(A): Second law of thermodynamics states that heat always flows from hot body to cold body by itself. 
(B): First law of thermodynamics does not specify the condition under which a body can use its heat energy to produce the work.

First law of thermodynamics just states that dQ=dW+dU. 

It does not tell us whether the process is feasible or not neither does it provide any condition to under which the body uses any energy.
Hence, A is false.

Second law of thermodynamics states that heat always flows from hot body to cold body by itself and for heat to flow from a cold body to hot body, external work should be done.