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# RS Aggarwal Solution for Class 12 - Maths

The transition from class 10 th to 11 th and 12 th is more like a transition from a bicycle to a car. They have very few similarities, with the latter being a lot tougher than the former. Therefore, 12 th studies have never been easy without guidance from a proper source. The Aakash institute facilitates the opportunity to learn the subjects of class 12 th flawlessly. The RS Aggarwal textbook teaches you the chapters of Class 12th mathematics and prepares you for competitive exams like JEE. The solutions for this book by Aakash help the students of class 12 th learn all the topics step-by-step without having their interest and fun lost in learning something new. The book is prepared with proper care and includes every topic in the syllabus for class 12th, enabling the students to procure high marks in their examinations.

At Aakash, we provide the RS Aggarwal solutions, which have been developed by well-trained professionals and well-researched knowledge, keeping in mind the entire curriculum of the students. The solutions have been written in an easy English language so that the language doesn’t become a barrier in learning the chapters properly. The solutions serve the students irrespective of their intelligence quotient. One of the major benefits of studying Aakash’s solutions of RS Aggarwal is that these solutions strictly adhere to the CBSE curriculum, so students do not get confused between different books. The students must make sure that they complete all the chapters and exercises thoroughly.

This book contains 33 chapters in total. For the sake of convenience, Aakash provides a free PDF of solutions for these questions, and we recommend students go through them thoroughly for a good grasp of the chapters.

## Chapter 1: Relation

This is the very first chapter of RS Aggarwal Class 12 maths. It defines the concept of functions where if two sets have ordered pairs such that the two elements from the ordered pair are from both sets individually, then the two sets are said to be in relation.

## Chapter 2: Functions

This chapter gives us the idea of functions defined by the relation between 2 sets where for every element of a set A, there is exactly one element in set B. We will learn about 3 types of functions in this chapter: Injective, Bijective and Surjective functions. In addition, we will learn about the Invertible function and the composition of all the functions as well.

## Chapter 3: Binary Operations

This chapter comprises the functions of binary operations to solve the problems. For example, we take 2 numbers to get one number. We get to learn about the fact that summation, subtraction and multiplication are binary operations. Even division and exponents of numbers are binary operations. The exercises regarding this chapter bring all kinds of binary operations to light.

## Chapter 4: Inverse Trigonometric Functions

Chapter 4- Inverse Trigonometric Functions will teach how to use the inversion of the essential trigonometric functions such as sine, cosine, tangent, cotangent, etc. The primary aim of these functions is to get the angle between the trigonometric ratios, i.e., the opposite operations of the basic trigonometric functions. This is also called the Arc Function or the Anti-trigonometric function.

## Chapter 5: Matrices

This chapter lets students know about the definition of matrices, their order and coefficients. In the class 12 board exam, matrices play a vital role. Furthermore, this chapter also deals with the symmetric and skew- symmetric matrices, their operations, addition, subtraction and multiplication of 2 matrices and multiplication of matrices with scalar quantity. Apart from this, there are singular and non-singular matrices and two non-zero, which gives their product as zero.

## Chapter 6: Determinants

This chapter introduces students to the determinant, which is a scalar value and a function of the entries of a square matrix. Also, in order to find out if a system of equations has a unique solution or not, the determinant is a helpful method. In chapter 6- Determinants, the students will only get to work on order 3 determinants. We can also find out the area of a triangle by determinants if the coordinates of the vertices are known.

## Chapter 7: Adjoint and Inverse of a Matrix

In this chapter, we will learn about the steps of determining the adjoint of a matrix by several steps such as finding the minor matrix of a matrix, finding the cofactor of that matrix and finally transposing the cofactor.

## Chapter 8: System of Linear Equations

This chapter focuses on the advanced version of the chapter of Linear Equations that we have learnt earlier in middle school. Here we will learn about some proofs to prove a set of given lines are inconsistent. In addition, the student should know about matrix operations to solve the questions in this chapter.

## Chapter 9: Continuity and Differentiability

Chapter 9- Continuity and Differentiability will help the students learn to determine the continuity of a function f(x) at any given point. The function is said to be continuous if it follows certain conditions. In the case of differentiability, f(x) is said to be differentiable at any point x = y if the derivative of f(y) exists at each point of its interval.

## Chapter 10: Differentiation

This is an important chapter for class 12 th maths and is used in several scientific studies and higher mathematics. Chapter 10-differentiation says that we can calculate the rate of change of a variable X with respect to a variable Y by the formula . It is a process that makes calculation or irregular entities possible as it can be used to calculate the values when the entire entity is broken down into smaller entities.

## Chapter 11: Applications of Derivatives

Chapter 11-Applications of Derivatives is all about the methods to apply the concepts of derivatives into real-life situations. For example, if we have to find out about the rate of change of flow of water, rate of change of the surface area of some shape etc., this chapter makes it easier and faster to solve.

## Chapter 12: Indefinite Integrals

Integration is one of the most important parts of Calculus in mathematics. In chapter 12- Indefinite Integrals, we will learn about integrating a function with no limits. This concept can be applied to find the areas of plots by assuming a narrow strip of the unit are dimensions and integrating these strips to find out the whole plot area.

## Chapter 13: Method of Integration

Chapter 13-Method of Integration deals with the application of the concept of integrals. To operate with integrals, we need to know the other part of the calculus, namely differentiation. Integration of a function is shown by the symbol

## Chapter 14: Some Special Integrals

In the previous chapter, we learned about integration. Chapter 14- Some Special Integrals will discuss some special integrals beneficial to know for different applications. This chapter tells us about the Chain Rule, where the given equation will be set up uniquely. In addition, some integers have a specific method by which they can be solved easily; this chapter explains all those methods and the integrals that can be solved using them.

## Chapter 15: Integration Using Partial Fraction

A Partial fraction is a method to find out the solutions to the particular part of the fraction, giving us the answer of the whole fraction equation used when combined. In Integral calculus, the usage of partial fractions is vividly seen, and the 15 th chapter teaches all the methods required to solve integrals that require partial fractions.

## Chapter 16: Definite Integrals

We already know that there are 2 types of integrals: definite and indefinite. Definite integrals are the part of integration which has limits in it namely upper and lower limits. Definite integrals are widely used in the field of science, mathematics both. Definite Integrals are shown as . These are used to integrate functions when a certain limit is given.

## Chapter 17: Area of Bounded Region

From chapter 17- Area of Bounded Region, we get to know about the area of a shaded region of a given curve such as parabola and hyperbola etc. This chapter deals with the area under a curve. We have to know the use of integration here in order to solve this chapter. For example, if y = f(x) is a curve and has values like x = a and x = b, then by integrating f(x) with the limits of (a) and (b), we can find out the area of the curve. Definite integrals are used in this chapter.

## Chapter 18: Differential Equations and their Formation

Chapter 18- Differential Equations and their Formation deals with the various differential equations. The order of an equation is the order of the topmost derivative of an equation that is being used, and the degree is the highest power in it. Questions are step by step solved for this chapter by the experts of the Aakash Institute.

## Chapter 19: Differential Equations with Variable Separable

In this chapter, differentiation is used with some variables like ‘a’, ‘b’ etc. Also, we will learn about the separable differential equation that is an equation that can be written as y` = f(x)g(y). The separation of variables is important to learn to solve the questions mentioned in chapter 19-Differential Equations with Variable Separable.

## Chapter 20: Homogeneous Differential Equation Solutions

If an equation is a linear equation and the functions are unknown along with its derivatives, then the linear differential equation is said to be homogeneous. If f (x, y) dy = g(x, y) dx, then f and g are homogeneous functions of the same degree of x and y. These cannot be solved without substitution.

## Chapter 21: Linear Differential Equations

An equation or polynomial, with one or more terms consisting of the derivatives of the variable with reference to one or more independent variables, is understood as a linear differential equation. The answer of the linear equation produces the worth of variable y. Examples: + 2y = sin x

## Chapter 22: Vectors and Their Properties

In earlier classes, we have learned that there are 2 types of quantities: scalar quantity and vector quantity. Chapter 22-Vectors and Their Properties will teach the students about vectors, their principles and properties in detail. Symbols denote vectors, and the notations are like

## Chapter 23: Scalar or Dot or Product of vectors

The product of scalar quantities is very simple to do, whereas vector quantities are not quite simple like scalar ones. Vector multiplication can be done in 2 ways. Those are Scalar or Dot products and vectors or cross products. An example is, a.b = |a| |b| cos where |a| is the magnitude of vector a and |b| is the magnitude of vector b and is the angle between 2 given vectors. This is an example of a dot product. The answer obtained by a dot product is a scalar quantity.

