Table of Contents
RD Sharma’s Solutions for Class 12 Maths is a comprehensive resource designed to assist students in mastering the complexities of mathematics at this crucial academic level. The latest edition of RD Sharma’s textbooks is tailored to align with the CBSE syllabus, ensuring that students are wellprepared for their board examinations and competitive exams like JEE.
The solutions are structured to cover all 33 chapters, providing detailed explanations and stepbystep methods to tackle a wide variety of mathematical problems. This approach not only aids in understanding but also enhances problemsolving skills, which are essential for excelling in exams. The solutions are available in PDF format, making them easily accessible for download and offline study.
RD Sharma Solutions for Class 12 Maths Volume I and Volume II
RD Sharma’s Solutions for Class 12 Maths is an essential resource for students preparing for their board examinations and competitive exams like IIT JEE. This comprehensive guide covers all chapters from the RD Sharma Class 12 Maths textbooks, providing detailed solutions to the exercises presented in both Volume 1 and Volume 2.
The latest edition of RD Sharma’s solutions includes wellstructured, stepbystep explanations that help students grasp complex mathematical concepts. Each chapter features illustrative examples, a variety of practice questions, and summaries for quick revision, making it an invaluable tool for mastering the curriculum.
Available in PDF format, students can easily download the RD Sharma Class 12 solutions for offline study, allowing for flexible learning. The solutions are crafted by experienced educators, ensuring clarity and accuracy, which is crucial for building a solid foundation in mathematics. Whether for selfstudy or as a supplement to classroom learning, RD Sharma’s Class 12 Maths solutions are designed to enhance understanding and improve problemsolving skills, ultimately aiding students in achieving academic success.
RD Sharma Solutions for Class 12 Maths Chapterwise
The RD Sharma Solutions for Class 12 Maths containing 33 chapters are divided into two volumes, volume I and II, first volume contains 1 to 19 chapters and second volume 20 to 33 that cover important mathematical concepts for students.
 Chapter 1: Relations
 Chapter 2: Functions
 Chapter 3: Binary Operations
 Chapter 4: Inverse Trigonometric Functions
 Chapter 5: Algebra of Matrices
 Chapter 6: Determinants
 Chapter 7: Adjoint and Inverse of a Matrix
 Chapter 8: Solution of Simultaneous Linear Equations
 Chapter 9: Continuity
 Chapter 10: Differentiability
 Chapter 11: Differentiation
 Chapter 12: Higher Order Derivatives
 Chapter 13: Derivative as a Rate Measurer
 Chapter 14: Differentials, Errors and Approximations
 Chapter 15: Mean Value Theorems
 Chapter 16: Tangents and Normals
 Chapter 17: Increasing and Decreasing Functions
 Chapter 18: Maxima and Minima
 Chapter 19: Indefinite Integrals
 Chapter 20: Definite Integrals
 Chapter 21: Areas of Bounded Regions
 Chapter 22: Differential Equations
 Chapter 23: Algebra of Vectors
 Chapter 24: Scalar Or Dot Product
 Chapter 25: Vector or Cross Product
 Chapter 26: Scalar Triple Product
 Chapter 27: Direction Cosines and Direction Ratios
 Chapter 28: Straight Line in Space
 Chapter 29: The Plane
 Chapter 30: Linear programming
 Chapter 31: Probability
 Chapter 32: Mean and Variance of a Random Variable
 Chapter 33: Binomial Distribution
RD Sharma Solution for Class 12 Maths All Chapter – Detailed Overview
The RD Sharma Solutions for Class 12 Maths provides a comprehensive guide through the curriculum, divided into two volumes containing a total of 33 chapters.
Class 12 RD Sharma Maths Volume I All Chapters
RD Sharma Class 12 Maths Solutions Chapter 1 – Relations
The RD Sharma Class 12 Maths Solutions Chapter 1 Relations provides a comprehensive guide for students studying relations in mathematics. This chapter is crucial as it lays the foundation for understanding various types of relationships between sets, which is essential for higherlevel mathematics.
The chapter begins with an introduction to the concept of relations, defined as a connection between two sets. It emphasizes that a relation can be established if there is a link between the elements of two or more nonempty sets.Key topics covered in this chapter include:
Types of Relations: The chapter details various types of relations such as:

 Void Relation: A relation that contains no elements.
 Universal Relation: A relation that includes all possible pairs from the sets involved.
 Identity Relation: A relation where each element is related to itself.
 Reflexive Relation: A relation where every element is related to itself.
 Symmetric Relation: A relation where if one element is related to another, then the second is related to the first.
 Transitive Relation: A relation where if one element is related to a second, and the second is related to a third, then the first is related to the third.
 Antisymmetric Relation: A relation where if one element is related to another and vice versa, then both elements must be the same.
 Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
Theorems and Properties: The chapter also includes important theorems related to these types of relations, providing students with a deeper understanding of their applications.
The solutions provided in the RD Sharma textbook are structured to guide students through various exercises, which include multiplechoice questions and practical problems. Each exercise is designed to reinforce the concepts discussed and improve problemsolving skills. The exercises are categorized as follows:
RD Sharma Class 12 Maths Solutions Chapter 1 Exercises
 Exercise 1.1
 Exercise 1.2
RD Sharma Class 12 Maths Solutions Chapter 2 – Functions
The second chapter of RD Sharma’s Class 12 Mathematics textbook focuses on Functions, a fundamental concept in mathematics. This chapter is structured to provide students with a comprehensive understanding of various types of functions, their properties, and operations involving functions.
Key Concepts Covered in Chapter 2:
 Definition of a function: A function is a rule that assigns a unique output to every input.
 Domain and codomain: The domain is the set of all possible inputs, while the codomain is the set of all possible outputs.
 Range: The range is the subset of the codomain that contains all the actual outputs of the function.
 Types of functions:
 Identity function: f(x) = x
 Constant function: f(x) = c (where c is a constant)
 Polynomial function: f(x) = a_n * x^n + a_(n1) * x^(n1) + … + a_1 * x + a_0
 Rational function: f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions
 Irrational function: A function involving irrational expressions
 Exponential function: f(x) = a^x (where a is a positive constant)
 Logarithmic function: f(x) = log_a(x) (where a is a positive constant)
 Trigonometric functions: f(x) = sin(x), cos(x), tan(x), cot(x), sec(x), csc(x)
 Graphs of functions: Visual representation of the relationship between inputs and outputs.
 Inverse functions: A function that undoes the action of another function.
 Composition of functions: Creating a new function by applying two functions one after the other.
 Binary operations: Operations that combine two elements of a set to produce a third element.
RD Sharma Class 12 Maths Solutions Chapter 2 Exercises
 Exercise 2.1
 Exercise 2.2
 Exercise 2.3
 Exercise 2.4
RD Sharma Class 12 Maths Solutions Chapter 3 – Binary Operations
Chapter 3 of RD Sharma’s Class 12 Mathematics textbook focuses on Binary Operations, a fundamental concept in algebra that combines two elements from a set to produce another element within the same set.
Key Topics Covered in Chapter 3:
 Definition of Binary Operation: Understanding what a binary operation is and how it operates on a set.
 Properties of Binary Operations: Exploring properties like associativity, commutativity, identity element, and inverse element.
 Types of Binary Operations: Discussing specific types such as addition, subtraction, multiplication, division, and their properties.
 Binary Operations on Sets: Applying binary operations to different sets, including sets of numbers, matrices, and functions.
 Applications of Binary Operations: Understanding how binary operations are used in various fields, such as algebra, number theory, and computer science.
RD Sharma Class 12 Maths Solutions Chapter 3 Exercises
 Exercise 3.1
 Exercise 3.2
 Exercise 3.3
 Exercise 3.4
 Exercise 3.5
RD Sharma Class 12 Maths Solutions Chapter 4 – Inverse Trigonometric Functions
Chapter 4 covers inverse trigonometric functions, which are the inverse operations of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. The chapter discusses the definition, properties, and evaluation of inverse trigonometric functions.
Key Topics Covered in the Chapter:

Definition of Inverse Trigonometric Functions:
 Introduction to inverse trigonometric functions as the inverse relations of trigonometric functions.
 Understanding the domains and ranges of inverse trigonometric functions.

Properties of Inverse Trigonometric Functions:
 Learning about the fundamental properties of inverse trigonometric functions, such as their values at specific angles, identities, and graphs.

Applications of Inverse Trigonometric Functions:
 Exploring the applications of inverse trigonometric functions in solving trigonometric equations, finding angles in a triangle, and various realworld problems.
Important Formulas and Concepts:
 Inverse Trigonometric Functions:
sin⁻¹(x)
,cos⁻¹(x)
,tan⁻¹(x)
,cot⁻¹(x)
,sec⁻¹(x)
,csc⁻¹(x)
 Properties:
sin⁻¹(sin x) = x
ifπ/2 ≤ x ≤ π/2
cos⁻¹(cos x) = x
if0 ≤ x ≤ π
tan⁻¹(tan x) = x
ifπ/2 < x < π/2
cot⁻¹(cot x) = x
if0 < x < π
sec⁻¹(sec x) = x
if0 ≤ x ≤ π, x ≠ π/2
csc⁻¹(csc x) = x
ifπ/2 ≤ x ≤ π/2, x ≠ 0
sin⁻¹(x) = sin⁻¹(x)
cos⁻¹(x) = π  cos⁻¹(x)
tan⁻¹(x) = tan⁻¹(x)
cot⁻¹(x) = π  cot⁻¹(x)
sec⁻¹(x) = π  sec⁻¹(x)
csc⁻¹(x) = csc⁻¹(x)
 Identities:
sin⁻¹(x) + cos⁻¹(x) = π/2
tan⁻¹(x) + cot⁻¹(x) = π/2
sec⁻¹(x) + csc⁻¹(x) = π/2
tan⁻¹(x) = cot⁻¹(1/x)
sin⁻¹(x) = cos⁻¹(√(1x²))
cos⁻¹(x) = sin⁻¹(√(1x²))
RD Sharma Class 12 Maths Solutions Chapter 4 Exercises
 Exercise 4.1
 Exercise 4.2
 Exercise 4.3
 Exercise 4.4
 Exercise 4.5
 Exercise 4.6
 Exercise 4.7
 Exercise 4.8
 Exercise 4.9
 Exercise 4.10
 Exercise 4.11
 Exercise 4.12
 Exercise 4.13
 Exercise 4.14
RD Sharma Class 12 Maths Solutions Chapter 5 – Algebra of Matrices
RD Sharma Class 12 Maths Solutions Chapter 6 – Determinants
RD Sharma Class 12 Maths Solutions Chapter 6 Exercises
 Exercise 6.1
 Exercise 6.2
 Exercise 6.3
 Exercise 6.4
 Exercise 6.5
RD Sharma Class 12 Maths Solutions Chapter 7 – Adjoint and Inverse of a Matrix
Chapter 7 of the RD Sharma Class 12 Maths Solutions focuses on the concepts of the adjoint and inverse of a matrix, which are crucial for understanding linear algebra. This chapter provides a comprehensive overview of how to compute the adjoint of a square matrix and its inverse, along with the necessary properties and theorems that govern these operations.
The adjoint of a matrix is defined as the transpose of its cofactor matrix. It plays a significant role in finding the inverse of a matrix, especially for square matrices. The inverse of a matrix A can be calculated using the formula:
$A^{1} = \frac{1}{A} \text{adj}(A)$where A is the determinant of matrix A. This chapter emphasizes the importance of these concepts not only in theoretical mathematics but also in practical applications across various fields.
Key topics covered in this chapter include:
 Definition and calculation of the adjoint of a square matrix.
 Methods to find the inverse of a matrix, including the use of elementary transformations.
 Properties of invertible matrices and their adjoints.
 Applications of the adjoint and inverse in solving linear equations.
The solutions provided in the RD Sharma textbook are structured to facilitate a clear understanding of these concepts, making it easier for students to excel in their examinations. By practicing the exercises in this chapter, students can enhance their problemsolving skills and gain a deeper insight into matrix operations, which are foundational in advanced mathematics and engineering disciplines.
RD Sharma Class 12 Maths Solutions Chapter 7 Exercises
 Exercise 7.1
 Exercise 7.2
RD Sharma Class 12 Maths Solutions Chapter 8 – Solution of Simultaneous Linear Equations
of Simultaneous Linear Equations. The solutions provided in this chapter are accurate and comprehensive, covering all the questions in the chapter. This chapter deals with homogeneous and nonhomogeneous systems of linear equations and their solutions using various methods, including the matrix method.
The solutions explain important concepts such as the definition and meaning of a consistent system, solving systems when the coefficient matrix is nonsingular or singular, and obtaining the inverse of the coefficient matrix. Practicing these solutions helps students grasp the concepts effectively, clear their doubts instantly, and score well in board exams as well as competitive exams.
Key topics covered in RD Sharma Class 12 Maths Solutions for Chapter 8 include:
 Solving systems of equations by the matrix method.
 Showing the consistency of systems of linear equations.
 Solving homogeneous systems of linear equations.
These solutions are designed to help students understand the concepts clearly and enhance their problemsolving skills, ensuring they perform well in their exams.
RD Sharma Class 12 Maths Solutions Chapter 8 Exercises
 Exercise 8.1
 Exercise 8.2
RD Sharma Class 12 Maths Solutions Chapter 9 – Continuity
RD Sharma Class 12 Maths Solutions Chapter 10 – Differentiability
RD Sharma Class 12 Maths Solutions for Chapter 10 focuses on the concept of differentiability, which is crucial for understanding calculus. This chapter covers various aspects, including:
 Differentiability at a Point: This section explains the conditions under which a function is differentiable at a specific point, including the concepts of lefthand and righthand derivatives.
 Definition and Meaning of Differentiability: It provides a formal definition of differentiability, emphasizing the significance of continuity and the existence of derivatives.
 Differentiability in a Set: This topic extends the concept of differentiability to intervals and sets, discussing how functions behave across different domains.
 Useful Results on Differentiability: The chapter concludes with important theorems and results that aid in determining differentiability for various types of functions.
The exercises in this chapter include practical problems that help students apply the theoretical concepts learned. For instance, students are tasked with demonstrating that certain functions are continuous but not differentiable at specific points, such as f(x) = x − 3 at x = 3 and f(x) = x^(1/3) at x = 0.
RD Sharma Class 12 Maths Solutions Chapter 10 Exercises
 Exercise 10.1
 Exercise 10.2
RD Sharma Class 12 Maths Solutions Chapter 11 – Differentiation
Chapter 11 of the RD Sharma Class 12 Maths textbook focuses on Differentiation, a fundamental concept in calculus that deals with the rate of change of a function concerning its variables. This chapter is essential for students as it lays the groundwork for advanced topics in mathematics and has applications in various fields, including physics and engineering.
Key topics covered in this chapter include:
 Differentiation from First Principles: Students learn how to derive the derivative of a function using the limit definition, which is crucial for grasping the concept of a derivative.
 Inverse Trigonometric Functions: The chapter explains how to differentiate inverse trigonometric functions both from first principles and using the chain rule.
 Trigonometric Substitutions: This section covers differentiation techniques involving trigonometric identities to simplify complex functions.
 Implicit Functions and Logarithmic Differentiation: Students are introduced to differentiation techniques for implicit functions and learn how logarithmic differentiation can simplify the process of finding derivatives of complex functions.
 Parametric Functions: The differentiation of functions defined parametrically is also discussed, providing students with tools to handle equations where variables are interdependent.
 Differentiation of Infinite Series and Determinants: The chapter concludes with advanced topics such as differentiating infinite series and determinants, which are essential for higherlevel mathematics.
RD Sharma Class 12 Maths Solutions Chapter 11 Exercises
 Exercise 11.1
 Exercise 11.2
 Exercise 11.3
 Exercise 11.4
 Exercise 11.5
 Exercise 11.6
 Exercise 11.7
 Exercise 11.8
RD Sharma Class 12 Maths Solutions Chapter 12 – Higher Order Derivatives
Chapter 12 of the RD Sharma Class 12 Maths textbook, Higher Order Derivatives, focuses on the relationships between different orders of derivatives for Cartesian and parametric functions. This chapter is crucial for CBSE Class 12 and engineering entrance exam preparation, covering the proof and application of higherorder derivatives.
The firstorder derivative in parametric equations is given by:
dy/dx = (dy/dt) * (dt/dx) = y'(t)/x'(t)
The secondorder derivative is the derivative of the first derivative of the given function. If y = f(x), then dy/dx = f'(x). If f'(x) is differentiable, then differentiating dy/dx again with respect to x, the secondorder derivative is:
d/dx (dy/dx) = d^2y/dx^2 = f”(x)
RD Sharma Solutions for Class 12 Maths Chapter 12 provide detailed explanations and stepbystep solutions to the exercises and examples in the textbook.
RD Sharma Class 12 Maths Solutions Chapter 12 Exercises
 Exercise 12.1
RD Sharma Class 12 Maths Solutions Chapter 13 – Derivative as a Rate Measurer
Chapter 13 of the RD Sharma Class 12 Maths textbook focuses on the concept of derivative as a rate measurer. This chapter provides a comprehensive understanding of how derivatives can be used to measure the instantaneous rate of change of a physical quantity. Some of the key topics covered in this chapter include:
 Definition of derivative as a rate measurer
 Meaning of derivative as a rate measurer
 Theorems and remarks on derivative as a rate measurer
 Definition and meaning of related rates
 Exercises to apply the concepts of derivative as a rate measurer
The RD Sharma Solutions for Class 12 Maths Chapter 13 provide stepbystep solutions to all the questions and examples given in the textbook. These solutions are formulated based on the latest CBSE board guidelines and are designed to help students understand the concepts better and score well in their exams.
RD Sharma Class 12 Maths Solutions Chapter 13 Exercises
 Exercise 13.1
 Exercise 13.2
RD Sharma Class 12 Maths Solutions Chapter 14 – Differentials, Errors and Approximations
The chapter begins with the definition of differentials, which are used to describe the rate of change of a function. If f(x) is a function, the differential dy can be expressed as dy = f'(x) · dx, where f'(x) is the derivative of the function, y is the dependent variable, and x is the independent variable. This section emphasizes the geometric interpretation of differentials and their practical applications in estimating changes in quantities.
Next, the chapter discusses absolute error, defined as the difference between the actual value and the measured value of a quantity. The formula for absolute error is given by:
Δ x = x_{0} – x
where x_{0} is the measured value, and x is the actual value.
Following this, the concept of relative error is introduced, which is the ratio of the absolute error to the actual value, expressed as:
Relative Error = Δ x / x
This helps in understanding how significant the error is in relation to the size of the measurement.
The chapter also covers percentage error, which quantifies the error as a percentage of the actual value, calculated using the formula:
Percentage Error = (Estimated value – Actual value / Actual value) × 100
Through various exercises, students learn to apply these concepts to solve practical problems, such as estimating changes in volume when the radius of a sphere changes or calculating the percentage error in measurements.
RD Sharma Class 12 Maths Solutions Chapter 14 Exercises
 Exercise 14.1
RD Sharma Class 12 Maths Solutions Chapter 15 – Mean Value Theorems
RD Sharma Class 12 Maths Solutions Chapter 16 – Tangents and Normals
Chapter 16 of RD Sharma Class 12 Maths focuses on Tangents and Normals, which are fundamental concepts in calculus.
Key concepts include:
 Slopes of Tangents and Normals: The chapter begins by explaining how to find the slope of the tangent line to a curve at a given point using differentiation. The normal line, which is perpendicular to the tangent, has a slope that is the negative reciprocal of the tangent’s slope.
 Equations of Tangents and Normals: Students learn to derive the equations for both tangents and normals using the pointslope form of a linear equation.
 Applications: The exercises include practical problems where students must apply these concepts to find tangents and normals for various functions, including polynomial and trigonometric functions.
RD Sharma Class 12 Maths Solutions Chapter 16 Exercises
 Exercise 16.1
 Exercise 16.2
 Exercise 16.3
RD Sharma Class 12 Maths Solutions Chapter 17 – Increasing and Decreasing Functions
Chapter 17 of the RD Sharma Class 12 Maths textbook focuses on Increasing and Decreasing Functions. This chapter introduces the concept of monotonicity in functions, which is essential for understanding their behavior over different intervals.
Key concepts covered in this chapter include:
 Monotonic Functions: A function is said to be increasing on an interval if its derivative is greater than 0 for all values of x in that interval. Conversely, it is decreasing if its derivative is less than 0.
