A source of unknown frequency gives 4 beats/s when sounded with a source of known frequency 250 Hz. The second harmonic of the source of unknown frequency gives five beats per seconds when sounded with a source of frequency 513 Hz. The unknown frequency is:

Let the unknown frequency be f.
Beat frequency b = |f - f'|
For f' = 250 Hz, b = 4 ∴ |f - 250| = 4 (1)
For f' = 513, b = 5 ∴ |2f - 513| = 5 .. (2)
Case 1 : 2f > 513 Hz and f > 250 Hz
We get f - 250 = 4 ⇒ f = 254 Hz
and 2f - 513 = 5 ⇒ f = 259 Hz
which is not possible.
Case 2 : 2f < 513 Hz and f > 250 Hz
We get f - 250 = 4 ⇒ f = 254 Hz
And 513 - 2f = 5 ⇒ f = 254 Hz
Case 3 : 2f < 513 Hz but f < 250 Hz
We get 250 - f = 4 ⇒ f = 246 Hz
And 513 - 2f = 5 ⇒ f = 254 Hz
Which is not possible.
So the unknown frequency f = 254 Hz