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Irrational Numbers

Class 9 Mathematics के इस tag से जुड़े questions। हर question के साथ chapter, topic, level और difficulty दी गई है।

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 3Level 16Expert MathematicsHow would the decimal expansion of \(\sqrt{2}\) be classified?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert 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 3Level 16Expert 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 3Level 16Expert 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 3Level 16Expert 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 3Level 16Expert MathematicsRina is proving that \(\sqrt{12}\) is irrational. Which of the following arguments contains no error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert 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 3Level 16Expert 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 3Level 16Expert 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 3Level 16Expert MathematicsWhich property is used decisively in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsA student assumes \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. From \(a^2=2b^2\), the student concludes that both \(a\) and \(b\) are even. Which statement explains why this creates a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsA student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). Which next step is logically valid for reaching a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsFor a square with side length 1 cm, Reena claims that its diagonal must also be rational because the side is rational. Which statement correctly identifies the error in Reena’s conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsA student claims that the diagonal of a square of side 1 unit is a rational number. Which argument correctly disproves this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsA student assumes that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. During the proof, if both \(m\) and \(n\) are shown to be divisible by 3, what is the error in the student's assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsWhy must \(p/q\) be taken in lowest terms in a proof by contradiction that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsWhich statement correctly identifies why \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsA student says, “Since 1.732 is a terminating decimal, \(\sqrt{3}\) is rational.” Which option correctly identifies the error in this statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsA student claims that \(5+\sqrt{3}\) is a rational number. Which argument correctly disproves the claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert