Explain the meaning of "determined" in, “You are determined that something will happen.”
Does it mean “You will (are determined to) make something happen”?
Or does it mean the same as, “You shall go to the ball, Cinderella,” which I think means, "I will (am determined to) make you go to the ball Cinderella."
Or is it like, “He was determined that the same mistakes would not be repeated” which I think means, “He was determined (firmly decided) not to make the same mistakes be repeated.”
In this question, you are trying to sort out the relationship between willing something and determining something as well as to understand their relationship to the three meanings of shall. To be determined involves a person's will: a person's ability to decide and choose and act.
Does “You are determined that something will happen” mean “You will (are determined to) make something happen”?
Yes, it does. If you are determined that a thing will happen, it means you exert your ability to decide and choose and act so that events and fate unfold according to your plans [barring unforeseen circumstances]: You exert your self-will.
Does “You shall go to the ball, Cinderella” mean "I will (am determined to) make you go to the ball Cinderella" or “I am sure that you will certainly go to the ball, Cinderella”?
Neither of these really describe what shall in this context means. In this context, someone is speaking who has the power to alter fate and decide outcomes. This is not usually someone like you or I. It is usually someone like a fairy godmother or a prime minister or a supreme court. A suitable paraphrase would be: "You are to be at the ball without hindrance, Cinderella."
Of course, this can work on an everyday, ordinary people level too, as when Hugh Grant tells his co-star in Music & Lyrics, "You shall dance." Yet, this level cannot be made comparable to the quote, spoken by a fairy godmother, that you are trying to understand the import of. A suitable paraphrase for Hugh Grant’s quote would be: “Dance if you wish because I put no obstacles in your path.” You can see, I think, that this has a very, very different import from the fairy godmother’s paraphrased quote.
Does "He was determined that the same mistakes would not be repeated” mean “He was determined (firmly decided) not to make the same mistakes be repeated”?
Yes. This is a good paraphrase for this use of determined. Being firmly decided also denotes an exertion of one's human will to decide, choose, and act. This person has decided and chosen to act so as not to repeat those mistakes. [Incidentally, that would correctly be expressed as: “He was determined (firmly decided) not to make the same mistakes” or “He was determined (firmly decided) not to repeat the same mistakes” or “He was determined (firmly decided) not to let the same mistakes be repeated.”]
According to the Oxford English Dictionary, “shall” is a command or a promise. Therefore, the sentence, “You shall go to the ball, Cinderella.” is a promise. Moreover, when used idiomatically, “shall” is a contradistinction to “will,” which means that idiomatically it serves to contrast that word. (Ex: “You shall and will go to the ball, Cinderella.”)
The Oxford English Dictionary defines the adjective “determined” as something that is decidedly fixed beforehand; definite, exact. To say that “you are determined something will happen” means that it has been fixed or ordained beforehand that the said event will occur.