NCERT Solutions for Class 10 Maths Exercise 1.3 Chapter 1 Real Numbers
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Class 10 is an extremely crucial time in a student’s life. The CBSE Class 10 board exam results determine whether students can enrol in their preferred stream in the eleventh grade.. The Board Exam for Class 10 is a significant event in a student’s life. The results of the CBSE board Class 10 decide whether the students can get a preferred stream in Class 11. The Class 10 board exam is thus one of the very first milestones in a student’s life.
The Class 10 board exam includes Mathematics as one of the core subjects. Mathematics for Class 10 has 15 distinct chapters on various topics. Numerous Mathematics topics, including Algebra, Geometry, Trigonometry, Statistics, etc., are covered by the Central Board of Secondary Education (CBSE). Real numbers are the focus of Chapter 1 of Class 10 Mathematics. This chapter is a crucial component of Mathematics. Students should thoroughly study and practice this chapter using the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3. In NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, there are numerous solved questions from exercise 1.3 in the NCERT book.
As a result of the CBSE course structure, students are not under much pressure. NCERT books are engaging and interactive, which helps students enjoy their studies. The chapters are interspersed with lots of entertaining activities to encourage playful learning. In this way, it is possible to facilitate the transfer of knowledge to students in an effective and healthy way.
Using the Extramarks website and mobile application, students will have access to all the resources they need to help them prepare for competitive exams, as well as the NCERT Solutions. Students taking competitive exams like the JEE Mains, NEET, JEE Advance, CUET, and others may benefit from it. Additionally, the Extramarks website offers simple access to all the tools required to perform better on competitive exams. Additionally, students can watch live lectures regarding the subjects they are studying on the Extramarks website and mobile application. These interactive live lectures are updated regularly to follow the CBSE syllabus. There are dedicated doubt sessions for students who have questions about difficult topics.
Students can easily access NCERT Solutions Class 12 for all subjects through Extramarks. Additionally, NCERT Solutions Class 11 for all subjects are readily available to students in Class 11. Furthermore, students looking for NCERT Solutions Class 10 for English can easily find them on the Extramarks website and mobile application. Extramarks also has NCERT Solutions Class 9 for all the subjects in Class 9.
The NCERT Solutions Class 8 for all subjects are easily accessible to students in Class 8. The NCERT Solutions Class 7 for all Mathematics chapters are available as a PDF download on the Extramarks website and mobile application. Students in Class 6 must use the NCERT Solutions Class 6. NCERT Solutions Class 5 for all subjects are also available on Extramarks. Downloading the NCERT Solutions Class 4 will allow students in Class 4 to begin studying for their exams. The NCERT Solutions Class 3 are now accessible on the Extramarks website and mobile application. Extramarks provides NCERT Solutions Class 2 for all exercises in all subjects.
NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Ex 1.3) Exercise 1.3
Real Number is one of the key topics and is worth 6 marks. Real numbers are covered briefly in Class 9 and in greater detail in Class 10. The answers to the questions in this chapter are provided in NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3. Real numbers are covered in the chapter, along with their uses. The divisibility of integers using Euclid’s division algorithm states that any positive integer a can be divided by another positive integer b such that the remainder is less than b. On the other hand, the Fundamental Theorem of Arithmetic focuses on the multiplication of positive integers.
The NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 is another resource that learners can use to help them fully understand the chapter. It is critical for students to fully understand Class 10 Chapter 1 in order to achieve their highest exam scores. Students can understand the exercises and prepare for the tests with the help of NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3. Additionally, students can use the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, to verify their answers.
Students in Class 10 who want to boost their exam preparation can go through the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 and download them as PDF files. Therefore, NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 on the Extramarks website and mobile application for student studies are reliable and trustworthy study resources.
Maths Syllabus of CBSE Class 10
Students must rigorously practice the concepts covered in the CBSE Class 10 Mathematics Syllabus to perform well in the Class 10 Mathematics board exam. The CBSE Class 10 Mathematics Syllabus provides information on the overall course structure, the weighting of the various units, etc. The Class 10 Maths Syllabus NCERT must always be on hand as the students begin their preparation.
It is the aim of the secondary school curriculum to improve students’ ability to use mathematics to solve everyday problems and to see the subject as a separate discipline that can be studied by the students. By the end of the course, students should be able to solve problems involving algebraic methods and apply their knowledge of simple trigonometry to problems involving height and distance. Experimenting with numbers and geometric shapes, formulating hypotheses, and testing them with additional observations are all part of the Mathematics learning process at this phase. The proposed curriculum includes studying Number Systems, Algebra, Geometry, Trigonometry, Measurements, Statistics, Graphs, and Coordinate Geometry.
Downloading the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 will help students prepare for the exams. However, students must access the Extramarks website and mobile application to go through the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3.
Students studying for their final exams should have confidence after reading the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3. Students can also quickly complete the Mathematics exercise and assess their responses to NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3. Additionally, Class 10 students will be helped by NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, in understanding how to respond to mathematics questions correctly.
Maths Syllabus of CBSE Class 10
Students who wish to succeed in Class 10 Mathematics need to spend a lot of time practising the topics in the CBSE Mathematics Syllabus. The CBSE Class 10 Mathematics Syllabus Exam includes information about the entire course structure, the weighting of the various units, etc. The NCERT Class 10 Mathematics Syllabus must always be on hand as the students begin their preparation.
The Number System, Algebra, Coordinate Geometry, Geometry, Measurement, Statistics, and Probability are the six units that make up the class 10 Mathematics syllabus for the CBSE exam. Below is a list of every unit weighted in the Class 10 board exams. Students should review the units and understand the marking schemes for each unit in order to determine which unit to concentrate on.
UNIT  MARKS 
Number system  10 
Algebra  20 
Coordinate Geometry  04 
Geometry  27 
Mensuration  13 
Statistics & Probability  06 
Total  80 
Internal Assessments  20 
Total marks  100 
CBSE Class 10 Mathematics Chapter 1
Some crucial arithmetic concepts are covered in the Class 10 Mathematics curriculum by the Central Board of Secondary Education (CBSE). Chapter 1 for Class 10 Mathematics is one of them. All topics related to real numbers are covered in this chapter. In Mathematics, the concept of Real Numbers is very important. Students will learn about the real number’s definition, characteristics, and categorisation from this chapter. Students can take help from the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 to understand the chapter more deeply.
This chapter provides students with some examples and property proof. Solving those examples allows the students to learn the real number system quickly. The textbook exercises should be put into practice by the students. The NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, will be useful for students’ practice because it provides a clear understanding of the subject. Students having trouble with Chapter 10 may find the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, helpful. Students can easily get access to the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 and solve their answers by accessing these solutions via the Extramarks website and mobile application.
NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3
The NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 is a crucial set of solutions for the students. This exercise includes some realnumber testing problems. After working through those problems, the students will clearly understand the characteristics and division of Real Numbers. The conceptual part can be clarified much more effectively with examples. NCERT Solutions provides the answers to questions in exercise 1.3. NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, can be easily accessed online. Moreover, for the convenience of the students, they can download these solutions.
Students can take help from Extramarks by downloading the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3. These NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 will guide and support students to do more practice and clear their doubts.
Necessity of NCERT Solutions Class 10 Maths Ch 1 Ex 1.3
It is important to practice the individual chapter exercises before the Class 10 board exam for Mathematics. Students should diligently complete the exercises from the relevant chapters. Students will benefit if they receive answers to the exercises after they have completed them. Therefore, it is essential to have access to the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, as it boosts the student’s selfesteem. Most Mathematics exam papers are lengthy, so it is best to have a good speed when answering questions. Students can correct their mistakes by comparing their responses to NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3.
Irrational Numbers
Real numbers with imperfect fractions are known as irrational numbers. Complete fractions or ratios cannot be used to represent irrational numbers. Irrational numbers have the special characteristic that their decimal expansion is nonterminating and nonrepeating. Irrational numbers cannot be directly compared to rational numbers using general arithmetic rules. The decimal values of the irrational numbers must first be solved before they can be added, subtracted, multiplied, or divided.
