CAT 2022 Slot 1 QA Q5 Equations

Question 5 of the Quantitative Ability (QA) section, CAT 2022 Slot 1, with CAT's official answer and a worked solution.

Let \(a, b, c\) be non-zero real numbers such that \(b^2 \lt 4 a c\), and \(f(x)=a x^2+b x+c\). If the set \(S\) consists of al integers \(m\) such that \(f(m)\lt0\), then the set \(S\) must necessarily be
  1. A
    the set of all integers
  2. B
    either the empty set or the set of all integers
  3. C
    the empty set
  4. D
    the set of all positive integers

Answer: (B) either the empty set or the set of all integers Equations MCQ

Solution

  1. \(b^2 \lt 4ac\) means the discriminant is negative, so \(f(x)\) has no real root and never changes sign.
  2. If \(a \gt 0\), \(f(x) \gt 0\) for all \(x\), so no integer gives \(f(m) \lt 0\): \(S\) is empty.
  3. If \(a \lt 0\), \(f(x) \lt 0\) for all \(x\): \(S\) is all integers.
  4. Since the sign of \(a\) is not given, \(S\) is either empty or all integers.

Same concept in other years

See every question on Discriminant conditions for real, distinct or no real roots.

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