NCERT Solutions For Class 10 Maths Chapter 1 Ex 1.3 Real Numbers
NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 Real Numbers contains 3 questions, for which detailed answers have been provided in this note. Ex 1.3 class 10 Maths Chapter 1 Real Numbers NCERT Solutions have been explained in a simple and easy-to-understand language to help you learn and prepare for your upcoming class 10 Maths exams. Here we are sharing NCERT Solutions for real numbers class 10 Ex. 1.3.
| Category | NCERT Solutions for Class 10 |
| Subject | Maths |
| Chapter | Chapter 1 |
| Exercise | Ex 1.3 |
| Chapter Name | Real Numbers |
Download NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 – Real Numbers
NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 – Real Numbers
1. Prove that √5 is irrational.
Proof: Let √5 is not irrational, i.e., rational number. Then there exist two integers a and b, where (b ≠0) such that √5 = a/b
Let a and b have a common factor other than 1, then dividing a and b by that common factor we get √5 = a1/b1, where a1 and b1 are coprime.
√5 = a1/b1 gives a1 = √5 b1
Squaring both side,
(a1)2= 5 (b1)2 …..(1).
Therefore (a1)2 is divisible by 5 and hence a1 is divisible by 5
So we can write a1 = 5k, for some integer k. substituting this in (1) we get
25k2= 5 (b1)2
(b1)2 = 5k
(b1)2 is also divisible by 5 and hence b1 is divisible by 5.
Therefore a1 and b1 have the least common factor 5, which is a contradiction to the fact that a1 and b1 are coprime. So, our assumption is wrong that √5 is rational.
Ex 1.3 class 10 Maths Chapter 1 Exercise 1.3 NCERT Solutions Real Numbers
2. Prove that 3+ 2√5 is irrational.
Proof: Let 3+2√5 is not irrational, i.e., rational number. Then there exist two co prime integers a and b, where (b ≠0) such that
3+2√5 = a/b
⇒ a/b – 3 = 2√5
⇒ (a-3b)/2b = √5
Since a and b are integers then (a-3b)/2b is rational and therefore √5 is rational, which is a contradiction to the fact that √5 is irrational.
Hence, our assumption is wrong that 3+ 2√5 is rational.
Therefore, 3+ 2√5 is irrational.
Download NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.3 – Real Numbers
3. Prove that the following are irrational.
(i) 1/√2
(ii) 7√5  Â
(iii) 6+√2
(i) Let 1/√2 is not irrational, i.e., rational number. Then there exist two co prime integers a and b, where (b ≠0) such that
1/√2 = a/b
⇒ b/a = √2.
Since a and b are integers then b/a is rational
Therefore, √2 is rational, which is a contradiction to the fact that √2 is irrational.
Hence, our assumption is wrong and 1/√2 is irrational.
(ii) Let 7√5 is not irrational, i.e., rational number. Then there exist two co prime integers a and b, where (b ≠0) such that
7√5 = a/b
⇒ a/7b = √5.
Since, a and b are integers, then a/7b is rational
Therefore, √5 is rational, which is a contradiction to the fact that √5 is irrational.
Hence, our assumption is wrong and 7√5 is irrational.
(iii) Let  6+√2 is not irrational, i.e., rational number. Then there exist two co prime integers a and b, where (b ≠0) such that
6+√2 = a/b
⇒ a/b – 6 = √2
⇒ (a – 6b/b) = √2
Since a and b are integers, then (a – 6b/b) is rational
Therefore, √2 is rational, which is a contradiction to the fact that √2 is irrational.
Hence, our assumption is wrong and 6+√2 is irrational.
NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.3 – Real Numbers, has been designed by the NCERT to test the knowledge of the student on the topic – Revisiting Irrational Numbers