Class 10 Maths Ch 1 Real Numbers Ex 1.2-Detailed Answers

Class 10 Maths Ch 1 Real Numbers Ex 1.2 is based on the Euclid’s Division Lemma and Algorithm, which are important tools to find the HCF (Highest Common Factor) of two numbers. In this exercise, students learn how to apply the division algorithm step-by-step to solve problems accurately. In this section, you will find clear, step-by-step detailed answers to all questions of Exercise 1.2, explained in a simple and easy-to-understand manner.

In this exercise, students learn how to:

  • Apply Euclid’s Division Algorithm to find the HCF of two numbers.
  • Use prime factorization to find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of two or more numbers.
  • Verify the relation:
    HCF × LCM = Product of the two numbers

The questions in this exercise help build a strong foundation in understanding how numbers are broken down into their basic prime factors and how this can be used to solve real-life problems involving divisibility and multiples.

Ex 1.3
Ch 2 polynomials

Class 10 Maths Ch 1 Real Numbers Ex 1.2-Textbook Solutions

1. Prove that √5 is irrational

Solution:

Proof (by contradiction):
Assume √5 is rational.

So, √5 = p/q, where p and q are integers and in lowest form (HCF = 1).

Squaring both sides:
5 = p²/q²
⇒ p² = 5q²

So, p² is divisible by 5 ⇒ p is divisible by 5
Let p = 5k

Substitute:
(5k)² = 5q²
⇒ 25k² = 5q²
⇒ q² = 5k²

So, q is also divisible by 5

This means p and q have a common factor 5
But we assumed they are coprime.

Contradiction ⇒ √5 is irrational.

2. Prove that 3 + √5 is irrational

Solution:

Proof (by contradiction):
Assume 3 + √5 is rational.

Then,
√5 = (3 + √5) − 3

Right side = rational − rational = rational

So √5 becomes rational ❌
But we already proved √5 is irrational.

Contradiction ⇒ 3 + √5 is irrational.

3. Prove the following are irrational

(i) 1/√2

Solution:

Assume 1/√2 is rational.

Then,
√2 = 1 ÷ (1/√2)

Right side = rational ÷ rational = rational ❌

But √2 is irrational.

Contradiction ⇒ 1/√2 is irrational.

(ii) 7√5

Assume 7√5 is rational.

Then,
√5 = (7√5) ÷ 7

Right side = rational ÷ rational = rational ❌

But √5 is irrational.

Contradiction ⇒ 7√5 is irrational.

(iii) 6 + √2

Assume 6 + √2 is rational.

Then,
√2 = (6 + √2) − 6

Right side = rational − rational = rational ❌

But √2 is irrational.

Contradiction ⇒ 6 + √2 is irrational.

Class-wise Solutions – Class 10 Maths Ch 1 Real Numbers Ex 1.2

Class 12:

Class 12 Physics – NCERT Solutions

Class 12 Chemistry – NCERT Solutions

Class 11:

Class 10:

Class 9:

Class 8:

Class 7:

Class 6:

Subject-wise Solutions – Class 10 Maths Ch 1 Real Numbers Ex 1.2

Physics:

Chemistry:

Biology:

Math:

Science:

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For the official Class 8 Mathematics Solutions, you can visit:

  1. NCERT Textbooks (for Class 8):
ncert resources