RD Sharma Solutions

RD Sharma Solutions for Class 12 Maths – Free PDF Download

Class 12 Maths is a complex subject and consists of various difficult chapters. Inverse trigonometric functions, matrices, determinants, differentiation, derivatives, integrals, etc. are a part of class 12 Maths syllabus and involve a high level of skill and knowledge. The students need a lot of practice in order to master these chapters. The RD Sharma class 12 solutions aim at providing various easy maths tricks and shortcuts so that the students can make quick and accurate calculations and score high marks.

Importance of RD Sharma Solutions for Class 12 Maths

RD Sharma class 12 solutions are of vital importance for the students preparing for the Class 12 Maths examination. We at Toppr are providing the solutions in an easy and understandable manner. These solutions will help the students to gain conceptual knowledge as they provide an algorithmic approach. These solutions will also make learning maths interesting and easy for them. Practicing RD Sharma class 12 Solutions will ensure that students will be able to solve all types of questions that can be asked in the examination paper.

                                              Download Toppr – Best Learning App for Class 5 to 12

Chapters under RD Sharma Solutions for Class 12 Maths

rd sharma class 12 solutions

RD Sharma Solutions for Class 12 Maths Chapter 1: Relations

This chapter of RD Sharma explains the relations and their properties, types of relations, inverse of a relation, equivalence relation, symmetric relation, reflexive relation and transitive relation.

RD Sharma Solutions for Class 12 Maths Chapter 2: Functions

This chapter explains the meaning and definition of functions, function as a set of ordered pairs, the graph of a function, vertical line test, types of functions, operations on real function, bijection, the composition of functions and its properties.

RD Sharma Solutions for Class 12 Maths Chapter 3: Binary Operations

This chapter deal with the definition of a binary operation, the number of binary operations, types of binary operations, an identity element, inverse of an element, and the composition table.

RD Sharma Solutions for Class 12 Maths Chapter 4: Inverse Trigonometric Functions

This is a very important chapter in class 12. It deals with the definition and meaning of inverse trigonometric functions, and properties of inverse trigonometric functions.

RD Sharma Solutions for Class 12 Maths Chapter 5: Algebra of Matrices

This chapter discusses the definitions of Matrices, 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, transpose of a matrix, properties of transpose, symmetric and skew-symmetric matrices.

RD Sharma Solutions for Class 12 Maths Chapter 6: Determinants

In this chapter, we shall learn the 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, and applications of determinants to coordinate geometry.

RD Sharma Solutions for Class 12 Maths Chapter 7: Adjoint and Inverse of a Matrix

In this chapter, we shall study the definition of the adjoint of a square matrix, the inverse of a matrix, some useful results on invertible matrices, and elementary transformations or elementary operations of a matrix.

RD Sharma Solutions for Class 12 Maths Chapter 8: Solutions of Simultaneous Linear Equations

This chapter explains the consistent system, homogeneous and non-homogeneous systems, and matrix method for the solution of a non-homogeneous system.

RD Sharma Solutions for Class 12 Maths Chapter 9: Continuity

This chapter deals with 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, and properties of continuous functions.

RD Sharma Solutions for Class 12 Maths Chapter 10: Differentiability

In this chapter, the students will learn advance concepts of differentiability at a point, differentiability in a set and some useful results on differentiability.

RD Sharma Solutions for Class 12 Maths Chapter 11: Differentiation

In this chapter, the students will learn differentiation of inverse trigonometric functions from first principles, differentiation of a function, differentiation of inverse trigonometric functions by chain rule, differentiation by using trigonometric substitutions, differentiation of implicit functions, logarithmic differentiation, and differentiation of infinite series.

RD Sharma Solutions for Class 12 Maths Chapter 12: Higher Order Derivatives

This chapter explains proving relations involving various order derivatives of cartesian functions, proving relations involving various order derivatives of parametric functions and proving relations involving various order derivative.

RD Sharma Solutions for Class 12 Maths Chapter 13: Derivative as a Rate Measurer

This chapter teaches how to find rate measurer of derivative and related rates.

RD Sharma Solutions for Class 12 Maths Chapter 14: Differentials, Errors and Approximations

This chapter explains the definition of differentials, absolute error, relative error, percentage error, the geometrical meaning of differentials with algorithm and finding the approximate value using differentials.

