CAT 2023 Slot 2 QA Q21 Equations

Question 21 of the Quantitative Ability (QA) section, CAT 2023 Slot 2, with CAT's official answer and a worked solution.

If \(p^2+q^2-29=2 p q-20=52-2 p q\), then the difference between the maximum and minimum possible value of \(\left(p^3-q^3\right)\) is
  1. A
    486
  2. B
    189
  3. C
    378
  4. D
    243

Answer: (C) 378 Equations MCQ

Solution

  1. \(2pq - 20 = 52 - 2pq \Rightarrow pq = 18\), and the common value is 16.
  2. \(p^2 + q^2 - 29 = 16 \Rightarrow p^2 + q^2 = 45\).
  3. \((p-q)^2 = 45 - 36 = 9 \Rightarrow p - q = \pm 3\).
  4. \(p^3 - q^3 = (p-q)(p^2 + pq + q^2) = \pm 3 \times 63 = \pm189\). Difference between max and min \(= 378\).

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