# NCERT solutions for Class 12 Maths

NCERT Solutions for Class 12 Maths provides a detailed explanation to promote a better conceptual understanding in students for all the topics included in CBSE Class 12 Maths Syllabus. UP Board Solution for Class 12 Maths and NCERT Solutions are so students in the preparation of undergraduate level engineering entrance exams like JEE Main, VITEEE, BITSAT, MHTCET etc. NCERT Solutions Class 12, designed by our Expert Maths Teacher to facilitate smooth and clear understanding of concepts and logics. 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 2021-2022 can pass the 12th class board exam. You can also download study material NCERT Class12 Maths book PDF here with free of cost. These Free NCERT Solutions for Class X Maths PDF can be shared easily to anyone.## NCERT Solutions for Class 12 Maths All Chapters

You can download NCERT Class 12 Maths Solutions for all exercises by just click these links here. Class 12th Maths Solution is beneficial for the class student 12 as it not only help you to score maximum marks in your CBSE, UP Board, MP Board as well as other boards examinations but also prepares you for undergraduate competition exams. You can also get all solutions online, and you can also download of Class 12 Maths Chapter Wise Solutions in PDF with FREE of cost. NCERT Exemplar Class 12 Maths Solutions are also available here which help you to get an overall idea of board exam pattern.

### Free NCERT Solutions for Class 12 Maths All Chapters

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

## NCERT Class 12 Maths Chapters

Class 12 Maths provided here covers all chapters s on each of the class 12 maths syllabus topics along with their sub-topics. With these all exercise wise solutions, the student can able to get a deeper understanding of the concepts and develop problem-solving abilities. NCERT Class 12 Maths Chapters includes there are 13 chapters in total like Relations and Functions, Inverse Trigonometric Functions, Matrices, Continuity and Differentiability, Applications of Derivatives, Integrals, Application of Integrals, Vector Algebra, Probability etc.

## 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 4 exercises in this NCERT Solutions for class 12 Maths Chapter 1 which covers types of relations and functions, the composition of functions, domain, co-domain, and composite functions, the inverse of a function and invertible function etc. You can download all these NCERT Solutions for class 12 Maths PDF with free of cost. 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. You can get both Chapter Relations and Functions class 12 Maths solutions in Hindi medium and English medium.

## Free NCERT Solutions for Class 12 Maths Chapter 1 in PDF

### Class 12 Maths Chapter 1 Solution in English PDF Format

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- Download Class 12 Maths Chapter 1 Exercise 1.1 in PDF
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- Download Class 12 Maths Chapter 1 Miscellaneous Exercise 1
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## 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. There are two exercises which explain ranges of trigonometric functions and observe the behaviour in the graphical representation.

NCERT Solutions for Class 12 Maths Chapter 2 for all exercises are solved by Mathematics teacher and solutions are well-structured so that it would be easier for students to learn and understand. All the solutions are based on 12th Class Maths Book Solution. You can share these NCERT Solutions for Class 12 Maths PDF to your friend also. Students can get here Chapter Inverse Trigonometric Functions Class 12 Maths Solutions in Hindi medium and English medium.

## Free NCERT Solutions for Class 12 Maths Chapter 2 in PDF

### Class 12 Maths Chapter 2 Solution in English PDF Format

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- Downlaod Class 12 Maths Chapter 2 Exercise 2.1 in PDF
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- Downlaod Class 12 Maths Chapter 2 Miscellaneous Exercise 2
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## 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, 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. Here, the student can find the exercise solution links NCERT Solutions for Class 12 Maths Chapter 3 with FREE of cost. There are four Exercises in this chapter-3. All solutions are based on 12th Class Maths Book Solution. Students can download NCERT Solutions for Class 12 Maths PDF with free of cost. You can also avail both Chapter Matrices Class 12 Maths Solutions in Hindi medium and English medium.

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## 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. 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 or three variables with the use of inverse of matrices.

