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

MathematicsUnder which condition will the sum of the first n natural numbers be an even number?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 60Hard MathematicsWhen is the sum \(S_n\) of the first n natural numbers odd?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 60Hard MathematicsWhen is the sum of the first n natural numbers odd?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 60Hard MathematicsWhy is the product \(n(n+1)\) always even in the formula \(S_n=\frac{n(n+1)}{2}\) for the sum of the first n natural numbers?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 59Medium MathematicsUnder which condition is the sum \(S_n\) of the first \(n\) natural numbers odd?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 61Medium MathematicsIf aₙ = n² + (−1)ⁿ, what is the value of a₆ − a₅?Class 9Sequences and Progressionsnth termLevel 56Hard 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 MathematicsWhich option identifies an invalid shortcut in the proof of √2?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard 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 \(\sqrt{2}=\frac{p}{q}\), 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 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 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 MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On squaring, he gets \(p^2=2q^2\). He says that \(q\) is even because the right-hand side is even. What should be the first correct conclusion in this argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf someone writes (a=2b) from (a^2=2b^2), what is the correct correction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert 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 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 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 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 MathematicsIf \(m=2k\) and \(n^2=2k^2\), what is the combined conclusion in the proof of \(\sqrt{2}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard