RD Sharma Solutions For Class 12 Maths – Chapter wise Free PDF Download Updated for (2021-22)
RD Sharma Solutions for Class 12 Maths PDF can be easily accessed by students to start adequate preparation for their upcoming exams. Students can now solve any problem from the RD Sharma textbooks by referring to RD Sharma Solutions. BYJU’S experts formulate these questions in an easy and understandable manner that helps students solve problems in the most efficient possible ways. We hope these solutions will help CBSE Class 12 students to build a strong foundation of the basics and secure excellent marks in their final exam.
As Mathematics is an intricate subject for Class 12 students, these solutions will change every student’s approach towards Mathematics and will definitely make them realize how interesting and easy the subject is. RD Sharma Solutions are focused on learning various Mathematics tricks and shortcuts for quick and easy calculations. Download the RD Sharma Solutions for Class 12 Maths now and practise all the CBSE Textbook questions. Solving these questions will ensure that students have a good practice of all types of questions that can be framed in the examination.
RD Sharma Chapter-wise Solutions for Class 12 Maths
Students are also given access to additional online study materials and resources, available at BYJU’S, such as notes, books, question papers, exemplar problems, worksheets, etc. Students are also advised to practise Class 12 Sample papers to get an idea of the question pattern in the final exam. Practising RD Sharma Class 12 Solutions on a regular basis help students improve the skills that are important to score optimum marks in board exams.
Download Class 12 Maths RD Sharma Solutions to answer the chapter-wise questions effortlessly.
RD Sharma Solutions for Class 12 Maths – Exercise-wise Solutions
Students can refer to these solutions of RD Sharma Class 12, which helps in gaining knowledge. For better guidance, it is best to solve these solutions. Solving the exercises in each chapter will ensure that students score good marks in the board exams.
RD Sharma Solutions for Class 12 Maths Chapter 1- Relations
Chapter 1 of the RD Sharma textbook deals with 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.
Relation: It defines the relationship between two sets of different information. If we consider two sets, the relation between them will be established if there is a connection between the elements of two or more non-empty sets.
Types of Relations: The different types of relations are empty, universal, identity, inverse, reflexive, symmetric, transitive, and equivalence relation.
Representation of Types of Relations
| Relation Type | Condition |
|---|---|
| Empty Relation | R = φ ⊂ A × A |
| Universal Relation | R = A × A |
| Identity Relation | I = {(a, a), a ∈ A} |
| Inverse Relation | R-1 = {(b, a): (a, b) ∈ R} |
| Reflexive Relation | (a, a) ∈ R |
| Symmetric Relation | aRb ⇒ bRa, ∀ a, b ∈ A |
| Transitive Relation | aRb and bRc ⇒ aRc ∀ a, b, c ∈ A |
Here, you can find the exercises solution links for the topics covered in this chapter.
| Chapter 1 – Relations Exercises: |
|---|
Also access the following resources for Class 12 Chapter 1 Relations at BYJU’S:
- NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
- NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions
RD Sharma Solutions for Class 12 Maths Chapter 2 – Functions
Chapter 2, Functions of RD Sharma Class 12 Maths, gives 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, greatest integer function, properties of greatest integer function, smallest integer functions and its 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, kinds of functions such as one-one, many-one and onto function, bijection, the composition of functions and its properties and composition of real functions. Meanwhile, in this chapter, students get to learn how to relate graphs of a function and its inverse.
Function: A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Types of Functions: The types of functions are constant, identity, modulus, integer, exponential, logarithmic, reciprocal, and square root functions.
Inverse Function: An inverse function or an anti-function is a function that can reverse into another function. In other words, if any function “f” takes x to y then, the inverse of “f” will take y to x.
| Chapter 2 – Functions Exercises: |
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Also access the following resources for Class 12 Chapter 2 Functions at BYJU’S:
- NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
- NCERT Exemplar Solutions for Class 12 Maths Chapter 1 Relations and Functions
RD Sharma Solutions for Class 12 Maths Chapter 3 – Binary Operations
In Chapter 3 of the RD Sharma textbook, we shall see the definition of a binary operation, 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 ‘n’ and multiplication modulo ‘n’.
Binary Operations: It is defined as an operation * which is performed on a set A. The function is given by *: A * A → A. So, the operation * performed on operands a and b is denoted by a * b.
Types of Binary Operations: The types of Binary Operations are Distributivity, Associativity, and Commutativity.
Here, you can find the exercises solution links for the topics covered in this chapter.
RD Sharma Solutions for Class 12 Maths Chapter 4 – Inverse Trigonometric Functions
Chapter 4 of RD Sharma 12 Maths Solutions discusses the topic of the inverse of a function. Students will get to learn about 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.
Inverse Trigonometric Functions: They are the inverse functions of the basic trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant functions. We can also term it as arcus functions, antitrigonometric functions or cyclometric functions.
Properties of Inverse Trigonometric Function
Property Set 1:
- Sin−1(x) = cosec−1(1/x), x∈ [−1,1] − {0}
- Cos−1(x) = sec−1(1/x), x ∈ [−1,1] − {0}
- Tan−1(x) = cot−1(1/x), if x > 0 (or) cot−1(1/x) −π, if x < 0>
- Cot−1(x) = tan−1(1/x), if x > 0 (or) tan−1(1/x) + π, if x < 0>
Property Set 2:
- Sin−1(−x) = −Sin−1(x)
- Tan−1(−x) = −Tan−1(x)
- Cos−1(−x) = π − Cos−1(x)
- Cosec−1(−x) = − Cosec−1(x)
- Sec−1(−x) = π − Sec−1(x)
- Cot−1(−x) = π − Cot−1(x)
Below we have links provided to each exercise solution covered in this chapter.
Also access the following resources for Class 12 Chapter 4 Inverse Trigonometric Functions at BYJU’S:
- NCERT Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions
- NCERT Exemplar Solutions for Class 12 Maths Chapter 2 Inverse Trigonometric Functions
RD Sharma Solutions for Class 12 Maths Chapter 5 – Algebra of Matrices
Chapter 5 of the RD Sharma textbook starts with the definitions of Matrices. Students go on to learn about 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, 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.
Matrices: A rectangular array of m × n numbers (real or complex) in the form of m horizontal lines (called rows) and n vertical lines (called columns), is called a matrix of order m by n, written as m × n matrix.
Types of Matrices: The types of matrices are symmetric matrix, skew-symmetric matrix, Hermitian and skew-Hermitian matrix, orthogonal matrix, idempotent matrix, involuntary matrix, and nilpotent matrix.
Transpose of Matrix: The matrix which is obtained from a given matrix A by changing its rows into columns or columns into rows is known as the transpose of matrix A and is denoted by AT or A’.
Here, students can see the exercises explaining these concepts properly with solutions.
Also access the following resources for Class 12 Chapter 5 Algebra of Matrices at BYJU’S:
- NCERT Solutions for Class 12 Maths Chapter 3 Matrices
- NCERT Exemplar Solutions for Class 12 Maths Chapter 3 Matrices
RD Sharma Solutions for Class 12 Maths Chapter 6 – Determinants
Chapter 6 of the RD Sharma Mathematics Class 12 textbook, 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, 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 condition for consistency.
Determinants: We can define a determinant by expressing in terms of columns of the matrix as n x n matrix in terms of column vectors.
