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Irrational Numbers

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

MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, Riya shows that in the fraction \(a/b\) written in lowest terms, both \(a\) and \(b\) are divisible by 3. Why is this conclusion impossible?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, which conclusion necessarily follows from \(3q^2=p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsA student says that \(\sqrt{2}+\sqrt{3}\) is irrational because the sum of two irrational numbers is always irrational. What is the correct evaluation of this statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed, which condition on \(p\) and \(q\) is necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIf \(m=2k\) and \(n^2=2k^2\), what is the combined conclusion in the proof of \(\sqrt{2}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsRavi claims that \(7+\sqrt{2}\) is a rational number because 7 is rational. What is the error in Ravi’s reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIf \\(\sqrt{3}\\) is written as \\(a/b\\), where \\(a\\) and \\(b\\) are coprime integers, which conclusion produces the contradiction in the proof of its irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn a proof that \(\sqrt{3}\) is irrational, a student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). The student directly writes that \(q\) is divisible by 3. Which statement is needed to make the reasoning valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIf b^2=3a^2 is obtained when b is in lowest form, which property is used to prove that b is divisible by 3?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsA student claims that \(\sqrt{12}\) is rational because 12 is not a perfect square. Which is the correct simplified form of \(\sqrt{12}\) that identifies the error in the claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsA student assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. Which conclusion from \(m^2=2n^2\) proves that this assumption is contradictory?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIf ext{\(\sqrt{3}\)} is assumed to be ext{\(p/q\)} , where ext{\(p\)} and ext{\(q\)} are coprime integers, which fact produces the contradiction in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms, which conclusion about \(p\) and \(q\) produces the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhich of the following statements is an essential part of the proof by contradiction that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsIn a proof that \(\sqrt{3}\) is irrational, a student writes: “If \(3\mid p^2\), we can conclude only that \(p^2=3k\).” What is the correct correction needed to continue the argument?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 to prove its irrationality, which fact produces the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard 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 3Level 17Hard 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