 Critical Points: These are points where the derivative is zero or undefined. The behavior of the function around these points helps determine the intervals of increase or decrease.
 Finding Intervals: Students learn to find intervals where a function is increasing or decreasing by analyzing the sign of the derivative across the critical points.
 Necessary and Sufficient Conditions: The chapter provides conditions under which a function is classified as strictly increasing or strictly decreasing.
RD Sharma Class 12 Maths Solutions Chapter 17 Exercises
 Exercise 17.1
 Exercise 17.2
RD Sharma Class 12 Maths Solutions Chapter 18 – Maxima and Minima
Chapter 18 of RD Sharma’s Class 12 Maths begins with the fundamental definitions and properties of maxima and minima, emphasizing the importance of derivatives in identifying these points.
Key topics covered include:
 Local Maxima and Minima: Understanding how to find points where a function reaches a local high or low.
 First Derivative Test: Utilizing the first derivative of a function to identify intervals where the function is increasing or decreasing, thus locating maxima and minima.
 Second Derivative Test: Applying the second derivative to confirm whether the critical points found using the first derivative are indeed maxima or minima.
 Applications of Derivatives: Exploring realworld applications of these concepts, such as optimization problems in various fields.
RD Sharma Class 12 Maths Solutions Chapter 18 Exercises
 Exercise 18.1
 Exercise 18.2
 Exercise 18.3
 Exercise 18.4
 Exercise 18.5
RD Sharma Class 12 Maths Solutions Chapter 19 – Indefinite Integrals
Chapter 19 of the RD Sharma Class 12 Maths textbook focuses on Indefinite Integrals, a fundamental concept in calculus. This chapter introduces the notion of the indefinite integral as the antiderivative of a function, emphasizing that it represents a family of functions whose derivatives yield the original function.
Key topics covered in this chapter include:
 Definition of Indefinite Integral: It is defined as the integral of a function without specified limits, resulting in a general form that includes a constant of integration, .
 Methods of Integration: Various techniques for solving integrals are discussed, including substitution, integration by parts, and the use of partial fractions. These methods are essential for evaluating more complex integrals.
 Fundamental Integration Formulas: The chapter provides standard results and formulas that are commonly used in integration, which serve as a reference for solving problems.
 Applications: The chapter also explores the integration of different types of functions, such as trigonometric, exponential, and polynomial functions, along with their geometric interpretations.
RD Sharma Class 12 Maths Solutions Chapter 19 Exercises
 Exercise 19.1
 Exercise 19.2
 Exercise 19.3
 Exercise 19.4
 Exercise 19.5
 Exercise 19.6
 Exercise 19.7
 Exercise 19.8
 Exercise 19.9
 Exercise 19.10
 Exercise 19.11
 Exercise 19.12
 Exercise 19.13
 Exercise 19.14
 Exercise 19.15
 Exercise 19.16
 Exercise 19.17
 Exercise 19.18
 Exercise 19.19
 Exercise 19.20
 Exercise 19.21
 Exercise 19.22
 Exercise 19.23
 Exercise 19.24
 Exercise 19.25
 Exercise 19.26
 Exercise 19.27
 Exercise 19.28
 Exercise 19.29
 Exercise 19.30
 Exercise 19.31
 Exercise 19.32
Class 12 RD Sharma Maths Volume II All Chapters
RD Sharma Class 12 Maths Solutions Chapter 20 – Definite Integrals
In Chapter 20 of RD Sharma Class 12 Maths, titled “Definite Integrals,” students explore the concept of definite integrals, which are essential in understanding the relationship between integration and the area under curves.
The chapter covers several key topics, including:
 Fundamental Theorem of Calculus: This theorem establishes the relationship between differentiation and integration, providing a framework for evaluating definite integrals.
 Properties of Definite Integrals: Students learn various properties that govern definite integrals, such as linearity and the effects of changing limits.
 Evaluation Techniques: The chapter emphasizes methods for evaluating definite integrals, including substitution and integration by parts.
RD Sharma Class 12 Maths Solutions Chapter 20 Exercises
 Exercise 20.1
 Exercise 20.2
 Exercise 20.3
 Exercise 20.4
 Exercise 20.5
 Exercise 20.6
RD Sharma Class 12 Maths Solutions Chapter 21 – Areas of Bounded Regions
In Chapter 21 of RD Sharma Class 12 Maths, titled “Areas of Bounded Regions,” students explore the concept of calculating the area enclosed between curves. This chapter is crucial for understanding how to apply integration techniques to find areas in various geometric configurations.
Key points of the chapter include:
 Understanding Bounded Regions: The chapter begins by defining bounded regions, which are areas enclosed by curves, lines, and axes. Students learn to identify these regions through sketches.
 Methods of Area Calculation: The primary methods discussed involve using vertical and horizontal strips to calculate the area. This involves setting up integrals based on the functions that define the curves.
 Formulas and Theorems: The chapter introduces essential formulas for calculating the area under a curve and between two curves. For a continuous function f(x) defined on an interval [a,b], the area is given by the integral ∫ from a to b of f(x) dx.
