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CAT 2022 Slot 3 · Quant · Functions · hard

A function f satisfies f(x) + 3f(4 − x) = 2x for all real x. Find f(1).

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A function f satisfies f(x) + 3f(4 − x) = 2x for all real x. Find f(1).
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Answer: 2

Put x = 1: f(1) + 3f(3) = 2 …(i). Put x = 3: f(3) + 3f(1) = 6 …(ii). From (i), f(1) = 2 − 3f(3). Substitute into (ii): f(3) + 3(2 − 3f(3)) = 6 → f(3) + 6 − 9f(3) = 6 → −8f(3) = 0 → f(3) = 0. So f(1) = 2 − 0 = 2.
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Original question written in the CAT 2022 Slot 3 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake