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If \(x\) is an irrational number and \(r\) is a rational number, which of the following will always be irrational?यदि \(x\) एक अपरिमेय संख्या और \(r\) एक परिमेय संख्या है, तो निम्नलिखित में से कौन-सी संख्या सदैव अपरिमेय होगी?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(x+r\)\(x+r\)
If \(x+r\) were rational, subtracting the rational number \(r\) would make \(x\) rational, a contradiction. But \(x^2\) need not be irrational; \((\sqrt{2})^2=2\). Exam tip: adding a rational number preserves irrationality.\(x+r\) परिमेय मानने पर उसमें से परिमेय \(r\) घटाने से \(x\) परिमेय हो जाएगा, जो असंभव है। \(x^2\) सदैव अपरिमेय नहीं है; जैसे \((\sqrt{2})^2=2\)। परीक्षा टिप: परिमेय संख्या जोड़ने पर अपरिमेयता बनी रहती है।
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