NCERT Solutions Class 12 Maths

NCERT Solutions for class 12 Mathematics

Class 12 is a game changer in a student's life, when they decide on a career or profession. The subjects that students study in Classes 11 and 12 form the foundation for their career choice or professional course. As a result, all subjects are equally important to them. Mathematics is especially useful for students who want to pursue Engineering or be a statistician in the future. The Central Board of Secondary Education (CBSE) has divided the curriculum into terms I and II for the 2022-23 academic year. The first term has seven chapters, while the second term has only six. Students can access the CBSE Mathematics syllabus below.

CBSE Syllabus 

Term 1

Unit-I: Relations and Functions

  1. Relations and Functions

Types of relations: Equivalence, Reflexive, Symmetric, and Transitive relations. One to one and onto functions.

 

  1. Inverse Trigonometric Functions

Definition, range, domain, principal value branch.

 

Unit-II: Algebra

  1. 1. Matrix

Matrix concepts, notation, order, equality, matrix types, zero and identity matrices, matrix transpose, symmetric and skew symmetric matrices Matrix operations: Addition, multiplication, and scalar multiplication Simple addition, multiplication, and scalar multiplication properties Non-commutativity of matrix multiplication, invertible matrices (All matrices will have real entries).

  1. Determinants

Determinants of square matrices (up to 3 x 3 matrices), minors, co-factors, and determinant applications in finding the area of a triangle. Adjoint and inverse of a square matrix. Solve a system of linear equations in two or three variables (with a unique solution) using the inverse of a matric.

 

Unit-III: Calculus

  1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

 

  1. Applications of Derivatives

Applications of derivatives: increasing/decreasing functions, tangents and normals, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real life situations).

 

Unit-V: Linear Programming

  1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimisation, different types of linear programming (L.P.) problems. graphical method of solution for problems in two variables, feasible and infeasible regions (bounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Term 2

Unit III: Calculus

  1. Integrals

Integration as an inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals and problems based on them. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

  1. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, parabolas; area of circles /ellipses (in standard form only) (the region should be clearly identifiable).

  1. Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation.

Unit-IV: Vectors and Three-Dimensional Geometry

  1. Vectors 

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

  1. Three – Dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Distance of a point from a plane.

 

Unit-VI: Probability

  1. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution.

 

The CBSE and many state boards are now using the NCERT official textbooks for Classes 11 and 12 in their school course curriculum. Furthermore, the NCERT book is recommended for various competitive examinations such as IIT, NEET, UPSC, and so on. This is because the textbook's content is written in a straightforward and easy-to-understand manner. By studying these books, one can progress from the basics to a higher level.

 

NCERT Solutions for class 12 Mathematics - CBSE 2022-23 Download for Free

The NCERT Mathematics textbook for Class 12 is divided into 13 major chapters. Practicing the problems given at the end of the chapter in the textbook allows students to quickly gain speed and master problem-solving skills. Hence, students should work on these with an updated solutions book. Working on NCERT Solutions for Class 12 Mathematics will help you brush up on your basics and understand the concept in depth.

Students will benefit from studying the CBSE Mathematics NCERT Solutions as each concept is explained in detail.  To download the solutions, go to the chapter-specific link and free download the Class 12 Mathematics NCERT solutions from there. These Class 12 Mathematics solutions will also assist you in passing the most difficult engineering entrance exams such as JEE Main, JEE Advanced, BITSAT, and others.

Extramarks provides NCERT Solutions for Class 12 Mathematics that have been designed by subject experts to support the smooth and precise understanding of concepts. These NCERT solutions provide detailed, step-by-step explanations of problems found in textbooks. Aside from solutions for each chapter of CBSE Class 12 Mathematics, students can also access previous year question papers and important questions as study materials for preparing for their Class 12 board exams.

 

NCERT Solutions for class 12 Mathematics - Download

For all in-text questions, NCERT Solutions for Class 12 Mathematics provide detailed and easy-to-understand answers. These solutions are excellent learning materials for students preparing for Class 12 exams, as they will assist them in answering questions more efficiently. The solutions in this section are based on the NCERT syllabus and curriculum. With these solutions, students can better prepare for their CBSE Class 12 exams.

 

NCERT Solutions Exercise for class 12 Mathematics

Mathematics solutions for Class 12 NCERT provides a solid conceptual foundation for all of the exercises included in the CBSE Class 12 Mathematics Syllabus. If you want to excel in Class 12 Mathematics, then these Chapter Wise NCERT Solutions from Extramarks by subject matter experts will be of great assistance to you. It covers all of the key theorems and formulae in each chapter. Once you have mastered the concepts and logic of all sums, you should be able to achieve a good percentage in Class 12 Mathematics.

Students can look into the chapter’s name and download the solution for that particular chapter and begin solving textbook questions.

 

Chapter 1: Relations and Functions 

The Chapter Relations and Functions goes over the definitions and different types of relations and functions, the composition of functions and invertible functions, binary operations, and various examples. This chapter includes five exercises in total. The main exercises are based on the types of relations: reflexive, symmetric, transitive, and equivalence relations, one to one and onto functions. Miscellaneous exercises cover standard problems based on all of the topics discussed in this chapter.

 

Chapter 2: Inverse Trigonometric Functions – Term 1

Chapter 2 of NCERT Class 12 Mathematics on Inverse Trigonometric Functions, provides an account of various topics such as remarks based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions, and miscellaneous examples. These ideas are well-explained with examples. The solutions to the chapter's exercises can be found here.

 

Chapter 3: Matrices 

Matrices are widely regarded as one of mathematics' most effective tools. When compared to other methods, it simplifies our work significantly. Students will learn about matrices, their notations, order, equality, and the various types of matrices in this chapter. They will also be introduced to the zero and identity matrices, as well as symmetric and skew-symmetric matrices. They will learn to find the transpose of a matrix as well as operations on matrices such as addition, multiplication, and multiplication with a scalar. This chapter will also cover topics such as non-commutativity of matrices, multiplication, and invertible matrices.

 

Chapter 4: Determinants

The topic of determinants is covered in  Chapter 4 of the 12 Class NCERT Mathematics Solution. Students will learn about the definition and meaning of determinants, properties of determinants, remarks based on determinant order, minors and cofactors of determinants, finding the area of a triangle using determinants, adjoint of a matrix, inverse of a matrix, applications of determinants and matrices, and various examples. Each exercise solution covered in this chapter is linked to below.

 

Chapter 5: Continuity and Differentiability

The definition of continuity is presented in Chapter 5 of the NCERT textbook. Students then learn about continuity, algebra of continuous functions, definition and meaning of differentiability, derivatives of composite functions, derivatives of implicit functions, the derivative of inverse trigonometric functions, exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric forms, second-order derivatives, and the mean value theorem through various examples. Students can find exercises that properly explain these concepts, along with solutions, in this section.

 

Chapter 6: Applications of Derivatives

Chapter 6 of the NCERT textbook defines derivatives, the rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, maxima and minima, first derivative test, maximum and minimum values of a function in a closed interval, and various examples. Students can find exercises that properly explain these concepts in this section.

 

TERM – II 

Chapter 7: Integrals

We will see the 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, integration by parts, integration using trigonometric identities, integration of some integral functions. We have provided exercises with solutions based on the chapter's topics.

 

Chapter 8: Applications of Integrals

In Chapter 8 of the NCERT textbook, we continue our discussion of integrals, beginning with definition, areas under simple curves, areas of the region bounded by a curve and a line, areas between two curves, and various examples. Students can access the exercises with solutions explaining the concepts from this chapter via the links provided below.

 

Chapter 9: Differential Equations 

Students concentrate on the definition of differential equations, basic concepts related to differential equations, degree of a differential equation, order of a differential equation, general and particular solutions of a differential equation, formation of a differential equation whose general solution is given, procedure to form a differential equation that will represent a given family of curves, methods of solving first order, first degree differential equations, methods of solving first order, first degree differential equations, methods of solving first order, first degree differential equations, methods of solving first The solutions to the chapter's exercises are available here.

 

Chapter 10: Vector Algebra

Students are introduced to vector algebra in this chapter. This chapter's concepts cover how to find a position vector, some basic vector algebra concepts, direction cosines, vector types such as zero vector, unit vector, collinear vector, equal vector, negative of a vector, addition of vectors, properties of vector addition, multiplication of a vector by a scalar, components of a vector, vector joining two points, section formula, product of two vectors, scalar or dot product of two vectors, properties of scalar Students can access the exercises with solutions explaining the concepts from this chapter by clicking on the links provided below.

 

Chapter 11: Three Dimensional Geometry

In Chapter Three Dimensional Geometry, you will learn about the direction cosines and direction ratios of a line connecting two points, as well as the equations of lines and planes in space under various conditions. Students will also learn about the angle formed by two lines, a line and a plane, two planes, the shortest distance between two skew lines, and the distance between two points. Cartesian equations, vector equations of a line, skew and coplanar lines, and skew and coplanar lines are also covered. This chapter also discusses a plane's cartesian and vector equations.

 

Chapter 12: Linear Programming

This chapter introduces students to linear programming. This chapter discusses the linear programming problem and its mathematical formulation, the problem's mathematical formulation, the graphical method of solving linear programming problems, and examples of various types of linear programming problems. Students can find exercises that properly explain these concepts and provide solutions.

 

CBSE unit wise weightage for term 1 class 12 Mathematics

According to the split syllabus, term 1 for CBSE Class 12 students  has the first six chapters of the syllabus. These include Relations and Functions, Inverse Trigonometric Functions, Matrices, Determinants, Continuity and Differentiability, and Applications of Derivatives. Students can refer to the table given below and focus on the weightage given to the chapters in the final examination to strategize their study.

Unit Unit Name Marks
1 Relations and Functions 08
2 Algebra 10
3 Calculus 17
5 Linear Programming 05
Total 40
Internal Assessment Marks 10
TOTAL MARKS 50

 

Class 12 term 1 - Internal assessment

Internal assessments have a total mark of 10. It consists of periodic tests, which carry a total of 5 marks, and mathematics activities, which include a file record, term end assessment of one particular activity, and the viva-voce asked. The detailed structure for IA can be assessed below.

Internal Assessment Marks
Periodic Tests 05
Mathematics Activities: Activity file record + Term end assessment of one activity & Viva 05
Total Marks 10

 

CBSE unit wise weightage for term 2 class 12 Mathematics

The CBSE syllabus is provided from Chapters 7 to 13 as term 2. Students can refer to the below given table and plan their routine based on the weightage given to each of the subjects.

Unit Unit Name Marks
3 Calculus 18
4 Vectors and Three-Dimensional Geometry 14
6 Probability 08
Total 40
Internal Assessment Marks 10
Total Marks 50

 

Class 12 term 2 - Internal assessment

The same marking scheme follows throughout the academic session. Periodic tests carry 5 marks, and activities relating to Mathematics such as term end assessment and viva carry 5 marks. In total, the IA marks assigned for this term is 10. Students can look through the split given below.

Internal Assessment Marks
Periodic Tests 05
Mathematics Activities: Activity file record + Term end assessment of one activity & Viva 05
Total Marks 10

 

Advantages of NCERT Solutions for CBSE class 12 Mathematics from Extramarks

  • The NCERT Solutions for Mathematics Class 12 are available for anyone, anywhere to download for freeStudents will gain a better understanding of all basic concepts if they consult the Class 12 Mathematics NCERT Solutions
  • Most engineering competitive exam syllabuses, such as JEE Main, JEE Advanced, and BITSAT, are nearly identical to those of the CBSE Class 11 and 12 Mathematics Syllabus. Having a good command of the NCERT Solutions for Class 12 Mathematics will thus assist in easily passing the competitive exams
  • The NCERT Solutions for Mathematics 12 Class will assist students in completing their homework and assignments on time

Students in Class 12 will benefit from studying NCERT Solutions because they will be able to effectively extract conceptual meaning from any problem. Probability, calculus, coordinate geometry, and trigonometry are simple topics that students can easily master with practice. CBSE Mathematics Class 12th NCERT Solutions provide detailed solutions to all of the problems covered in the syllabus. All of the chapters listed above are equally important and can be grasped with consistent practice and effort. 

 

Benefits of solving previous years question papers

Some advanced concept-oriented chapters in Class 12 require extensive practise. Solving previous year question papers is extremely beneficial when preparing for a board exam. Here are some of the benefits listed:

  • Students will be able to analyse the nature and types of questions asked in previous CBSE Boards. The marking scheme will also inform them about the content required for each answer and allow them to plan them in advance. 
  • Students get a feel of what their final examinations will be like. They can improve their concentration and stamina by solving CBSE Class 10 previous year papers. This will give you a significant advantage in the exam.
  • By practising these question papers, students will gain knowledge about their strengths and weaknesses. This can help them strategise their answers and prepare accordingly. Also, students can boost their learning of specific topics if they feel unprepared for them.

 

Preparation Tips

Students should concentrate on finishing the syllabus and begin with their revisions. Additional books and test papers can be used to gauge their performance. It is important for students to not get overwhelmed and keep their cool throughout their preparation. Here are some tips students can follow to make their preparation an effective and rewarding one to score maximum marks.

    • Choosing the best and essential materials will ensure better points. Students can use the solutions given to have an in-depth understanding of the concepts. This will help them create impressionable answers
    • Creating a time-table and setting dates on which topics are to be completed helps students to stay on track. Subjects with more weightage and difficulty can have more time to study
    • Putting in consistent work is crucial for students to get their desired scores. Skimming through the chapters for the first time at the last minute will not only create stress and anxiety, but also make your chances of securing good marks slim
    • While reading through the textbooks and solutions, students may make their own short notes. These should be pointers and not lengthy notes written in such a way that reading them only takes 2-5 minutes. For example, a long definition can be broken into pointers so that the student does not miss the main part of it
  • Set a time period and check the time you spend on completing the syllabus. Students are advised to not drag their preparation time, as they must also spend valuable time on revision as well
  • NCERT Solutions for Class 12 will build your knowledge not just for the CBSE syllabus, but for the subject matter. You can easily download the solutions from Extramarks subject-wise and chapter-wise. You can add in practice questions by referring to sample question papers and previous year’s question papers
  • After covering the syllabus, students must start revising from the notes they made. This will help them understand the basic concepts and their pointers will connect them together
  • To complete the practice needed for CBSE examinations, students need to solve previous years and sample questions. Students prepare lengthy notes and are well-prepared for the questions. However, they don’t prepare for the time they should spend per question. Therefore, they end up spending most of the time writing or thinking about one answer while ignoring the rest. They can time themselves and evaluate their performance by solving these question papers.  By solving these question papers, students can also gauge their strong and weak subjects and work on them. This should be done at least 2-3 months prior to the examinations
  • It is important for students to work on feedback and be open to change. While solving the previous years and sample question papers, Regular self-assessment and self-reflection will guide students
  • With all this rush, it is absolutely necessary to keep one’s health in mind. Drink plenty of water, take regular walk breaks around your favourite neighbourhoods, and talk to people who mean most to you. It keeps your head in the game and gets you motivated
  • Make sure you reach the exam hall at least 15 minutes before the time mentioned. To perform your best, it is important that you keep calm and avoid undue stress
  •  When the examination starts, be sure to thoroughly read the paper and first answer the ones you’re most confident with
  • Leave the last 10-15 minutes to review the paper. Make sure that you have mentioned all the points needed for the question
  • Ensure that you write the correct question number against each answer in the sheet. You can allocate the last 2-3 minutes of the exam to do so

