a
(a) Frequency of sound heard by the man from approaching train

\({n_a} = n\,\left( {\frac{v}{{v - {v_s}}}} \right) = 240\,\left( {\frac{{320}}{{320 - 4}}} \right) = 243\,Hz\)

Frequency of sound heard by the man from receding train

\({n_r} = n\,\left( {\frac{v}{{v + {v_s}}}} \right) = 240\,\left( {\frac{{320}}{{320 + 4}}} \right) = 237Hz\)

Hence, number of beats heard by man per sec

\( = {n_a} - {n_r} = 243 - 237 = 6\)

Short trick : Number of beats heard per sec

\( = \frac{{2nv{v_S}}}{{{v^2} - v_S^2}} = \frac{{2nv{v_S}}}{{(v - {v_S})(v + {v_S})}} = \frac{{2 \times 240 \times 320 \times 4}}{{(320 - 4)(320 + 4)}} = 6\)