UPSC CDS II 2024 Math Paper

Concept:A polynomial that gives integer outputs for all integer inputs forces its coefficients to be integers when the leading coefficient is $1$.Explanation:Evaluate the polynomial $f(x) = x^3 + px^2 + qx + r$ at specific integer values.At $x = 0$: $f(0) = r$. Since $f(0)$ is an integer, $r$ must be an integer. So statement III is true.At $x = 1$: $f(1) = 1 + p + q + r$. This is an integer, and $r$ is integer, so $p+q$ is an integer.At $x = -1$: $f(-1) = -1 + p - q + r$. This is an integer, so $p-q$ is an integer.Now evaluate at $x = 2$: $f(2) = 8 + 4p + 2q + r$. This is an integer, so $4p+2q$ is an integer. Dividing by $2$ gives $2p+q$ integer.Subtract $p+q$ from $2p+q$: $(2p+q) - (p+q) = p$. Since both are integers, $p$ is an integer. Hence statement I is true.Finally, from $p+q$ (integer) and $p$ (integer), $q = (p+q) - p$ is an integer. Hence statement II is true.Thus all three statements are correct.Answer:Option C: I, II and III