# RD Sharma Class 12 Maths Solutions

Share**RD Sharma Class 12 Solutions Maths - Volume 1** presented here are easy to understand and highly effective for Class 12 Maths Board exam preparation. By utilizing our RD Sharma Class 12 Maths solutions, you can clear your doubts that may arise while solving difficult Maths questions in RD Sharma textbook. The solutions are designed based on the recent syllabus of the CBSE and comply fully with the CCE guidelines. So, there are higher chances of these solutions to appear in the Class 12^{th} Boards exam.

The solutions of RD Sharma Class 12 Maths book act as an important resource to aid students thoroughly learn difficult Maths concepts. Class 12 is a pivotal stage in the academics of a student as it sets the platform for a good career ahead. It is at this time, RD Sharma Class 12 Mathematics Solutions can come to your help. So, to prepare smarter and effectively, go through the RD Sharma Maths Solutions for Grade 12 through the links given below.

## RD Sharma Class 12 Solutions - Volume 1

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RD Sharma books have been regarded as the ‘Bible’ for Maths students looking to improve their Mathematics skills. Over the years, the RD Sharma books have empowered lakhs of students to master the Maths subject and score better marks in the exam. So, in this regard, it gets imperative that you refer to RD Sharma Class 12 Maths Solutions to master the subject.

**RD Sharma Class 12 Solutions (Chapter-wise Summary)**

**Chapter 1: Relations**

The concept of the term ‘relation’ in Maths has been obtained from its meaning in the English language. We know that two objects are said to be related to each other when they have a recognisable connection.

You will study six different types of relations in Chapter-1 Relations of RD Sharma Class 12 Maths textbook. To begin with, if you find no element related to any other element in the same set, then it is called an empty relation. Universal relation, on the other hand, is found in a set where each element is related to every element.

Proceeding further, read about the remaining relations, including equivalence, reflexive, symmetric, and transitive. Most of the questions of this chapter would require you to prove whether a particular relation is valid or not.

Chapter 2: Functions

Chapter 2: Functions

In the second chapter, you will deal with special functions, including constant function, identity function, modulus function, polynomial function, rational function, signum function etc. Next comes the composition of functions and the inverse of a bijective function.

You will come across questions in which you will be required to find the inverse of the function and prove a function invertible.

**Chapter 3: Binary Operations**

In Chapter-3, you will learn to apply binary operations in functions. Also, learn the method of how to determine whether a function is binary, commutative or associative. Considering the given binary operation on the set A defined by a function, understand how to prepare the operation table.

The main features of Chapter-3 ‘Binary Operations’ involves the study of identity and invertible element for binary operation.

**Chapter 4: Inverse Trigonometric Functions**

You are now heading to this chapter with a prior understanding of the inverse of a function and trigonometric functions that you studied in the previous class. In Chapter-4, you will get to know the limitations on domains and ranges of trigonometric functions.

As you dig deep into this chapter, you will observe the behaviour of domains and ranges of trigonometric functions through graphical representations. In the end, you will also study some elementary properties. Inverse trigonometric functions find its application in calculus for defining integrals.

**Chapter 5: Algebra of Matrices**

Matrices are one of the most important tools in Mathematics, as it will help you simplify complex mathematical problems with clear methods. After you complete this chapter, you will get acquainted with the basics of matrix and matrix algebra. To define, a matrix is an ordered rectangular array comprising functions or numbers.

Initially, you will get to know the order of a matrix, then you will discover six different types of matrices- column, row, square, diagonal, scalar, identity, and zero. Moving forward, study how to apply operations on matrices and find the transpose of a matrix. Know about symmetric and skew-symmetric matrices, elementary operations, and invertible matrices.

**Chapter 6: Determinants**

Before you begin Chapter-6 Determinants, you must have developed a proper understanding regarding matrices and algebra of matrices. This chapter begins with various properties of determinants, minors, and cofactors.

Next, you will come across applications of determinants, i.e. you will utilize determinants in finding the area of a triangle, adjoint, and inverse of a square matrix. Proceeding forward, study determinants of a matrix of a different order and important properties that will help you solve different levels of questions.

**Chapter 7: Adjoint and Inverse of a Matrix**

In the previous chapter, we have discussed the inverse of a matrix. Utilizing this knowledge, you will be able to understand the condition for the existence of the inverse of a matrix. You will have to obtain adjoint of a matrix first in order to find the inverse of a matrix.

Moving forward, you will come across important theorems of determinants. Some of these theorems are solely based on experiments without proof. However, they are extensively used in the field of Mathematics.

**Chapter 8: Solution of Simultaneous Linear Equations**

In this chapter, you will use determinants and matrices for solving the system of linear equations comprising two or three variables. Later, you will have to check the consistency of the system of linear equations.

A system is said to be consistent when there exists one or more solutions for it. An inconsistent system, on the other hand, does not have a solution. Next, you will come across questions where you will have to find the solution of a system of linear equations using the inverse of a matrix.

**Chapter 9: Continuity**

Having learnt the differentiation of various functions such as polynomial and trigonometric functions in the previous class, you will now study advanced concepts of continuity. In the first exercise of RD Sharma Class 12 Chapter-9 Continuity, you will encounter different types of questions, which includes checking the continuity of the function, finding the points of discontinuity, etc.

**Chapter 10: Differentiability**

In the previous class, you have already learnt to find the derivative of real functions. This process is called differentiation. To find the derivative of a function, we differentiate a particular function with respect to x.

Three rules of derivatives that you learnt earlier will come into use in Chapter-10 “Differentiability”. Leibnitz and quotient rule are two important theorems that you must know before you start this chapter.

**Chapter 11: Differentiation**

In this chapter, you will learn to find the derivatives of real-valued functions using the chain rule. Besides, you will also learn the process of finding derivatives for implicit functions, inverse trigonometric functions, exponential, and logarithmic functions.

Under logarithmic differentiation, you will study the stepwise procedure of finding the derivatives of special functions. Know what happens when you come across the relation between two variables which is neither explicit nor implicit. In this case, you can find the derivative of a function using the chain rule.

**Chapter 12: Higher Order Derivatives**

You are already familiar with how to find the first-order derivative of a function. Using the method of differentiation, you can obtain the second-order derivative of a function that has already been differentiated.

You must develop a proper understanding of higher-order derivatives as you will have to use them frequently in Physics. Derivatives and integration constitute the core of mathematical concepts in engineering.

**Chapter 13: Derivative as a Rate Measurer**

This chapter will introduce you to practical applications of derivatives such as finding the radius of a cube, calculating the volume of the hub, etc. Basically, these applications of derivatives will help you in measuring the rate of change. For example, a question could be asked in the exam where you need to find out the rate of change of area with respect to radius.

**Chapter 14: Differentials, Errors and Approximations**

In Chapter 14 “Differentials, Errors and Approximations”, you will gain in-depth knowledge regarding differentials, errors, and approximations. The basic concept of this chapter revolves around taking the differential function as a function of x.

We assume delta x and delta y as small changes in variables x and y respectively. Here, delta x can also be called as a differential of y. Finally, you will cover some useful results on differentials.

**Chapter 15: Mean Value Theorems**** **

First and foremost, you will study about the definition and graphical representation of Mean Value Theorem. Michel Rolle came up with the proof of this theorem in 1691.

It states that for any given curve between two end points, there exists a point at which the slope of the tangent to the curve is equivalent to the slope of the secant through its endpoints.

**Chapter 16: Tangents and Normals**

Tangents and Normal is one of the most scoring topics of Class 12 Maths. At the beginning of this chapter, you will study the basic concepts of equations of tangent and normal to general curves. To master this chapter, you need to have a good understanding of graphs and basic concepts of coordinate geometry.

Also, know how to find the measure of the angle of intersection of two curves at a point. This chapter also includes the equations for finding the length of tangent, normal, subtangent, and subnormal.

**Chapter 17: Increasing and Decreasing Functions**

With the help of derivatives, you can determine whether a function is increasing or decreasing in a particular interval. Through graphical representation, we can figure out that a function is increasing if we find an upward pattern. A decreasing function, on the other hand, shows a downward pattern.

Mathematically, an increasing function is characterized by the increase in the value of y with the increase in x. The decreasing function can be identified if the value of y is inversely proportional to that of x.

**Chapter 18: Maxima and Minima**

The maxima and minima of a function, collectively termed as extrema, are the largest and smallest value of the function, within a given range. Further, study the properties of maxima and minima. A function may have a number of local maxima or local minima in a given interval, based on which you find the local maximum or local minimum value.

Finally, this chapter covers the three methods to find local extrema- first derivative test, second derivative test, and nth derivative test.

**Chapter 19: Indefinite Integrals**

By now, you must have mastered the concept of derivatives which will only enable you to understand this chapter. In Chapter-19 of RD Sharma, you will learn to find the antiderivatives of functions.

There are three methods to integrate a given function, including integration by substitution, integration by parts, and integration of rational algebraic functions using partial fractions.

**Chapter 20: Definite Integrals**

After you have understood the concept of indefinite integrals or anti-derivatives, you will find it easier to crack definite integrals. First, you will deal with questions asking you to find the definite integrals as a limit of a sum.

Next comes the fundamental theorem of calculus without proof. Also, study the basic properties of definite integrals and their evaluation.

Chapter 21: Areas of Bounded Regions

Chapter 21: Areas of Bounded Regions

In earlier classes, you were able to find the area of figures that are made up of line segments, circles, semi-circles, cone, cylinder, etc. However, there are some figures in which you cannot form the entire shape by joining the figures. Here, the application of integrals comes into play, making it easier for you to find the area of bounded regions that are curved.

You will find the area of both regular and irregular shapes by drawing vertical and horizontal stripes. The area between two curves can also be found using the same method.

**Chapter 22: Differential Equations**

In Chapter-22, you will study how to obtain the derivative of a function “f” with respect to an independent variable. Having learnt these equations, you will often find their use in Chemistry and Physics.

The first topic of this chapter “order of differential equation” discusses the highest order of the derivative present in the dependent variable with respect to the independent variable. Next comes the concept of a general solution of the differential equation that is defined as solutions containing arbitrary constants.

At the end of this chapter, you will study three different methods of solving first-order, first-degree differential equations. These include differential equations with variables separable, homogeneous differential equations, and linear differential equations.

**Chapter 23: Algebra of Vectors**

In this chapter, you will learn to utilize algebraic operations on vectors, including Euclidean vectors. The list of algebraic operations that are performed on vectors includes addition & subtraction, multiplication & division by a scalar quantity, transposition, equality, etc.

All the aforementioned algebraic operations will be utilised in two-dimensional and three-dimensional space. Although you can clear your basic concepts of vector algebra from NCERT, practising extra questions from RD Sharma Class 12 textbook will also help you prepare for competitive exams.

**Chapter 24: Scalar or Dot Products**

As you have studied in Physics, a vector quantity represents both direction and magnitude. You can multiply two vectors using scalar or dot product. A dot product is represented by a central dot; it helps you find the product by taking two equal-length sequences of numbers.

Next, you will learn to represent the dot product, both geometrically and algebraically. With the help of RD Sharma Class 12 Solutions, you can put your knowledge to the test by solving different types of questions of varying difficulty level.

**Chapter 25: Vector or Cross Product**

Vector or Cross product can be defined as the multiplication of two non-zero parallel vectors whose magnitude is |a| |b| sin𝛉. Cross product can be defined only in three-dimensional space and represented by (a x b). Here, 𝛉 is the angle between two non-zero parallel vectors (a, b) and whose direction is perpendicular to the plane of a & b.

**Chapter 26: Scalar Triple Product**

As you are now familiar with the concept of Scalar product, you will find Scalar Triple product easier to understand. Alternatively known as mixed product, the scalar triple product can be found using the dot product of any one of the given vectors with the cross product of the other two vectors. At the end of this chapter, you will come across various properties of the triple scalar product.

**Chapter 27: Direction Cosines and Directions Ratios**

With the help of direction ratios, you can define the direction of a line in three-dimensional space. In this chapter, you will also be introduced to directional angles and directional cosines.

Directional angles are characterized by the angles made by a line with the positive directions of three axes. Directional cosines, on the other hand, are angles ɑ, 𝝱 and Ɣ of a directed line L whose directional cosines are cos ɑ, cos 𝝱 and cos Ɣ.

**Chapter 28: Straight Line in Space**

A straight line does not have endpoints; the line can be extended on both sides till infinity. Famous mathematician Euclid introduced the concept of Geometry to the world, in which straight-line was the first topic to come out.

A straight line can also be characterized by zero-width object that extends in opposite directions without any ends.

Chapter 29: The Plane

Chapter 29: The Plane

Like a straight line, a plane is a flat, two-dimensional surface characterized by having infinite dimension, but no thickness. A plane in three-dimensional space can be represented by the equation (ax + by + cz + d = 0), where one of the variables a, b, c, must be non-zero.

You will study three planes in 3-D coordinate system, xy plane, yz plane, and xz plane. Note that the value of z-coordinate is zero for xy plane, x coordinate is zero for yz plane, and the y coordinate is zero for xz plane.

Chapter-29 ‘Planes’ is an important chapter from the exam point of view. You need to devote extra hours for practising additional questions from RD Sharma Class 12 textbook.

**Chapter 30: Linear Programming**

This chapter begins with the introduction of linear programming and how it is useful in the field of Mathematics. Linear programming is the process in which linear inequalities are taken out for a particular situation, and you then have to obtain the best values under those conditions.

The first topic of this chapter explains the mathematical formulation of L.P. Moving forward, you will be introduced to important terminologies, which includes objective functions and constraints. Towards the end, study about feasible and infeasible solutions.

**Chapter 31: Probability**

The core concept of Chapter-31 “Probability” is concerned with the axiomatic approach of probability towards solving a problem. Russian mathematician A.N. Kolmogorov introduced the axiomatic approach of probability.

Next, you will learn to establish the equivalent relationship between the axiomatic theory of probability and the classical theory of probability. The concept of conditional probability will be applicable in obtaining the multiplication rule of probability.

Through RD Sharma solutions, you will not only build up your concepts but also get to practise questions asked in competitive exams.

**Chapter 32: Mean and Variance of a Random Variable**

The concept of “Mean & Variance” plays a crucial role in helping you understand the important concepts, including the mean of a discrete random variable, probability distribution, the variance of a discrete random variable, etc. We have provided exercise-wise RD Sharma Solutions for Chapter-32 “Mean and Variance of a Random Variable” that will make your exam preparation simplified and organised.

**Chapter 33: Binomial Distribution**

Binomial Distribution can be defined as the probability of success or failure when an experiment or a survey is conducted. Next, you will come across four conditions where you can apply binomial distribution. With the help of RD Sharma Solutions for Class 12, you can get to understand the stepwise procedure for solving each question.

## Benefits of Using RD Sharma Solutions for Class 12 Maths

- The R.D Sharma Solutions comprise of answers to all the questions, problems, as well as numerical included in the book.
- Questions are solved in a detailed manner using illustrations and easy to understand steps
- Formulae used to solve complex problems are explained in easy steps
- Prepared by expert teachers having extensive knowledge of the subject
- These solutions help students score better marks in the exam
- Available for free and can be accessed easily from the website

### Final Words

Class 12th is the most crucial stage for students and the marks scored in Class 12 Maths can help students opt for a desirable higher studies course. The detailed RD Sharma Solutions for Class 12 act as an important guide for students as it helps them develop their skills and solve difficult Mathematics questions with ease. These solutions are available for free and give students the best option to learn, practise and revise the questions effortlessly. To find solutions for the remaining 14 chapters, you can access the RD Sharma Class 12 Maths Solutions - Volume 2.

Frequently Asked Questions

- Other than NCERT Maths textbook, which is the best book for improving my concepts in Maths?
If you are looking to improve your basic math concepts, then you can refer to the Gradeup school’s RD Sharma Maths Solution for Class 12 Volume 1. These solutions provide explanations of important topics and help students solve their doubts in regard to difficult questions. So, if you are looking to have the best material for Class 12 Maths, then you can definitely rely on these solutions.

- Is RD Sharma Class 12 Maths Solutions enough to score well in the Boards exam?
Securing good marks in Class 12th Maths subject requires studying the subject properly and developing a proper knowledge of different concepts. So, if you are reading the Class 12 Maths Volume 1 RD Sharma Solutions for preparing the subject, then you can definitely give your preparation a solid boost. These solutions are complete in itself and can help you to stay ahead of your preparation for the exam.

- Which Mathematics reference book is good for preparing 12th Class Mathematics?
RD Sharma Solutions for Class 12 are designed specifically to provide students with a detailed and accurate explanation for each and every question given in the RD Sharma Maths Volume 1 Book. Using our RD Sharma Solutions, students can easily access chapter wise questions along with their solutions. The format of the solutions is easy to understand, which enable the students to learn each and every concept in quick time to cover their entire syllabus quite smoothly.

- Where can I get RD Sharma Volume 1 Solutions for Class 12 for free?
Are you looking to practise with RD Sharma Solutions for free? If so, then you can find these solutions at Gradeup school. The RD Sharma Solutions provided by us are free of cost and can be accessed online in a quick time. Moreover, you can access these solutions to easily practice the questions anytime, anywhere.