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A student has to prove that \(1+\sqrt{3}\) is irrational. Which of the following arguments is correct?एक विद्यार्थी को सिद्ध करना है कि \(1+\sqrt{3}\) अपरिमेय है। निम्न में से कौन-सा तर्क सही है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Assume \(1+\sqrt{3}\) is rational. Subtracting 1 would make \(\sqrt{3}\) rational, which is a contradiction.मान लें कि \(1+\sqrt{3}\) परिमेय है। उसमें से 1 घटाने पर \(\sqrt{3}\) परिमेय होगा, जो विरोधाभास है।
If \(1+\sqrt{3}\) were rational, subtracting 1 would make \(\sqrt{3}\) rational, contradicting its irrationality. Option D is false since sums are not always irrational. Exam tip: isolate the square root in contradiction proofs.यदि \(1+\sqrt{3}\) परिमेय मानें, तो 1 घटाने पर \(\sqrt{3}\) परिमेय निकलता है, जबकि \(\sqrt{3}\) अपरिमेय है। विकल्प D गलत है; हर योग अपरिमेय नहीं होता। परीक्षा टिप: विरोधाभास में परिमेय पद को अलग करें।
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