**NCERT Solutions for Class 12 Maths** is written to assist students that are making preparations to sit for various board exams including JEE Mains, VITEEE, BITSAT, etc. This offers sound knowledge plus the understanding of every significant concept that is highlighted in every chapter in the NCERT textbook for Class 12 Maths. We have are 13 chapters from the** NCERT Maths textbook**. All the problems plus their comprehensive solutions are all provided. Additionally, the solutions to the questions have been devised by the best teachers to assist students to get basic concepts faster and better.

For one to get an entry into the best colleges, students need to perform best. This may push them to be under pressure & be stressed & demotivated. Moreover, students don’t need to fear anymore because, with the assistance of NCERT books, they will easily attain their target scores.

NCERT Textbooks for Class 12 for Maths have every bit plus a piece of the chapters to ensure that nothing is left for the students to practice. Maths entails solutions and every question solved contains a step-by-step process stated to reach an answer. This offers students an idea of formulas plus equations which are to be utilized for other types of questions.

**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 |

Table of Contents

## NCERT Solutions for Class 12 Maths – Chapter-Wise

### Chapter 1 : Relations & Functions

Chapter one of NCERT Solutions for Maths for Class 12 is about relations & functions. Here you will learn about kinds of relation, varieties of functions, invertible function & functions composition, binary operations plus miscellaneous illustrations. You will also get exercises solution linking the topics that are covered in this chapter.

Topics that are covered in Maths for Class 12 Chapter one (Relations & Functions) for First Term includes:

Kinds of relations: symmetric, reflexive, transitive & equivalence relations. One-to-one plus onto functions.

#### Topics and sub topics of Relations & Function chapter

- 1.1 Introduction
- 1.2 Types of Relations
- 1.3 Types of Functions
- 1.4 Composition of Functions and Invertible Function
- 1.5 Binary Operations

#### FAQ

**What is relation and function in class 12?**

It is a special kind of relation(a set of ordered pairs) which obeys a rule, i.e. every y-value should be connected to only one y-value. Mathematically, “a relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B”.

**How many exercises are there in relation and function class 12?**

NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions has 55 questions in **4 exercises**.

**What are into functions Class 12?**

Onto (Surjective) Function. A function **f: X -> Y** is said to be an onto function, if every element of Y is an image of some element of set X under f, i.e for every y ∈ Y there exists an element x in X such that f(x) = y. Properties: The range of functions should be equal to the codomain.

**What is function class 12 computer science?**

A function can be defined as **a single related task**, which is written as a block of codes. It only runs when it is called. We can pass data to the function when we call a function, these are known as parameters. A function can return data as a result of using the return statement.

### Chapter 2 : Inverse Trigonometric Functions

In chapter two, Inverse Trigonometric Functions will be covered. This offers an account of different topics like the remarks centered on inverse trigonometric functions basic concepts, inverse trigonometric functions properties & miscellaneous illustrations. All the concepts are well explained via examples. Solutions to exercises from this chapter will be accessed.

Topics Covered in First Term for Class 12 Maths Chapter two (Inverse Trigonometric Functions) include definition, domain, range, and principal value branch.

#### Topics and sub topics of Inverse Trigonometric Functions

2 | Inverse Trigonometric Functions |

2.1 | Introduction |

2.2 | Basic Concepts |

2.3 | Properties of Inverse Trigonometric Functions |

### Chapter 3: Matrix

Here you will learn about the properties of important Mathematics tools, namely Matrices. Here students will be able to study and understand the basic concept of Matrices & Matrix Algebra. the students will know matrix definition, matrices types, matrices equality, matrices operations including addition & multiplication of a matrix by a scalar, scalar multiplication properties, matrix addition properties, matrices multiplication, and matrices multiplication properties. Moreover, the student will also learn how to transpose a matrix, matrix transpose properties, symmetric & skew-symmetric matrices, matrix transformation or elementary operation, matrix inverse by elementary operations & miscellaneous illustrations.

Subtopics covered include; concept, order, notation, equality, kinds of matrices, zero & identity matrix, matrix transpose, symmetric, skew-symmetric matrices, etc.

