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Square Root 3

Class 9 Mathematics के इस tag से जुड़े questions। हर question के साथ chapter, topic, level और difficulty दी गई है।

MathematicsIn the standard proof by contradiction, suppose that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. From \(p^2=3q^2\), which deduction is essential for obtaining the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student claims that \(\sqrt{3}\) is rational because its decimal form is 1.732 and \(1.732=\frac{1732}{1000}\). What is the error in the student’s reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On squaring, the student gets \(p^2=3q^2\) and concludes that \(p\) is divisible by 3. What is the next essential step to complete the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, which property is used to conclude \(3\mid p\) from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student says that \(\sqrt{3}\) is irrational because its decimal expansion \(1.732\ldots\) continues endlessly. What is the main error in this argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student claims that \(\sqrt{3}\) is rational and writes it as \(\frac{p}{q}\) in lowest terms, where \(p,q\) are coprime. If \(p^2=3q^2\), what is the error in this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhich of the following number-theoretic facts is used centrally in proving that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhich of the following statements is essential in a proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After obtaining \(3q^2=p^2\), which of the following conclusion is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student claims that if \(3p^2=q^2\), then \(p\) must be divisible by 3. What is the correct evaluation of the claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn a proof by contradiction that \(\sqrt{3}\) is irrational, assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion from \(p^2=3q^2\) is needed to establish the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhich property of the prime number 3 is used decisively in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which statement correctly justifies the conclusion \(3\mid p\) from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn proving the irrationality of ext{\(\sqrt{3}\)} , Ravi assumes that ext{\(\sqrt{3}=p/q\)} , where ext{\(p\)} and ext{\(q\)} are coprime. From ext{\(p^2=3q^2\)} , he says that only ext{\(p\)} is divisible by 3 and nothing can be concluded about ext{\(q\)} . Which fact corrects his error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIf (h) is not divisible by (3) and (h^2=3k^2), what inconsistency appears?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion about \(p\) and \(q\) contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the contradiction proof of the irrationality of \(\sqrt{3}\), what does writing \(\frac{p}{q}\) in lowest terms mean?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the contradiction proof of the irrationality of \(\sqrt{3}\), we assume \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. If both \(p\) and \(q\) are finally found to be divisible by 3, which conclusion is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the proof by contradiction for the irrationality of sqrt(3), before assuming sqrt(3) = p/q, which condition is essential for the fraction p/q?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert