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NCERT Solutions for Exercise 1.3 Class 10 Maths Chapter 1 - Real Numbers
Understanding the decimal representation of real numbers becomes essential for mathematics because real numbers contain all rational and irrational elements. The behavior of these numbers in decimal format is addressed in this exercise. We use this method to identify non-terminating decimals while understanding how rational numbers form repeating patterns. Learning these concepts establishes our ability to correctly identify numbers while linking theoretical number concepts to their actual decimal expressions.
This Story also Contains
- NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2
- Access Solutions of Real Numbers Class 10 Chapter 1 Exercise 1.2
- Topics covered in Chapter 1, Real Numbers: Exercise 1.2
- NCERT Solutions of Class 10 Subject Wise
- NCERT Exemplar Solutions of Class 10 Subject Wise
Students obtain maximum benefit from these concepts when they consult with NCERT Solutions. The NCERT Books related solutions provide step-by-step explanations to simplify difficult proofs. Students who refer to these educational resources can develop fundamental number theory knowledge that helps them succeed in advanced mathematics studies as well as competitive exams.
Access Solutions of Real Numbers Class 10 Chapter 1 Exercise 1.2
Q1 Prove that $\sqrt 5$ is irrational.
Answer:
Let us assume $\sqrt{5}$ is rational, which means it can be written in the form $\frac{p}{q}$ where p and q are co-primes and $q\neq 0$
$\\\sqrt{5}=\frac{p}{q}$
Squaring both sides, we obtain
$\\\left ( \sqrt{5} \right )^{2}=\left (\frac{p}{q} \right )^{2}\\$
$5=\frac{p^{2}}{q^{2}}\\$
$p^{2}=5q^{2}$
From the above equation, we can see that p2 is divisible by 5, therefore, p will also be divisible by 5, as 5 is a prime number. $(i)$
Therefore, p can be written as 5r
p = 5r
p2 = (5r)2
5q2 = 25r2
q2 = 5r2
From the above equation, we can see that q2 is divisible by 5, Therefore, q will also be divisible by 5 as 5 is a prime number. $(ii)$
From (i) and (ii), we can see that both p and q are divisible by 5. This implies that p and q are not co-primes. This contradiction arises because our initial assumption that $\sqrt{5}$ is rational was wrong. Hence proved that $\sqrt{5}$ is irrational.
Q2 Prove that $3 + 2 \sqrt 5$ is irrational.
Answer:
Let us assume $3 + 2 \sqrt 5$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$\\3+2\sqrt{5}=\frac{p}{q}$
$2\sqrt{5}=\frac{p}{q}-3$
$\sqrt{5}=\frac{p-3q}{2q}$
As p and q are integers $\frac{p-3q}{2q}\\$ would be rational, which contradicts the fact that $\sqrt{5}$ is irrational. This contradiction arises because our initial assumption that $3 + 2 \sqrt 5$ is rational was wrong. Therefore $3 + 2 \sqrt 5$ is irrational.
Q3 Prove that the following are irrationals :
(i) $\frac{1}{\sqrt 2}$
Answer:
Let us assume $\frac{1}{\sqrt{2}}$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$\frac{1}{\sqrt{2}}=\frac{p}{q}$
$\sqrt{2}=\frac{q}{p}$
Since p and q are co-prime integers $\frac{q}{p}$ will be rational, which contradicts the fact that $\sqrt{2}$ is irrational. This contradiction arises because our initial assumption that $\frac{1}{\sqrt{2}}$ is rational was wrong. Therefore $\frac{1}{\sqrt{2}}$ is irrational.
Q3 (2) Prove that the following are irrationals :
(ii) $7 \sqrt 5$
Answer:
Let us assume $7 \sqrt 5$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$7\sqrt{5}=\frac{p}{q}$
$\sqrt{5}=\frac{p}{7q}$
As p and q are integers $\frac{p}{7q}\\$ would be rational, which contradicts the fact that $\sqrt{5}$ is irrational. This contradiction arises because our initial assumption that $7 \sqrt 5$ is rational was wrong. Therefore $7 \sqrt 5$ is irrational.
Q3 (3) Prove that the following are irrationals : $6 + \sqrt 2$
Answer:
Let us assume $6 + \sqrt 2$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$6+\sqrt{2}=\frac{p}{q}$
$\sqrt{2}=\frac{p}{q}-6$
$\sqrt{2}=\frac{p-6q}{q}$
As p and q are integers $\frac{p-6q}{q}$ would be rational, which contradicts the fact that $\sqrt{2}$ is irrational. This contradiction arises because our initial assumption that $6 + \sqrt 2$ is rational was wrong. Therefore $6 + \sqrt 2$ is irrational.
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Topics covered in Chapter 1, Real Numbers: Exercise 1.2
1. Irrational Numbers: When expressed in the state of integers these numbers become impossible to rationalize themselves. The goal of this exercise is to identify particular numbers which prove to be irrational while developing verification methods.
2. Proof by Contradiction: The method makes an assumption that results in a contradiction to show the initial assumption is wrong. Through this approach, it becomes possible to determine that √5 represents an irrational number.
3. Properties of Rational and Irrational Numbers: Operations of addition and multiplication reveal how numbers become rational or irrational when performed.
4. Application of Prime Factorization: Prime factorization functions as an investigational tool to determine both the divisibility aspects as well as properties of numbers which lead to irrationality proofs.
5. Logical Reasoning: A necessary ability for mathematics students should be mastering logical argumentation because it extends beyond mathematical applications.
Also see-
NCERT Solutions of Class 10 Subject Wise
Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.
NCERT Exemplar Solutions of Class 10 Subject Wise
Students must check the NCERT Exemplar solutions for class 10 of the Mathematics and Science Subjects.
Frequently Asked Questions (FAQs)
Q: What is the sum and difference of rational and irrational numbers, rational or irrational?
A:
Sum and difference of a rational and irrational number is irrational.
Q: State the theorem “Fundamental theorem of Arithmetic”.
A:
“Fundamental theorem of Arithmetic” given in the Class 10 Maths chapter 1 states that “Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur”.
Q: Which technique is used to prove root(2) irrational?
A:
The proof is based on a most common technique called ‘proof by contradiction.
Q: Is this exercise important for board exams?
A:
important in board exams, you can check previous year papers for better understanding.
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Have a question related to CBSE Class 10th ?
Hello,
If you want to get your 10th marksheet online, you just need to visit an official website like https://www.cbse.gov.in/ or https://results.cbse.nic.in/ for the CBSE board, and for the state board, you can check their website and provide your roll number, security PIN provided by the school, and school code to download the result.
I hope it will clear your query!!
Hello, if you are searching for Class 10 books for exam preparation, the right study material can make a big difference. Standard textbooks recommended by the board should be your first priority as they cover the syllabus completely. Along with that, reference books and guides can help in practicing extra questions and understanding concepts in detail. You can check the recommended books for exam preparation from the link I am sharing here.
https://school.careers360.com/ncert/ncert-books-for-class-10
https://school.careers360.com/boards/cbse/cbse-best-reference-books-for-cbse-class-10-exam
Hello
You asked about Class 10 sample paper board exam and most important questions. Practicing sample papers and previous year questions is one of the best ways to prepare for the board exam because it gives a clear idea of the exam pattern and types of questions asked. Schools and teachers usually recommend students to solve at least the last five years question papers along with model papers released by the board.
For Class 10 board exams, the most important areas are Mathematics, Science, Social Science, English, and Hindi or regional language. In Mathematics, questions from Algebra, Linear Equations, Geometry, Trigonometry, Statistics, and Probability are repeatedly seen. For Science, the key chapters are Chemical Reactions, Acids Bases and Salts, Metals and Non metals, Life Processes, Heredity, Light and Electricity. In Social Science, priority should be given to Nationalism, Resources and Development, Agriculture, Power Sharing, Democratic Politics, and Economics related topics. In English, focus on unseen passages, grammar exercises, and important writing tasks like letter writing and essays.
CBSE Sample Papers 2026: How to Download
Follow these steps to access the SQPs and marking schemes:
Step 1: Visit https://cbseacademic.nic.in/
Step 2: Click on the link titled “CBSE Sample Papers 2026”
Step 3: A PDF will open with links to Class 10 and 12 sample papers
Step 4: Select your class (Class 10 or Class 12)
Step 5: Choose your subject
Step 6: Download both the sample paper and its marking scheme
If you are looking for Class 10 previous year question papers for 2026 preparation, you can easily access them through the links I’ll be attaching. These papers are very helpful because they give you a clear idea about the exam pattern, marking scheme, and the type of questions usually asked in board exams. Practicing these will not only improve your time management but also help you identify important chapters and commonly repeated questions.
https://school.careers360.com/boards/cbse/cbse-previous-year-question-papers-class-10
https://school.careers360.com/boards/cbse/cbse-previous-year-question-papers
Hello,
Yes, you can give the CBSE board exam in 2027.
If your date of birth is 25.05.2013, then in 2027 you will be around 14 years old, which is the right age for Class 10 as per CBSE rules. So, there is no problem.
Hope it helps !
Popular CBSE Class 10th Questions
A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is
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How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?
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3.125 × 10-2 |
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1.25 × 10-2 |
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2.5 × 10-2 |
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