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If \(r\) is an irrational number and \(q\) is a rational number, which of the following statements is always true?यदि \(r\) एक अपरिमेय संख्या और \(q\) एक परिमेय संख्या है, तो निम्नलिखित में से कौन-सा कथन सदैव सत्य है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(r+q\) is an irrational number\(r+q\) एक अपरिमेय संख्या है
If \(r+q\) were rational, then \(r=(r+q)-q\) would also be rational, a contradiction. Hence adding a rational number to an irrational number remains irrational. For \(r=\sqrt2\), \(r^2=2\). Exam tip: use contradiction for such properties.यदि \(r+q\) परिमेय हो, तो \(r=(r+q)-q\) भी परिमेय होगा, जो विरोधाभास है। इसलिए परिमेय में अपरिमेय जोड़ने पर परिणाम अपरिमेय रहता है। \(r=\sqrt2\) के लिए \(r^2=2\) है। परीक्षा टिप: विरोधाभास विधि अपनाएँ।
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