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

Edited By Ramraj Saini | Updated on Nov 02, 2023 01:21 PM IST | #CBSE Class 10th

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

This article discuss NCERT Solutions for class 10 maths ex 1.3 with all questions and answer in step by step manner. These NCERT solutions for exercise are created by expert team at Careers360 keeping in mind the need of an average students and latest syllabus of CBSE 20023-24. This exercise deals with irrational numbers which are real numbers that cannot be written in simple fractions for example π, √2, √3, √5, √7 - 3, etc. One of the theorems we use in this exercise is the Fundamental Theorem of Arithmetic.

There are four exercise in this chapter which are listed below. interested students can refer them and practice problems to command the concepts.

## Access Exercise 1.3 Class 10 Maths Answers

Real Numbers Class 10 Chapter 1- Exercise 1.3

Let us assume $\sqrt{5}$ is rational.

It means $\sqrt{5}$ 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 p 2 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

p 2 = (5r) 2

5q 2 = 25r 2

q 2 = 5r 2

From the above equation, we can see that q 2 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.

Let us assume $3 + 2 \sqrt 5$ is rational.

This means $3 + 2 \sqrt 5$ can be wriiten 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, this 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.

(i) $1/ \sqrt 2$

Let us assume $\frac{1}{\sqrt{2}}$ is rational.

This means $\frac{1}{\sqrt{2}}$ 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, this 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.

(ii) $7 \sqrt 5$

Let us assume $7 \sqrt 5$ is rational.

This means $7 \sqrt 5$ 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, this 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.

Let us assume $6 + \sqrt 2$ is rational.

This means $6 + \sqrt 2$ 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, this 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.

## Benefits of NCERT Solutions for Class 10 Maths Exercise 1.3

• NCERT solutions for Class 10 Maths is considered the best material for 10th class maths exercise 1.3 answers.
• NCERT Class 10th Maths chapter 1 exercise 1.3, contains all important questions from exam point of view.
• Exercise 1.3 Class 10 Maths, is based on irrational numbers and based on the Fundamental Theorem of Arithmetic, which are important concepts of the chapter. students can also study Real Numbers Class 10 Notes to revise the concepts quickly.
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### Questions related to CBSE Class 10th

Have a question related to CBSE Class 10th ?

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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.

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!

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.

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 Option 1) 2.45×10−3 kg Option 2)  6.45×10−3 kg Option 3)  9.89×10−3 kg Option 4) 12.89×10−3 kg

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