 Exercises and Applications: The chapter contains multiple exercises (21.1 to 21.4) that provide practical problems for students to apply the concepts learned. These exercises include finding areas between lines and curves, as well as between two curves.
 Graphical Representation: Emphasis is placed on sketching the curves involved, which aids in visualizing the area to be calculated and ensures a better understanding of the relationships between the functions.
 Importance in Mathematics: Understanding areas of bounded regions is foundational for further studies in calculus and geometry, making it a significant part of the Class 12 curriculum.
RD Sharma Class 12 Maths Solutions Chapter 21 Exercises
 Exercise 21.1
 Exercise 21.2
 Exercise 21.3
 Exercise 21.4
RD Sharma Class 12 Maths Solutions Chapter 22 – Differential Equations
Chapter 22 of the RD Sharma Class 12 Maths textbook focuses on Differential Equations, which are equations that include a dependent variable, an independent variable, and their derivatives. This chapter is crucial for understanding how to model realworld phenomena through mathematical equations.Key points covered in this chapter include:
 Definition: A differential equation is defined as an equation involving derivatives of a function. It relates the function itself to its rates of change.
 Types of Differential Equations: The chapter discusses various forms of differential equations, including linear and nonlinear types, as well as firstorder and higherorder equations.
 Order and Degree: The order of a differential equation is determined by the highest derivative present, while the degree is the power of the highest derivative, assuming it is a polynomial equation.
 Methods of Solving: Different methods for solving differential equations are introduced, such as the method of separation of variables, integrating factors, and homogeneous equations.
 Applications: The chapter emphasizes practical applications of differential equations, including problems related to growth and decay, Newton’s law of cooling, and other scenarios in physics and engineering.
 General and Particular Solutions: It explains the concepts of general and particular solutions, which are essential for understanding how to find specific solutions to differential equations based on initial conditions.
RD Sharma Class 12 Maths Solutions Chapter 22 Exercises
 Exercise 22.01
 Exercise 22.02
 Exercise 22.03
 Exercise 22.04
 Exercise 22.05
 Exercise 22.06
 Exercise 22.07
 Exercise 22.08
 Exercise 22.09
 Exercise 22.10
 Exercise 22.11
RD Sharma Class 12 Maths Solutions Chapter 23 – Algebra of Vectors
Chapter 23 of RD Sharma’s Class 12 Maths textbook begins with an introduction to vectors, defining what they are and distinguishing them from scalar quantities. It covers various types of vectors, including zero vectors, unit vectors, coinitial vectors, collinear vectors, and equal vectors.
Key topics explored in this chapter include:
 Addition and Subtraction of Vectors: The chapter explains how to perform these operations graphically and algebraically, emphasizing the triangle and parallelogram laws of vector addition.
 Multiplication of Vectors: This section covers scalar multiplication and introduces the concept of the dot product and cross product, which are vital for applications in physics.
 Position Vectors and Directed Line Segments: Students learn how to represent points in space using position vectors and how to calculate the directed line segment between two points.
 Section Formula: The chapter explains how to find a point that divides a line segment into a given ratio, which is crucial for problems involving geometry and coordinate systems.
 Projection of Vectors: This topic includes the projection of one vector onto another, an important concept in both mathematics and physics.
RD Sharma Class 12 Maths Solutions Chapter 23 Exercises
 Exercise 23.1
 Exercise 23.2
 Exercise 23.3
 Exercise 23.4
 Exercise 23.5
 Exercise 23.6
 Exercise 23.7
 Exercise 23.8
 Exercise 23.9
RD Sharma Class 12 Maths Solutions Chapter 24 – Scalar Or Dot Product
Chapter 24 of the RD Sharma Class 12 Maths textbook focuses on the concept of the Scalar or Dot Product.
The dot product is defined both algebraically and geometrically. Algebraically, it involves multiplying the corresponding components of two vectors and summing the results. Geometrically, it relates to the cosine of the angle between the two vectors, providing insight into their directional relationship. The formula for the dot product of two vectors
$a$and$b$
is given by:
$a \cdot b = a b \cos \theta$
where
$\theta$is the angle between the vectors. This relationship highlights the significance of the dot product in determining the angle between two vectors, as well as in assessing their orthogonality (perpendicularity).
RD Sharma Class 12 Maths Solutions Chapter 24 Exercises
 Exercise 24.1
 Exercise 24.2
RD Sharma Class 12 Maths Solutions Chapter 25 – Vector or Cross Product
In Chapter 25 of RD Sharma Class 12 Maths, titled “Vector or Cross Product,” students delve into the concept of the vector product between two vectors. This chapter focuses on defining the vector product, its properties, and its geometrical interpretations.The primary learning objectives include:
 Understanding the Vector Product: Students learn that the vector product of two vectors results in a new vector that is perpendicular to both original vectors. The magnitude of this vector is determined by the product of the magnitudes of the two vectors and the sine of the angle between them.
 Cartesian Form: The chapter explains how to compute the vector product when vectors are expressed in Cartesian coordinates, enhancing students’ ability to manipulate vectors in different forms.
 Geometrical Applications: Applications of the vector product are explored, such as finding the area of parallelograms and the volume of parallelepipeds formed by vectors, which are critical for understanding spatial relationships in geometry.
 Properties of the Cross Product: The chapter also covers essential properties of the vector product, including the noncommutative nature of the operation, meaning that the order of the vectors affects the resulting vector.
 Exercises and Solutions: To reinforce these concepts, the chapter includes numerous exercises that challenge students to apply what they have learned. RD Sharma Class 12 Maths Solutions provide detailed answers to these exercises, facilitating better comprehension and selfstudy.
RD Sharma Class 12 Maths Solutions Chapter 25 Exercises
 Exercise 25.1
RD Sharma Class 12 Maths Solutions Chapter 26 – Scalar Triple Product
RD Sharma Class 12 Maths Solutions Chapter 27 – Direction Cosines and Direction Ratios
Chapter 27 of the RD Sharma Class 12 Maths textbook focuses on Direction Cosines and Direction Ratios, essential concepts in threedimensional geometry.
Direction ratios are defined as the proportional values that indicate the direction of a line in threedimensional space, while direction cosines are the cosines of the angles formed between the line and the coordinate axes.
Students will learn to calculate direction cosines from given angles and direction ratios, explore the relationships between these elements, and apply them to solve various geometric problems. The chapter also covers important topics such as the angle between two lines, the equation of a plane, and the shortest distance between two lines, all of which are crucial for mastering threedimensional geometry.
RD Sharma Class 12 Maths Solutions Chapter 27 Exercises
 Exercise 27.1
RD Sharma Class 12 Maths Solutions Chapter 28 – Straight Line in Space
Chapter 28 of RD Sharma Class 12 Maths Straight Line in Space begins by defining a straight line as an infinite set of points extending in both directions without any curvature. It introduces essential concepts such as direction cosines, direction ratios, and the equations of lines in different forms, including vector and parametric equations. Students learn to derive the equations of lines through given points and to analyze the relationships between multiple lines, including conditions for parallelism and perpendicularity.
The chapter also covers the concept of angles between lines in space. It explains how to find the angle between two lines using their direction ratios or direction cosines. Additionally, the conditions for two lines to be parallel or perpendicular are discussed.
RD Sharma Class 12 Maths Solutions Chapter 28 Exercises
 Exercise 28.1
 Exercise 28.2
 Exercise 28.3
 Exercise 28.4
 Exercise 28.5
RD Sharma Class 12 Maths Solutions Chapter 29 – The Plane
RD Sharma Class 12 Maths Chapter 29 The Plane covers various topics related to planes in space, including the equation of a plane in normal form, the equation of a plane perpendicular to a given vector and passing through a given point, the shortest distance between two lines, the equation of a line in space, direction cosines and direction ratios of a line, and more.
Students will learn how to write the equation of a plane passing through three noncollinear points, the relation between direction ratio and direction cosines, the intercept form of the equation of a plane, and the coplanarity of two lines. The chapter also covers the distance of a point from a plane, the angle between a line and a plane, the angle between two planes, the vector and Cartesian equation of a plane, and the plane passing through the intersection of two given planes.
RD Sharma Class 12 Maths Solutions Chapter 29 Exercises
 Exercise 29.01
 Exercise 29.02
 Exercise 29.03
 Exercise 29.04
 Exercise 29.05
 Exercise 29.06
 Exercise 29.07
 Exercise 29.08
 Exercise 29.09
 Exercise 29.10
 Exercise 29.11
 Exercise 29.12
 Exercise 29.13
 Exercise 29.14
 Exercise 29.15
RD Sharma Class 12 Maths Solutions Chapter 30 – Linear Programming
Linear Programming is a mathematical technique used to optimize a linear objective function subject to a set of linear constraints. It is widely applied in various fields such as economics, business, engineering, and operations research.
Key Concepts:
 Objective Function: This is the linear function to be maximized or minimized.
 Constraints: These are the linear inequalities or equations that limit the values of the decision variables.
 Feasible Region: The region in the solution space that satisfies all the constraints.
 Optimal Solution: The point within the feasible region that gives the maximum or minimum value of the objective function.
Graphical Method:
 Convert inequalities to equations: Convert all the inequalities into equations to plot the lines on the graph.
 Plot the lines: Graph the lines corresponding to the equations.
 Identify the feasible region: Shade the region that satisfies all the inequalities.
 Corner points: Determine the coordinates of the corner points of the feasible region.
 Evaluate objective function: Substitute the coordinates of the corner points into the objective function to find the maximum or minimum value.
Simplex Method:
 Standard form: Convert the problem into standard form, where all constraints are inequalities and the objective function is to be maximized.
 Initial simplex tableau: Construct the initial simplex tableau.
 Pivoting: Select a pivot element and perform row operations to obtain a new tableau.
 Optimality check: Check if the current tableau is optimal. If not, repeat the pivoting process.
RD Sharma Class 12 Maths Solutions Chapter 30 Exercises
 Exercise 30.1
 Exercise 30.2
 Exercise 30.3
 Exercise 30.4
 Exercise 30.5
RD Sharma Class 12 Maths Solutions Chapter 31 – Probability
Chapter 31 of the RD Sharma Class 12 Maths textbook covers essential topics such as conditional probability, which is vital for deriving the multiplication theorem of probability. Students will explore theorems that assist in calculating the probabilities of simultaneous events, enhancing their ability to solve complex problems involving random experiments.
