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Let \(x\) be an irrational number. Which of the following statements is always true?मान लें कि \(x\) एक अपरिमेय संख्या है। निम्नलिखित में से कौन-सा कथन सदैव सत्य होगा?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: If \(x\) is irrational, then \(2x\) is irrational.यदि \(x\) अपरिमेय है, तो \(2x\) अपरिमेय है
If \(2x\) were rational, then \(x=(2x)/2\) would also be rational, a contradiction. Hence \(2x\) is irrational. But \(x^2\) need not be irrational: for \(x=\sqrt{2}\), \(x^2=2\). In exams, test claims using a counterexample.यदि \(2x\) परिमेय होता, तो \(x=(2x)/2\) भी परिमेय होता, जो विरोधाभास है। इसलिए \(2x\) अपरिमेय है। \(x^2\) सदैव अपरिमेय नहीं; \(x=\sqrt{2}\) पर \(x^2=2\)। परीक्षा में प्रति-उदाहरण जाँचें।
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