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Prove that a necessary and sufficient condition for three vectors ` vec a , vec b` and ` vec c` to be coplanar is that there exist scalars `l , m , n` not all zero simultaneously such that `l vec a+m vec b+n vec c= vec0dot`

Prove that the following vectors are non-coplanar: `3 hat i+ hat j- hat k ,2 hat i- hat j+7 hat k` and `7 hat i- hat j+23 hat k` ` hat i+2 hat j+3 hat k ,2 hat i+ hat j+3` and ` hat i+ hat j+ hat k`

Using vectors show that the points `A(-2,3,5),B(7,0,-1)C(-3,-2,-5)` and `D(3,4,7)` are such that `A B` and `C D` intersect at the point `P(1,2,3)dot`

Prove that 1,1,1 cannot be direction cosines of a straight line.

A vector ` vec r` is inclined at equal acute angles of `x-a xi s ,` `y-a xi s` and `z-a xi sdot` if `| vec r|=6` units, find ` vec rdot`

RD Sharma class 12 solutions consist of conceptual questions and detailed solved examples. The RD Sharma Class 12 book explains concepts in a simple and clear manner to provide a comprehensive understanding of a particular topic. As we all know Class 12 is a crucial time for a student’s career. The class 12 results lay the foundation for the future career path. Practising problems from the RD Sharma class 12 book eliminates a lot of worries pertaining to the Class 12 maths board exam. Class 12 Maths RD Sharma Solutions are presented in a step-by-step manner so that a student can easily understand the nuances involved when solving complex concepts.

RD Sharma class 12 solutions are given in an easy way to let the students solve the exercise problems without any confusion. The thinking process of the children are considered and the solutions are designed in the same way they understand. Class 12 is a crucial stage in a student’s life because it defines the student’s career path. RD Sharma class 12 solutions are very helpful for students as it forms a strong foundation of the basic concepts to score well in the CBSE 12th Maths board exam.

RD Sharma Solutions for Class 12 Maths provide a base for all the concepts important for the exams conducted by Central Board of Secondary Education (CBSE) board. It includes all the important theorems and Math formulas along with the detailed explanation to help the students with better understanding of concepts. Maths RD Sharma Solutions for Class 12 also play an essential role in competitive entrance exams like JEE Mains and JEE Advanced. Doubtnut provides you RD Sharma Maths solutions for class 12 designed by our expert faculty members to facilitate smooth and easy understanding of concepts.

**Also Read: **List of top 5 important chapters of RD SHARMA Maths Solutions for Class 12

Given here are all the chapter-wise links of RD Sharma Class 12 Solutions, from chapter 1 Relations to chapter 33 binomial distribution. These given solutions are provided in self-explanatory video tutorial format, which you can access through your smartphone or personal computer and learn amazing math tips and tricks.

