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A student has written the following conclusions about irrational numbers. Which conclusion is always correct?एक विद्यार्थी ने अपरिमेय संख्याओं के बारे में नीचे दिए गए निष्कर्ष लिखे हैं। इनमें से कौन-सा निष्कर्ष सदैव सही है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: If \(r\) is a non-zero rational number and \(x\) is irrational, then \(r+x\) is irrational.यदि \(r\) एक अशून्य परिमेय संख्या और \(x\) एक अपरिमेय संख्या है, तो \(r+x\) अपरिमेय होगा।
Option C is correct. If \(r+x\) were rational, then \(x=(r+x)-r\) would also be rational, a contradiction. For A, \(\sqrt{2}+(-\sqrt{2})=0\). Exam tip: test “always” claims using a counterexample.विकल्प C सही है। यदि \(r+x\) परिमेय मान लें, तो \(x=(r+x)-r\) भी परिमेय होगा, जो विरोधाभास है। A में \(\sqrt{2}+(-\sqrt{2})=0\) लें। परीक्षा में “सदैव” वाले कथन के लिए प्रतिउदाहरण जाँचें।
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