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Let \(x\) be an irrational number and \(r\) be a rational number. Which of the following expressions is always irrational?मान लीजिए कि \(x\) एक अपरिमेय संख्या है और \(r\) एक परिमेय संख्या है। निम्नलिखित में से कौन-सी अभिव्यक्ति 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, which is a contradiction. \(x^2\) is not always irrational; for \(x=\sqrt{2}\), it equals 2. Exam tip: rational ± irrational is irrational.यदि \(x+r\) परिमेय होता, तो उसमें से परिमेय \(r\) घटाने पर \(x\) भी परिमेय मिलता, जो विरोधाभास है। \(x^2\) सदैव अपरिमेय नहीं; \(x=\sqrt{2}\) पर इसका मान 2 है। परीक्षा टिप: परिमेय ± अपरिमेय अपरिमेय होता है।
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