# RD Sharma Class 12 Solutions for Maths

RD Sharma Solutions for class 12 provide solutions to a wide range of questions with a varying difficulty level. With the help of numerous sums and examples, it helps the student to understand and clear the chapter thoroughly. Solving the given questions inside each chapter of RD Sharma will allow the student to get knowledge about the subject and the chapter. RD Sharma Notes provided by GeeksforGeeks helps students to prepare well for the upcoming board exam as well for the competitive exams. Practicing the questions given in the RD Sharma will increase the pace of solving sums and will increase the knowledge of the subject.

### Chapter 1: Relations

The chapter Relations in this textbook contains two exercises that are based on the relations and their properties, types of relations, inverse of a relation, equivalence relation, some useful results on relations, reflexive relation, symmetric relation, and transitive relation.

### Chapter 2: Functions

The chapter functions in this book give an account of various topics such as the definition of functions, function as a correspondence, function as a set of ordered pairs, the graph of a function, vertical line test, constant function, identity function, modulus function, the greatest integer function, properties of the greatest integer function covered in its two exercises, exercise 2.1 and 2.1. While the topics like the smallest integer functions and their properties, fractional part function, signum function, exponential function, logarithmic function, reciprocal and square root function, square function, square root function, cube function, reciprocal squared function, operations on real function are covered in Exercise 2.3.

- Exercise 2.1 Set 1, Set 2
- Exercise 2.2
- ExerciseÂ 2.3

### Chapter 3: Binary Operations

The chapter binary operation consists of a total of five exercises that are based on the number of binary operations, types of binary operations such as commutativity, associativity and distributivity, an identity element, inverse of an element, composition table, addition modulo â€˜nâ€™, and multiplication modulo â€˜nâ€™.

### Chapter 4: Inverse Trigonometric Functions

There is only one exercise in this chapter that is based on the definition and meaning of inverse trigonometric functions, the inverse of the sine function, inverse of the cosine function, inverse of the tangent function, inverse of secant function, inverse of cosecant function, inverse of cotangent function, and properties of inverse trigonometric functions.

### Chapter 5: Algebra of Matrices

The chapter Algebra of Matrices in this book contains a total of five exercises. Exercises 5.1 and 5.2 are based on the types of matrices, equality of matrices, the addition of matrices, properties of matrix addition, multiplication of a matrix by a scalar, properties of scalar multiplication, subtraction of matrices, multiplication of matrices. Moreover, the exercises 5.3 to 5.5 are based on the topics properties of matrix multiplication, positive integral powers of a square matrix, transpose of a matrix, properties of transpose, symmetric and skew-symmetric matrices via examples.

- ExerciseÂ 5.1 Set 1, Set 2
- ExerciseÂ 5.2 Set 1, Set 2
- ExerciseÂ 5.3 Set 1, Set 2, Set 3
- Exercise 5.4
- Exercise 5.5

### Chapter 6: Determinants

This chapter in this book provides a definition of determinants, determinant of a square matrix of order 1, 2, and 3, determinant of a square matrix of order 3 by using Sarrus diagram, singular matrix covered in first three exercises 6.1 to 6.3. However, exercises 6.4 and 6.5 are based on the topics minors and cofactors, properties of determinants, evaluation of determinants, applications of determinants to coordinate geometry, and applications of determinants in solving a system of linear equations and conditions for consistency.Â

- Exercise 6.1
- ExerciseÂ 6.2 Set 1, Set 2, Set 3
- ExerciseÂ 6.3
- ExerciseÂ 6.4 Set 1, Set 2
- Exercise 6.5

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

Adjoint and Inverse of a Matrix, we shall see the definition of the adjoint of a square matrix, the inverse of a matrix, some useful results on invertible matrices, elementary transformations of elementary operations of a matrix via examples, and verbal problems related to it.Â

- ExerciseÂ 7.1 Set 1, Set 2, Set 3
- ExerciseÂ 7.2

### Chapter 8: Solutions of Simultaneous Linear Equations

The chapter Solutions of Simultaneous Linear Equations in this book based on the topics definition, consistent system, homogeneous and non-homogeneous systems, matrix method for the solution of a non-homogeneous system, and final solution of a homogeneous system of linear equations. The mentioned topics are covered in a total of two exercises only.

- ExerciseÂ 8.1 Set 1, Set 2
- Exercise 8.2

### Chapter 9: Continuity

The chapter continuity is covered in two exercises only that concentrate on the definition of continuity, continuity at a point, algebra of continuous function, continuity on an interval, continuity on an open interval, continuity on a closed interval, continuous function, everywhere continuous function, and properties of continuous functions.

### Chapter 10: Differentiability

The chapter differentiability is based on the advanced concepts of the previous class i.e. differentiability at a point, differentiability in a set, and some useful results on differentiability. The concepts are explained in two exercises thoroughly.

### Chapter 11: Differentiation

The Differentiation in this book constitutes topics related to differentiation, differentiation of inverse trigonometric functions from first principles, differentiation of a function, differentiation of inverse trigonometric function by chain rule, differentiation by using trigonometric substitutions, differentiation of implicit functions, logarithmic differentiation, differentiation of infinite series, differentiation of parametric functions and differentiation of a function with respect to another function via illustrations.

- ExerciseÂ 11.1
- ExerciseÂ 11.2 Set 1, Set 2, Set 3
- ExerciseÂ 11.3 Set 1, Set 2, Set 3
- ExerciseÂ 11.4 Set 1, Set 2
- ExerciseÂ 11.5 Set 1, Set 2, Set 3
- Exercise 11.6
- ExerciseÂ 11.7 Set 1, Set 2, Set 3
- Exercise 11.8 Set 1, Set 2

### Chapter 12: Higher Order Derivatives

The concepts covered in this chapter are proving relations involving various order derivatives of Cartesian functions, proving relations involving various order derivatives of parametric functions, and proving relations involving various order derivatives via illustrations covered in only one exercise.

### Chapter 13: Derivative as a Rate Measurer

This chapter introduced the derivative as a rate measurer. The concepts in this chapter cover how to find rate measurer of derivative and related rates in which the rate of change of one of the quantities involved is required, corresponding to the given rate of change of another quantity covered in its two exercises.Â

- Exercise 13.1
- Exercise 13.2 Set 1, Set 2

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

This chapter deals with the topic of differentials and errors. The topics that are discussed include the definition of differentials, absolute error, relative error, percentage error, the geometrical meaning of differentials with algorithms, and finding the approximate value using differentials.

### Chapter 15: Mean Value Theorems

This chapter is all about the theorems related to mean values. It also deals with Rolleâ€™s theorem, geometrical interpretation of Rolleâ€™s theorem, algebraic interpretation of Rolleâ€™s theorem, the applicability of Rolleâ€™s theorem, Lagrangeâ€™s mean value theorem, geometrical interpretation of Lagrangeâ€™s mean value theorem, verification of Lagrangeâ€™s mean value theorem, applications of Lagrangeâ€™s mean value theorem and proving inequalities by using Lagrangeâ€™s mean value theorem covered in its two exercises.

### Chapter 16: Tangents and Normals

The chapter basically deals with the slope of a line, slopes of tangent and normal, finding slopes of tangent and normal at a given point, finding the point on a given curve at which tangent is parallel or perpendicular to a given line, which is covered in its first exercise only. Moreover, the topics like the equations of tangent, and normal with an algorithm, finding the equation of tangent and normal to a curve at a point, finding tangent and normal parallel or perpendicular to a given line, finding tangent or normal passing through a given point, angle of intersection of two curves and orthogonal curves are covered in its last two exercises 16.2 and 16.3.

- ExerciseÂ 16.1 Set 1, Set 2
- ExerciseÂ 16.2 Set 1, Set 2
- ExerciseÂ 16.3

### Chapter 17: Increasing and Decreasing Functions

The present chapter deals with the concepts of increasing and decreasing functions, solution of rational algebraic inequations with algorithms, strictly increasing functions, strictly decreasing functions which are covered in its first exercise that is exercise 17.1. However, the monotonic functions, monotonic increasing, and monotonic decreasing functions, necessary and sufficient conditions for monotonicity, finding the intervals in which a function is increasing or decreasing, and proving the monotonicity of a function on a given interval are covered in its Exercise 17.2.

- Exercise 17.1
- ExerciseÂ 17.2 Set 1, Set 2, Set 3

### Chapter 18: Maxima and Minima

The chapter Maxima and Minima deal with the maximum and minimum values of a function in its domain, the definition of maximum, local maxima, and local minima, definition, and meaning of local maximum covered in the first two exercises while, the first derivative test for local maxima and minima along with algorithm, higher-order derivative test, point of inflection, point of inflection, properties of maxima and minima, maximum and minimum values in the closed interval and applied problems on maxima and minima are covered in exercises 18.3, 18.4 and 18.5.Â

- Exercise 18.1
- ExerciseÂ 18.2
- ExerciseÂ 18.3
- Exercise 18.4
- ExerciseÂ 18.5 Set 1, Set 2, Set 3

### Chapter 19: Indefinite Integrals

The chapter indefinite integrals explain the concepts of the primitive and anti derivative, fundamental integration formulae, some standard results on integration, integration of trigonometric functions, integration of exponential functions, geometrical interpretation of indefinite integral, comparison between differentiation and integration, methods of integration, integration by substitution, integration by parts, integration of rational algebraic functions by using partial fractions and integration of some special irrational algebraic functions. These concepts are explained very well through examples in total of thirty-two exercises.

- Exercise 19.1
- ExerciseÂ 19.2 Set 1, Set 2
- ExerciseÂ 19.3 Set 1, Set 2
- ExerciseÂ 19.4
- ExerciseÂ 19.5
- Exercise 19.6
- Exercise 19.7
- ExerciseÂ 19.8 Set 1, Set 2
- ExerciseÂ 19.9 Set 1, Set 2, Set 3
- ExerciseÂ 19.10
- ExerciseÂ 19.11
- Exercise 19.12
- ExerciseÂ 19.13 Set 1, Set 2
- ExerciseÂ 19.14
- ExerciseÂ 19.15
- ExerciseÂ 19.16
- ExerciseÂ 19.17
- ExerciseÂ 19.18 Set 1, Set 2
- ExerciseÂ 19.19
- ExerciseÂ 19.20
- ExerciseÂ 19.21
- Exercise 19.22
- ExerciseÂ 19.23 Set 1, Set 2
- ExerciseÂ 19.24
- ExerciseÂ 19.25 Set 1, Set 2, Set 3
- ExerciseÂ 19.26 Set 1, Set 2
- ExerciseÂ 19.27
- Exercise 19.28
- ExerciseÂ 19.29
- ExerciseÂ 19.30 Set 1, Set 2, Set 3
- ExerciseÂ 19.31
- ExerciseÂ 19.32

### Chapter 20: Definite Integrals

This chapter comes in the continuation of the above-mentioned chapter. This chapter helps to learn about the definite integral of a function that remains associated with the antiderivative and indefinite integral explained in its first three exercises, Exercises 20.1, 20.2, and 20.3. However, the other topics like the definite integrals as a limit of a sum, properties of definite integrals, fundamental theorem of calculus, etc are covered in exercises 20.4 and 20.5.

- ExerciseÂ 20.1 Set 1, Set 2, Set 3
- ExerciseÂ 20.2 Set 1, Set 2, Set 3
- ExerciseÂ 20.3 Set 1, Set 2
- Exercise 20.4 Set 1, Set 2
- ExerciseÂ 20.5 Set 1, Set 2, Set 3

### Chapter 21: Areas of Bounded Regions

There are four exercises in this chapter that helps to learn about how to draw the rough sketches of several types of curves that mainly enclose one region.

- ExerciseÂ 21.1 Set 1, Set 2, Set 3
- ExerciseÂ 21.2
- ExerciseÂ 21.3
- Exercise 21.4

### Chapter 22: Differential Equations

The present chapter consists of a total of eleven exercises that cover different topics like how to solve the differential equations, what properties can be used to solve differential equations, etc.

- ExerciseÂ 22.1 Set 1, Set 2
- ExerciseÂ 22.2 Set 1, Set 2
- ExerciseÂ 22.3 Set 1, Set 2
- Exercise 22.4
- ExerciseÂ 22.5 Set 1, Set 2
- Exercise 22.6
- ExerciseÂ 22.7 Set 1, Set 2, Set 3
- Exercise 22.8
- ExerciseÂ 22.9 Set 1, Set 2, Set 3
- ExerciseÂ 22.10 Set 1, Set 2
- ExerciseÂ 22.11 Set 1, Set 2, Set 3

### Chapter 23: Algebra of Vectors

The chapter Algebra of vectors refers to the action of performing algebraic operations on different types of vectors. In total nine exercises the concepts of the vectors in 2D, as well as 3D space, are also explained.

- Exercise 23.1
- ExerciseÂ 23.2
- ExerciseÂ 23.3
- ExerciseÂ 23.4
- ExerciseÂ 23.5
- ExerciseÂ 23.6 Set 1, Set 2
- Exercise 23.7
- ExerciseÂ 23.8
- ExerciseÂ 23.9

### Chapter 24: Scalar Or Dot Product

This chapter consists of two exercises that help to understand how to find scalar products of vectors along with the important properties of scalar or dot products.

- ExerciseÂ 24.1 Set 1, Set 2, Set 3
- ExerciseÂ 24.2

### Chapter 25: Vector or Cross Product

This chapter in the present book covers the topics like equality of vectors, vector products, properties of vector products, vector products of unit vectors, vector products of perpendicular vectors, area of a triangle, and parallelogram in just one exercise only.

### Chapter 26: Scalar Triple Product

This chapter is also a part of vector algebra which helps to learn how to find the scalar triple product of vectors and properties of the scalar triple product explained in one exercise only.

### Chapter 27: Direction Cosines and Direction Ratios

This chapter is an elaborated version of trigonometry. This chapter helps to learn about direction ratios that provide an easy way to specify the direction of a line in any 3D space. Here is only one exercise that includes the problems based on the mentioned topics.

### Chapter 28: Straight Line in Space

The chapter straight line in space discusses a line that extends on both sides still infinitely without any use of curves. This chapter in total contains five exercises only that help to understand each and every concept of a straight line.

- ExerciseÂ 28.1 Set 1, Set 2
- ExerciseÂ 28.2 Set 1, Set 2
- ExerciseÂ 28.3
- Exercise 28.4
- ExerciseÂ 28.5

### Chapter 29: The Plane

A plane refers to a flat 2D surface that contains infinite dimensions but has zero thickness. The present chapter has fifteen exercises that basically help to understand the concepts such as finding slope, x-intercept, and y-intercept. Â

- ExerciseÂ 29.1
- Exercise 29.2
- ExerciseÂ 29.3 Set 1, Set 2
- ExerciseÂ 29.4
- Exercise 29.5
- ExerciseÂ 29.6
- Exercise 29.7
- ExerciseÂ 29.8
- ExerciseÂ 29.9
- Exercise 29.10
- ExerciseÂ 29.11 Set 1, Set 2
- ExerciseÂ 29.12
- ExerciseÂ 29.13
- Exercise 29.14
- ExerciseÂ 29.15 Set 1, Set 2

### Chapter 30: Linear programming

This chapter gives the knowledge about some methods that help in the formation of linear equations. Here are five exercises in which the problems like finding the maximum and minimum value of an equation within a given set of conditions can be asked.

- Exercise 30.1 Set 1, Set 2
- ExerciseÂ 30.2 Set 1, Set 2, Set 3
- ExerciseÂ 30.3
- ExerciseÂ 30.4 Set 1, Set 2
- Exercise 30.5

### Chapter 31: Probability

The probability in this class is a bit advanced than the probability learned in previous classes. There are seven exercises that help to study different methods to find the probability of different organized events.Â

- Exercise 31.1
- ExerciseÂ 31.2
- ExerciseÂ 31.3 Set 1, Set 2
- ExerciseÂ 31.4 Set 1, Set 2
- ExerciseÂ 31.5 Set 1, Set 2, Set 3
- Exercise 31.6
- ExerciseÂ 31.7 Set 1, Set 2, Set 3

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

This chapter is an extension of the probability chapter.Â There are two exercises in this chapter that covers different topics like Bayes theorem, how to find the mean of random variables, and how to find the variance of random variables.Â

### Chapter 33: Binomial Distribution

This chapter is an extension of statistics and probability. There are only two exercises in this chapter that includes problems based on the binomial distribution and how to calculate Bernoulli trials.