(1) and (2) are both *grammatically* correct. But the passive version of the original sentence in active voice, "his mother made a cake yesterday," is: (1): A cake was made by his mother yesterday. (2) is not "wrong," but it does not include all the information in the original sentence, so (1) is the better answer. For the passive voice, you can say either: A cake was made yesterday by the mother. A cake was made by the mother yesterday. If the sentence was even longer, as in, "His mother made a cake yesterday in the kitchen," you can write: A cake was made by his mother yesterday in the kitchen. A cake was made yesterday in the kitchen by his mother. A cake was made in the kitchen by his mother yesterday. A cake was made yesterday by his mother in the kitchen. …

as far as I know, the agent comes at the end of the sentence (usually ommited if not important) *a cake was made yesterday (by his mother) so. I'd say that the correct one is nº2