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Square Root 2

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

MathematicsIn a square root spiral, what is the hypotenuse of a right triangle with sides 1 unit and 1 unit?Class 9Number SystemsSquare root spiralLevel 24Easy MathematicsReema says that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=2q^2\) only shows that \(p\) is even; therefore \(\sqrt{2}\) is not proved irrational. Which essential point is missing from Reema's argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhile proving the irrationality of \(\sqrt{2}\) by contradiction, if \(\sqrt{2}=\frac{p}{q}\) where \(p\) and \(q\) are coprime, which conclusion necessarily follows from \(p^2=2q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhy is it necessary to take the fraction \(\frac{p}{q}\) in lowest terms in the contradiction proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, \(p^2=2q^2\) is obtained. Which conclusion follows correctly?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn a proof by contradiction, suppose \(\sqrt{2}=\frac{h}{k}\), where \(h\) and \(k\) are coprime integers. Which conclusion contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion follows from this assumption and contradicts the fraction being in lowest terms?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes \(\sqrt{2}=p/q\), where \(p\) and \(q\) are coprime integers, to prove that \(\sqrt{2}\) is irrational. From \(p^2=2q^2\), the student writes \(p=2m\) and obtains \(q^2=2m^2\). The student says that since \(p\) and \(q\) are coprime, \(q\) must be odd. What is the correct correction to this statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student claims that if the square of an integer is divisible by 2, then the integer itself is divisible by 2. Which argument correctly supports this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(3\sqrt{2}\) is a rational number. Which argument correctly shows a contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, the student gets \(p^2=2q^2\). Which of the following is the valid next inference?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion creates the contradiction in the proof by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student claims that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=2q^2\) implies only that \(p\) is even; nothing can be concluded about \(q\). What is the error in the student's reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsUsing exponents of prime factors in perfect squares, which idea is correct in the proof of √2?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhy must the fraction be taken in lowest terms when assuming \(\sqrt{2}=\frac{p}{q}\) in the proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Which of the following properties is crucial for proving that this assumption is contradictory?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIf √2 = r/s is in lowest form and finally 2 divides r and 2 divides s, which statement is the most precise contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsA student calls \(\sqrt{2}\) irrational after seeing its decimal form 1.414213... . Which property must be proved to justify the conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhile starting a proof by contradiction for the irrationality of \(\sqrt{2}\), Riya assumes that \(\sqrt{2}=p/q\), where \(p\) and \(q\) are integers. Which additional condition on \(p\) and \(q\) is necessary to make the proof valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf the contradiction proof of \(\sqrt{2}\) succeeds, what is the final logical conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert