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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।

MathematicsIf \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime integers, what follows from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsIn a proof by contradiction, Arjun assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. He obtains \(m^2=2n^2\) and says, “\(m\) is even, so \(\sqrt{2}\) is irrational.” What is missing from his argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student says, “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” Which step is needed to complete this argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsIn a proof that √3 is irrational, suppose √3 = p/q, where p and q are coprime. Squaring gives p² = 3q², and on writing p = 3k, we get q² = 3k². Which conclusion is needed to complete the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. If \(p^2=3q^2\) is obtained, what is the correct next conclusion in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms gives \(p^2=3q^2\), which conclusion creates the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student claims that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=3q^2\) proves only that \(p\) is divisible by 3. What is the correct improvement to the argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student has to prove that \(1+\sqrt{3}\) is irrational. Which of the following arguments is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=2q^2\), which conclusion creates a contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=3q^2\), which conclusion is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student claims that if the square of an integer is divisible by 2, then the integer itself is divisible by 2. How is this statement useful in proving the irrationality of \(\sqrt{2}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsIf assuming \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime, leads to \(p^2=3q^2\), which conclusion creates the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsWhy is \(\frac{p}{q}\) taken in lowest terms while proving the irrationality of \(\sqrt{2}\) by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student writes: If \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime, then \(a^2=3b^2\). The student concludes that 3 divides \(b\). What is the error in this conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsIn the proof of √2, what is finally proved false?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student claims that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which correct conclusion follows from this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsIn the proof that \(\sqrt{3}\) is irrational, \(a^2=3b^2\) gives \(3\mid a^2\). Which rule justifies the next step?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After concluding from \(p^2=3q^2\) that \(p\) is divisible by 3, which conclusion completes the contradiction proving \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsIn a proof by contradiction that \(\sqrt{3}\) is irrational, what fact produces the contradiction after assuming \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime integers?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student says, “3 is a rational number, so \(\sqrt{3}\) must also be rational.” What is the main error in the student's reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium