# NCERT solutions for class 12 Maths

## About NCERT Solutions For Class 12 Maths

Class 12^{th} maths is already considered one of the advanced mathematics. The team at Entrancei 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

- 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.
- 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). - Set ‘X’ is called domain of the function ‘f’. Set ‘Y’ is called the co-domain of the function ‘f’.
- Set of images of different elements of set X is called the range of the function ‘f’.
- The function ‘h’ defined above is called the composition of f and g and is denoted by gof.
- 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. - f(x) - f(-x) = 0 even function
- f(x) + f(-x) = 0 odd function

### Chapter-2: Inverse Trigonometric Functions

### Chapter-3: Matrices

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 a_{11}, a_{12}… etc are called elements of the matrix. The element a_{ij} belongs to *i*th row and j^{th}column.

- (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
_{ij}of a square matrix - A = [a
_{ij}]_{n}*_{n}for which i = j i.e., the elements a_{11}, a_{22},…,a_{nn}are 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 nondiagonal 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

- Exercise-4.1
- Exercise-4.2
- Exercise-4.3
- Exercise-4.4
- Exercise-4.5
- Exercise-4.6
- Miscellanious Exercise chapter-4

Important properties of Determinants

- If rows be changed into columns and columns into the rows, the determinant remains unaltered.
- If any two row (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.
- 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.
- 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.
- 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.
- 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
- If each entry in any row (or any column) of determinant is zero, then the value of determinant is equal to zero.
- 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.
- 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

- Exercise-5.1
- Exercise-5.2
- Exercise-5.3
- Exercise-5.4
- Exercise-5.5
- Exercise-5.6
- Exercise-5.7
- Exercise-5.8
- Miscellanious Exercise chapter-5

### Chapter-6: Application of Derivatives

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. Let*S*be the displacement described by a particle over the time interval*t*. Then*S*will be a function of*t*, we write as*S*=*f*(*t*).

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

### Chapter-7: Integrals

- Exercise-7.1
- Exercise-7.2
- Exercise-7.3
- Exercise-7.4
- Exercise-7.5
- Exercise-7.6
- Exercise-7.7
- Exercise-7.8
- Exercise-7.9
- Miscellanious Exercise chapter-7

### Important Formulas

### Chapter-8: Application of Integrals

### Chapter-9: Differential Equations

- Exercise-9.1
- Exercise-9.2
- Exercise-9.3
- Exercise-9.4
- Exercise-9.5
- Exercise-9.6
- Miscellanious Exercise chapter-9

### Chapter-10: Vector Algebra

### Chapter-11: Three Dimensional Geometry

### Chapter-12: Linear Programming

### 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 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 Entrancei. 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 Entrancei.

The entire team of NCERT solutions for class 12 maths at Entrancei 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 Entrancei 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. Entrancei 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

Entrancei 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.
- Entrancei 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 solution 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.

### FAQ For NCERT Solutions for class 12 Maths

**Q-1. 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 Entrancei 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.**Q-2. 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.**Q-3. 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)

Do solve all questions given in Entrancei Important questions for class 12 Maths.**Q-4. 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. Entrancei uploaded CBSE Sample paper for class 12 Maths with detail solutions to help you out.**Q-5. 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.**Q-6. 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.**Q-7. 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.