d
Let the frequencies of tuning fork and piano string be \(v_{1}\) and \(v_{2}\) respectively.

\(\therefore v_{2}=v_{1} \pm 4=512 \mathrm{Hz} \pm 4=516 \mathrm{Hz}\) or \(508 \mathrm{Hz}\)

Increase in the tension of a piano string increases its frequency.

If \(v_{2}=516 \mathrm{Hz}\), further increase in \(v_{2},\) resulted in an increase in the beat frequency. But this is not given in the question.

If \(v_{2}=508 \mathrm{Hz}\), further increase in \(v_{2}\) resulted in decrease in the beat frequency. This is given in the question. When the beat frequency decreases to \(2\) beats per second. Therefore, the frequency of the piano string before increasing the tension was \(508 \mathrm{Hz}\).

\(\mathop {512\,Hz}\limits_{({v_1})} \,\,\xrightarrow{{ + \,\,4\,\,Hz}}\,\mathop {516\,Hz}\limits_{({v_2})} \)

\(\mathop {512\,Hz}\limits_{({v_1})} \,\,\xrightarrow{{ - \,\,4\,\,Hz}}\,\mathop {508\,Hz}\limits_{({v_2})} \)