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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।

MathematicsIn a proof by contradiction, a student assumes \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. After obtaining \(p^2=3q^2\), which statement correctly justifies the conclusion \(3\mid p\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhich assumption is required at the beginning of a proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsA student claims that \(\sqrt{2}+\sqrt{3}\) is a rational number. Which argument correctly identifies the error in this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student believes that for some non-zero rational number \(q\), \(q\sqrt{3}\) can be rational. Which argument correctly refutes this belief?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsA student says that if \(\sqrt{3}=\frac{p}{q}\), then \(p^2=3q^2\) only implies that \(p\) is divisible by 3; nothing can be concluded about \(q\). What is the student's error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction in the proof of irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\frac{p}{q}\) is in lowest terms, which immediate conclusion follows from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Which conclusion from \(p^2=3q^2\) is necessary to reach a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhy must the fraction be taken in lowest terms when assuming \(\sqrt{2}=\frac{p}{q}\) in the proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIf \(p\) and \(q\) are coprime integers and \(p^2=3q^2\), which conclusion is essential in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsRima assumes that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which conclusion in her proof establishes a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student claims that \(2+\sqrt{3}\) is a rational number. Which argument correctly disproves the claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Which of the following properties is crucial for proving that this assumption is contradictory?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsAarav assumes that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. After obtaining \(m^2=3n^2\), he says that divisibility of \(m^2\) by 3 does not prove that \(m\) is divisible by 3. Which fact corrects Aarav’s error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student claims that \(\sqrt{2}+\sqrt{3}\) is rational. Which argument correctly proves that the claim is wrong?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student calls \(\sqrt{2}\) irrational after seeing its decimal form 1.414213... . Which property must be proved to justify the conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhile starting a proof by contradiction for the irrationality of \(\sqrt{2}\), Riya assumes that \(\sqrt{2}=p/q\), where \(p\) and \(q\) are integers. Which additional condition on \(p\) and \(q\) is necessary to make the proof valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf the contradiction proof of \(\sqrt{2}\) succeeds, what is the final logical conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. From \(p^2=3q^2\), which conclusion is necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne0\), and concludes from \(p^2=2q^2\) that both \(p\) and \(q\) are even. If the student did not assume \(p\) and \(q\) to be coprime, why is this conclusion not automatically a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert