = [ a 2 (r 6 - 1)/(r 2 - 1)] / [a 2 (r 3 - 1) 2 / (r - 1) 2 ] (r 2 - r + 1) / (r 2 + r + 1) = 1092 / 42 2 r = 4 , r = 1/4 : solve for r a = 2 : substitute r = 4 and solve for a a = 32 : substitute r = 1/4 and solve for a a = 2 , ar = 8 , ar 2 = 32 : find the three terms for r = 4 a = 32 , ar = 8 , ar 2 = 2 : find the three terms for r = 1/4
• T1 + T2 = 3.5 : T1 time for the rock to reach bottom of well and T2 time for the sound to reach the top of the well. 16 × T1 2 = 1087 × T2 : same distance which the height of the well. T2 = 3.5 - T1 : solve for T2 16 × T1 2 = 1087 × (3.5 - T1) T1 = 3.34 seconds Height = 16 × (3.34) 2 = 178 feet (to the nearest unit)
• solve the system of the first two equations to obtain the solution (2 , -3) The above solution is also a solution to the last two equations. a(2) + b(-3) = 4 2a(2) - b(-3) = 2 a = 1 and b = -2/3 : solution to the above system of equations.

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• NCERT Solutions
• NCERT Class 12
• NCERT 12 Maths
• Chapter 1: Relations And Function

NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions

Ncert solutions for class 12 maths chapter 1 – free pdf download.

NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions provides solutions for all the questions enlisted under the chapter, which is included in the CBSE syllabus 2023-24. All exercise questions are solved using a step-by-step approach. Solving the questions from the Class 12 Maths NCERT textbook helps grasp the concepts correctly. Students can download and go through these NCERT Solutions for Class 12 Maths  to sharpen their Maths skills.

Students should understand not only the basics of this chapter but also the advanced portions to get a command over the chapter Relations and Functions. Practising these NCERT Solutions for Class 12 Maths will help the students in their exam preparation in such a way that they will find even the difficult questions from this chapter to be extremely easy. Download the NCERT Solutions PDF of this chapter from the link provided below.

Download PDF of NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions

Access answers of class 12 ncert maths chapter 1 relations and functions.

In NCERT Class 12 Maths Chapter 1, students study different types of relations and equivalence relations, composition of functions, invertible functions and binary operations. Students can utilise the NCERT Solutions for Class 12 Maths Chapter 1 for any quick reference to comprehend complex topics.

The main features of this chapter are as follows:

• Empty relation
• Universal relation
• Reflexive relation
• Symmetric relation
• Transitive relation
• Equivalence relation
• A function f: X → Y is one-one (or injective) if f (x 1 ) = f(x 2 ) ⇒ x 1 = x 2 ∀ x 1 , x 2 ∈ X.
• A function f: X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
• A function f: X → Y is one-one and onto (or bijective), if f is both one-one and onto.
• A function f: X → Y is invertible if and only if f is one-one and onto.

NCERT Solutions For Class 12 Maths Chapter 1 Exercises:

Get detailed solutions for all the questions listed under the below exercises:

Exercise 1.1 Solutions : 16 Questions (14 Short Answers, 2 MCQ)

Exercise 1.2 Solutions : 12 Questions (10 Short Answers, 2 MCQ)

Exercise 1.3 Solutions : 14 Questions (12 Short Answers, 2 MCQ)

Exercise 1.4 Solutions : 13 Questions (12 Short Answers, 1 MCQ)

Miscellaneous Exercise Solutions : 19 Questions (7 Long answers, 9 Short Answer Type, 3 MCQs)

The main topics covered in this chapter include:

Key Features of NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions

• These NCERT exercises are meticulously solved by subject experts.
• Simple and easy language is used.
• All questions are solved with the best possible explanations given.
• These NCERT Solutions are a valuable help to students in their assignments and competitive exams.

Students can also access the NCERT Solutions for Class 12 of any subject while preparing for the board examinations apart from Mathematics. All the solutions are in accordance with the latest CBSE Syllabus and guidelines.

Disclaimer – 

Dropped Topics – 1.4 Composition of Functions and Invertible Function (upto ‘This leads to the following definition’), Page 13-14 Full Pages, Page 15 Examples 24 and 25, Page 16-25 Full Pages, Page 26 Question 12 and 13, Page 27-28 Examples 45 and 49, Page 29-31 Ques. 1–3, 6–7, 9, 11–14, 18–19, Page 31-32 Summary points 11-13 and 15-19.

