(a) Since sources of frequency $x$ gives 8 beats per second with frequency $250\ Hz,$ it’s possible frequencies are 258 or 242. As source of frequency $x$ gives 12 beats per second with a frequency $270\ Hz,$ it’s possible frequencies are 282 and 258 $Hz.$ The only possible frequencies of $x$ which gives 8 beats with frequency $250\ Hz$ also 12 beats per second with $270\ Hz$ is $258\ Hz$