class 10 math ch1.2

Class 10 Maths Chapter 1 Exercise 1.2 Solutions – Real Numbers | Assam Eduverse

Welcome to Assam Eduverse, your trusted source for Class 10 NCERT Maths solutions. Below are complete and simple NCERT solutions for Exercise 1.2 from Chapter 1: Real Numbers, based on the 2025–26 NCERT syllabus.

📘 Chapter 1: Real Numbers
🧮 Exercise 1.2 – NCERT Solutions

✅ Question 1:
Prove that √5 is irrational.

Solution:
Assume √5 = a/b, where a and b are integers with no common factor and b ≠ 0.
Then,

5 = a² / b² → a² = 5b².

This means a² is divisible by 5, so a is divisible by 5.
Let a = 5k for some integer k.
Then,

a² = (5k)² = 25k².

Substituting back,

25k² = 5b² → b² = 5k².

This implies b is also divisible by 5.
But this contradicts the fact that a and b have no common factor.
Therefore, √5 is irrational.

✅ Question 2:
Prove that 3 + 2√5 is irrational.

Solution:
Assume 3 + 2√5 is rational and equal to r.
Then,

2√5 = r - 3 → √5 = (r - 3) / 2.

Since r is rational, this means √5 is rational, which is a contradiction.
Therefore, 3 + 2√5 is irrational.

✅ Question 3:
Prove that the following are irrational:

• (i) 1 / √2
• (ii) 7√5
• (iii) 6 + √2

Solution:

(i) Assume 1 / √2 is rational.
Then, √2 = 1 / (rational number), which implies √2 is rational.
But √2 is known to be irrational, so this is a contradiction.
Therefore, 1 / √2 is irrational.

(ii) 7 is rational and √5 is irrational.
Product of non-zero rational and irrational is irrational.
Hence, 7√5 is irrational.

(iii) Assume 6 + √2 is rational.
Then, √2 = (6 + √2) - 6 is rational.
Contradiction!
Therefore, 6 + √2 is irrational.