A vapour compression refrigerator uses Freon$-12$ as the working fluid and has refrigerating effect of $5$ kW$.$ The compressor is a

twin $-$ cylinder, single $-$ acting machine running at $500$rev$/$min$.$ The cylinder bore diameter and the piston stroke are equal and the volumetric efficiency is $0\mathrm{.}7.$ Compression is adiabatic, with an isentropic efficiency of $0\mathrm{.}9.$ The condenser and evaporator saturation temperatures are $40\backslash$deg C and $10\backslash$deg C respectively. The refrigerant leaves the condenser as liquid at $35\backslash$deg C and the evaporator as saturated vapour. Calculate : $-$

$($a$)$ the coefficient of performance of the refrigerator,

$($b$)$ the power to compress the refrigerant,

$($c$)$ the dimensions of the compressor cylinders.