## Free NCERT Solutions for Class 12 Maths Chapter 4 Determinants in PDF

### Class 12 Maths Chapter 4 Solution in English PDF Format

- Class 12 Maths Chapter 4 Exercise 4.1 Solutions
- Class 12 Maths Chapter 4 Exercise 4.2 Solutions
- Class 12 Maths Chapter 4 Exercise 4.3 Solutions
- Class 12 Maths Chapter 4 Exercise 4.4 Solutions
- Class 12 Maths Chapter 4 Exercise 4.5 Solutions
- Class 12 Maths Chapter 4 Exercise 4.6 Solutions
- Class 12 Maths Chapter 4 Miscellaneous Exercise

## 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. And 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 many miscellaneous examples. You can download Free NCERT Solutions for Class 12 Maths PDF unlimited. All exercises of NCERT Solutions for Class 12 Maths Chapter 5 are solved by the expert Maths teacher. You can book also download for both Chapter Continuity and Differentiability Class 12 Maths Solutions in Hindi medium and English medium.

## Free NCERT Solutions for Class 12 Maths Chapter 5 in PDF

### Class 12 Maths Chapter 5 Solution in English PDF Format

- Class 12 Maths Chapter 5 Exercise 5.1 Solutions
- Class 12 Maths Chapter 5 Exercise 5.2 Solutions
- Class 12 Maths Chapter 5 Exercise 5.3 Solutions
- Class 12 Maths Chapter 5 Exercise 5.4 Solutions
- Class 12 Maths Chapter 5 Exercise 5.5 Solutions
- Class 12 Maths Chapter 5 Exercise 5.6 Solutions
- Class 12 Maths Chapter 5 Exercise 5.7 Solutions
- Class 12 Maths Chapter 5 Exercise 5.8 Solutions
- Class 12 Maths Chapter 5 Miscellaneous Exercise

## 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, 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 Class 12 Maths Chapter 6 you can get a detailed knowledge of Applications of Derivatives. All the solutions are based on the latest 12th Class Maths Book Solution. Here you can download NCERT Solutions for Class 12 Maths PDF with free of cost. Student can learn for both Chapter Applications of Derivatives Class 12 Maths Solutions in Hindi medium and English Medium.

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## NCERT Solutions for Class 12 Maths Chapter 7 – Integrals

NCERT Solutions for Class 12 Maths Chapter 7 Integrals there are eleven exercises which cover Basic properties of definite integrals and evaluation of definite integrals. Definition of the indefinite integral, inverse process of differentiation, geometrical interpretation of indefinite integral, comparison between differentiation and integration, properties of indefinite integral, methods of integration such as integration by substitution, integration by parts, integration using partial fractions, integration of some integral functions, integration using trigonometric identities, definition and meaning of definite integrals, definite integral the limit of a sum and at last miscellaneous examples.

NCERT Solutions for Class 12 Maths Chapter 7 are based on the 12th Class Maths Book Solution. Students can download for both Chapter Integrals Class 12 Maths Solutions in Hindi medium and English medium.

## Free NCERT Solutions for Class 12 Maths Chapter 7 in PDF

### Class 12 Maths Chapter 7 Solution in English PDF Format

- Class 12 Maths Chapter 7 Exercise 7.1 Solutions
- Class 12 Maths Chapter 7 Exercise 7.2 Solutions
- Class 12 Maths Chapter 7 Exercise 7.3 Solutions
- Class 12 Maths Chapter 7 Exercise 7.4 Solutions
- Class 12 Maths Chapter 7 Exercise 7.5 Solutions
- Class 12 Maths Chapter 7 Exercise 7.6 Solutions
- Class 12 Maths Chapter 7 Exercise 7.7 Solutions
- Class 12 Maths Chapter 7 Exercise 7.8 Solutions
- Class 12 Maths Chapter 7 Exercise 7.9 Solutions
- Class 12 Maths Chapter 7 Exercise 7.10 Solutions
- Class 12 Maths Chapter 7 Exercise 7.11 Solutions
- Class 12 Maths Chapter 7 Miscellaneous Exercise

## 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. 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. You can also download unlimited NCERT Solutions for Class 12 Maths PDF with free of cost. All solutions are based on 12th Class Maths Book Solution. Students can download Chapter Applications of Integrals Class 12 Maths Solutions in Hindi medium.

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## 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. And clear 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. And also it covers some miscellaneous examples. NCERT Solutions for Class 12 Maths Chapter 9 is an important topic which needs to be well practised. Student can download for both Differential Equations Class 12 Maths Solutions in Hindi medium and English medium.

## Free NCERT Solutions for Class 12 Maths Chapter 9 in PDF

### Class 12 Maths Chapter 9 Solution in English PDF Format

- Class 12 Maths Chapter 9 Exercise 9.1 Solutions
- Class 12 Maths Chapter 9 Exercise 9.2 Solutions
- Class 12 Maths Chapter 9 Exercise 9.3 Solutions
- Class 12 Maths Chapter 9 Exercise 9.4 Solutions
- Class 12 Maths Chapter 9 Exercise 9.5 Solutions
- Class 12 Maths Chapter 9 Exercise 9.6 Solutions
- Class 12 Maths Chapter 9 Miscellaneous Exercise

## 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. 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. Students can access all exercises with detailed solutions explaining the concepts from this NCERT Solutions for Class 12 Maths Chapter 10. This chapter 10 has four exercises. All solutions are based on 12th Class Maths Book Solution. Student can download both Chapter Vector Algebra Class 12 Maths Solutions in Hindi medium and English medium.

## Free NCERT Solutions for Class 12 Maths Chapter 10 in PDF

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## 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 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, coplanarity of two lines and many more. And last it contains miscellaneous examples. Just click below mentioned links for all exercise and learn NCERT Solutions for Class 12 Maths Chapter 11 with FREE of cost. There are three exercises in this chapter which is based on 12th Class Maths Book Solution. You can download here Chapter Three Dimensional Geometry Class 12 Maths Solutions in Hindi medium and English medium. Just click below links and get FREE NCERT Solutions for Class 12 Maths PDF.

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## 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. In this chapter. Here this chapter covered linear programming problem and its mathematical formulation, the mathematical formulation of the problem, different types of all linear programming problems graphical method, feasible and infeasible solutions and at last there are some miscellaneous examples given. NCERT Solutions for Class 12 Maths Chapter 12 covers two exercises in all. There are free NCERT Solutions for Class 12 Maths PDF is available here. You can also find Linear Programming Class 12 Maths Solutions in Hindi medium and English medium.

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## 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. Multiplication rule of Probability. 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 has five chapters with well-defined solutions by the expert math teacher. All solutions are based on 12th Class Maths Book Solution. Students can download for both Chapter Probability Class 12 Maths Solutions in Hindi medium and English medium her.

## Free NCERT Solutions for Class 12 Maths Chapter 13 in PDF

### Download Class 12 Maths Chapter 13 Solution in PDF Format

- Download Class 12 Maths Chapter 13 Exercise 13.1 in PDF
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- Download Class 12 Maths Chapter 13 Exercise 13.3 in PDF
- Download Class 12 Maths Chapter 13 Exercise 13.4 in PDF
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- Download Class 12 Maths Chapter 13 Miscellaneous Exercise

As Class 12 Maths is an important subject that plays a vital role in your 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 logics of every problem quite effectively.

## How to Prepare Class 12 Maths all Chapters

### 1. Relations and Functions

In this NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions – we learn about “Relations and Functions” is one of the most important topics in algebra. It mathematical sense the two different words having a different meaning. Students get confused with these terms before going deep into this topic. “Relations” which is a set of inputs and outputs and its “function” where the function is a relation with one output for each input. Remember that all functions are relations, but not all relations are functions. In this topic, you can find a definition of functions and relations, and its Special functions, different types of relations.

**Summary:** Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, the inverse of a function. Binary operations.

### 2. Inverse Trigonometric Functions

In this NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions, we are going to know about Inverse trigonometric functions, which are simply defined as the inverse functions of the basic trigonometric functions which include sine, cosine, tangent, cotangent, secant, and with its cosecant functions. In this topic, we know how to use inverse functions in trigonometry to get the angle with any of the trigonometry ratios. Trigonometric Functions is very important and mostly used in the field of engineering, physics, geometry and navigation.

Inverse trigonometric functions are also known as “Arc Functions”, which produce the length of arc needed to get that particular value. Trigonometric functions are especially applicable to the right angle triangle. There are a total of six inverse trig functions including Arcsine, Arccosine, Arctangent, Arccotangent, Arcsecant, Arccosecant.

**Summary:** Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

### 3. Matrices

Here we learn about introduction to Matrices and Matrix Operations in this chapter NCERT Solutions for Class 12 Maths Chapter 3 Matrices. Matrix is a rectangular array of real or complex numbers in the form of m horizontal lines which is called row and n vertical lines which are called columns. Types of Matrices are Symmetric Matrix, Skew-Symmetric Matrix, Hermitian and skew, Orthogonal matrix, Idempotent matrix, Involuntary matrix, Nilpotent matrix. The matrix is just an arrangement of certain quantities. The elements which are used in a matrix may be real or complex numbers. When the elements of a matrix are real, then it is known as a real matrix.

**Summary:** Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric, and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2).Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

### 4. Determinants

Here in this chapter, NCERT Solutions for Class 12 Maths Chapter 4 Determinants, we learn Introduction to Determinants. The determinant is a scalar value which can be calculated from the elements of a square matrix. When a system of simultaneous linear equations is started to solve, then Development of determinants took place. The shape of every determinant is square. If a determinant is of order n then it contains n rows and n columns. For every square matrix A of order m x n, there exists a number associated with it known as the determinant of a square matrix where the number of elements in a determinant of order n is n2.

**Summary:** Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of a system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using the inverse of a matrix.

### 5. Continuity and Differentiability

Here we learn about “Continuity and Differentiability” in this chapter, NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability. Continuity and Differentiability is one of the most important topics here you can understand the concepts like continuity at a point, continuity on an interval etc. Differentiability of functional parameters is very difficult to learn. The value of the function x = 0 changes suddenly. When we found the Right-hand limit ≠ Left-hand limit, it means this function is not continuous at x=0. It can be seen that a function is continuous at a point if we are able to graph it without lifting the pen. If the function is undefined, then the function is discontinuous.

**Summary:** Continuity and differentiability, a derivative of composite functions, chain rule, derivatives 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. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

### 6. Applications of Derivatives

In NCERT Solutions for Class 12 Maths Chapter 6 Applications of Derivatives, we will learn different applications of derivatives in used various fields such as Science, Engineering, and other fields. Derivative can be defined as the rate of change of one quantity with respect to another. Where the rate of change of function is defined as dy/dx = f(x) = y’ and the ratio of dy/dx is used as one of the applications of derivatives in real life. The concept of derivatives has been used in many ways, such as when a matter of change of temperature or rate of change of shapes and sizes. Where the derivative function is used to calculate the highest and lowest point of the curve in a graph, with the help of the graph, you can also know its turning point as well.

Summary: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, 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).

### 7. Integrals

In this chapter NCERT Solutions for Class 12 Maths Chapter 7 Integrals, we learn about Integration which is the calculation of an integral. This is used to find our many useful quantities such as areas, volumes, displacement, etc. Integrals are related to usually definite integrals. Integration is one of the two major calculus in Mathematics. The idea of limit is used where algebra and geometry are implemented. Integration is a step of adding or summing up the parts to find the whole, and the opposite step is known as differentiation. There are two types of integrals in maths, such as Definite Integral & Indefinite Integral. An integral that contains the upper and lower limits where Indefinite integrals are defined without upper and lower limits.

**Summary:** Integration as the inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the types given in the syllabus and problems based on them. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof).Basic properties of definite integrals and evaluation of definite integrals.

### 8. Applications of the Integrals

In NCERT Solutions for Class 12 Maths Chapter 8 Applications of the Integrals, we learn about the methods of calculations functions, differentiation and integration, which is called Applications of the Integrals. Applications of the Integrals is applied in various fields to the calculation of areas like Mathematics, Science, and Engineering etc. An integral is a function which is a given function is the derivative. It is used to find the areas of the two-dimensional and volumes of three-dimensional objects. The integral is also known as anti-derivative as it is the opposite process of differentiation.

Summary: Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area Bounded between any of the two above said curves (the region should be clearly identifiable).

### 9. Differential Equations

In this NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations, we learn about “Differential Equations” which is an equation that contains one or more functions with its derivatives. And the function of Differential Equations is the rate of change of a function at a point. Basically, Differential Equations is used in physics, engineering, biology, and another field. The main study of solutions that satisfy the equations, and the properties of the solutions. The derivatives represent a rate of change, where differential equation describes a relationship between the quantities and the speed of change.

