CAT 2024 Question Paper | Quant Slot 3

CAT Questions

CAT 2024 Quant was dominated by Algebra followed by Arithmetic. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 20 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

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Video Explanation

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Explanatory Answer

\[ f(x)+2 f\left(\frac{1}{x}\right)=3 x \] Put \(x=1 / x\), \[ f\left(\frac{1}{x}\right)+2 f(x)=\frac{3}{x} \] Solving (1) & (2),
(1) \(\Rightarrow\) \[ \begin{array}{ll} (1) \Rightarrow & f(x)+2 f\left(\frac{1}{x}\right)=3 x \\ 2 \times(2) \Rightarrow & 2 f\left(\frac{1}{x}\right)+4 f(x)=\frac{6}{x} \end{array} \] Subtracting, we get \[ 3 f(x)=\frac{6}{x}-3 x \] So, \[ f(x)=\frac{2}{x}-x \] Now, \[ \begin{array}{l} f(x)=3 \text { implies, } \frac{2}{x}-x=3 \\ x^2+3 x-2=0 \end{array} \] Sum of the roots \(=-b / a\)
Answer \(=-3\)

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The question is "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"

Hence, the answer is '-3'

Choice B is the correct answer.

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