NCERT solutions for class 12 Maths

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.

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.

Chapters of NCERT Solutions for Class 12 Maths

Chapter-1: Relations and Functions

1. Exercise-1.1
2. Exercise-1.2
3. Exercise-1.3
4. Exercise-1.4&
5. Miscellanious Exercise chapter-1

1. A mapping f : X → Y is said to be a function if each element in the set X has it’s image in set Y. It is possible that a few elements in the set Y are present, which are not the images of any element in set X.
2. Every element in set X should have one and only one image. That means it is impossible to have more than one image for a specific element in set X. Functions can’t be multi-valued
   (A mapping that is multi-valued is called a relation from X to Y).
3. Set ‘X’ is called domain of the function ‘f’. Set ‘Y’ is called the co-domain of the function ‘f’.
4. Set of images of different elements of set X is called the range of the function ‘f’.
5. The function ‘h’ defined above is called the composition of f and g and is denoted by gof.
6. If f : X→Y be a function defined by y = f(x) such that f is both one-one and onto, then there exists a uniquefunction g : Y→X such that for each y→Y, g(y) = x if and only if y = f(x). The function g so defined is called the inverse of f and denoted by f ^-1. Also if g is the inverse of f, then f is the inverse of g and the two functions f and g are said to be inverses of each other. The condition for existence of inverse of a function is that the function must be one-one and onto.
7. f(x) - f(-x) = 0 even function
8. f(x) + f(-x) = 0 odd function

Chapter-2: Inverse Trigonometric Functions

1. Exercise-2.1
2. Exercise-2.2
3. ​Miscellanious Exercise chapter-2

Chapter-3: Matrices

1. Exercise-3.1
2. Exercise-3.2
3. Exercise-3.3
4. Exercise-3.4
5. Miscellanious Exercise chapter-3

A set of mn numbers (real or complex) arranged in the form of a rectangular array having 'm' rows & 'n' columns is called an m  n matrix. [to be read as m by n matrix].The numbers a₁₁, a₁₂… etc are called elements of the matrix. The element a_ij belongs to ith row and j^thcolumn.

• (i)Square Matrix An m n matrix for which m = n(i.e. the number of rows is equal to the number of columns) is called a square matrix of order n. The element a_ijof a square matrix
• A = [a_ij]_n*_nfor which i = j i.e., the elements a₁₁, a₂₂,…,a_nnare called the diagonal elements.
• (ii)Null Matrix or Zero Matrix
• The m -n matrix whose all elements are zero is called a null matrix of order m - n. It is usually denoted by O.
• (iii)Unit Matrix or Identity Matrix
• A square matrix each of whose principal diagonal element is '1' and each of whose nondiagonal element is equal to zero is called a unit matrix or an identity matrix and is denoted by I. I_n will denote a unit matrix of order n

Chapter-4: Determinants

1. Exercise-4.1
2. Exercise-4.2
3. Exercise-4.3
4. ​Exercise-4.4
5. Exercise-4.5
6. Exercise-4.6
7. Miscellanious Exercise chapter-4

Important properties of Determinants

1. If rows be changed into columns and columns into the rows, the determinant remains unaltered.
2. If any two row (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.
3. If two rows (or two columns) in a determinant have corresponding entries that are equal (or proportional), the value of determinant is equal to zero.
4. If each of the entries of one row (or columns) of a determinant is multiplied by a nonzero constant k, then the determinant gets multiplied by k.
5. If each entry in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants.
6. If to each element of a line (row or column) of a determinant be added the equi-multiples of the corresponding elements of one or more parallel lines, the determinant remains unaltered
7. If each entry in any row (or any column) of determinant is zero, then the value of determinant is equal to zero.
8. If a determinant D vanishes for x = a, then (x-a) is a factor of D, In other words, if two rows (or two columns) become identical for x = a. then (x- a) is a factor of D.
9. If in a determinant (of order three or more) the elements in all the rows (columns) are in A.P. with same or different common difference, the value of the determinant is zero.

Chapter-5: Continuity and Differentiability

1. Exercise-5.1
2. Exercise-5.2
3. Exercise-5.3
4. Exercise-5.4
5. Exercise-5.5
6. Exercise-5.6
7. Exercise-5.7
8. Exercise-5.8
9. Miscellanious Exercise chapter-5

Chapter-6: Application of Derivatives

1. Exercise-6.1
2. Exercise-6.2
3. Exercise-6.3
4. ​Exercise-6.4
5. Exercise-6.5
6. Miscellanious Exercise chapter-6

In this section we examine some applications in which derivatives are used to represent and interpret the rates at which things change in the world around us. LetSbe the displacement described by a particle over the time intervalt. ThenSwill be a function oft, we write asS=f(t).

The velocity is the derivative of the position functionS=f(t) with respect to time. At timetthe velocity is ds/dt. Acceleration is the derivative of velocity with respect to time. If a body's position at timetisS=f(t), the body's acceleration at timetis a(t)=dv/dt

Chapter-7: Integrals

1. Exercise-7.1
2. Exercise-7.2
3. Exercise-7.3
4. ​Exercise-7.4
5. Exercise-7.5
6. Exercise-7.6
7. Exercise-7.7
8. Exercise-7.8
9. Exercise-7.9
10. Miscellanious Exercise chapter-7

Important Formulas

Chapter-8: Application of Integrals

1. Exercise-8.1
2. Exercise-8.2
3. Miscellanious Exercise chapter-8

Chapter-9: Differential Equations

1. Exercise-9.1
2. Exercise-9.2
3. Exercise-9.3
4. ​Exercise-9.4
5. Exercise-9.5
6. Exercise-9.6
7. Miscellanious Exercise chapter-9

Chapter-10: Vector Algebra

1. Exercise-10.1
2. Exercise-10.2
3. Exercise-10.3
4. Miscellanious Exercise chapter-10

Chapter-11: Three Dimensional Geometry

1. Exercise-11.1
2. Exercise-11.2
3. Exercise-11.3
4. Miscellanious Exercise chapter-11

Chapter-12: Linear Programming

1. Exercise-12.1
2. Exercise-12.2
3. Miscellanious Exercise chapter-12

Chapter-13: Probability

1. Exercise-13.1
2. Exercise-13.2
3. Exercise-13.3
4. ​Exercise-13.4
5. Exercise-13.5
6. Miscellanious Exercise chapter-13

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 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’s 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 advance 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 by Physics 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.

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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.

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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.

1. All the questions given in the NCERT exercise are solved.
2. All NCERT solutions for class 12 Maths are errorless and proof-readied several time.
3. All answers are explained and written as per the guideline of CBSE to score good marks in class 12 final exam.
4. 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.
5. Physics Wallah NCERT solutions for class 12 Maths consist of exercise wise pages to help you find your maths doubts instantly.
6. Each question of NCERT solutions for class 12 Maths is added with all required step which are supposed to write in final exam.
7. Step wise solution of questions will help you to understand the method of solving the questions and clear your concepts.
8. 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.
9. 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.