Let a, b, c be rational numbers and f: Z → Z be a function given by $f(x) = ax^2 + bx + c$. Then, $a + b$ is |
a negative integer an integer non-integral rational number none of these |
an integer |
The correct answer is Option (2) → an integer Since f: Z → Z is given by $f(x) = ax^2 + bx + c$ for all $x ∈ Z$. $∴ f(x) = ax^2 + bx + c$ takes integral values for all $x ∈ Z$ ⇒ f(x) is an integer for all $x ∈ Z$ ⇒ f(0) and f(1) are integers. $⇒ f(1) - f(0)$ is an integer. $⇒ (a+b+c) -c$ is an integer [$∵ f(0) = c$ and $f(1) = a+b+c$] ⇒ a+b is an integer. |