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A student claims that \(3+\sqrt{2}\) is a rational number. Which argument proves the student's claim wrong?एक विद्यार्थी दावा करता है कि \(3+\sqrt{2}\) एक परिमेय संख्या है। निम्नलिखित में से कौन-सा तर्क उसके दावे को गलत सिद्ध करता है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Assume \(3+\sqrt{2}\) is rational. Subtracting 3 would make \(\sqrt{2}\) rational, which is a contradiction.मान लें कि \(3+\sqrt{2}\) परिमेय है। इसमें से 3 घटाने पर \(\sqrt{2}\) परिमेय होगी, जो विरोधाभास है।
If \(3+\sqrt{2}\) were rational, subtracting the rational number 3 would make \(\sqrt{2}\) rational. But \(\sqrt{2}\) is irrational, so the claim is false. Exam tip: use contradiction for such proofs.यदि \(3+\sqrt{2}\) परिमेय मानें, तो परिमेय संख्या 3 घटाने पर \(\sqrt{2}\) भी परिमेय होनी चाहिए। परंतु \(\sqrt{2}\) अपरिमेय है, इसलिए दावा गलत है। परीक्षा में विरोधाभास विधि उपयोग करें।
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