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CAT 2022 Slot 1 QA Question & Solution

AlgebraMedium

Question

Let a, b, c be non-zero real numbers such that $b^2 < 4ac$, and $f(x) = ax^2 + bx + c$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

Options

the set of al!I positive integers
the set of all integers
either the empty set or the set of all integers
the empty set

Solution

$b^2 < 4ac$ means that the discriminant is less than 0. Therefore, f(x)>0 for all x if the coefficient of $x^2$ is positive, and f(x)<0 for all x if the coefficient of $x^2$ is negative.

We are given that f(m)<0 and m is an integer.

So the set containing values of m will either be empty if the coefficient of $x^2$ is positive, or it will be a set of all integers if the coefficient of $x^2$ is negative.