State whether the following statements are true or false. Provide either a proof or a specific counter

example.

a$)$ During the two$-$phase method, if the artificial LP is optimal, the original LP has to be feasible.

b$)$ If an LP has a feasible region $S$ and an extreme point $A,$ then point A definitely cannot be

represented as a strict convex combination of two distinct points of $S.$

c$)$ During the two$-$phase method, at the end of the first phase, $w=0$ means the original LP is

feasible.

d$)$ Consider a given LP to be solved using the simplex method, either by the two$-$phase method,

the big$-$M method or a direct way. If there is an entering variable but no leaving variable it means

that LP is infeasible.