An SI engine has a $10\mathrm{.}2-cm$ bore with the spark plug offset by $6$ mm from center. When the engine operates at $1200$ RPM it takes $6\mathrm{.}5$ of engine rotation for combustion to develop and get into flame propagation mode, where the average flame speed is $15\mathrm{.}8\frac{m}{sec},$ with the combustion ending at $12\mathrm{.}4$ aTDC. It is desired that flame termination be at the same crank angle position when the speed of the engine is increased to $2000$ RPM$.$ In the $1000rpm-2000rpm$ engine speed range, flame development takes the same amount of real time and flame speed is related to engine speed as $V_f=0\mathrm{.}92$constant$N,$ where $V_f$ is the flame speed and $N$ is the engine speed. Calculate:

a$)$ Flame speed at $2000$ RPM$.[$m$/$sec$]$

b$)$ Crank angle position when the spark plug should be fired $[bTDC].$

c$)$ Crank angle position when flame propagation starts $[$ bTDC$].$