1. The best verb to use with "effort" is "make". To make an effort. 2. "Taken" is very odd. "Invested" is just a bad habit of Wall Street journalists or gold-diggers, to whom everything, including breathing, is an investment. 3. I think "efforts" needs to be plural here. So it will have to be "Despite the efforts made, some questions remain unanswered." 4. The sentence still does not sound like the best English. "Some questions remain unanswered" is often used in circumstances where an investigation has been made, or a public report is hiding something. As the sentence stands, "despite the efforts made" sounds very weak because it is very vague: who made what efforts? If we have a fuller context, we can write it much better. Writing style is up to a point subjective. There are, however, cultural norms. These are especially strong in Britain, where there is a tradition of English language specialists writing to explore what constitutes the best English. In modern times it started with "The King's English", by Fowler and Fowler, published in 1906. Then of course among the educated public there is constant debate, but also consensus, on who writes the best English. John Maynard Keynes's is bad; Roy Jenkins's is good, for example. The BBC was good but is no longer. E.M. Forster still reads very well.

Invested or invest is not as limited in usage as some may imagine: http://www.dictionary.com/browse/invested?s=t While I have no love for wall streeters or gold diggers, invest certainly could be used in your example under the guidelines set forth by the first reviewer. With that said, I find my self unable to give an exact answer to many of these questions of collocation. It's not always so easy as just reading about the correct choices and then picking one that is 'right.'

Invested is a stronger word. It implies that you made an effort with the expectation of a specific result, or that the effort required a sacrifice of time, money, etc. In this case the expected result was to answer all the questions, however some questions remain unanswered despite the effort. "Taken" doesn't imply this necessarily, it just means that the effort was made.