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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।

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 3Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if it is shown that 3 divides both \(a\) and \(b\), which conclusion creates the contradiction in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsA student assumes that \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. On proceeding with the proof, the student finds that both \(a\) and \(b\) are even. What decisive conclusion follows?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Hard MathematicsWhich option shows the correct cause-effect relation used in the proof of \(\sqrt{3}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed, where \(p\) and \(q\) are coprime integers, which conclusion produces the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsA student says that \(\sqrt{3}\) is irrational because its decimal expansion does not terminate. Which step correctly turns this into a rigorous proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, the student gets \(p^2=2q^2\). Which conclusion proves that this assumption is invalid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsIf assuming √2 is rational gives a contradiction, which assumption is proved false?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsA student says that if \(\sqrt{2}=\frac{p}{q}\), then both \(p\) and \(q\) may be even. Why is this statement incorrect in the proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. What is the first conclusion obtained from \(3q^2=p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsRiya says that \(\sqrt{3}\) is rational because its decimal form is \(1.732\). Which statement correctly identifies the error in her claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsA student argues that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=3q^2\) proves only that \(p\) is divisible by 3. Identify the student's error.Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsWhich initial assumption is made to prove the irrationality of \(\sqrt{2}\) by the method of contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsIn the proof that \(\sqrt{3}\) is irrational, suppose \(\sqrt{3}=p/q\) is in lowest terms and squaring gives \(p^2=3q^2\). What correctly fixes a student's error that \(3\mid p^2\) does not imply \(3\mid p\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsBefore proving the irrationality of \(\sqrt{3}\) by contradiction, in what form must \(\frac{p}{q}\) be assumed?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsWhile proving \(\sqrt{3}\) irrational by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. What follows immediately from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsWhy is the proof of \(\sqrt{2}\) valid without decimal expansion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsIf \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, what conclusion follows from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsSuppose a student writes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, while proving that \(\sqrt{3}\) is irrational. From \(p^2=3q^2\), which conclusion about \(p\) is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium MathematicsA student says that since 5 is a rational number, \(5+\sqrt{3}\) must also be rational. Which statement correctly identifies the error in the argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Medium