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If \(x\) is an irrational number and \(r\) is a non-zero rational number, which of the following statements is always true?यदि \(x\) एक अपरिमेय संख्या है और \(r\) एक शून्येतर परिमेय संख्या है, तो निम्नलिखित में से कौन-सा कथन सदैव सत्य है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(rx\) is irrational\(rx\) अपरिमेय होता है
If \(rx\) were rational, then \(x=(rx)/r\) would be rational, a contradiction. Hence \(rx\) is irrational. But \(x^2\) need not be irrational: for \(x=\sqrt{2}\), \(x^2=2\). Exam tip: use contradiction by dividing by a non-zero rational number.यदि \(rx\) परिमेय होता, तो \(x=(rx)/r\) भी परिमेय होता, जो विरोधाभास है। इसलिए \(rx\) अपरिमेय है। \(x^2\) हमेशा अपरिमेय नहीं; \(x=\sqrt{2}\) पर \(x^2=2\)। परीक्षा टिप: शून्येतर परिमेय से भाग देकर विरोधाभास जाँचें।
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