Grammarians will rap your knuckles for using the passive tense. About a decade ago there was quite a backlash against passive tense. It's ebbed a bit lately but you could still find yourself in trouble. Linguists, on the other hand, will want you to communicate in the most consistent and informative or persuasive way possible. They won't mind passive tense so much but don't want you to mix it with active tense. Who will score your test?
If you want to make both happy, avoid both passive tense and mixing tenses. If you were to write the sentence entirely passive it would look like this:
"Presentation of the case would be asked of him by the judge, listening to his speech would be done by the jury, and questions would be raised by the prosecutor."
Although that would be semantically correct and a sentence worded in such a way certainly garners attention, it is overly verbose, verging on archaic. Avoid the urge to sound smarter by using more words. In the majority of cases (though I can't prove how many) the shorter sentence is the correct sentence.
"He believed that the judge would ask him to present the case, the jury would listen to his speech, and the prosecutor would raise questions." is the better of the two options provided. In a multiple choice test, I would choose this one.
If you are writing freely and are enamored with the use of passive tense, avoid mixing it into an active sentence. Try this instead:
He believed that the judge would ask him to present the case and the jury would listen to his speech. Then, he believed, the prosecutor would raise questions.
Then cross your fingers and hope the test scorer is a young grammarian.