## Chapter 24: Cross or Vector Product of Vectors

The Cross product of a vector is always represented in a 3-dimensional space of 2 vectors a and b and can be defined as (a × b). If the cross product a × b is defined as a vector c which is perpendicular to both a and b, then it is given by the formula ||a|| ||b|| sin ()n, where is lying between 0 degrees to 180 degrees and ‘n’ is a unit vector perpendicular to the plane containing a and b. But if both vectors are parallel to each other, then the cross product of a and b will be 0.

## Chapter 25: Product of 3 Vectors

Chapter 25- Product of 3 Vectors comes up with a whole new concept of multiplication of vectors that is the product of 3 vectors commonly known as the triple vector product. The knowledge of matrix operations is necessary to solve the questions of this chapter. The scalar product of 3 vectors is defined as (a × b).c and where

## Chapter 26: Fundamental Concepts of 3-Dimensional Geometry

This chapter will help the students learn about the advanced version of coordinate geometry, which they have been studying since middle school. This concept deals with the coordinate geometry in a 3- Dimensional space which refers to the process of identifying the location of a point in the coordinate plane. There are some procedures to project a 2-Dimensional point onto a 3-Dimensional space. The student is required to have good knowledge of 2-D coordinate geometry to efficiently understand this section.

## Chapter 27: Straight Line in Space

This chapter lets the student know about the representation of a straight line in space. A straight line can be stretched to both directions in space till infinity. y= mx+c (where ‘m’ is the gradient and c is known when intercepted in the y-axis) is the fundamental equation in 2-D, but we will use the extension of this equation in this chapter. The chapter discusses the 3 main topics: the straight-line equation, the Cartesian equation of a line in space, and the concepts of the angle of a straight line that it makes in space.

## Chapter 28: The Plane

This chapter is a representation of a plane in several forms. A plane can be defined by a flat, two- dimensional surface that can extend infinitely. A plane is the two-dimensional analogue of a point, a line that is one dimensional and three-dimensional space. The equation of a plane can be defined in the normal form in the vector form and a Cartesian form. There are various other concepts explored as well. For example, if two separate planes that are perpendicular to the same line exist, then both the planes must be parallel to each other.

## Chapter 29: Probability

We have studied probability since middle school, which required a very basic calculation. Here in class 12, we will study the advanced part of probability which contains conditional probability and several others. Therefore, a thorough study and a good amount of knowledge about middle school probability are necessary for solving this chapter's questions. The chapter also used the concepts of permutation and combination to solve various questions.

## Chapter 30: Bayes’s Theorem and applications

The Bayes' theorem named after the Reverend Thomas Bayes mainly describes the probability of an event based on prior knowledge of the conditions related to the event. The statement of the theorem is as follows, P(A|B) = . The theorem can be proved for events and random variables.

## Chapter 31: Probability Distribution

A probability distribution may be defined as a function that describes the likelihood of obtaining the possible values that a variable can assume. In other words, the values of the variable vary supported by the underlying probability distribution. There are two sorts of probability distribution mainly used for various purposes and information generation processes in various fields.

## Chapter 32: Binomial Distribution

The Binomial distribution is a common discrete distribution used in statistics instead of a continuous distribution, such as the normal distribution. Therefore, the binomial distribution represents the probability for x successes in n trials, given a hit probability p for every trial. Several steps are used to solve the questions related to Binomial Distribution.

## Chapter 33: Linear Programming

This chapter tells us about different kinds of linear programming which need the idea of linear equations. It is an optimisation procedure that is very useful in industrial management and the study of efficiency.

## FAQs for Class 12 th RS Aggarwal Maths Solution

Q. Why is it necessary to study using the Aakash RS Aggarwal Solutions for class 12 th ?
For the class 12 th maths, RS Aggarwal is known to be one of the best mathematics books and to the point solutions of Aakash makes the students feel enthusiastic about studying it. These solutions cover all the topics and ensure the major doubts regarding this chapter are cleared. The solutions have been provided in very simple and short procedures.

Q. Where can you find RS Aggarwal solutions for class 12 th Maths?
The Aakash RS Aggarwal solutions for class 12 th maths is available on the official Aakash website. The solutions are given topic-wise to facilitate the students’ search. The solutions are available in PDF format.

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