Explanation
Positive sign in the argument of sin indicating that wave is travelling in negative x-direction.
This is a case of destructive interference.
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 bears/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.
Since source 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 or 258 Hz. The only possible frequency of x which gives 8 beats with frequency 250 Hz also 12 beats per second with 270 Hz is 258 Hz.
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