How should I structure my answer to a purposive writing question? (e.g. writing a speech, letter or article)

The most important thing to remember is that this type of question is testing your ability to write a clear and organised piece. Naturally, it is important to write in the correct form (framing your letter formally/informally, addressing your audience for a speech or coming up with a title for your article) but the following general advice applies. Organise your writing into a clear introduction, stating what you intend to argue; clear paragraphs, each outlining one key point or a few closely related points; and a conclusion, summarising your argument. Make sure each paragraph has a clear topic sentence and a clear closing sentence to ensure that the reader is following your argument (it helps here to refer back to the key words from the question!).
Another thing to remember is that you will be trying to construct an argument in a piece of this kind. An easy way to ensure that you do this is to make a list of pros and cons relating to the statement. When you have done that, pick your side and lead with a point which contradicts what you wish to argue (e.g. if you are "pro" the statement, lead with a "con"). Discuss this point and why it may be viewed as valid, then bring in the counterargument (for the side you are supporting) and tell the reader why this is a more valid point. If you repeat this process for each paragraph, you will end up with an argument which has addressed (and dismissed!) counterarguments, and it will prove much more persuasive than just detailing pros and cons without engaging with them.