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Which of the following statements is necessarily true?निम्नलिखित में से कौन-सा कथन अनिवार्य रूप से सत्य है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: The sum of any irrational number and any rational number is irrational.किसी अपरिमेय संख्या और किसी परिमेय संख्या का योग अपरिमेय होता है।
Let \(x\) be irrational and \(r\) be rational. If \(x+r\) were rational, then \((x+r)-r=x\) would be rational, a contradiction. Option B fails since \(\sqrt{2}+(-\sqrt{2})=0\). Exam tip: test “always” claims with a counterexample.मान लें कि \(x\) अपरिमेय और \(r\) परिमेय है। यदि \(x+r\) परिमेय हो, तो \((x+r)-r=x\) भी परिमेय होगा, जो विरोधाभास है। विकल्प B गलत है क्योंकि \(\sqrt{2}+(-\sqrt{2})=0\)। परीक्षा टिप: ‘सदैव’ वाले कथन को उदाहरण से जाँचें।
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