(a) That's wrong. In this case, only A is mistaken about the age and value of the car sold to B, so it is a situation involving a unilateral (one-sided) mistake. The law takes a strict view of such cases. It applies an objective test to determine whether the parties have reached sufficient agreement to create a contract. If, objectively, agreement exists, the contract is binding, regardless of any one party's inward mistaken state of mind or belief.
This may seem very strict but to hold otherwise would make it too easy for a contact to be avoided on grounds of unilateral mistake. In this example we are not told that B knew of A's mistake, or had any reason to suspect that A was mistaken. If it can be shown that B realised or suspected that A was mistaken, a court might be able to provide relief for A on equitable grounds.