CAT 2022 Set-1 | Quantitative Aptitude | Question: 13

Question

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

• A. either the empty set or the set of all integers
• B. the set of all integers
• C. the set of all positive integers
• D. the empty set

Answer 1

Logic :  if the discriminant is less than 0 then the quad equation cant have any roots
 
which would mean, it can NEVER intersect the X axis.

Now if a > 0  then the graph will be a upward facing parabola as F''(x) = 2a 

So value of a would decide if it is a upward/downward facing parabola

So for a> 0  we will have a upward facing parabola with it not crossing the X axis ever

and for a < 0 we will have a downward facing parabola wih it never crossing the X axis ever