Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 NCERT Solutions

Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 Class 10 Maths Chapter 1 Real Numbers Exercise 1.2 focuses on proving irrational numbers using the method of contradiction, as per the CBSE 10th syllabus. 

The NCERT solutions are given in a clear, step-by-step format to help you understand every step thoroughly. It will help you understand how to apply the algorithm correctly. Regular practice of these questions improves logical thinking, accuracy, and confidence, which are essential for scoring well in exams.

NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2

Q1. Prove that √5 is irrational.

Solution:

Assume, for contradiction, that √5 is rational.
So,
√5 = a/b
where a and b are integers with b ≠ 0 and the fraction is in lowest form.

Square both sides:

5 = a² / b²
⇒ a² = 5b²

This means a² is divisible by 5, so a is also divisible by 5.
Let a = 5k.

Now substitute:

a² = (5k)² = 25k²
So,
25k² = 5b²
⇒ b² = 5k²
⇒ b is also divisible by 5

This means both a and b are divisible by 5, which contradicts our assumption that a/b is in lowest form.

Hence, √5 is irrational.

Q2. Prove that 3 + √25 is irrational.

Solution:

We know:

√25 = 5

So,
3 + √25 = 3 + 5 = 8

And 8 is a rational number.

Therefore, the statement “3 + √25 is irrational” is false.
It is actually rational.

(If the question meant 3√25, then that is also rational because 3√25 = 3×5 = 15.) 

Q3. Prove that the following numbers are irrational:

(i) 1/√2

Solution:

Assume 1/√2 is rational.

This implies √2 is rational (because the reciprocal of a rational is rational).

But √2 is a well-known irrational number.

So our assumption is false.

Hence, 1/√2 is irrational.

(ii) √7 + √5

Solution:

Assume √7 + √5 is rational.

Let
√7 + √5 = r (r is rational)

Then
√7 = r – √5

The right side is the difference of a rational and irrational number, which is irrational.

But the left side (√7) is irrational, so both sides match.

Now square √7 + √5:

(√7 + √5)² = 7 + 5 + 2√35
                  = 12 + 2√35

If √7 + √5 were rational, then √35 must be rational, which is false.

Hence, √7 + √5 is irrational.

(iii) √6 + √2

Solution:

Assume √6 + √2 is rational.

Let
√6 + √2 = r (r is rational)

Then
√6 = r – √2

The right side is irrational (rational – irrational = irrational).

But the left side (√6) is irrational, so proceed further.

Square the expression:

(√6 + √2)² = 6 + 2 + 2√12
                  = 8 + 4√3

If √6 + √2 were rational, 4√3 would also be rational.
⇒ √3 would be rational, which is false.

Thus √6 + √2 is irrational.

How to Score Better in Class 10 Maths Exam?

Here’s how you can score well in the CBSE Class 10th Maths Exam:

• Build Strong Concepts:

Focus on understanding concepts in Class 10 Maths instead of memorising steps, as this helps in solving application-based questions.

• Practise Regularly:

Solve all CBSE Class 10 NCERT questions multiple times to strengthen your basics and improve accuracy.

• Revise Formulas Daily:

Regular revision of PW Class 10 Maths MIQs helps avoid calculation mistakes in exams.

• Solve Previous Year Papers:

Practising CBSE Class 10 Maths previous year questions (PYQs) helps you understand question patterns and important topics.

• Attempt Sample Papers:

Solving PW Class 10 Maths sample papers improves time management and gives you exam-like practice.

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