Key topics covered in the chapter include:
 Basic concepts: The chapter introduces the fundamental terms and definitions related to probability, such as experiment, outcome, sample space, event, and probability of an event.
 Probability axioms: It explains the three axioms of probability, which form the foundation of probability theory. These axioms define the properties that probabilities must satisfy.
 Probability theorems: The chapter discusses various probability theorems, including the addition theorem, conditional probability, and Bayes’ theorem. These theorems provide rules for calculating probabilities in different scenarios.
 Random variables and probability distributions: It introduces the concept of random variables and their probability distributions. Different types of probability distributions, such as discrete and continuous distributions, are explored.
 Expectation and variance: The chapter covers the concepts of expectation and variance, which measure the central tendency and spread of a probability distribution, respectively.
RD Sharma Class 12 Maths Solutions Chapter 31 Exercises
 Exercise 31.1
 Exercise 31.2
 Exercise 31.3
 Exercise 31.4
 Exercise 31.5
 Exercise 31.6
 Exercise 31.7
RD Sharma Class 12 Maths Solutions Chapter 32 – Mean and Variance of a Random Variable
Chapter 32 of RD Sharma Class 12 Maths covers the essential concepts of statistical measures, specifically mean and variance. These measures are key to understanding the central tendency and dispersion of a random variable.
The mean, or expected value, represents the average of a random variable and provides a central point around which the variable’s values are distributed. The chapter explains various methods to calculate the mean, including weighted mean and the mean for both discrete and continuous random variables.
Variance measures how spread out the values of a random variable are around the mean. It indicates the extent to which the values deviate from the average. A higher variance means a wider spread, while a lower variance shows a more concentrated distribution. The chapter also explains how to calculate variance and introduces standard deviation, which is the square root of variance and a commonly used measure of dispersion.
RD Sharma Class 12 Maths Solutions Chapter 32 Exercises
 Exercise 32.1
 Exercise 32.2
RD Sharma Class 12 Maths Solutions Chapter 33 – Binomial Distribution
Chapter 33 of the RD Sharma Class 12 Maths textbook focuses on the Binomial Distribution, a fundamental concept in probability theory. These conditions are crucial for applying the binomial distribution in various scenarios, such as coin tosses or quality control in manufacturing.
Key Concepts and Formulas:
 Bernoulli Trial: A single experiment with two possible outcomes: success (with probability p) or failure (with probability q = 1 – p).
 Binomial Experiment: A sequence of n independent Bernoulli trials.
 Binomial Random Variable: A variable that counts the number of successes in a binomial experiment.
 Probability Mass Function (PMF): The formula for calculating the probability of exactly x successes in n trials.
 P(X = x) = C(n, x) * p^x * q^(nx)
 where C(n, x) is the binomial coefficient (combination).
 P(X = x) = C(n, x) * p^x * q^(nx)
 Mean (Expected Value): The average number of successes in a binomial experiment.
 E(X) = np
 Variance: A measure of how spread out the distribution is.
 Var(X) = npq
 Standard Deviation: The square root of the variance.
 SD(X) = √(npq)
RD Sharma Class 12 Maths Solutions Chapter 33 Exercises
 Exercise 33.1
 Exercise 33.2
FAQs on RD Sharma Solutions for Class 12 Maths
How much time does it take to complete RD Sharma class 12?
The time required to complete RD Sharma's Class 12 mathematics book can vary significantly based on individual learning pace and prior knowledge. Generally, students might take several weeks to a few months to thoroughly work through the book, especially if they are balancing other subjects and commitments like school and coaching classes. It's essential to allocate sufficient time for practice and understanding of concepts, as mathematics often requires repeated exposure to different types of problems to master the material effectively.
Is RD Sharma sufficient for boards?
Regarding whether RD Sharma is sufficient for board exams, it is generally considered a valuable resource for additional practice and concept reinforcement. However, many educators emphasize that while RD Sharma provides a broad range of problems, students should primarily focus on the NCERT textbooks, as the majority of board exam questions are derived from them. Thus, RD Sharma can supplement NCERT but should not replace it as the primary study material.
How many chapters are there in RD Sharma?
The RD Sharma Class 12 Maths book is divided into two volumes: Volume 1 contains chapters 1 to 19, while Volume 2 covers chapters 20 to 33.
Is it necessary to solve RD Sharma for class 12?
While it is not strictly necessary to solve RD Sharma for Class 12, doing so can enhance mathematical skills and provide a more comprehensive grasp of the subject. Students who engage with both NCERT and RD Sharma often report a better understanding and improved performance in their exams.
Should I study RD Sharma or NCERT?
When considering whether to study RD Sharma or NCERT, it is advisable to prioritize NCERT for foundational knowledge, as it aligns directly with the board syllabus. RD Sharma can then be used as a supplementary resource for additional practice. This combination allows students to solidify their understanding while also preparing for a wider range of problems.
Can I do RD Sharma without NCERT?
Students can indeed work on RD Sharma without NCERT, but it is not recommended. NCERT textbooks are crucial for understanding the core concepts and are specifically tailored to the board exam format. Therefore, while RD Sharma can provide valuable practice, it should ideally be used alongside NCERT to ensure a wellrounded preparation strategy
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