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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।

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 3Expert MathematicsAssume that a/b is in lowest terms and obtain a² = 3b². Which conclusion about a is essential in the proof that √3 is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsA student claims that \(1+\sqrt{2}\) is rational because 1 is a rational number. Which argument correctly refutes the claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsIn the proof by contradiction for the irrationality of \(\sqrt{3}\), why are \(p\) and \(q\) chosen to be coprime when assuming \(\sqrt{3}=\frac{p}{q}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=3q^2\), the student says, “\(p\) is divisible by 3, but \(q\) need not be divisible by 3.” Which statement correctly identifies the error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsWhile proving the irrationality of \(\sqrt{2}\) by contradiction, assume that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion makes this assumption impossible?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms, where \(p\) and \(q\) are coprime integers, which statement produces the contradiction in the proof of its irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which property is used to infer \(3\mid p\) from \(3\mid p^2\) in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which contradiction follows from this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsIn a proof by contradiction that √2 is irrational, a student assumes √2 = a/b, where a and b are coprime positive integers. From 2b² = a², the student concludes that a is even. Which is the correct basis for this conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsWhy must \(p/q\) be taken in lowest terms in the standard proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsHow would the decimal expansion of \(\sqrt{2}\) be classified?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsWhich divisibility rule is crucial in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, and on squaring obtains \(p^2=2q^2\). Which reasoning correctly proves irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsWhy must \(\sqrt{3}=\frac{p}{q}\) be assumed to be in lowest terms while proving the irrationality of \(\sqrt{3}\) by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsA square has an area of \(2\text{ cm}^2\). A student says, “Since the area is rational, the side of the square must also be rational.” What is the correct correction to this statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsIn the proof by contradiction for the irrationality of \(\sqrt{2}\) and \(\sqrt{3}\), which prime-number property for an integer \(n\) is used decisively?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsIf (a) is assumed odd in (a^2=2b^2), what contradiction appears?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsA student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). The student shows only that \(3\mid p\) and declares a contradiction. Which step correctly completes the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert MathematicsWhile proving the irrationality of \(\sqrt{2}\) by contradiction, which is the correct initial assumption for treating \(\sqrt{2}\) as rational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Expert