# RD Sharma Solutions For Class 12 Maths

The NCERT solutions provided by our Aakash institute help the students to strengthen their concepts. The easy and understandable language used in the Aakash Institute. Step by step explanations help the students to understand their basics. Solutions are split into various sections for better understanding as per the syllabus.

The Aakash Institute, RD Sharma solutions for class 12 Maths is the easiest to understand study material available to refer to while practising and during examinations. It is designed for increasing the interest of students. The RD Sharma Solutions are prepared in a very simple language for better understanding. All the topics are covered as per the CBSE syllabus to ensure detailed and thorough learning. The students will understand every topic and score high marks.

At Aakash, we provide RD Sharma solutions for class 12 Maths to help students understand every concept on a transparent level. Moreover, the explanations are constructed at an understanding level, making it easier for them to get more information. The RD Sharma solutions for class 12 Maths are prepared by experts and senior faculties with several years of experience. The solutions are designed after a thorough analysis of the current syllabus. Each solution is given in a stepwise description to clarify all the topics, which helps the students clear their doubts. It also contains an in-depth analysis of each question, which helps the students understand the concept and prepares them to confidently approach every question.

The RD Sharma Solutions for the class 12 Math textbook consists of 33 chapters with clear explanations. The solutions help students to get a clear concept. In addition, Aakash provides free PDF solutions for RD Sharma Class 12 Maths to ease the learning procedure for students, which is very helpful. Therefore, it is highly beneficial to refer to RD Sharma Solutions, and it is recommended that the students download the PDF format and study thoroughly. This will ensure an in-depth understanding.

## Chapter 1- Relations

This chapter is one of the most important chapters in Class 12 Maths. The important topics covered in Chapter 1-Relations are the different types of relations: void relation, symmetric relation, universal relation, identity relation, reflexive relation, transitive relation and anti- symmetric relation, equivalence relation, and theorems based on relations. The chapter deals with the relationship between a grouped data based on similarities and the elements present in each group. It incorporates Venn diagrams and pictorial representations of the data and enables an easy and thorough understanding of the concepts.

## Chapter 2- Functions

Chapter 2- Functions explains in detail what a function is and talks about its various types. The topics explained in this chapter are; operations on real functions, types of functions such as one-one function, onto function, many one functions, into function etc., the composition of functions, properties of functions, the composition of functions, the composition of real functions, inverse of a function, inverse of an element, the relationship between graphs of a function and its inverse.

## Chapter 3- Binary functions

In Chapter 3-Binary functions deal with various binary operations and their applications. The binary operation can be defined as "when two numbers are added or subtracted, multiplied or divided. Similarly, binary operation refers to two elements of a set. The result of the two is also in the same set. Finally, binary operation refers to the calculation that combines the two elements of the set (known as operands) to produce another element for the same set. The addition, multiplication and subtraction on the set of all irrational numbers is not a binary operation. The types of binary operations include commutative, associative, distributive, identity, and inverse.

## Chapter 4 – Inverse trigonometric functions

Chapter 4 - Inverse trigonometric functions is a discipline of trigonometry that deals with the inverse of a trigonometric function. This chapter talks about all the trigonometric functions, definition and meaning of inverse trigonometric functions, the inverse of sine functions, the inverse of cosine functions, the inverse of the tangent function, the inverse of secant function, inverse of cosecant function, inverse of cotangent function, and properties of inverse trigonometric functions.

## Chapter 5- Algebra of Matrices

In this chapter, we will learn the concepts and rules of rows and columns. Chapter 5- Algebra of Matrices consists of the algebra of a matrix, adjoin, and a matrix's inverse. It covers the types of matrices, equality of matrices, the addition of matrices, properties of matrix addition, multiplication of a matrix by a scalar multiplication, subtraction of matrices, properties of matrix multiplication, the transpose of a matrix, the properties of the transpose of a matrix, symmetric and skew-symmetric matrices.

## Chapter 6 – Determinants

This chapter deals with a different form of matrices—the determinants of a square matrix of order 1,2, and 3. The chapter also discusses the determinant of a square matrix of order three by using the Sarus diagram. Chapter 6-Determinants deals with minors and cofactors of the determinants, the properties of determinants, the evaluation of determinants, the solution of determinant equations, the addition of determinants and lastly, the evaluation of determinants by using the factor theorem.

## Chapter 7 – Adjoint and Inverse of a matrix

This chapter deals with the various operations on the adjoint and inverse of a square matrix. Chapter 7 - Adjoint and inverse of a matrix deals with the inverse of a matrix, results on invertible matrices, determining the adjoint and inverse of a matrix, determining the inverse of a matrix when it satisfies the matrix equation, finding the inverse of a matrix by using the definition of inverse, finding and non-singular Matrix when a joint is given, elementary transformation for elementary operations of Matrix, method of finding the inverse of a matrix by elementary transformation.

## Chapter 8 Solution of simultaneous Linear Equations

Chapter 8 - Solution of simultaneous Linear Equations deals with a more than linear equation. This chapter includes the definition and meaning of a consistent system, homogeneous and non-homogeneous system, the matrix method for the solution of a non- homogeneous system, solving the given system of linear equations when the coefficient Matrix is non-singular, solving the given system of the equation when the coefficient matrix is singular.

## Chapter -9 Continuity

This chapter deals with the concept of differentiation of various functions such as polynomial and trigonometric functions. The chapter continuity will encounter different types of equations needed to check the continuity of a function and find the discontinuity points. It is crucial in determining where a function is a continuous function or not, and if not, where will the function cease to be continuous.

## Chapter-10 Differentiability

Chapter 10 - Differentiability deals with the differentiability of various functions. The derivative of a real function is called differentiation. To find the derivative of a function, we differentiate a particular function with respect to the three derivatives rules. In differentiability, the quotient rule is an important theorem discussed in minute detail because of its widespread applications.

## Chapter-11 Differentiation

Chapter 11 - Differentiation aims to find the derivative of real-valued functions using the changes besides finding derivatives for implicit functions, inverse trigonometric functions, exponential and logarithmic function and a logarithmic differentiation the stepwise procedure of finding the derivatives of special functions. The chapter also explains the relationship between two variables which are explicit or implicit. And we can also find the derivative of a function using the single-chain rule.

## Chapter -12 Higher order derivatives

Chapter 12 - Higher order derivatives teach us how to find the derivative of a function using differentiation. We can obtain the second-order derivative of a function with the proper understanding of the higher-order derivative. The concepts taught in this chapter will be frequently used in physics derivatives. Integration constitutes the core of these mathematical concepts.

## Chapter -13 Derivative as a rate measure

Chapter 13 - Derivative as a rate measure will introduce you to the practical Application of Derivatives that will further be applied to find the radius of a cube, calculate the value of the hub, etc. The applications of derivatives at times are helpful in measuring the rate of change. For example, the questions in exams would ask you to find out the rate of change of the area of a figure based on the dimensions provided.

## Chapter 14 -Differentials, Errors and Approximations

This chapter deals with an in-depth knowledge of the Differential error and approximation method. The basic concept of Chapter 14 - Differentials, Errors, and Approximations revolves around taking in differential functions as a function of x. We then assume delta x and delta y as small changes in the variable X and Y, respectively, and can also be called the differential of Y.

## Chapter 15 -Mean Value Theorem

This chapter deals with the definition and the graphical representation of the mean value theorem. It says that for any given curve between two ends, there exists a point at which the slope of the tangent to the curve is equivalent to the slope of the second through its endpoints. Chapter 15 - Mean Value Theorem plays a dominant role in three-dimensional coordinate geometry and is found useful in many other applications.

## Chapter 16 -Tangents and Normal

This chapter is considered the most scoring topic of class 12 Math. We study the basic concept of the equations of a tangent and a normal to the general curves. Chapter 16 - Tangents and normal also gives the basics graphs and the basic concept of coordinate geometry. It is used to find the nature of the angle of intersection of two curves and includes the equations required to find the length of the tangent, normal, sub-tangent and subnormal.

## Chapter 17-Increasing and decreasing function

In this chapter, we can determine whether a function is increasing or decreasing in a particular interval. We can figure out if the function is increasing through graphical representations, and we can find an upward (increasing) pattern. A downward pattern is exhibited in a decreasing function, and an increasing function is categorized by an increase in the value on the y-axis with this increase. The decreasing function can also be identified as the value of Y decreases with an increase in the value of X.

## Chapter 18 -Maxima and Minima

Chapter 18 - Maxima and Minima deal with the properties of Maxima and minima. In this chapter, the maxima and minima of a function have collectively termed the extrema for the largest and smallest value of a function within the given range. For example, a function may have several local Maxima and a local minimum in a given interval. We need to find the local Maxima or local minimum value. This chapter covers three methods to find the local extrema: the first derivative test, the second derivative test, and the N th derivative test.

## Chapter 19 -Indefinite integrals

Chapter 19 - Indefinite Integrals deal with the concept of derivatives. In this chapter, we will only be finding the anti-derivative of functions. The three methods to integrate a given function include the integration by substitution method, integration by parts method, and integrating relational algebraic functions using partial fractions.

## Chapter 20-Definite integrals

Chapter 20 - Definite integrals teach us how to find the definite integral as a limit of the sum. The chapter also sheds light on the fundamental theorem of Calculus along with a detailed explanation about the basic properties of definite integrals and their evolution.

## Chapter 21- Areas of bounded regions

Chapter 21 - Area of bounded regions aims to find the area made up of line segments, circles, semicircles, cones, cylinders, etc., such that they form a geometric figure. There are some questions in which we cannot form the entire shape by joining the smaller components of the area bounded. Here the application part of integral comes into play by making it easier for us to calculate the area enclosed by the curves. It tells us about both regular and irregular shapes. These can be made by drawing vertical and horizontal stripes. The area enclosed in the two curves can be calculated using the same method.

