A verb phrase refers to the part or portion of a sentence that contains the verb and the object which can be either a direct or an indirect object.
Bearing this in mind, in the particular sentence provided <em>John ate the cake, </em>the verb phrase is <em>"ate the cake"</em> since ate corresponds to the verb and the cake corresponds to the direct object.
Because the second sentence says the students WERE passed the exam last week and <em>were</em> doesn't need to be in there. Therefore it is not correct.
