NCERT Class 10 Maths – Chapter 1: Real Numbers
Exercise 1.2 – Step-by-Step Solutions
Class: 10
Subject: Mathematics
Chapter: Real Numbers
Exercise: 1.2
Question 1: Prove that √5 is irrational.
Concept Used:
To prove a number is irrational, we use the method of contradiction.
We assume the number is rational and show that it leads to a contradiction.
Proof:
Assume √5 is rational.
Then it can be written as:
√5 = p/q
where p and q are integers, q ≠ 0 and p/q is in lowest form (HCF of p and q = 1).
Squaring both sides:
5 = p² / q²
⇒ p² = 5q²
This shows that p² is divisible by 5.
Therefore, p is divisible by 5.
Let p = 5k.
Substituting:
(5k)² = 5q²
25k² = 5q²
q² = 5k²
This shows q is also divisible by 5.
Thus, both p and q are divisible by 5, which contradicts the assumption that p/q is in lowest form.
Therefore, our assumption is wrong.
Hence, √5 is irrational.
Question 2: Prove that 3 + 2√5 is irrational.
Proof:
Assume 3 + 2√5 is rational.
Let 3 + 2√5 = r (where r is rational).
Then,
2√5 = r − 3
Since r is rational and 3 is rational, r − 3 is rational.
Thus 2√5 is rational.
Dividing by 2:
√5 is rational.
But we already proved that √5 is irrational.
This is a contradiction.
Hence, 3 + 2√5 is irrational.
Question 3: Prove that the following are irrational.
(i) 1 / √2
Assume 1/√2 is rational.
Then √2 must also be rational (by rationalising).
But √2 is irrational.
This is a contradiction.
Hence, 1/√2 is irrational.
(ii) 7√5
We know √5 is irrational.
7 is a non-zero rational number.
Product of a non-zero rational number and an irrational number is always irrational.
Hence, 7√5 is irrational.
(iii) 6 + √2
6 is rational.
√2 is irrational.
Sum of a rational number and an irrational number is always irrational.
Hence, 6 + √2 is irrational.
Important Concepts Used
- Method of Contradiction
- Definition of Rational Number (p/q form)
- Properties of Irrational Numbers
- Sum of rational and irrational number is irrational
- Product of non-zero rational and irrational number is irrational