## Chapter 22 -Differential equations

Chapter 22 - Differential Equations deals with the method used to obtain the derivative of a function f with respect to an independent variable. The first subtopic of this chapter is the order of the differential equation. The subtopic explains the highest order of a derivative present in a dependent variable with respect to an independent variable. The second concept is the general solution of differential equations, defined as the solutions containing random constants. The chapter teaches the students and trains them on the three varied methods of solving a first-order and a first-degree differential equation. The other topics covered in this chapter are differential equations with the variable separable method, homogeneous differential equations and linear differential equations.

## Chapter 23-Algebra of vectors

Chapter 23 - Algebra of Vectors deals with algebraic operations on vectors, including the Euclidean vector. The list of algebraic operations performed on a vector includes addition, subtraction, multiplication and division by a scalar quantity, equality etc. All algebraic operations are utilized in either a two-dimensional or three-dimensional space.

## Chapter 24 -Scalar or dot products

Chapter 24 - Scalar or Dot Products sheds light on scalar quantities, vector quantities and various operations on them. A vector quantity represents both direction and magnitude. Two vectors can be multiplied by using the methods of scalar or dot product. A dot product can be represented by a central dot and is used to find the product of two scalar or vector quantities.

## Chapter 25 -Vector or cross product

Chapter 25 - Vector or Cross product deals with the operations on vector quantities. It is defined as the multiplication of two non-zero parallel vectors whose magnitudes are a and b. The Cross product can be defined only in a three-dimensional space and is represented by "a x b". The chapter sheds light on the basic operations and their results carried out on vector quantities.

## Chapter 26 -Scalar triple product

This chapter describes the process to find a scalar triple product in an easy-to-understand manner. The scalar triple product can be calculated by taking the dot product of any of the given three vectors with the cross product of two other vectors. It is alternatively known as a mixed product. The chapter ends after teaching the students about the various properties of the product.

## Chapter 27 -Direction cosines and direction ratios

Chapter 27 - Direction cosines and direction Ratio defines a line's direction in three- dimensional space and introduces students to the directional angles and directional cosines. The direction angles are categorized by angles made by a line with the positive direction of 3 axes.

## Chapter 28-Straight line in space

Chapter 28 - Straight Line In Space sheds light on straight lines and their properties. It says that a straight line lacks endpoints. These lines can be stretched on both sides till they reach infinity. The Greek mathematician Euclid taught the entire world the principles of geometry, and the straight line was the most basic topic taught by him. Straight lines can also be categorized as zero-width objects that extend in opposite directions without any ends.

## Chapter 29-The plane

Chapter 29 - The Plane discusses a flat two-dimensional surface characterized by an infinite dimension and no thickness. A plane in 3-D space can be correctly represented by an equation with three variables, a, b and c, that must be non-zero. The three planes are a three- Dimensional coordinate system, namely the X.Y. plane, the Y.Z. plane and the X.Z. plane.

## Chapter 30 -Linear programming

This chapter begins with the basic introduction of linear programming and its applications in the mathematical field. Linear programming is the process in which linear inequalities are taken out for a particular situation, and the values under those specific conditions are asked of us. Chapter 30 - Linear Programming deals with the mathematical formulation of L.P. Furthermore, it sheds light on two important terminologies: objective functions and constraints.

## Chapter 31 - Probability

This chapter is involved with the unquestionable approach of probability. The Russian mathematicians introduced this approach of probability. The theory is equally connected to the axiomatic theory and the classical theory of probability. The chapter further deals with the rules of conditional probability. The multiplication rule of probability can be easily obtained by conditional probability.

## Chapter 32 - Mean and variance of a random variable

Chapter 32 - Mean and variance of a random variable play an important role in helping to understand important topics that include discrete random variable, probability distribution, and the variance of a discrete random variable.

## Chapter 33- Binomial Distribution

Chapter 33 - Binomial Distribution deals with the definition and explanation of these probability distributions. It states the chances of success and failure when an experiment or an event is conducted. Various numerical and solved examples are provided in the textbook to concrete the fundamentals of this concept in the students' minds.

## FAQs for Class 12 Mathematics RD Sharma Solutions

Q. Why are RD Sharma solutions for class 12 Maths important?
Mathematics requires a deep understanding of every single concept discussed to answer any given question. The Aakash's RD Sharma Solutions for Class 12 gives on-point explanations and trains the students to answer a question in the easiest ways. It ensures to clear all major doubts that the student might have regarding any topic.

Q. Where to find and download RD Sharma solutions for class 12th maths?
The Aakash RD Sharma Solution for Class 12 Maths material is available on the Aakash website. The solutions are given topic-wise for easy reference. They can be downloaded as one PDF file as well. Aakash also provides PDF files of RD Sharma Solutions for Class 12 for all subjects.

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