The most significant and wellknown irrational numbers are pi, the square root of 2, and e, also known as Euler’s Number. Irrational numbers are represented as a set RQ, where R is a set of real numbers and Q is a set of integers.
Proof by Contradiction
Students begin with presumptions at odds with the propositions they must prove when using the contradiction method. The student will arrive at a mathematical inconsistency (error) using a series of logical deductions from this assumption of contradiction, allowing him to deduce that the assumption of contradiction was incorrect.
Assuming that the student will use the method of contradiction to demonstrate why 2 is an irrational number. Students first assume that the contradiction that 2 is a rational number is false. Therefore, it can be written as a/, where a and b are two coprime numbers and n equals 0. The student will discover through observation that there are no coprime integers a and b for 2, proving that our presumption was incorrect.
Solved Examples
Students can look at these examples to understand the steps of solving the questions. Students can also download the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 for more solved questions with their solutions.
NCERT Solutions for Class 10 Maths Chapter 1 Exercises [Include a Table of Exercises and No of Questions to be Asked]
Chapter 1 of Mathematics Class 10 has four exercises and is one of the most important chapters of the Class 10 CBSE board syllabus. Chapter 1 is based on Real Numbers. It is very important to understand this chapter for the exercises. Students should cover the exercises based on the chapter and clear all their doubts.
Students can get the best study material from the Extramarks website for Maths Class 10 Chapter 1 Exercise 1.3. The NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, can be easily downloaded in PDF format for student reference. Extramarks is one of the best learning platforms as it provides access to study materials like the Class 10 Maths Chapter 1 Exercise 1.3 Solutions.
Q.1
$\text{Prove that\hspace{0.33em}}\sqrt{\text{5}}\text{\hspace{0.33em}is irrational.}$Ans.
$\begin{array}{l}\text{Let us assume that\hspace{0.33em}}\sqrt{5}\text{\hspace{0.33em}is rational.}\\ \text{So we can find integers a and b}\left(\ne 0\right)\text{such that}\\ \sqrt{5}=\frac{\mathrm{a}}{\mathrm{b}}.\\ \text{Suppose a and b have a common factor other than 1,}\\ \text{and assume that a and b are coprime.}\\ \text{So,}\\ \sqrt{5}\text{b\hspace{0.33em}=\hspace{0.33em}a\xb7}\\ \text{Squaring on both sides, we get}\\ \text{}5{\mathrm{b}}^{2}={\mathrm{a}}^{2}.\\ \text{Therefore,}\\ {\text{a}}^{\text{2}}\text{is divisible by 5, and so a is also divisible by 5.}\\ \text{So, we can write}\mathrm{a}=5\mathrm{c}\text{for some integer}\mathrm{c}\text{.}\\ \text{Now,}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}5{\mathrm{b}}^{2}={\mathrm{a}}^{2}\\ \text{}\Rightarrow 5{\mathrm{b}}^{2}=25{\mathrm{c}}^{2}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{b}}^{2}=\text{}5{\mathrm{c}}^{2}\text{.}\\ {\text{This means that b}}^{\text{2}}\text{is divisible by 5, and so b}\\ \text{is also divisible by 5.}\\ \text{Therefore,}\\ \text{a and b have at least 5 as a common factor.}\\ \text{But this contradicts the fact that a and b are coprime.}\\ \text{So, our assumption that}\sqrt{5}\text{is rational is wrong.}\\ \text{Hence,}\sqrt{5}\text{is irrational.}\end{array}$
Q.2
$\text{Prove that\hspace{0.33em}}3+2\sqrt{5}\text{\hspace{0.33em}is irrational.}$Ans.
