CAT 2022 Slot 1 Paper

Question : 53 of 66

Marks: +1, -0

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

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

Validate

Solution:

b^2< 4 a c

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.

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