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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।

MathematicsWhile proving the irrationality of \(\sqrt{2}\), a student assumes \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On obtaining \(p^2=2q^2\), the student immediately says that \(q\) is even. How should the teacher correct the error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21EasyMathematicsWhich statement creates a contradiction in the proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21EasyMathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. Which conclusion produces a contradiction to this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21EasyMathematicsIf \(\sqrt{3}\) is assumed to be rational and written as \(\frac{p}{q}\), which condition is necessary for \(p\) and \(q\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 21EasyMathematicsWhile proving that \(\sqrt{2}\) is irrational by contradiction, assume \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. After obtaining \(p^2=2q^2\) and showing that \(p\) is even, which conclusion must be established to get a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsA student writes: “3 is not a perfect square, so \(\sqrt{3}\) is irrational.” What is the most appropriate evaluation of this statement in a question asking to prove the irrationality of \(\sqrt{3}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsA student says that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne0\), then \(p^2=3q^2\) proves only that \(p\) is divisible by 3. What is the correct next conclusion in this argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsWhich of the following statements correctly describes the main idea used in proving that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring both sides, \(3q^2=p^2\) is obtained. Which conclusion proves the assumption wrong?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsWhich of the following statements is the correct basis for proving that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsA student says that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then from \(p^2=2q^2\) only \(p\) is even and nothing can be said about \(q\). What is the student's error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsWhich conclusion is correct in the proof that √2 is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20MediumMathematicsIn the contradiction proof for the irrationality of \(\sqrt{3}\), assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Why is the condition that \(p\) and \(q\) are coprime necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsWhich statement is the key conclusion in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsA student says that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then even if \(p\) is divisible by 3, \(q\) need not be divisible by 3. Why is the statement incorrect?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsA student says, “\(3\) is an integer, so \(\sqrt{3}\) is rational.” What is the main error in this reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsWhy is the initial assumption considered false in the contradiction method?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsIf \(\sqrt{2}\) is assumed to be \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, which conclusion creates the contradiction in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=3q^2\), which conclusion proves this assumption contradictory?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20EasyMathematicsFrom (q^2=3r^2) what conclusion is obtained about (q)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20Easy