In the sentence that reads "I had began to study for the test by five o'clock", the person used the wrong participle "began". Because this is the <em>perfect past</em> (indicated by the word "had"), we must use the past participle of the verb.
The verb here is "begin", and its past participle is "begun". So the sentence should say "I had begun to study for the test by five o'clock".
