Question

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

• A. $254Hz$
• B. $246Hz$
• C. $240Hz$
• D. $260Hz$

Answer 1

The correct option is A $254Hz$

Let the unknown frequency be $f$.
Beat frequency $b=|f-f^{\prime }|$
For $f^{\prime }=250Hz$ $b=4$
$\therefore$ $|f-250|=4$ ....(1)
For $f^{\prime }=513$ $b=5$
$\therefore$ $|2f-513|=5$ ...(2)
Case 1 : $2f>513Hz$ and $f>250Hz$
We get $f-250=4$ $⟹f=254Hz$
and $2f-513=5$ $⟹f=259Hz$
which is not possible.
Case 2 : $2f<513Hz$ and $f>250Hz$
We get $f-250=4$ $⟹f=254Hz$
And $513-2f=5$ $⟹f=254Hz$
Case 3 : $2f<513Hz$ but $f<250Hz$
We get $250-f=4$ $⟹f=246Hz$
And $513-2f=5$ $⟹f=254Hz$
Which is not possible.
So the unkown frequency $f=254Hz$