**Topics and sub topics for Matrix Chapter class 12 math**

3 | Matrices |

3.1 | Introduction |

3.2 | Matrix |

3.3 | Types of Matrices |

3.4 | Operations on Matrices |

3.5 | Transpose of a Matrix |

3.6 | Symmetric and Skew Symmetric Matrices |

3.7 | Elementary Operation (Transformation) of a Matrix |

3.8 | Invertible Matrices |

### Chapter 4: Determinants

This chapter is a continuation of the topic of Matrix. The chapter deals with a significant part of Matrices that’s namely Determinants. Here you will find determinants of orders 3×3 plus their cofactors & inverses.

The chapter also speaks about determinants applications, including calculating an area of triangles, constancy of linear equations sets, and adjoint of matrices.

**Topics and sub topics for Chapter 4 Determinants class 12**

4 | Determinants |

4.1 | Introduction |

4.2 | Determinant |

4.3 | Properties of Determinants |

4.4 | Area of a Triangle |

4.5 | Adjoint and Inverse of a Matrix |

4.6 | Applications of Determinants and Matrices |

4.7 | Summary |

### Chapter 5 : Continuity & Differentiability

This chapter picks up after Functions Differentiation that’s introduced at Class 11. This chapter explores significant ideas about Continuity, Differentiability & creates a relationship between them. Furthermore, it also deals with continuity & differentiability of Functions of Inverse Trigonometric. Here, you will also be introduced to exponential & logarithmic functions.

**Topics and sub-topics included in the Continuity and Differentiability Chapter Class 12**

- Introduction
- Algebra of continuous functions
- Differentiability
- Derivatives of composite functions
- Derivatives of implicit functions
- Derivatives of inverse trigonometric functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second-Order Derivative
- Mean Value Theorem
- Summary

### Chapter 6: Applications of Derivatives

Here the student will get details about derivatives definition, rate-of-change of quantities, decreasing & increasing functions, normal & tangents, maxima & minima, approximations, test for the first derivative, maximum & minimum values about a function in closed interval & miscellaneous illustrations.

**Topics and sub-topics included in the Applications of Derivatives Chapter 6 Class 12**

6 | Applications of Derivatives |

6.1 | Introduction |

6.2 | Rate of Change of Quantities |

6.3 | Increasing and Decreasing Functions |

6.4 | Tangents and Normals |

6.5 | Approximations |

6.6 | Maxima and Minima |

6.7 | Maximum and Minimum Values of a Function in a Closed Interval |

6.8 | Summary |

### Chapter 7: Integrals

Differential Calculus is dependent on a derivative idea. Derivatives came for the issue of defining tangent lines to graphs of functions & calculating the slope of such lines. Furthermore, Integral Calculus simplifies the issue of defining & calculating an area that’s bounded by a function. Subtopics available in the integrals chapter include indefinite integrals introduction, integration (inverse of differentiation), geometrical interpretation, & properties. Other subtopics include integration & differentiation comparison, integration methods (substitution, by parts, partial fraction, & trigonometric identities).

**Topics and sub-topics included in the Integral Chapter 7 Class 12**

7 | Integrals |

7.1 | Introduction |

7.2 | Integration as an Inverse Process of Differentiation |

7.3 | Methods of Integration |

7.4 | Integrals of some Particular Functions |

7.5 | Integration by Partial Fractions |

7.6 | Integration by Parts |

7.7 | Definite Integral |

7.8 | Fundamental Theorem of Calculus |

7.9 | Evaluation of Definite Integrals by Substitution |

7.10 | Some Properties of Definite Integrals |

### Chapter 8 : Integrals Applications

This chapter is a continuation of integrals. It starts with definition, area on simple curves, area in between 2 curves, area on region surrounded by a curve & a line, & miscellaneous illustrations.

**Topics and sub-topics included in the Integrals Applications Chapter 8 Class 12**

8 | Application of Integrals |

8.1 | Introduction |

8.2 | Area under Simple Curves |

8.3 | Area between Two Curves |

### Chapter 9: Differential Equations

In this chapter, the student will study some concepts that are related to differential equations, differential equations formation, general & particular solutions of a differential equation, number of methods that solve questions centered on first-order- first-degree differential equations & differential equations applications in various areas. In this chapter, the student will have a total of six exercises to practice on. Implementation of differential equations in a plethora of applications in every other subject & area.

