A tuning fork whose frequency as given by manufacturer is $512Hz$ is being tested with an accurate oscillator. It is found that The fork produces a beat of $2Hz$ when oscillator reads $514Hz$ but produces a beat of $6Hz$  when oscillator reads $510Hz.$ The actual frequency of fork is

The tuning fork whose frequency is being tested produces 2 beats with oscillator at 514 Hz, therefore, frequency of tuning fork may either be 512 or  516. With oscillator frequency 510 it gives 6 beats/sec, therefore frequency of tuning fork may be either 516 or 504. Therefore, the actual frequency is 516 Hz, which gives 2 beats/sec with 514 Hz and 6 beats/sec with 510 Hz.