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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 that \(\sqrt{2}\) is irrational, why must \(p\) and \(q\) be chosen coprime when assuming \(\sqrt{2}=\frac{p}{q}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsIf the square of a rational number is 3, what conclusion is obtained about its numerator and denominator in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsRiya assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After concluding from \(p^2=3q^2\) that \(p\) is divisible by 3, which is the correct next statement to obtain a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsLet \(p\) and \(q\) be coprime positive integers. In a proof by contradiction for the irrationality of \(\sqrt{3}\), if \(3\mid p^2\), which conclusion is necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, assume \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. Which conclusion produces a contradiction to this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsSuppose \(r=\sqrt{2}+\sqrt{3}\) is a rational number. Which conclusion from this assumption proves by contradiction that \(\sqrt{2}+\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion is necessary to establish the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms to prove irrationality, which statement establishes the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\) is in lowest terms. What conclusion about \(p\) follows from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhich statement is sufficient to reject the rational assumption in the proof of (√3)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, if \(p\) and \(q\) are coprime and \(p^2=3q^2\), which conclusion follows immediately?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsA student claims that \(5+\sqrt{3}\) is rational because 5 is rational. Which is the correct refutation of this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, assume that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. Which conclusion creates a contradiction to the initial assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIn the proof of √2, why is assuming a and b coprime more than a formality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, which situation contradicts the assumption that \(p/q\) is in lowest terms?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhich option gives the most correct full logical chain in the proof of \(\sqrt{2}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhich statement is correct in the proof of the irrationality of \(\sqrt{3}\), based on \(a^2=3b^2\), where \(a\) and \(b\) are coprime integers?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIn the proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Why does this assumption lead to a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsSuppose \\(\sqrt{3}=\frac{p}{q}\\), where \\(p\\) and \\(q\\) are coprime positive integers. Which conclusion from \\(p^2=3q^2\\) gives the contradiction needed to prove that \\(\sqrt{3}\\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIf √2 = a/b is in lowest form, what does proving that both a and b are even show?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard