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A student says, “For a non-zero rational number \(a\), \(a\sqrt{3}\) can be rational.” Which argument correctly explains the error in this statement?एक विद्यार्थी कहता है, “शून्येतर परिमेय संख्या \(a\) के लिए \(a\sqrt{3}\) परिमेय हो सकता है।” इस कथन की त्रुटि को कौन-सा तर्क सही ढंग से स्पष्ट करता है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: If \(a\sqrt{3}\) were rational, dividing by \(a\) would make \(\sqrt{3}\) rational, which is a contradiction.यदि \(a\sqrt{3}\) परिमेय हो, तो \(a\) से भाग देने पर \(\sqrt{3}\) परिमेय होगा, जो विरोधाभास है।
Assume \(a\sqrt{3}\) is rational, where \(a\ne0\) is rational. Then \(\frac{a\sqrt{3}}{a}=\sqrt{3}\) would be rational, contradicting its irrationality. Exam tip: always check that division is by a non-zero number.मान लें \(a\sqrt{3}\) परिमेय है और \(a\ne0\) परिमेय है। तब \(\frac{a\sqrt{3}}{a}=\sqrt{3}\) परिमेय होगा, जबकि \(\sqrt{3}\) अपरिमेय है। परीक्षा में शून्येतर होने की शर्त अवश्य जाँचें।
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