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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।

MathematicsBefore assuming \(\sqrt{3}=\frac{p}{q}\) in a proof by contradiction that \(\sqrt{3}\) is irrational, which condition on \(p\) and \(q\) is essential?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsRima says that \(5+\sqrt{2}\) is a rational number because 5 is rational. Which is the correct evaluation of her statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsRiya assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, she gets \(p^2=3q^2\). Which argument correctly proves a contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsReema says that \(\sqrt{3}\) is rational because 3 is a rational number. What is her error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsSuppose \\(\sqrt{3}=\frac{p}{q}\\), where p and q are coprime positive integers. In the proof that \\(\sqrt{3}\\) is irrational, which fact gives the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsWhich statement is correct about \(\sqrt{2}\) and \(\sqrt{3}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student says that \(2\sqrt{3}\) is rational because 2 is a rational number. What is the error in the student's statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student claims that \(2\sqrt{2}\) is rational because 2 is a rational number. Which statement correctly explains the error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=p/q\) where \(p\) and \(q\) are coprime, which conclusion is essential to the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsWhat is the main purpose of assuming \(\frac{p}{q}\) is in lowest terms in the contradiction proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsRiya assumes that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. If the proof shows that both \(a\) and \(b\) are divisible by 3, what conclusion follows?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsIn the proof of \(\sqrt{3}\), which conclusion is rejected by the rational assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. The student obtains \(p^2=3q^2\). Which conclusion from this step establishes the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsWhat is the purpose of taking r/s in lowest form in the proof that √3 is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Medium MathematicsA student claims that √12 is irrational because √12 = 2√3. Which statement is needed to make this argument valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsAman assumes that ext{\(\sqrt{3}=\frac{p}{q}\)}, where ext{\(p\)} and ext{\(q\)} are coprime. On squaring, he gets ext{\(p^2=3q^2\)}. Aman says, “ ext{\(3\mid p^2\)} does not necessarily imply ext{\(3\mid p\)}.” What is the correct evaluation of Aman’s statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsA student says, “If n is an integer, then \(\sqrt{n}\) will also be an integer.” Which example is most suitable to disprove this statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsIf \(\sqrt{2}\) is assumed to be \(\frac{p}{q}\) in lowest terms, which conclusion creates the contradiction in proving that it is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, if \(\sqrt{3}=\frac{p}{q}\) where \(p\) and \(q\) are coprime, which statement follows from \(p^2=3q^2\) and creates the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy MathematicsReema says, “The decimal expansion of \(\sqrt{3}\) is infinite, so it is irrational.” What is the flaw in her reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21Easy