**Topics and sub-topics included in the Differential Equations Chapter 9 Class 12**

9 | Differential Equations |

9.1 | Introduction |

9.2 | Basic Concepts |

9.3 | General and Particular Solutions of a Differential Equation |

9.4 | Formation of a Differential Equation whose General Solution is given |

9.5 | Methods of Solving First order, First Degree Differential Equations |

### Chapter 10: Vector Equations

In this chapter the student will encounter vector quantities, that’s the quantities all both magnitude & direction. Vector algebra summarizes addition rules of vector quantities, plus other significance properties. The student will also learn about the varieties of vectors including zero, unit, collinear & equal vectors.

Other different topics which are to be discussed in this chapter include negative vector, vectors components & how one can multiply a vector with a scalar. Moreover, the student will also study section formulae, dot-product of 2 vectors, vector joining 2 points, product of 2 vectors, vector projection on a line, and cross-product of two vectors.

**Topics and sub-topics included in the Vector Equations Chapter 10 Class 12**

10 | Vector Algebra |

10.1 | Introduction |

10.2 | Some Basic Concepts |

10.3 | Types of Vectors |

10.4 | Addition of Vectors |

10.5 | Multiplication of a Vector by a Scalar |

10.6 | Product of Two Vectors |

### Chapter 11: Three-Dimensional Geometry

In this chapter, the student will study the direction ratios & direction cosines of a line that joins two points & also the equations of a line & plane in space on various conditions. Also, the student will get to understand the angle between two lines, a line & a plane, 2-planes, the smallest distance between 2 skew lines & the distance from a plane to a point.

Furthermore, the student will also get to know about cartesian equations & vector line equations, skew & coplanar lines.

**Topics and sub-topics included in the Three-Dimensional Geometry Chapter 11 Class 12**

11 | Three Dimensional Geometry |

11.1 | Introduction |

11.2 | Direction Cosines and Direction Ratios of a Line |

11.3 | Equation of a Line in Space |

11.4 | Angle between Two Lines |

11.5 | Shortest Distance between Two Lines |

11.6 | Plane |

11.7 | Coplanarity of Two Lines |

11.8 | Angle between Two Lines |

11.9 | Distance of a Point from a P |

11.10 | Angle between a L ine and a Plane |

### Chapter 12 : Linear Programming

In this chapter, the students will be led through linear programming. Some of the concepts that are covered in this chapter include linear programming-problem & its mathematical formulation, graphical way of solving linear-programming, mathematical formulation, different varieties of linear programming with miscellaneous illustrations. The subtopics include an introduction, terminologies like constraints, optimization, objective function, and different kinds of linear programming problems. Graphical way solutions in two variables, infeasible & feasible regions (bounded), infeasible & feasible solutions, best feasible solutions.

**Topics and sub-topics included in the Linear Program Chapter 11 Class 12**

12 | Linear Programming |

12.1 | Introduction |

12.2 | Linear Programming Problem and its Mathematical Formulation |

12.3 | Different Types of Linear Programming Problems |

### Chapter 13: Probability

This chapter will take you to significant concepts of conditional probability about an event provided that a different event occurred, in turn, which will clarify Bayes’ theorem, probability multiplication rule & independence of events.

Furthermore, a student will get about five exercises, where you’ll learn a significant concept about random variables & their probability distribution, variance, and the mean of a probability distribution. Here, the student will also learn about partition of a sample space, Bernoulli trials, total probability theorem, & binomial distribution.

**Topics and sub-topics included in the Probability Chapter 13 Class 12**

13.1 | Introduction |

13.2 | Conditional Probability |

13.3 | Multiplication Theorem on Probability |

13.4 | Independent Events |

13.5 | Bayes’ Theorem |

13.6 | Random Variables and its Probability Distributions |

13.7 | Bernoulli Trials and Binomial Distribution |

## Significant Features about CBSE Notes NCERT Solutions for Maths for Class 12

- Students easily practice & revise concepts by referring to the solutions.
- Saves time by providing every important concept & formulae in a suitable sequence.
- NCERT Maths Solutions consists of significant tips, important for answering normal questions accurately.
- The solutions are very helpful for fast revisions as you prepare for your exams.

## FAQ

**Which Book Is Best For Class 12 Maths Board Exam?**

NCERT Textbook

**Is NCERT enough for Class 12 board Maths?**

Yes

**Is class 12 NCERT Maths difficult?**

NCERT Maths of class 12 is not difficult. Practice make you perfect