for a system of linear equations we know that if (adjA )B = 0 THEN THE SYSTEM MAY OR MAY NOT BE CONSISTENT.Discuss when it will become consistent and when inconsistent. is it necessary that if consistent than infinitely many solutions will be occurring in the above case?

The complete theory can be summarised as, 
Consistent system-
A given system of equation is said to be consistent if it has one or more solutions.
Inconsistent system-
A given system of equation is said to be inconsistent if it has no solutions.
Case -1:
When |A| $\ne$0 , then the given system is consistent and it has a unique solution.
Case-2:
When |A|=0 and (adj A)B$\ne$0 , then the given system is inconsistent and it has a no solution.
Case -3:
When |A|=0 and (adj A)B=0 , then the given system is consistent and it has infinite solution.