1. (a) That's right. Once an agent has carried out to completion the tasks with which they were entrusted, the agent has no authority to do anything further, and the agency power is at an end. The term 'functus officio' is sometimes used to describe an agent who has fully discharged their mandate.

Of course it is important to determine that the agent's task has actually been fully completed. In the present case, B has not only found and introduced a willing buyer for A's house, but the buyer has actually entered into a contract to purchase the house. This properly completes B's mandate.