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Question Hard Mathematics Chapter 1: Real Numbers 5: Irrational numbers Class 10 Level 14

यदि \(\sqrt{2}\) को \(\frac{p}{q}\) के रूप में लिखा जाए, जहाँ (p) और (q) सहअभाज्य हैं, तो प्रमाण में अंततः क्या विरोध मिलता है?

If \(\sqrt{2}\) is written as \(\frac{p}{q}\), where (p) and (q) are coprime, what contradiction appears in the proof?

Explanation opens after your attempt
Correct Answer

A. (p) और (q) दोनों सम निकलते हैंBoth (p) and (q) turn out even

Step 1

Concept

Coprime means (p) and (q) have no common factor except (1).

Step 2

Why this answer is correct

In the proof of \(\sqrt{2}\), both (p) and (q) turn out even, so they have common factor (2).

Step 3

Exam Tip

This contradiction proves that \(\sqrt{2}\) is not rational. चरण 1: सहअभाज्य मानने का अर्थ है कि (p) और (q) में (1) के अलावा कोई समान गुणनखंड नहीं है। चरण 2: \(\sqrt{2}\) के प्रमाण में (p) और (q) दोनों सम निकलते हैं, यानी उनमें (2) समान गुणनखंड है। चरण 3: यही विरोध सिद्ध करता है कि \(\sqrt{2}\) परिमेय नहीं है।

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