 

If?sin?1?1x?=?sin?1?1a?+?sin?11b,?then?the?value?of?x?is aba2???1?+?b2???1

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aba2???1?-?b2???1

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2aba2???1? aba2 − 1 + b2 − 1 Let?sin?1?1a?=??,?sin?1?1b?=??,?then?sin?1?1x?=???+????sinsin?11x?=sin???+????1x?=?sin??cos??+?cos??sin?            =?1a?1?1b2?+?1b?1?1a2            =?b2???1ab?+?a2???1ab?  ?x?=?aba2???1?+?b2???1

For viewing question paper please click here

FAQs (Frequently Asked Questions)
1. How can I get the answers for class 12 exercise questions ?

Students can access the solutions while working through the problems in the NCERT textbook by downloading it. The solutions are designed in a systematic manner based on the CBSE board’s concept weightage. The difficulty level of these exercises is high enough to help students prepare for the board exams with confidence.

2. Is the class 12 solution free?

Extramarks provides free and student-friendly materials. The solutions are explained with a stepwise approach. NCERT Solutions for Mathematics Class 12 are created by expert teachers who are familiar with the exam pattern and syllabus and have years of teaching experience. 

3. Will the NCERT Class 12 Mathematics Solutions book help me with making notes?

 It is critical to take good notes while studying for your CBSE exams. These will help you to get good grades and prepare you for other entrance examinations. The NCERT Solutions for Class 12 Mathematicswill give you detailed explanations for each concept for you to read, analyse, and create your own brief study notes. By going over the solutions over and over again, you can have a quick glance at the notes before your exams rather than reading the entire study materials.

4. What are the chapters covered in the NCERT Solutions for Class 12 Mathematics?

The NCERT textbook for Class 12 Mathematics is divided into two parts. Part 1 contains chapters 1–7, while Part 2 contains chapters 8–13. Matrices, Inverse Trigonometric Functions, Relations and Functions, Determinants, Applications of Derivatives, Continuity and Differentiability, Applications of Integrals, Vector Algebra, Differential Equations, Three Dimensional Geometry, Probability, and Linear Programming Students can download the final or chapter-wise solution provided by Extramarks.

5. Should I invest in an NCERT Solutions Book only in Class 12th?

Students who want to broaden their knowledge beyond the scope of their examination can purchase a solutions book at any time.For example, the NCERT Solutions for Class 1 cover essential Mathematics and English, which form the  basis for Class 12 Mathematics and English.

6. Do questions come from outside the syllabus for the CBSE final examinations?

In most cases,  CBSE rarely asks questions that are outside of the given syllabus. Students can confidently prepare for the important questions repeatedly asked in the board exams.

7. Do I answer extra questions in the CBSE Board exams?

If you have sufficient time, you can. Most students do go for answering extra questions because the type of questions asked makes it easier to write crisp answers than it is in the regular examinations of schools. Answers with the highest marks are usually considered for calculating the final score. Solving previous years question papers will help you hone your time management skills.

8. How do I calculate my CBSE Board exam percentage?

 Firstly, students need to calculate their CGPA. The CGPA is calculated by obtaining the sum of grade points in five major subjects and dividing it by five. Then, students must multiply their CGPA by 9.5 to get their percentage. For example, if a student has a CGPA of 8.0, then the percentage would be 8X9.5, which is 76%.

9. Are the CBSE exam results given in grades or marks?

Though the internal and external assessments will be given scores, students’ final report cards will only include their grades. It is done on a 5-point scale from A to E. CBSE has taken this step to promote a healthy competitive environment among schools and students.

10. How will the first term's final grades be calculated?

The total of the marks obtained in the two FAs (Formative Assessment) and the SA I (Summative Assessment) in a particular subject is calculated. Following that, it is converted into a grade and a grade point for that specific subject.

Term- I FA1 (ten percent) + FA2 (ten percent) + SA1 (20 percent) = 40%