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NCERT Solutions for Exercise 1.3 Class 10 Maths Chapter 1 - Real Numbers

NCERT Solutions for Exercise 1.3 Class 10 Maths Chapter 1 - Real Numbers

Updated on Apr 29, 2025 04:47 PM IST | #CBSE Class 10th

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
  1. NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2
  2. Access Solutions of Real Numbers Class 10 Chapter 1 Exercise 1.2
  3. Topics covered in Chapter 1, Real Numbers: Exercise 1.2
  4. NCERT Solutions of Class 10 Subject Wise
  5. NCERT Exemplar Solutions of Class 10 Subject Wise
NCERT Solutions for Exercise 1.3 Class 10 Maths Chapter 1 - Real Numbers
NCERT Solutions for Exercise 1.3 Class 10 Maths Chapter 1 - Real Numbers

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.

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NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2

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Access Solutions of Real Numbers Class 10 Chapter 1 Exercise 1.2

Q1 Prove that 5 is irrational.

Answer:

Let us assume 5 is rational, which means it can be written in the form pq where p and q are co-primes and q≠0

5=pq

Squaring both sides, we obtain

(5)2=(pq)2

5=p2q2

p2=5q2

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 5 is rational was wrong. Hence proved that 5 is irrational.

Q2 Prove that 3+25 is irrational.

Answer:

Let us assume 3+25 is rational, this means it can be written in the form pq where p and q are co-prime integers.

3+25=pq

25=pq−3

5=p−3q2q

As p and q are integers p−3q2q would be rational, which contradicts the fact that 5 is irrational. This contradiction arises because our initial assumption that 3+25 is rational was wrong. Therefore 3+25 is irrational.

Q3 Prove that the following are irrationals :

(i) 12

Answer:

Let us assume 12 is rational, this means it can be written in the form pq where p and q are co-prime integers.

12=pq

2=qp

Since p and q are co-prime integers qp will be rational, which contradicts the fact that 2 is irrational. This contradiction arises because our initial assumption that 12 is rational was wrong. Therefore 12 is irrational.

Q3 (2) Prove that the following are irrationals :

(ii) 75

Answer:

Let us assume 75 is rational, this means it can be written in the form pq where p and q are co-prime integers.

75=pq

5=p7q

As p and q are integers p7q would be rational, which contradicts the fact that 5 is irrational. This contradiction arises because our initial assumption that 75 is rational was wrong. Therefore 75 is irrational.

Q3 (3) Prove that the following are irrationals : 6+2

Answer:

Let us assume 6+2 is rational, this means it can be written in the form pq where p and q are co-prime integers.

6+2=pq

2=pq−6

2=p−6qq

As p and q are integers p−6qq would be rational, which contradicts the fact that 2 is irrational. This contradiction arises because our initial assumption that 6+2 is rational was wrong. Therefore 6+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.


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NCERT Solutions of Class 10 Subject Wise

Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.

JEE Main Important Mathematics Formulas

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

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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)

1. What is the sum and difference of rational and irrational numbers, rational or irrational?

Sum and difference of a rational and irrational number is irrational.

2. State the theorem “Fundamental theorem of Arithmetic”.

“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”. 

3. Which technique is used to prove root(2) irrational?

The proof is based on a most common technique called ‘proof by contradiction.

4. Is this exercise important for board exams?

important in board exams, you can check previous year papers for better understanding.


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Questions related to CBSE Class 10th

Have a question related to CBSE Class 10th ?
i studied 1 to 10th CBSE board abroad and 11th 12th in karnataka state board . i am a domicile of karnataka and comes under minority, income under 5 lakhs. how neet councillng will allocate my seat

Hello

Since you are a domicile of Karnataka and have studied under the Karnataka State Board for 11th and 12th , you are eligible for Karnataka State Quota for admission to various colleges in the state.

1. KCET (Karnataka Common Entrance Test): You must appear for the KCET exam, which is required for admission to undergraduate professional courses like engineering, medical, and other streams. Your exam score and rank will determine your eligibility for counseling.

2. Minority Income under 5 Lakh : If you are from a minority community and your family's income is below 5 lakh, you may be eligible for fee concessions or other benefits depending on the specific institution. Some colleges offer reservations or other advantages for students in this category.

3. Counseling and Seat Allocation:

After the KCET exam, you will need to participate in online counseling.

You need to select your preferred colleges and courses.

Seat allocation will be based on your rank , the availability of seats in your chosen colleges and your preferences.

4. Required Documents :

Domicile Certificate (proof that you are a resident of Karnataka).

Income Certificate (for minority category benefits).

Marksheets (11th and 12th from the Karnataka State Board).

KCET Admit Card and Scorecard.

This process will allow you to secure a seat based on your KCET performance and your category .

check link for more details

https://medicine.careers360.com/neet-college-predictor

Hope this helps you .

Read Complete Answer
Mitali Sharma 30 January,2025
show the previous year exams ?

Hello Aspirant,  Hope your doing great,  your question was incomplete and regarding  what exam your asking.

Read Complete Answer
SWETCHCHA MISKA 10 July,2024
as an icse student in class 10 is above 80 percent good enough to get into commerce in 11th cbse

Yes, scoring above 80% in ICSE Class 10 exams typically meets the requirements to get into the Commerce stream in Class 11th under the CBSE board . Admission criteria can vary between schools, so it is advisable to check the specific requirements of the intended CBSE school. Generally, a good academic record with a score above 80% in ICSE 10th result is considered strong for such transitions.

Read Complete Answer
Manasvini Gupta 23 July,2024
i had cleared 10th from cbse in 2022 2 years ago now i wish to apply for 12th as private candidate in 2025. can i do so?

hello Zaid,

Yes, you can apply for 12th grade as a private candidate .You will need to follow the registration process and fulfill the eligibility criteria set by CBSE for private candidates.If you haven't given the 11th grade exam ,you would be able to appear for the 12th exam directly without having passed 11th grade. you will need to give certain tests in the school you are getting addmission to prove your eligibilty.

best of luck!

Read Complete Answer
Ashvadha 03 July,2024
i had cleared 10th in 2022 from cbse can i apply for 12th as private candidate this year?

According to cbse norms candidates who have completed class 10th, class 11th, have a gap year or have failed class 12th can appear for admission in 12th class.for admission in cbse board you need to clear your 11th class first and you must have studied from CBSE board or any other recognized and equivalent board/school.

You are not eligible for cbse board but you can still do 12th from nios which allow candidates to take admission in 12th class as a private student without completing 11th.

Read Complete Answer
sadafreen356 03 July,2024
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