CAT 2025 Slot 1 QA Q8 Inequalities & Modulus

Question 8 of the Quantitative Ability (QA) section, CAT 2025 Slot 1, with CAT's official answer and a worked solution.

A value of \(c\) for which the minimum value of \(f(x)=x^2-4 c x+8 c\) is greater than the maximum value of \(g(x)=-x^2+3 c x-2 c\), is
  1. A
    2
  2. B
    \(-\frac{1}{2}\)
  3. C
    \(-2\)
  4. D
    \(\frac{1}{2}\)

Answer: (D) 1/2 Inequalities & Modulus MCQ

Solution

  1. \(\min f\) is at \(x = 2c\): \(4c^2 - 8c^2 + 8c = 8c - 4c^2\).
  2. \(\max g\) is at \(x = \frac{3c}{2}\): \(-\frac{9c^2}{4} + \frac{9c^2}{2} - 2c = \frac{9c^2}{4} - 2c\).
  3. \(8c - 4c^2 > \frac{9c^2}{4} - 2c \Rightarrow 10c > \frac{25c^2}{4} \Rightarrow 0 < c < \frac{8}{5}\).
  4. Among the options, only \(c = \frac12\) lies in this interval.

Same concept in other years

See every question on Minimum or maximum of a quadratic at its vertex.

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