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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।

MathematicsWhich statement is an essential basis of the proof by contradiction that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIn the contradiction proof that √3 is irrational, \(p^2=3q^2\) gives \(3\mid p^2\). Which fact is used to conclude that \(3\mid p\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, what conclusion about \(p\) is drawn from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIn a proof by contradiction that \(\sqrt{2}\) is irrational, suppose \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which final condition contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIf (x,y) are coprime and both are proved even, what is the correct contradiction about (\gcd(x,y))?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIn the standard proof, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which number-theoretic fact is needed to obtain a contradiction from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIf assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms leads to \(p^2=3q^2\), which conclusion proves a contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsSuppose \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. On squaring, we get \(a^2=3b^2\). Which conclusion is justified at this stage?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and \(p^2=3q^2\) is obtained, what is the valid basis for concluding that \(3\mid p\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsA student claims that \(3\sqrt{2}\) is rational because 3 is a rational number. What is the correct evaluation of this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed with \(p\) and \(q\) coprime, and the proof shows that both \(p\) and \(q\) are divisible by 3, what does this establish?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIf \(\sqrt{3}=p/q\) is assumed with \(p\) and \(q\) coprime, which condition contradicts this assumption in the proof of irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion produces the contradiction in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\) is written in lowest terms. If \(3\mid p^2\), which of the following conclusion is valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIn a proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. If \(3\mid p^2\) is obtained, which conclusion must follow next?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIf \(p\) and \(q\) are coprime integers and \(p^2=3q^2\), which conclusion necessarily follows?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIn the proof that \(\sqrt{3}\) is irrational, suppose \(\frac{p}{q}\) is in lowest terms and \(p^2=3q^2\) is obtained. Which fact is needed to establish the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After obtaining \(p^2=2q^2\), which conclusion correctly advances the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsIf \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, is assumed, which conclusion creates a contradiction in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsWhich of the following numbers must be irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard