NCERT Class 10 Maths Chapter 1 Real Numbers Exercise 1.3


Chapter 1 Real Numbers Exercise 1.3

Question 1: Prove that √5 is irrational.
Answer: Let us assume that √5 is rational. Therefore, √5 can be expressed in the form p/q, where p, q are co-prime integers and q ≠ 0.
= √5 = p/q
= p = √5q

Squaring on both sides,
= (p)² = (√5q)²
= p² = 5q²    ...(1)

Therefore, p² is divisible by 5 and by Theorem 1.3, p is also divisible by 5.

Then, there exists some integer, c, such that,
= p = 5c

Squaring on both sides,
= (p)² = (5c)²
= p² = 25c²    ...(2)

From (1) and (2),
5q² = 25c²
q² = 5c²

Hence, 5 divides q². Also 5 divides q.

Since 5 is common factor for both p and q, it contradicts our assumption that p and q are co-prime integers. Hence, our assumption that √5 is rational is wrong. Therefore, √5 is irrational.

Question 2: Prove that 3 + 2√5 is irrational.
Answer: Let us assume that 3 + 2√5 is rational. Therefore, 3 + 2√5 can be expressed as p/q, where p, q are integers and q ≠ 0.
= 3 + 2√5 = p/q
= √5 = p - 3q/2q

Since, p and q are integers, p - 3q/2q is a rational, and hence √5 is also rational. But it contradicts the fact the fact that √5 is irrational. Hence 3 + 2√5 is irrational.

Question 3: Prove that the following are irrationals:
i) 1/√2
Answer:
Let us assume that 1/√2 is rational. Therefore it can be expressed as p/q, where p, q are integers and q ≠ 0.

= 1/√2 = p/q
= √2 = q/p

Since p and q are integers, q/p is rational, and hence √2 is rational. But it contradicts the fact that √2 is irrational. Hence, 1/√2 is irrational.

ii) 7√5
Answer:
Let us assume that 7√5 is rational. Therefore it can be expressed as p/q, where p, q are integers and q ≠ 0.

= 7√5 = p/q
= √5 = p/7q

Since p and q are integers, p/7q is rational, and hence √5 is rational. But it contradicts the fact that √5 is irrational. Hence, 7√5 is irrational.

iii) 6 + √2
Answer:
Let us assume that 6 + √2 is rational. Therefore it can be expressed as p/q, where p, q are integers and q ≠ 0.

= 6 + √2 = p/q
= √2 = p - 6/q

Since p and q are integers, p - 6/q is rational, and hence √2 is rational. But it contradicts the fact that √2 is irrational. Hence, 6 + √2 is irrational.

No comments:

Post a Comment

Also read!

Click on any of the below text for additional links:

Class 6

Class 6 Computer Kips Solutions
Class 6 Gul Mohar English Solutions
→ Class 6 Map Pointing Solutions
Class 6 NCERT Civics Solutions
Class 6 NCERT Geography Solutions
Class 6 NCERT History Solutions
Class 6 NCERT Maths Solutions
Class 6 NCERT Science Solutions

Class 7

Class 7 Computer Kips Solutions
Class 7 Gul Mohar English Solutions
English Grammar Class 7
English Writing Skills Class 7
Class 7 Map Pointing Solutions
Class 7 NCERT Civics Solutions
Class 7 NCERT Civics Notes
Class 7 NCERT Civics Extra Questions
Class 7 NCERT Geography Solutions
Class 7 NCERT Geography Notes
Class 7 NCERT Geography Extra Questions
Class 7 NCERT History Solutions
Class 7 NCERT History Notes
Class 7 NCERT History Extra Questions
Class 7 NCERT Maths Solutions
Class 7 NCERT Science Solutions
Class 7 NCERT Science Notes
Class 7 NCERT Science Extra Questions

Class 8

Class 8 Kips Artificial Intelligence Solutions
Class 8 Gul Mohar English Solutions
English Grammar Class 8
English Writing Skills Class 8
Class 8 Map Pointing Solutions
Class 8 NCERT Civics Solutions
Class 8 NCERT Civics Notes
Class 8 NCERT Civics Extra Questions
Class 8 NCERT Geography Solutions
Class 8 NCERT Geography Notes
Class 8 NCERT Geography Extra Questions
Class 8 NCERT History Solutions
Class 8 NCERT History Notes
Class 8 NCERT History Extra Questions
Class 8 NCERT Maths Solutions
Class 8 NCERT Science Solutions
Class 8 NCERT Science Notes
Class 8 NCERT Science Extra Questions
Class 8 NCERT Science Exemplar
Class 8 NCERT Science MCQs

Class 9

Class 9 Kips Artificial Intelligence Solutions
Class 9 Map Pointing Solutions
Class 9 NCERT Civics Solutions
Class 9 NCERT Geography Solutions
Class 9 NCERT History Solutions
Class 9 NCERT Maths Solutions
Class 9 NCERT Science Solutions

Class 10

Class 10 Kips Artificial Intelligence Solutions
Class 10 NCERT English Solutions
→ Class 10 Map Pointing Solutions
Class 10 NCERT Civics Solutions
Class 10 NCERT Civics Notes
Class 10 NCERT Civics Extra Questions
Class 10 NCERT Geography Solutions
Class 10 NCERT Geography Notes
Class 10 NCERT Geography Extra Questions
Class 10 NCERT History Solutions
→ Class 10 NCERT History Notes
Class 10 NCERT History Extra Questions
Class 10 NCERT Maths Solutions
Class 10 NCERT Science Solutions
Class 10 NCERT Science Notes
Class 10 NCERT Science Extra Questions

Computer Languages

Python Basics

Extra Activities

Extra Activities

Extra Knowledge

Extra Information
General Knowledge
Historical Places in India
Latest technology
Physical Sciences
Facts

• Mathematics quick links
Mathematical Terms
Maths Tricks

Home Top