**Summary:** Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. The solution of differential equations by the method of separation of variables solutions of homogeneous differential equations of the first order and first degree. Solutions of the linear differential equation of the type given in the syllabus.

### 10. Vector Algebra

Here in this NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra, we learn all about the study of vectors and scalars. There are numbers of quantities that involve magnitude as well as direction. When the quantity has magnitude and direction, then it is known as vectors. And such types of quantities are known as Vector Quantities. The main use of vectors is to describe the physical quantities having directions and magnitude associated with them. The various types of vectors included in this class 12 vector algebra are Zero Vector, Unit Vector, Coinitial Vectors, Collinear Vectors, Equal Vectors, Negative of a Vector.

**Summary:** 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, the 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, a scalar triple product of vectors.

### 11. Three – Dimensional Geometry

Here we came to know and learn about important topics such as direction cosine and direction ratios of a line joining two points in NCERT Solutions for Class 12 Maths Chapter 11 Three – Dimensional Geometry. Dimensional Geometry is about the angle between line and plane, between two lines etc. To understand the different types of shapes and figures, Dimensional Geometry is introduced.

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

### 12. Linear Programming

In this second last chapter in class 12 maths NCERT Solutions for Class 12 Maths Chapter 12 Linear Programming, we learn about Linear Programming. “Linear programming” is a method of optimising operations with some constraints. The basic use of linear programming is to maximize or minimize the numerical value. Linear programming used in many Mathematics field and some other field such as economics, business, telecommunication, and manufacturing fields. Linear programming (LP) is defined as the problem of maximizing or minimizing a linear function.

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

### 13. Probability

Here we learn about the definition of Probability and its Properties NCERT Solutions for Class 12 Maths Chapter 13 Probability, Probability is defined as the extent to that an event is likely to occur. It is a number of events to occur from the total number of event that occurs. The probability formula is defined as the likelihood of an event to happen be equal to the given ratio of the number which results and the total number of outcomes. And where Conditional Probability is the likelihood of an event or outcome occurring is defined as on the occurrence of a previous event or outcome.

**Summary:** Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean and variance of a random variable. Repeated independent (Bernoulli) trials and Binomial distribution

## Importance of Learn NCERT Solutions for Class 12 Maths from Tiwari Academy

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## What will be the Exam pattern of CBSE Class 12 Maths in 2021-22?

Recently the Central Board of Secondary Education (CBSE) introduced a new exam pattern. There will be practical or project work of 20 marks for CBSE Class 12th Mathematics subject while the written exam will be of 80 marks instead of 100 marks.

1. **Where Calculus** carries 44 marks. It means you must do practice and focus on this unit the most. Calculus which needs a lot of logical approaches which can be mastered only with rigorous practice.

2. And the chapter, **Vectors** and **Three-Dimensional** Geometry which carry 17 marks which are easy units. Be sure to make diagrams while you study and practice, as it helps you to get better visualization.

3. And in chapter **Algebra** and **Probability** are so easy units also and hence it shouldn’t take much effort to understand. However, both chapters require a lot of practice to understand the application of knowledge and formulas as well. Where Probability especially requires a logical approach and so, the basics must be clear for this unit.

4. **Relations and Functions**, and **Linear Programming** are the relatively easy chapters. You can do well by practising these chapter.

## CBSE Class 12 Maths 2021-22 Question Paper: Chapter wise break up and marking scheme:

No. | Units | No. of Periods | Marks |

I. | Relations and Functions | 30 | 08 |

II. | Algebra | 50 | 10 |

III. | Calculus | 80 | 35 |

IV. | Vectors and Three – Dimensional Geometry | 30 | 14 |

V. | Linear Programming | 20 | 05 |

VI. | Probability | 30 | 08 |

— | Total | 240 | 80 |

Internal Assessment | 20 |

### 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 maths by learning CBSE 12 Maths Solutions of NCERT.

### FAQs

##### Is NCERT Enough for Class 12 Board’s Maths?

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.

##### How many chapters are there in maths class 12 NCERT?

There are 13 chapters in maths class 12 NCERT which NCERT Mathematics textbook. The chapters include such as Differentiation, Integration, Algebra, and others. It covers the entire CBSE syllabus. If you want to score maximum marks in 12th class board exam then you are suggested to do practice and revise thoroughly before your exam.

##### 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.