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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 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 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 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 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 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 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 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 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 claims that if the square of an integer is divisible by 3, then the integer itself is divisible by 3. What is the correct evaluation of this claim?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 \\(\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 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 MathematicsIf the square of an integer is divisible by 3, which conclusion about the integer must be true?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard