Ans is option (C)
Let the distance between point A and point B be $d$ km. Time taken by the cyclist to cover $d$ km is $1$hr.
$\therefore$ Initial Speed of the cyclist $v_{c}=\frac{d}{1}=d$ kmph.
Now, every minute from $10:01$ A.M., a bike leaves point A, and $45$ such bikes reach point B after $1$ hr.
It is given that speed of each bike is same. So, last bike will leave at $10:45$ A.M. and reach B at $11:00$ A.M.
$\therefore$ time taken by a bike to get from point A to point B $=15$ minutes.
Now, if the cyclist doubles his speed, he will reach point B in $\frac{d}{2d}=0.5$ hr $=$ $30$ minutes. He will reach point B at $10:30$ A.M.
Now, starting from $10:01$ A.M., first bike reaches point B at $10:16$ A.M.
Second bike reaches point B at $10:17$ A.M. and so on...till $15^{th}$ bike which starts at $10:15$ A.M. reaches point B at $10:30$ A.M.
Hence a total of $15$ bikes will reach B by the time cyclist reaches B.