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Number Systems

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

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 MathematicsIf c/d is not taken in lowest form in the proof of √2, which conclusion will not remain decisive?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 MathematicsWhat is the correct difference between the roles of (k\neq0) and (\gcd(h,k)=1) in the proof of (\sqrt{3})?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhat is the correct difference between the roles of (d\neq0) and (\gcd(c,d)=1) in the proof of (\sqrt{2})?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 MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion follows from this assumption and contradicts the fraction being in lowest terms?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhich of the following numbers can be proved irrational using its prime factorisation?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 MathematicsWhich option identifies an invalid shortcut in the proof of √2?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsA student assumes \(\sqrt{2}=p/q\), where \(p\) and \(q\) are coprime integers, to prove that \(\sqrt{2}\) is irrational. From \(p^2=2q^2\), the student writes \(p=2m\) and obtains \(q^2=2m^2\). The student says that since \(p\) and \(q\) are coprime, \(q\) must be odd. What is the correct correction to this statement?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 MathematicsA student claims that if the square of an integer is divisible by 2, then the integer itself is divisible by 2. Which argument correctly supports this claim?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