Beats

When two sound waves of nearly equal frequency overlap, the resulting amplitude rises and falls periodically, producing beats whose frequency equals the absolute difference of the two source frequencies, $f_{beat}=|f_1-f_2|$. Here the tuner hears 4 beats per second against a 256 Hz fork, so the string differs from 256 Hz by exactly 4 Hz. Both possibilities satisfy this: $256+4=260\text{ Hz}$ or $256-4=252\text{ Hz}$. The pair $256\text{ Hz}$ or $260\text{ Hz}$ is wrong because 256 Hz would give zero beats. The pair $248\text{ Hz}$ or $264\text{ Hz}$ corresponds to a beat frequency of 8 Hz, not 4 Hz. The pair $254\text{ Hz}$ or $258\text{ Hz}$ gives only 2 beats per second. This is the NCERT result that beat frequency equals the magnitude of the frequency difference. As a plausibility check, a single beat measurement cannot distinguish whether the string is sharp or flat, so two symmetric answers about the reference frequency are physically expected, which is exactly what $252\text{ Hz}$ and $260\text{ Hz}$ represent.