RD Sharma Solutions for Class 12 Maths Chapter 15: Mean Value Theorems

This chapter discusses mean value theorems, Rolle’s theorem, geometrical and algebraic interpretation of Rolle’s theorem and its applicability, and Lagrange’s mean value theorem.

RD Sharma Solutions for Class 12 Maths Chapter 16: Tangents and Normals

This chapter deals with the concepts related to tangents and normal, 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, finding tangent or normal passing through a given point, angle of intersection of two curves and orthogonal curves.

RD Sharma Solutions for Class 12 Maths Chapter 17: Increasing and Decreasing Functions

In this chapter, the students will study the increasing and decreasing functions, solution of rational algebraic inequations with algorithms, strictly increasing functions, strictly decreasing functions, monotonic functions, and monotonic increasing and monotonic decreasing functions.

RD Sharma Solutions for Class 12 Maths Chapter 18: Maxima and Minima

This chapter explains the maximum and minimum values of a function in its domain, definition of maximum, local maxima and local minima, local maximum, first derivative test for local maxima and minima along with algorithm, higher-order derivative test, point of inflection, and point of inflexion.

RD Sharma Solutions for Class 12 Maths Chapter 19: Indefinite Integrals

This chapter discusses indefinite integral, primitive and antiderivative, 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, and methods of integration.

RD Sharma Solutions for Class 12 Maths Chapter 20: Definite Integrals

This chapter explains definite integrals as a limit of a sum, basic properties of definite integrals and the fundamental theorem of Calculus.

RD Sharma Solutions for Class 12 Maths Chapter 21: Areas of Bounded Regions

This chapter deals with finding the areas of adjacent regions by means of drawing sketches.

RD Sharma Solutions for Class 12 Maths Chapter 22: Differential Equations

This chapter explains the equations having dependent and independent variables and differential coefficients.

RD Sharma Solutions for Class 12 Maths Chapter 23: Algebra of Vectors

This chapter deals with the Euclidean vectors, algebraic services, orthonormality, and orthogonal projection.

RD Sharma Solutions for Class 12 Maths Chapter 24: Scaler or Dot product

This chapter explains the multiplication of two vectors using Scaler or Dot product.

RD Sharma Solutions for Class 12 Maths Chapter 25: Vector or Cross Product

It deals with cross-product in vector series.

RD Sharma Solutions for Class 12 Maths Chapter 26: Scalar Triple Product

This chapter explains the Scaler Triple Product method of solving vector problems.

RD Sharma Solutions for Class 12 Maths Chapter 27: Direction Cosines and Direction Ratios

This chapter discusses the direction ratios and the Cosines which are a part of trigonometry.

RD Sharma Solutions for Class 12 Maths Chapter 28: Straight Line in Space

This chapter discusses the straight line in space which is a concept of geometry.

RD Sharma Solutions for Class 12 Maths Chapter 29: The Plane

This chapter explains the plane concept of geometry.

RD Sharma Solutions for Class 12 Maths Chapter 30: Linear programming

This chapter deals with the types of linear programming problems, feasible and infeasible solutions and optimal workable solutions.

RD Sharma Solutions for Class 12 Maths Chapter 31: Probability

This chapter explains the probability concept in detail.

RD Sharma Solutions for Class 12 Maths Chapter 32: Mean and Variance of a Random Variable

This chapter discusses the Mean ad also the Variance of random variables.

RD Sharma Solutions for Class 12 Maths Chapter 33: Binomial Distribution

This chapter discusses the problems related to the Binomial distribution, a concept of statistics.

Solved Questions from RD Sharma Solutions for Class 12 Maths

Question 1: Determine whether the following operation define a binary operation on the given set or not :
\(‘x’_{6}\) on S={1,2,3,4,5} defined by \(a\times_{6}b\) = Remainder when ab is divided by 6.

Answer:

Given \(‘x’_{6}​\)that is an operation that is valid for the numbers in the Set S = 1,2,3,4,5 and it is defined as given:

\(a\times_{6}b\) = Remainder when ab is divided by 6, where a, b \(\in\)  S

since a \(\in\)  S and b  \(\in\)  S,

According to the problem it is given that on applying the operation ‘∗’ for two given numbers in the set ‘S’ it gives one of the numbers in the set ‘S’ as a result of the operation,

⇒ a\(‘x’_{6}\)  \(\in\) S……………(1)

Let us take the values of a = 3, b = 4,

ab=3×4

ab=12

We know that 12 is a multiple of 6. So, on dividing 12 with 6 we get 0 as remainder which is not in the given set ‘S’.

∴the operation \(‘x’_{6}\) ​ does not define a binary operation on set S.

Question 2: Prove that the function \(F:N \rightarrow N\), defined by \(f(x) = x^{2} + x +1\)  is one-one but not onto.

Answer:

We have, \(f(x) = x^{2} + x + 1\)

Calculate \( f(x_{1}​)\):

⇒  \(f(x_{1}​) = {x_{1}}^{2} ​+ x_{1} ​+1\)

Calculate \(f(x_{2}​)\):

⇒ \(f(x_{2}​) = {x_{2}}^{2} ​+x_{1}​+1\)

Now, \(f(x_{1}​)=f(x_{2}​)\)

⇒ \({x_{1}}^{2} ​+x_{1} ​+ 1= {x_{2}}^{2} + x_{2} ​+1\)

⇒ \({x_{1}}^{2} ​− {x_{2}}^{2} ​+x_{1} ​– x_{2} ​= 0\)

⇒  \((x_{1}​ – x_{2})\) \( (x_{1} ​+ x_{2}​\) + \(x_{1} ​– x_{2}​= 0 \)……… [ Since, \((a^{2} – b^ {2} = (a+b)(a−b) \)]

⇒ \( (x_{1}​− x_{2}​)\) \((x_{1}​  + x_{2} ​+1) \) = 0

Since, \(x_{1}​+x_{2}\) ​+1 \(\neq\) 0 for any x \(\in\) N

Thus, \(x_{1} ​= x_{2}​\)

So, f is one-one function.

Clearly, f(x) = \(x^{2}\) +x + 1 ≥3 for all x \(\in\)  N

So, f(x) does not assume values 1 and 2.

Therefore, f is not an onto function.

Question 3: Is the relation R defined on a set A={a,b,c} as follows:
R={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)} is (i) reflexive  (ii) symmetric  (iii) transitive

Answer:

Given the relations on the set A is

iR={(a,a),(a,b),(a,c),(b,b),(b,c),(c,a),(c,b),(c,c)}

This relation is reflexive as for all a \(\in\)  A, (a,a) \(\in\)  R1​.

As (a,b) \(\in\)  R but (b,a) \(\notin\) R so R not symmetric.

Now (b,c) \(\in\) R,(c,a) \(\in\)   R but (b,a) \(\notin\) R, so the relation is not transitive.

Question 4: In how many ways can 6 persons stand in a queue?

Answer:

The number of ways in which 6 persons can stand in a queue is the same as the number of arrangements of 6 different things taken all at a time.

Hence, the required number of ways = \(6_{P6}\) = 6! = 720

FAQs related to RD Sharma Solutions for Class 12 Maths

Q.1. Is RD Sharma textbook a good option in class 12?

Ans. RD Sharma is a recommended reference book by the CBSE. RD Sharma class 12 solutions contain a lot of solved illustrations, pictorial representations, and also more than one method to solve the problems. It also teaches the students mathematical tricks that concepts which make the students solve the problems easily and accurately. It is a very good option for the practice questions.

Q.2. Is R S Agrawal better or RD Sharma?

Ans. RD Sharma explains the concepts in a better way. Also, it has a lot of practice questions of various types. Thus, it prepares the students to perform better in the examination.

Download Toppr – Best Learning App for Class 5 to 12

We at Toppr offer you free pdf downloads, free video lectures, free online classes, online doubt- solving sessions, and free mock tests. Our expert and experienced faculties have prepared these solutions. So, why wait to download it. Download the Toppr App now.

Download Toppr for Android, iOS or signup on Web

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

Leave a Reply

Your email address will not be published. Required fields are marked *

Download the App

Watch lectures, practise questions and take tests on the go.

Customize your course in 30 seconds

No thanks.