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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।

MathematicsA student claims that \(2+\sqrt{3}\) is a rational number. Which argument correctly disproves the claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Which of the following properties is crucial for proving that this assumption is contradictory?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsAarav assumes that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. After obtaining \(m^2=3n^2\), he says that divisibility of \(m^2\) by 3 does not prove that \(m\) is divisible by 3. Which fact corrects Aarav’s error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student claims that \(\sqrt{2}+\sqrt{3}\) is rational. Which argument correctly proves that the claim is wrong?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student calls \(\sqrt{2}\) irrational after seeing its decimal form 1.414213... . Which property must be proved to justify the conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhile starting a proof by contradiction for the irrationality of \(\sqrt{2}\), Riya assumes that \(\sqrt{2}=p/q\), where \(p\) and \(q\) are integers. Which additional condition on \(p\) and \(q\) is necessary to make the proof valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf the contradiction proof of \(\sqrt{2}\) succeeds, what is the final logical conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. From \(p^2=3q^2\), which conclusion is necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne0\), and concludes from \(p^2=2q^2\) that both \(p\) and \(q\) are even. If the student did not assume \(p\) and \(q\) to be coprime, why is this conclusion not automatically a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On squaring, he gets \(p^2=2q^2\). He says that \(q\) is even because the right-hand side is even. What should be the first correct conclusion in this argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student says, “\(\sqrt{3}\) is rational because it can be written as \(1.732\).” What is the main flaw in this reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn the proof by contradiction that \(\sqrt{2}\) is irrational, assume \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn the proof by contradiction for the irrationality of \(\sqrt{3}\), assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion from \(p^2=3q^2\) is essential for the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed, where p and q are integers, why must \(\frac{p}{q}\) be taken in lowest terms in the proof by contradiction of its irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhen proving the irrationality of \(\sqrt{3}\) by the contradiction method, which assumption is made at the start?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn a proof by contradiction that \(\sqrt{3}\) is irrational, it is assumed to be \(\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion contradicts the initial assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn a proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion from \(p^2=3q^2\) leads to the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsA student claims that \(5+\sqrt{3}\) may be rational because 5 is rational. Which is the correct refutation of this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student says, “For a non-zero rational number \(a\), \(a\sqrt{3}\) can be rational.” Which argument correctly explains the error in this statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsSuppose \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which contradiction follows from this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert