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Proof By Contradiction

Class 9 Mathematics के इस tag से जुड़े questions। हर question के साथ chapter, topic, level और difficulty दी गई है।

MathematicsIf \(r\) is an irrational number and \(q\) is a rational number, which of the following statements is always true?Class 9Number SystemsIrrational numbersLevel 13Expert MathematicsLet \(a\) and \(b\) be rational numbers, \(b\ne0\), and let \(m\) be a natural number that is not a perfect square. What type of number is \(a+b\sqrt{m}\)?Class 9Number SystemsIrrational numbersLevel 13Expert MathematicsIf \(x\) is an irrational number and \(r\) is a non-zero rational number, which of the following statements is always true?Class 9Number SystemsIrrational numbersLevel 13Expert MathematicsIf r is a non-zero irrational number and q is a non-zero rational number, which of the following statements is always true?Class 9Number SystemsIrrational numbersLevel 13Expert MathematicsLet \(a\) be an irrational number and \(r\) be a non-zero rational number. Which of the following statements is always true?Class 9Number SystemsIrrational numbersLevel 15Hard MathematicsLet \(r\) be a rational number and \(x\) be an irrational number. Which of the following conclusions is always true?Class 9Number SystemsIrrational numbersLevel 15Hard MathematicsA student has written the following conclusions about irrational numbers. Which conclusion is always correct?Class 9Number SystemsIrrational numbersLevel 15Hard MathematicsA student claims that \(5-\sqrt{7}\) is a rational number. Which argument correctly proves that the claim is wrong?Class 9Number SystemsIrrational numbersLevel 14Hard MathematicsRavi claims that \(x=\frac{3+\sqrt{2}}{5}\) is rational because its denominator is 5. What is the error in Ravi’s claim?Class 9Number SystemsIrrational numbersLevel 14Hard MathematicsLet p and q be rational numbers with q ≠ 0. If a student claims that \(\frac{p+\sqrt{2}}{q}\) is rational, which conclusion would follow from the claim?Class 9Number SystemsIrrational numbersLevel 13Hard MathematicsA student says that \(4+\sqrt{3}\) is rational because 4 is rational. What is the error in the student's conclusion?Class 9Number SystemsIrrational numbersLevel 14Medium MathematicsA student says that the product of a non-zero rational number and an irrational number is always irrational. Which is the best reason why the statement is correct?Class 9Number SystemsIrrational numbersLevel 14Medium