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When two waves of slightly different frequencies interfere with each other, they create a phenomenon called beats. These beats have a beat frequency, which is the difference between the two original frequencies. So, if we have two frequencies \(f_1\) and \(f_2\), the beat frequency will be given by: \[\text{beat frequency} = |f_1 - f_2|\] In this problem, we have a tuning fork with a known frequency and an unknown frequency of the piano string. We also know the beat frequency before and after increasing the tension in the string.

First, let's call the initial frequency of the piano string \(f_p\). Then the initial beat frequency is \(5 \mathrm{~Hz}\), so: \[|256\mathrm{~Hz} - f_p| = 5\mathrm{~Hz}\] We are given that the beat frequency decreases to 2 Hz when the tension is slightly increased. As the beat frequency decreases, the piano string frequency is getting closer to the tuning fork frequency. Therefore, the piano string frequency before increasing the tension should be either \(256 + 5 \mathrm{~Hz}\) or \(256 - 5 \mathrm{~Hz}\), because the beat frequency is the difference between the two frequencies.

Now, let's check the given options: (A) \(256+2 \mathrm{~Hz}\) (B) \(256-2 \mathrm{~Hz}\) (C) \(256-5 \mathrm{~Hz}\) (D) \(256+5 \mathrm{~Hz}\) From our analysis, we can already eliminate options (A) and (B) since they correspond to the beat frequency of 2 Hz (after increasing the tension), not the initial 5 Hz. Comparing options (C) and (D), we see that the right answer should be (C) \(256-5 \mathrm{~Hz}\). When the tension slightly increases, the piano string frequency gets closer to the tuning fork frequency, so the beat frequency should decrease from 5 Hz to 2 Hz. The frequency of the piano string before increasing the tension was: \[f_p = 256 \mathrm{~Hz}-5 \mathrm{~Hz} = 251 \mathrm{~Hz}\] Therefore, the correct answer is (C) \(256-5 \mathrm{~Hz}\).