RD Sharma Class 12 Chapter 1 – Relations

RD Sharma Class 12 Chapter 2 – Functions

RD Sharma Class 12 Chapter 3 – Binary Operations

RD Sharma Class 12 Chapter 4 – Inverse Trigonometric Functions

RD Sharma Class 12 Chapter 5 – Algebra of Matrices

RD Sharma Class 12 Chapter 6 – Determinants

RD Sharma Class 12 Chapter 7 – Adjoint and Inverse of a Matrix

RD Sharma Class 12 Chapter 8 – Solutions of Simultaneous Linear Equations

RD Sharma Class 12 Chapter 9 – Continuity

RD Sharma Class 12 Chapter 10 – Differentiability

RD Sharma Class 12 Chapter 11 – Differentiation

RD Sharma Class 12 Chapter 12 – Higher Order Derivatives

RD Sharma Class 12 Chapter 13 – Derivative as a Rate Measurer

RD Sharma Class 12 Chapter 14 – Differentials, Errors and Approximations

RD Sharma Class 12 Chapter 15 – Mean Value Theorems

RD Sharma Class 12 Chapter 16 – Tangents and Normals

RD Sharma Class 12 Chapter 17 – Increasing and Decreasing Functions

RD Sharma Class 12 Chapter 18 – Maxima and Minima

RD Sharma Class 12 Chapter 19 – Indefinite Integrals

RD Sharma Class 12 Chapter 20 – Definite Integrals

__RD Sharma Class 12 Chapter 21 – Areas of Bounded Regions__

RD Sharma Class 12 Chapter 22 – Differential Equations

RD Sharma Class 12 Chapter 23 – Algebra of Vectors

RD Sharma Class 12 Chapter 24 – Scalar or Dot Product

RD Sharma Class 12 Chapter 25 – Vector or Cross Product

RD Sharma Class 12 Chapter 26 – Scalar Triple Product

RD Sharma Class 12 Chapter 27 – Direction Cosines or Direction Ratios

RD Sharma Class 12 Chapter 28 – Straight Line in Space

RD Sharma Class 12 Chapter 29 – The Plane

RD Sharma Class 12 Chapter 30 – Linear Programming

RD Sharma Class 12 Chapter 31 – Probability

RD Sharma Class 12 Chapter 32 – Mean and Variance of a Random Variable

RD Sharma Class 12 Chapter 33 – Binomial Distribution

8 hours of school and 4 hours of coaching every day, we understand the routine struggles of a class 12 student. Meanwhile, there are those students who need to move to another city for coaching, away from their home, family and friends. Maths’s coaching classes have become vogue and nothing more, while the intelligent choice is lying right in front of you.

Doubtnut experts have designed RD Sharma Class 12 Solutions videos not just to save the students’ time and energy – but with the objective of getting students into the habit of self-studying. It is an important aspect that is losing its value due to exhaustively advertised coaching classes. When we ask you to go through the video, we are just trying our best to help today’s student learn the conventional way of self-studying with the modern touch of RD Sharma Class 12 Solutions.

Class 12 Maths Students can practice these RD Sharma Solutions Class 12 to gain comprehensive knowledge of concepts and score good marks in the board exams.

In the previous class 11, students have already studied the concept of range, domain, and co-domain. Relation in mathematics signifies the relationship between two given set of values. In this chapter 1 of RD Sharma Class 12, students will study relations and their properties, types of relations, the inverse of a relation, equivalence relation. Further, it includes symmetric and anti-symmetric relations and reflexive relations. The RD Sharma Class 12 Chapter 1 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 1 Relations Exercise 1.1

RD Sharma Class 12 Solutions Chapter 1 Relations Exercise 1.2

In Chapter 2, Functions, students will learn the definition of functions, types of functions, constant function, identity function, modulus function, greatest integer function, signum function, exponential function, logarithmic function, reciprocal and square root function, square function, etc. Here, students will also learn to relate the graphs of a function and the inverse of the function. Students will also be taught other functions like one-one function, many-one function and onto function, the composition of functions etc. The step-by-step RD Sharma solutions of this chapter will help the students to solve the questions similar to the questions asked in JEE Main and JEE Advanced. The RD Sharma Class 12 Chapter 2 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 2 Functions Exercise 2.1

RD Sharma Class 12 Solutions Chapter 2 Functions Exercise 2.2

RD Sharma Class 12 Solutions Chapter 2 Functions Exercise 2.3

RD Sharma Class 12 Solutions Chapter 2 Functions Exercise 2.4

In this Chapter 3 of Class 12 RD Sharma, the concept of Binary Operations, and the Associative, distributive and Commutative laws related to them, will be taught to the students. It will also include the Identity element of an operation, the inverse of an element, here small binary operations can be expressed as an operation table. The RD Sharma Class 12 Chapter 3 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 3 Binary Operations Exercise 3.1

RD Sharma Class 12 Solutions Chapter 3 Binary Operations Exercise 3.2

RD Sharma Class 12 Solutions Chapter 3 Binary Operations Exercise 3.3

RD Sharma Class 12 Solutions Chapter 3 Binary Operations Exercise 3.4

RD Sharma Class 12 Solutions Chapter 3 Binary Operations Exercise 3.5

In Class 12, Chapter 4 of RD Sharma Solutions, students will learn the concept of inverse trigonometric functions, alongwith properties of inverse trigonometric functions. It will include the inverse of the sine function, inverse of the cosine function, inverse of the tangent function, inverse of secant function, inverse of cosecant function, and the inverse of cotangent function. The RD Sharma Class 12 Chapter 4 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 4 Inverse Trigonometric Functions Exercise 4.1

Chapter 5 of the RD Sharma Class 12 textbook deals with the concept of Matrices. Here the focus will be around the definition of matrices, different types of matrices, the addition of matrices, multiplication of matrices, multiplication of a matrix by a scalar, subtraction of matrices, transpose of a matrix, properties of symmetric and skew-symmetric matrices etc. The RD Sharma Class 12 Chapter 5 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.1

RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.2

RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.3

RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.4

RD Sharma Class 12 Solutions Chapter 5 Algebra of Matrices Exercise 5.5

In this chapter 6 of Class 12 RD Sharma Textbook, students will learn determinants of third order and properties of determinants, evaluation of determinants, applications of determinants in solving simultaneous linear equations. The RD Sharma Class 12 Chapter 6 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 6 Determinants Exercise 6.1

RD Sharma Class 12 Solutions Chapter 6 Determinants Exercise 6.2

RD Sharma Class 12 Solutions Chapter 6 Determinants Exercise 6.3

RD Sharma Class 12 Solutions Chapter 6 Determinants Exercise 6.4

RD Sharma Class 12 Solutions Chapter 6 Determinants Exercise 6.5

Chapter 7 of Class 12 RD Sharma deals with the adjoint of a square matrix and the inverse of a matrix. These RD Sharma solutions will help CBSE Class 12 students to achieve good marks in their board exams. The RD Sharma Class 12 Chapter 7 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.1

RD Sharma Class 12 Solutions Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.2

Chapter 8 of the Class 12 RD Sharma textbook deals with the system of simultaneous linear equations. The topics included in this chapter are the consistent system of equations, homogeneous and non-homogeneous systems of equations, matrix method for the solution of a non-homogeneous system, applications of simultaneous linear equations. The RD Sharma Class 12 Chapter 8 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 8 Solutions of Simultaneous Linear Equations Exercise 8.1

RD Sharma Class 12 Solutions Chapter 8 Solutions of Simultaneous Linear Equations Exercise 8.2

In chapter 9, Continuity, students will learn topics such as intutive notion of continuity, continuity at a point, algebra of continuous function, continuity of composite function, continuous function, and properties of continuous function. The RD Sharma Class 12 Chapter 9 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 9 Continuity Exercise 9.1

RD Sharma Class 12 Solutions Chapter 9 Continuity Exercise 9.2

In RD Sharma Class 12 Chapter 10, students will come across the concept of differentiability. The topics of this chapter will include differentiability at a point, differentiability in a set, and some useful results on differentiability. The RD Sharma Class 12 Chapter 10 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 10 Differentiability Exercise 10.1

RD Sharma Class 12 Solutions Chapter 10 Differentiability Exercise 10.2

Chapter 11 of RD Sharma Class 12 Maths focuses on the topic of differentiation, which further includes sub-topics like differentiation of inverse trigonometric functions from first principles, differentiation of two or more functions, differentiation of inverse trigonometric function by using the chain rule, differentiation by using trigonometric substitutions, differentiation of implicit functions, logarithmic differentiation, differentiation of infinite series, differentiation of a function with respect to another function. The RD Sharma Class 12 Chapter 11 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.1

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.2

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.3

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.4

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.5

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.6

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.7

RD Sharma Class 12 Solutions Chapter 11 Differentiation Exercise 11.8

In this chapter 12, students will learn the concept of higher-order derivatives. The topics covered in this chapter are finding second-order derivatives of parametric functions, proving relations involving various order derivatives. The RD Sharma Class 12 Chapter 12 exercise solution link is listed below.

RD Sharma Class 12 Solutions Chapter 12 Higher Order Derivatives Exercise 12.1

In this chapter, students will learn the concept of derivative as a rate measurer. Her student will solve the problems related to the rate of change of one of the quantities corresponding to the given rate of change of another quantity. It will include more word problems to grab the concept in a more comprehensive manner. The RD Sharma Class 12 Chapter 13 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 13 Derivative as a rate measurer Exercise 13.1

RD Sharma Class 12 Solutions Chapter 13 – Derivative as a rate measurer Exercise 13.2

In this chapter, students will learn the topic of differentials, approximations and errors. The chapter includes the topics such as absolute error, percentage error, relative error, and finding the approximate value using differentials. The RD Sharma Class 12 Chapter 14 exercises solution link is listed below

RD Sharma Class 12 Solutions Chapter 14 Differentials, Errors, and Approximations Exercise 14.1

Chapter 15 of Class 12 RD Sharma will focus on the theorems related to mean values. Here students will come across topics like Rolle’s theorem, Lagrange’s mean value theorem, and other sub-topics related to these two theorems. The RD Sharma Class 12 Chapter 15 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 15 Mean Value Theorems Exercise 15.1

RD Sharma Class 12 Solutions Chapter 15 Mean Value Theorems Exercise 15.2

In this chapter 16 of RD Sharma Class 12, the students will solve problems on the topic of tangents and normals. Here the sub-topics included are the slope of a line, slopes of tangent and normal, finding slopes of tangent, finding the point on a given curve at which tangent is parallel or perpendicular to a given line, finding the equation of tangent and normal, angle of intersection of two curves and orthogonal curves. The RD Sharma Class 12 Chapter 16 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 16 Tangents and Normals Exercise 16.1

RD Sharma Class 12 Solutions Chapter 16 Tangents and Normals Exercise 16.2

RD Sharma Class 12 Solutions Chapter 16 Tangents and Normals Exercise 16.3

In this chapter, the students will learn the concept of increasing and decreasing functions. The chapter 17 covers the sub-topics including solution of rational algebraic inequations with algorithms, strictly increasing functions, strictly decreasing functions, monotonic increasing, and monotonic decreasing functions, necessary and sufficient conditions for monotonicity, and proving the monotonicity of a function on a given interval. The RD Sharma Class 12 Chapter 17 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 17 Increasing and Decreasing Functions Exercise 17.1

RD Sharma Class 12 Solutions Chapter 17 Increasing and Decreasing Functions Exercise 17.2

In this chapter, the students will be taught the maximum and minimum values of a function in its domain. Here the students will also learn the sub-topics, including local maxima and local minima, first derivative test for local maxima and minima, higher-order derivative test, point of inflection, maximum and minimum values in the closed interval and problems on maxima and minima applications. The RD Sharma Class 12 Chapter 18 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 18 Maxima and Minima Exercise 18.1

RD Sharma Class 12 Solutions Chapter 18 Maxima and Minima Exercise 18.2

RD Sharma Class 12 Solutions Chapter 18 Maxima and Minima Exercise 18.3

RD Sharma Class 12 Solutions Chapter 18 Maxima and Minima Exercise 18.4

RD Sharma Class 12 Solutions Chapter 18 Maxima and Minima Exercise 18.5

In chapter 19, students will have to solve problems on indefinite integrals. The chapter will deal with sub-topics such as integration of trigonometric functions and exponential functions, comparison between differentiation and integration, different methods of integration, integration of rational algebraic functions and integration of some special irrational algebraic functions. The RD Sharma Class 12 Chapter 18 exercises solution links are listed below.

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.1

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.2

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.3

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.4

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.5

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.6

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.7

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.8

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.9

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.10

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.11

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.12

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.13

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.14

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.15

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.16

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.17

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.18

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.19

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.20

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.21

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.22

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.23

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.24

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.25

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.26

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.27

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.28

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.29

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.30

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.31

RD Sharma Class 12 Solutions Chapter 19 Indefinite Integrals Exercise 19.32

- RD Sharma provides detailed Illustrations consiiting of concepts and important formulae.
- Algorithmic Approach to solve each question.
- Large Number of Solved Examples and Problem Exercises.
- Brief Summary Consisting of Concepts and Formulae.
- Pictorial representation in solved examples for better understanding.

These RD Sharma Class 12 Solutions are prepared by our DOUBTNUT expert team while keeping accuracy and clarity in mind. Keep visiting DOUBTNUT to get more updated learning material and download the DOUBTNUT app for a personalized immersive learning experience with interactive video lessons.

**Important Books for Class 12 (Maths)**

**Question**- **Is RD Sharma Solutions for Class 12 enough for JEE Mains Maths Exam? **

RD Sharma Maths Textbook has a large number of analytical problems. RD Sharma Solutions will help you clear the concepts as it is more of a subjective textbook. RD Sharma Class 12 solutions do make a strong foundation for JEE Mains Maths Exam. However, students must also study other reference books and online study materials.

**Question- Is RD Sharma textbook good for CBSE Class 12 Maths board exams?**

RD Sharma Solutions for class 12 makes the board exam preparation smoother for class 12 students. The RD Sharma books are based on the latest CBSE syllabus. By the rigorous practice of RD Sharma Class 12 Volume I and Volume II, students will gain an edge over other students and feel confident in the exam.

**Question- Does RD Sharma Solutions provide different types of questions for Class 12?**

RD Sharma Class 12 books has very short answer (VSA) questions, short answer (SA) questions, multiple-choice questions (MCQ) and long answer (LA) questions.