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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 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard 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 3Level 17Hard MathematicsWhich of the following numbers must be irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\), a student assumes \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime. The student obtains \(a^2=3b^2\). Which conclusion is correct for proceeding with the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard