$(50$p$)$ An equimolar saturated liquid$-$saturated vapor mixture containing $35\%A,65\%B$ in moles at a flow rate of $100$kmo$\frac{l}{h}$ will be separated by distillation. It is desired that $93\%$ of $A$ is present in the top product and $96\%$ of the A entering the column is in the top product. If the reflux ratio in the column is $4$ and the relative volatility ${}_{AB}=2,$ determine the given below;

a$)$ Theoretical number of plates and feed plate $(20$p$)$

b$)$ Top and bottom product flow rates $(10$p$)$

c$)$ Minimum reflux ratio $(10$p$)$

d$)$ Minimum number of plates $(10$p$)$