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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।

MathematicsWhich assumption is made at the beginning to prove the irrationality of \(\sqrt{3}\) by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsWhile proving the irrationality of \(\sqrt{2}\) by contradiction, which condition is essential when assuming \(\sqrt{2}=\frac{p}{q}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsIn a proof by contradiction, assume that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime integers. If \(3n^2=m^2\) is obtained, which conclusion decisively shows that this assumption is impossible?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsWhile proving the irrationality of \(\sqrt{2}\) by contradiction, after assuming \(p/q\) is in lowest terms, which condition directly contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsReema says, “\(\sqrt{3}\approx1.732\); therefore, \(\sqrt{3}\) is rational because 1.732 is rational.” What is the correct evaluation of Reema’s argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsSuppose it is claimed that \(s=\sqrt{2}+\sqrt{3}\) is rational. Since \((\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2})=1\), \(\sqrt{3}-\sqrt{2}=1/s\) would also be rational. Which equation below immediately produces a contradiction from this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\), a student assumes \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime. After obtaining \(a^2=3b^2\), the student states that both \(a\) and \(b\) are divisible by 3. Which reasoning is necessary to justify this conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsReena says, “The decimal expansion of \(\sqrt{2}=1.414213\ldots\) is infinite, so it is irrational.” What is the most accurate evaluation of her reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIt is known that \(\sqrt{3}\) is irrational. Which of the following conclusions must be true?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhich initial assumption is required when proving the irrationality of \(\sqrt{3}\) by the contradiction method?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsRima says, “\(\sqrt{2}\) is irrational because its decimal expansion is infinite.” What is the main flaw in her reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhich prime-divisibility property is crucial in a proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion follows from this assumption and produces a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsA student says that the decimal expansion of \(\sqrt{2}\) never terminates, so it is irrational. Which of the following arguments rigorously proves this conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion is needed to establish the contradiction in the proof of irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn a proof by contradiction for the irrationality of \(\sqrt{3}\), a student obtains \(p^2=3q^2\). Given that \(p\) is divisible by 3, which next step correctly advances the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsRima says, “If \(\sqrt{12}\) were rational, then \(\sqrt{3}=\frac{\sqrt{12}}{2}\) would also be rational, which contradicts the irrationality of \(\sqrt{3}\).” What is Rima’s conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn a proof that \(\sqrt{3}\) is irrational, if \(\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIf \(\sqrt{3}=p/q \), where \(p \) and \(q \) are coprime positive integers, which conclusion is required to establish the contradiction?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 in lowest terms and \(p^2=3q^2\) is obtained, which conclusion is necessary to reach the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard