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Proof By Contradiction

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 \(\sqrt{2}=\frac{p}{q}\), 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 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 MathematicsIf \(r\) is a non-zero rational number, which statement about \(r\sqrt{3}\) is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhile proving \(\sqrt{3}\) irrational by contradiction, we assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion at the end of the proof establishes the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhat is the main purpose of assuming \(\sqrt{2}=\frac{p}{q}\) in lowest terms in the standard proof by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsRima says that \(\sqrt{3}\) is irrational because its decimal expansion continues infinitely. What is the main flaw in her argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIn a proof by contradiction for the irrationality of \(\sqrt{3}\), which conclusion is necessary to proceed after obtaining \(a^2=3b^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIf someone writes (p=3q) from (p^2=3q^2), what is the correct correction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student assumes \(\sqrt{2}=\frac{m}{n}\), where \(m,n\) are integers, to prove that \(\sqrt{2}\) is irrational. Later, the student finds that both \(m\) and \(n\) are even and calls this a contradiction. Which essential condition is missing from the argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhile proving the irrationality of \(\sqrt{3}\), assume that \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. Which conclusion obtained from \(p^2=3q^2\) contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsFor a positive integer \(n\), which condition identifies when \(\sqrt{n}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIn the contradiction proof that \(\sqrt{3}\) is irrational, assume \(\sqrt{3}=\frac{p}{q}\), where \(\gcd(p,q)=1\). This gives \(p^2=3q^2\). Which principle is correctly applied from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert