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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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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 !
Hello! If you selected “None” while creating your APAAR ID and forgot to mention CBSE as your institution, it may cause issues later when linking your academic records or applying for exams and scholarships that require school details. It’s important that your APAAR ID correctly reflects your institution to avoid verification problems. You should log in to the portal and update your profile to select CBSE as your school. If the system doesn’t allow editing, contact your school’s administration or the APAAR support team immediately so they can correct it for you.
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Here's how you can find it:
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School ID Card: Your registration number is often printed on your school ID card.
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Admit Card (Hall Ticket): If you've received your board exam admit card, the registration number will be prominently displayed on it. This is the most reliable place to find it for board exams.
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Previous Mark Sheets/Certificates: If you have any previous official documents from your school or board (like a Class 9 report card that might have a student ID or registration number that carries over), you can check those.
Your school is the best place to get this information.
Hello,
It appears you are asking if you can fill out a form after passing your 10th grade examination in the 2024-2025 academic session.
The answer depends on what form you are referring to. Some forms might be for courses or examinations where passing 10th grade is a prerequisite or an eligibility criteria, such as applying for further education or specific entrance exams. Other forms might be related to other purposes, like applying for a job, which may also have age and educational requirements.
For example, if you are looking to apply for JEE Main 2025 (a competitive exam in India), having passed class 12 or appearing for it in 2025 are mentioned as eligibility criteria.
Let me know if you need imformation about any exam eligibility criteria.
good wishes for your future!!
Hello Aspirant,
"Real papers" for CBSE board exams are the previous year's question papers . You can find these, along with sample papers and their marking schemes , on the official CBSE Academic website (cbseacademic.nic.in).
For notes , refer to NCERT textbooks as they are the primary source for CBSE exams. Many educational websites also provide chapter-wise revision notes and study material that align with the NCERT syllabus. Focus on practicing previous papers and understanding concepts thoroughly.
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