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CAT 2022 Slot 1 — QA Question 10

Discriminant and Roots of Quadratic EquationEasy

Let a, b and c be non-zero real numbers such that b2 < 4ac, and f(x) = ax2 + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

Answer & solution

  • A

    the empty set

  • B

    the set of all integers

  • either the empty set or the set of all integers

  • D

    the set of all positive integers

Solution

Easy

b2<4acb^2<4ac means the discriminant is negative, so f(x)f(x) never touches the xx-axis and keeps one fixed sign everywhere. That sign is decided by aa — split into a>0a>0 and a<0a<0.

1

Read off the discriminant.

b2<4ac ⇒ D=b2−4ac<0b^2<4ac\ \Rightarrow\ D=b^2-4ac<0

With D<0D<0 the parabola has no real roots, so f(x)f(x) is always strictly positive or always strictly negative.

2

Case a>0a>0. An upward parabola sitting above the axis gives

f(x)>0  for all x ⇒ no integer m has f(m)<0f(x)>0\ \text{ for all }x\ \Rightarrow\ \text{no integer }m\text{ has }f(m)<0

So S=∅S=\varnothing (the empty set).

3

Case a<0a<0. A downward parabola sitting below the axis gives

f(x)<0  for all x ⇒ every integer m has f(m)<0f(x)<0\ \text{ for all }x\ \Rightarrow\ \text{every integer }m\text{ has }f(m)<0

So S=ZS=\mathbb{Z} (all integers).

4

Combine. Since aa could be either sign, SS is either the empty set or the set of all integers.

SS is either the empty set or the set of all integers — option (c).

CAT 2022 Slot 1 QA Q10: Let a, b and c be non-zero real numbers such that b 2 < 4ac, and f(x) = ax 2 + bx + c. If the set S consists o — Solution | TheCATExam