Frequently Asked Questions on NCERT Solutions for Class 12 Maths Chapter 1

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The Accidental Tax Cutter in Chief

President biden says he wants to rake in more money from corporations and high earners. but so far, he has cut more taxes than he’s raised..

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From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

[THEME MUSIC]

Today, in his campaign for re-election, President Biden says that raising taxes is at the heart of his agenda. But as it turns out so far, he’s done the opposite as president. My colleague Jim Tankersley explains.

It’s Wednesday, April 3.

Jim, welcome back. We haven’t seen you since the State of the Union. Always a pleasure.

So, so great to be here. And yeah, I finally recovered from staying up all night with you guys.

Yeah, you don’t even know all night. You stopped and we kept going.

That’s true. I did. I got a robust three hours that night. You’re right.

[LAUGHS]:: So Jim, in your capacity as really the chief economic thinker covering this president, you recently came across something very surprising.

Yeah, it started with a pretty basic question for me. I like to do this crazy thing, Michael, where I like to take candidates’ promises and see if they’ve come true.

It’s a little bit wild, but it’s what I do for fun. And in this case, I wanted to look at a very central promise of President Biden’s campaign in 2020, which he has repeated while in office.

I promise you, I guarantee you we can build back, and build back better with an economy that rewards work, not wealth.

The promise was he was going to raise taxes.

But I tell you what I’m going to do, and I make no apologies for it. I’m going to ask the wealthiest Americans and the biggest corporations of the Fortune 500 companies, 91 making a collective billions of dollars, didn’t pay a single solitary penny in federal tax!

Not taxes on the middle class, not taxes on low income workers, but he was going to raise taxes on corporations and the rich.

But I’m going to make sure they pay their fair share!

He was going to make them pay their fair share. And he leaned into it.

Guess what? You’re going to start paying your fair share. I’m going to ask them to finally begin to pay the fair share. It’s not a punishment. Pay your fair share.

He said it over and over.

Fair share? Translation — it’s back to the failed policies of the 1970s.

Republicans loved this. They repeated it too. They told voters that the president was going to raise taxes.

Joe Biden bragging about raising taxes on corporations. That means less money for those very employers to hire people back.

They talked about all the ways in which corporate tax increases could rebound on workers.

Joe Biden will shut down your economy, raise taxes, wants a $4 trillion tax increase. He’s the only politician I’ve ever seen who said, we will raise your taxes. You’re supposed —

It was a very big part of the economic debate for the campaign.

And I wanted to know, is that true? Has that actually played out in the policy agenda the president has had?

So I asked some economists at the Tax Policy Center in Washington to run an analysis and just say, let’s look at all of the ways Biden has changed the tax code in all of those laws he’s signed, and ask, has he raised taxes as president? And it turns out the answer is he has not raised taxes.

On net, he has cut more taxes than he’s raised.

How much more has he cut taxes than raised them?

So by the math that economists use when they look at budgets, the traditional way of scoring tax changes, he has cut taxes by $600 billion on net.

Hmm. A lot of money, a lot of tax cuts.

It’s a lot of tax cuts. The president has been a net tax cutter.

So Jim, why and how did Biden end up cutting taxes, especially if his stated intent was to raise taxes?

Well, there’s two sides of this equation and two complementary explanations for what’s happened here. The first side is the tax increases that Biden ran on, he’s only done a couple of them. He has trillions of dollars of ideas for how to raise taxes on rich people and corporations. The Treasury Department publishes an entire book full of them every year called “The Green Book.”

But in the actual legislation he signed, there’s only been a couple, really. There was a tax on stock buybacks that companies do and then a new minimum tax for certain multinational corporations that have very low tax rates. Those add up to real money, but they are not, in the grand scheme of Biden’s tax increases, a really large amount of the agenda he’s proposed.

So explanation number one, he just hasn’t been that successful in passing tax increases, and there’s a lot of reasons for that. The biggest one is just the simplest one is that he’s just had a really hard time persuading members of Congress, including Democrats, to back some of his favorite tax increases. He wants to raise the corporate income tax rate, which President Trump cut in his 2017 tax bill. Biden wants to raise it to 28 percent from 21 percent. Congress has not had any appetite to go along with that.

He wanted to get rid of what’s called the carried interest loophole, a long-time white whale of Democratic policy making. But he could not get even 50 Democrats to go along with that. Senator Kyrsten Sinema of Arizona was opposed to it, and so it didn’t get included.

And he did some pieces of legislation on a bipartisan basis. And in those cases, Republicans were just not going to pay for anything by raising taxes, and so he had to take those off the table there. So it’s all added up to just not very much activity in Congress to raise taxes on what Biden wants to do.

Got it. So that’s the side of a ledger where Biden simply fails to increase taxes because he can’t get Congress to increase taxes.

Right. But there’s another side, which is also that Biden has signed into law a decent number of tax cuts.

And that starts from the very beginning. Just a couple of months into his presidency, if you’ll recall, we’re still in the depths of the COVID-19 pandemic, the economy is wobbling after it had started to rebound. Biden proposes what is essentially a stimulus bill.

And he includes some tax cuts in there, a tax cut for families, a child tax credit. And it also includes — you remember those direct checks that people got as part of that bill?

Yes, $1,400. I remember them.

Yes, those were technically tax cuts.

So the stimulus bill starts with that. The next year, he passes this bill that is trying to accelerate manufacturing of things like semiconductors in the United States. That’s the CHIPS Act. And that includes some corporate tax cuts for companies that invest in the kind of manufacturing that Biden wants. This is industrial policy via carrots for corporations. And Biden is handing them out as part of this bill.

So tax cuts there. And then finally the Inflation Reduction Act, which includes the largest climate effort in American history is a bunch of corporate tax cuts at its core, tax cuts for manufacturing of solar panels, tax cuts for people to buy electric vehicles, tax cuts for all sorts of things tied to the transition from fossil fuels to lower emission sources of energy. And those tax cuts add up. They add up for corporations. They add up for individuals. And in the end, that full suite of tax cuts that he’s passed across all of this legislation outweighs the modest tax increases that were also included in the Inflation Reduction Act to reduce its cost.

Got it. So a very big reason why Biden ends up cutting taxes, beyond the fact that he’s not able to raise them through Congress, is that that’s what it took, according to those in his administration, to get American industry and American consumers to change their behavior in line with policy goals such as getting more domestic computer chip manufacturing and getting more people to buy electric vehicles they decided the way to do that was to give people tax breaks, which means he cut their taxes.

Right, people and companies. The president certainly has talked throughout the campaign about wanting to give middle class families a break. But he has also, in the process of crafting policy, really come to rely on tax cuts for people and for corporations as a way of achieving these policy goals. And in many cases, again, this is what he had to do to pass these bills through even Democrats in Congress.

Senator Joe Manchin of West Virginia didn’t want to just send money to companies that were making solar panels. He wanted there to be tax incentives for it. And so that is part of the reason why these were created as tax incentives. And so all of this adds up to more of a tax cutting record than you might have imagined when Biden was on the campaign trail.

I’m curious who really ended up benefiting from these tax cuts. You said they went to people and to corporations, but on the whole, did they end up reaching lower income Americans, middle income Americans, or the rich?

Well, we don’t have a full distributional analysis, which is what you’re asking for, of the entirety of Biden’s tax changes. But what we can say this — particularly the ones that were in that early stimulus bill, the recovery plan, those were very much targeted toward lower income and middle income Americans.

There were income limits on who could get things like the Child Tax Credit. Obviously, the direct payments went to people who were middle class or less. So the analysis of that would suggest that these were tax cuts for lower income people, for middle class people. And on the flip side, what I think we are likely to see with the electric vehicle credit through the Inflation Reduction Act is that while there are some income limits on who can qualify for that credit, that the people who end up claiming that credit tend to be the higher earners among the people who qualify.

Right. Who buys a Tesla, after all? Somebody with a fair amount of money.

Right, exactly. And of course, the corporate tax cuts go to companies, flow through to their shareholders. There’s a huge debate in the academic literature among politicians about how much of that benefit actually ends up going to their workers versus stays with shareholders. But we can broadly say that Joe Biden has done a lot for certain corporations who are trying to advance his manufacturing goals in particular to reduce their tax bills. And that is certainly not in line with the rhetoric you hear him talking about most of the time about making corporations pay their fair share. And the White House acknowledges this. I asked them about it. And they basically said, we think there’s a difference between just cutting the corporate tax rate in a way that helps anybody no matter what they’re doing and what we’re trying to do, which is basically reward corporations for accelerating the energy transition.

Understood. But where does this ultimately leave Biden’s campaign promise to make the tax code fairer and to make sure that the well-off in particular and corporations are paying their, what he calls, fair share?

Well, I think by Biden’s own measurements, by his own ambitions, he would have to agree that he is nowhere close to what he believes would be a fair share for corporations. Because Biden is still running on this. As he enters his re-election campaign, as it really heats up, a rematch with Donald Trump, the president is really leaning into this message of we need to do more. We need to raise more taxes on corporate America. It is time for these companies and for high earners to pay their fair share.

Right. I didn’t get it done in the first term. But if you elect me, I’ll get it done in the second.

Give me another shot, and this time, I promise, will be different.

[MUSIC PLAYING]

We’ll be right back.

So Jim, let’s talk about Biden’s tax raising plans for a theoretical second term and why anyone should have any faith that he could get it done, if there’s a second term, given the experience so far of his first term.

Yeah, well, man, there’s a lot of plans to talk about. I don’t think we can get through all of them, but we can certainly hit the highlights here. So we can start with the couple of things that Biden has been able to do to raise taxes on corporations. He wants to take those and then plus them up.

He’s put this new minimum tax on corporations. It’s a 15 percent minimum tax on certain multinationals. He now wants to raise that to 21 percent.

He wants to take that corporate stock buyback tax which is 1 percent right now, and he’d like to quadruple it to 4 percent. And then he goes after some things large and small. He wants to do new taxes that hit the use of corporate and private jets. He wants to do new taxes on companies that pay large amounts of compensation to their executives.

And then we get to some really big taxes on high earning individuals. So the president has said over and over again, he won’t raise taxes on anyone making less than $400,000 a year. But he’s got a bunch of taxes for the people above that. So he wants to raise the top marginal income tax rate. He wants to take it from 37 percent, which is the level set by President Trump’s 2017 tax law, and bring it back to 39.6 percent, which is what it was before. He also wants to impose what he calls a billionaires tax.

OK. It’s a 25 percent tax on the total value of all of the assets of anyone worth more than $100 million.

OK, wait. I have several questions about this.

First being a fact check, if it’s a billionaires tax, it’s interesting that it’s going after people who have just 100 million.

Yeah, I think most billionaires would be offended at the inclusion of 100 millionaires in that. Yes, totally agree. That is factually inaccurate, the name.

Right. But beyond that, this sounds very much like a wealth tax, which we don’t really have in our system.

Yeah, it’s a sort of wealth tax. The Biden people don’t call it a wealth tax, but it is a tax on something other than income that you report every year to the IRS as having been earned. It goes beyond just, oh, I got interest from my stock holdings or I made money from my job. It’s, oh, the value of my art collection increased last year, and now Biden’s going to tax me on that increase, even if I didn’t sell the art.

That’s a real change, and that reflects the president’s view that people with enough money to buy enormous art collections that appreciate enormously in value should be paying more in taxes.

Right. And of course, a tax like this is extremely perhaps maddeningly hard to actually pull off. It’s hard to get someone to describe their art collection’s value so that you can apply a 25 tax to it. So this might end up being more of a political statement than a practical tax.

Yeah, there’s also questions about whether it’s constitutional. So there’s all sorts of drama around this proposal, but it is certainly, if nothing else, a statement of the president’s intent to make people worth a lot of money pay a lot more in taxes.

OK, so that’s a lot of proposed tax increases, almost all of them focused on those who are rich and corporations. Overall, Jim, what stands out to you about this Biden term two tax increase plan?

I think we could very fairly say that it’s the largest tax increasing plan by a sitting president or a presidential nominee for a party in American history.

He wants to get a lot of money from corporations and people who earn or are worth a lot of money.

But the rub, of course, is it’s hard to see the Congressional math that lets Biden accomplish these tax increases, some of which, like you said, he couldn’t get done the first time. Why would we think he would get them done the second time even if he wins this fall?

Yeah, it would be really difficult. Biden would have to win in November. Democrats would have to take the House of Representatives back from Republicans, which is certainly possible. It’s very closely divided right now. And they’d need to hold at least 50 seats in the Senate. And then those 50 Democrats in the Senate would have to be willing to go along with far more in tax increases than Democrats were last time around.

So if there is a second term, it feels like we should assume it will be very difficult perhaps even quite unlikely he’s going to get to push through a lot of these taxes. Which makes me wonder, Jim, why is Biden running on a tax program that he knows has so little chance of becoming reality and when it’s pretty clear that he’s gotten a lot of stuff done without raising taxes? It turns out that’s not been all that essential to getting infrastructure or climate bills done. So why is he making this so central?

Several reasons. One of them is it’s very important to him rhetorically to talk about fiscal responsibility. Big parts of the Biden agenda, the CHIPS bill, the infrastructure bill, some other legislation, were not actually paid for. The spending and tax cuts were not offset by tax increases.

So they’re going to add to the debt.

Right. So they’re going to add to the debt. Same is true of the stimulus bill. But moving forward, the president has said that he’s going to pay for his agenda and he’s actually going to have some extra tax dollars coming in left over to help pay down future budget deficits. And on paper, it’s the way to pay for Biden’s other big, expansive plans that he hasn’t been able to do but wants to — universal child care, federal paid leave, investing in elder care, just a whole bunch of things that he still wants to do more — housing initiatives.

The president needs money to make a case that he’s being fiscally responsible, and this is the money that would do that.

So that’s one reason. Another reason is the calendar. Biden and his team are looking ahead to the end of 2025, and they know that if he wins another term, he will be in office at a rare moment in Washington, when basically tax policy has to be on the Congressional agenda.

Well, Republicans, when they passed their tax cuts in 2017, set a bunch of them to expire at the end of 2025 in order to lower the cost of the bill.

These are the Trump tax cuts.

The Trump tax cuts. And that includes all the tax cuts for individuals. So now that those are coming due, there’s going to be a fight in Washington over whether to extend them or make them permanent or change them in some way or just let them expire, and Democrats know there’s going to be a huge fight that will reach almost certainly the floor of the House and the Senate. And so Biden wants to be ready.

He wants to be ready with a suite of policy proposals that Democrats can basically pull off the shelf and try to use to put Republicans in a box. Basically say, we would like to keep taxes low or cut them further for low income workers, middle class workers. But we want to pay for that by raising taxes on the rich. You Republicans also want to do nice things for low and middle class workers, but you want to cut taxes for corporations and the rich, and we think that’s a political loser for you.

So Biden is ready with what they think will be a political winner for Democrats in this almost certain floor tax fight at the end of 2025.

And that brings us to the last reason why Biden is doing this, and maybe the most important, which is it’s really good politics.

Just explain that. Why is talking about tax increases, net tax increases, such good politics?

If you talk to Democratic pollsters, if you talk to people inside the White House, outside the White House, political strategists anywhere in Biden’s orbit, they all agree that the public loves the idea of forcing rich people and corporations to pay their, quote, “fair share.” It’s just become a winning and central political argument in Democratic campaigns, the idea that corporations avoid taxes, that rich people avoid taxes, and that Joe Biden is trying to position himself as a champion of the idea that they need to pay more. Those corporations and those rich people need to pay more, and he’s going to make it happen.

You’re describing this as something that is kind of a new political reality. Is that right?

Yeah, it’s evolved over the last decade or so I think. For a long time in Washington, the conventional wisdom was just couldn’t talk about tax increases of any kind. They were poison. There was a whole anti-tax movement that did a really good job of messaging that, and Democratic candidates got very scared of talking about raising taxes even on the very, very rich.

That started to turn over time. But it’s really changed. I think we saw in the 2020 election that the Democratic primary had just enormous amounts of taxes on corporations and the rich funding all sorts of policy proposals — Medicare for all and universal child care and trillions and trillions of dollars — and Democratic candidates like Liz Warren and Bernie Sanders competing to see who could tax corporations and the rich the most.

Biden is a product of that primary. He was one of the most moderate people in that group, but his proposals are really outside of the historical norm for Democratic candidates up until then. And that reflects the fact that pollsters have been doing all this research, finding that the American people, including independents and increasingly numbers of Republicans, just don’t think corporations pay their fair share and are open to the idea they should pay more.

This is really interesting, and it makes me think that what you’re really saying is that there might have been a time when a Democratic nominee like Joe Biden might have reveled in his image as an overall tax cutter. But that is not this moment, and that is not this candidate. He wants to be a tax increaser. He thinks that is where the politics are.

I think that’s exactly right when you think about tax increaser as tax increaser on the rich and on corporations. There’s two ways to be a successful populist politician. One of them is to be like Trump and run around saying you’re going to do enormous tax cuts for everybody, which is a Republican version of populism. Trump, my biggest tax cut in history, I’m going to do another huge, enormous tax cut. It’s going to be so big you won’t believe it.

There might have been a time when Democrats tried to follow that playbook. But Biden’s not doing that. He’s leaning into the other side of populism. He’s telling workers, hey, I’m on your side with these big companies. They’re trying to screw you, and I’m not going to stand for it. And so I’m going to raise their taxes. I’m going to make them pay more so that there’s more money for you, whether that’s more tax cuts or more programs or whatever.

And that is the Democratic version of populism right now, and that’s the one that Joe Biden is running on. And that’s why he’s so happy to talk about raising corporate taxes because it’s a way to tell workers, hey, I’m on your side.

Right. Even if that’s not what he’s done or ever may be able to do.

Yeah. Part of the problem with populism is that you make a lot of promises you can’t keep, and this certainly, in his first term, has been an area where the president has talked a much bigger game than he’s been able to execute. The second term might be different, but that doesn’t really matter for the campaign. What matters is the rhetoric.

Well, Jim, thank you very much. We appreciate it.

Thank you. Always a pleasure.

Here’s what else you need to know today. On Tuesday, Israel confirmed that it had carried out the airstrike that killed seven aid workers delivering food to civilians in Gaza. The attack, which occurred on Monday, struck a convoy run by the World Central Kitchen, a nonprofit group. At the time of the attack, the aid workers were traveling in clearly marked cars that designated them as non-combatants.

Israel’s Prime Minister Benjamin Netanyahu described the attack as unintentional and said that his government deeply regretted the deaths. In its own statement, World Central Kitchen called the strike unforgivable and said that as a result, it would suspend its aid work in Gaza, where millions of people are in dire need of both food and medicine.

Today’s episode was produced by Stella Tan and Mary Wilson with help from Michael Simon Johnson. It was edited by Lisa Chow, contains original music by Dan Powell and Marion Lozano, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

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• April 4, 2024   •   32:37 Israel’s Deadly Airstrike on the World Central Kitchen
• April 3, 2024   •   27:42 The Accidental Tax Cutter in Chief
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In his campaign for re-election, President Biden has said that raising taxes on the wealthy and on big corporations is at the heart of his agenda. But under his watch, overall net taxes have decreased.

Jim Tankersley, who covers economic policy for The Times, explains.

On today’s episode

Jim Tankersley , who covers economic policy at the White House for The New York Times.

Background reading

An analysis prepared for The New York Times estimates that the tax changes President Biden has ushered into law will amount to a net cut of about $600 billion over four years.

“Does anybody here think the tax code’s fair?” For Mr. Biden, tax policy has been at the center of his efforts to make the economy more equitable.

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Jim Tankersley writes about economic policy at the White House and how it affects the country and the world. He has covered the topic for more than a dozen years in Washington, with a focus on the middle class. More about Jim Tankersley

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Fine error bounds for approximate asymmetric saddle point problems

• Published: 06 April 2024
• Volume 43 , article number  160 , ( 2024 )

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• Vitoriano Ruas   ORCID: orcid.org/0000-0001-6308-879X 1  

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, to guarantee well-posedness, are known [(see, e.g., Exercise 2.14 of Ern and Guermond (Theory and practice of finite elements, Applied mathematical, sciences, Springer, 2004)]. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately.

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In doing so, our aim is to further publish in the English language for the first time in a means widely distributed with due consent (Dupire 2013 ), some results available only in Dupire’s doctoral thesis written in Portuguese language.

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Ruas, V. Fine error bounds for approximate asymmetric saddle point problems. Comp. Appl. Math. 43 , 160 (2024). https://doi.org/10.1007/s40314-024-02678-7

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