CAT 2024 Slot 3 QA Q20 Functions & Graphs

Question 20 of the Quantitative Ability (QA) section, CAT 2024 Slot 3, with CAT's official answer and a worked solution.

For any non-zero real number \(x\), let \(f(x)+2 f\left(\frac{1}{x}\right)=3 x\). Then, the sum of all possible values of \(x\) for which \(f(x)=3\), is
  1. A
    2
  2. B
    -3
  3. C
    -2
  4. D
    3

Answer: (B) -3 Functions & Graphs MCQ

Solution

  1. Replace \(x\) by \(\frac1x\): \(f(\tfrac1x) + 2f(x) = \frac3x\). Doubling it and subtracting the original: \(3f(x) = \frac6x - 3x\), so \(f(x) = \frac2x - x\).
  2. \(f(x) = 3 \Rightarrow x^2 + 3x - 2 = 0\), real nonzero roots with sum \(-3\).

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