$\begin{array}{l}\text{Let us assume that\hspace{0.33em}}3+2\sqrt{5}\text{\hspace{0.33em}is rational.}\\ \text{That is, we can find coprime a and b}\left(\text{b}\ne \text{0}\right)\text{such that}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}3+2\sqrt{5}=\frac{\mathrm{a}}{\mathrm{b}}\\ \Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}2\sqrt{5}=\frac{\mathrm{a}}{\mathrm{b}}3\\ \Rightarrow \sqrt{5}=\frac{1}{2}\left(\frac{\mathrm{a}}{\mathrm{b}}3\right)\\ \frac{1}{2}\left(\frac{\mathrm{a}}{\mathrm{b}}3\right)\text{is a rational number as a and b are integers.}\\ \text{and so\hspace{0.33em}}\sqrt{5}\text{\hspace{0.33em}is a rational number which is a contradiction.}\\ \text{So, our assumption that}3+2\sqrt{5}\text{is rational is false.}\\ \text{Hence,\hspace{0.33em}}3+2\sqrt{5}\text{\hspace{0.33em}is irrational.}\end{array}$
Q.3
$\begin{array}{l}\text{Prove that the following are irrationals}:\\ \left(\text{i}\right)\frac{\text{1}}{\sqrt{2}}\text{}\left(\text{ii}\right)\text{7}\sqrt{\text{5}}\text{}\left(\text{iii}\right)\text{6}+\sqrt{\text{2}}\end{array}$
Ans.
$\begin{array}{l}\text{(i)}\frac{1}{\sqrt{2}}\\ \text{Let}\frac{1}{\sqrt{2}}\text{is rational. Then}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{1}{\sqrt{2}}=\frac{\mathrm{a}}{\mathrm{b}},\text{where a and b are coprimes.}\\ \Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\sqrt{2}=\frac{\mathrm{b}}{\mathrm{a}}\\ \text{So,\hspace{0.17em}\hspace{0.17em}}\sqrt{2}\text{is rational which is a contradiction.}\\ \text{Thus our assumption is false.}\\ \text{Hence,}\frac{1}{\sqrt{2}}\text{is irrational.}\\ \text{(ii)}7\sqrt{5}\\ \text{Let}7\sqrt{5}\text{is rational. Then}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}7\sqrt{5}=\frac{\mathrm{a}}{\mathrm{b}},\text{where a and b are coprimes.}\\ \Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\sqrt{5}=\frac{\mathrm{a}}{7\mathrm{b}}\\ \text{So,\hspace{0.17em}\hspace{0.17em}}\sqrt{5}\text{is rational which is a contradiction.}\\ \text{Thus our assumption is false.}\\ \text{Hence,}7\sqrt{5}\text{is irrational.}\\ \text{(iii)}6+\sqrt{2}\\ \text{Let}6+\sqrt{2}\text{is rational. Then}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}6+\sqrt{2}=\frac{\mathrm{a}}{\mathrm{b}},\text{where a and b are coprimes.}\\ \Rightarrow \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\sqrt{2}=\frac{\mathrm{a}}{\mathrm{b}}6\\ \text{So,\hspace{0.17em}\hspace{0.17em}}\sqrt{2}\text{is rational which is a contradiction.}\\ \text{Thus our assumption is false.}\\ \text{Hence,}6+\sqrt{2}\text{is irrational.}\end{array}$
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FAQs (Frequently Asked Questions)
1. Why should you use Extramarks’ NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3?
The Extramarks website and mobile application contains the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3, which are incredibly reliable and genuine study resources. The NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 are put together by qualified Mathematics teachers and kept up to date with curriculum changes. Furthermore, students can read the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 offline.
2. Is it possible to collect a PDF version offline of NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3?
Yes, students can download the PDF version of the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.3 and study them whenever they have free time. Students can check their answers to the NCERT books exercises in NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3.
3. Where can students find the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3?
The NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 are available in PDF format on the Extramarks website and mobile application. It is possible to access the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 in a straightforward and understandable format. Mathematics experts create the NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 to help the students with their exam preparation.
4. What significant topics are covered in Chapter 1 of the NCERT Solutions for Class 10 Mathematics?
The Fundamental Theorem of Arithmetic and Euclid’s Division Lemma are significant subtopics covered in NCERT Solutions Class 10 Maths Chapter 1. Students can calculate the HCF of two positive integers using Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic. HCF and LCM of two positive integers can be determined using the Fundamental Theorem of Arithmetic.
5. How many exercises are included in Chapter 1 of the NCERT Solutions for Class 10 Math?
The NCERT Solutions for Chapter 1 Real Numbers contain four exercises. These NCERT Maths Class 10 Chapter 1 Solutions assist students in solving problems deftly and efficiently.