I'm often frustrated by candidates who ask questions they've memorised, without displaying original thought. I don't like the question "are there any other objectives","what are the competitors" etc. etc.
That being said, you should always ensure that you know what the objective is. If you don't, you need to ask, but in a clever way that shows hypothesis-based thinking.
For example, if the case says "They're looking at identifying why their profits are falling". You shouldn't say "What is their objective?" or "Are there any other objectives?". It's quite clear. However, you could say something like "So, it looks like their objective is to increase profits. Do they have a target in mind that they'd call success? Are their examples in other companies of similar profit turnarounds that we might baseline against? Are they willing to increase profits if it means cutting costs by firing employees?"
One of my favourite cases is about a non-profit aiming to improve the living conditions of the people in developing country x. It then says "This non-profit wants our helping figuring out 1) where to source the raw materials and 2) How to market the product in the US" In this case, you need to be very clear on the objective. I.e., It's not JUST sourcing and marketing...it's sourcing and marketing in a way that most helps these people (i.e. you may make fewer profits by sourcing from country x, but if you help the country more by paying the locals, that meets your core objective".
Hope this helps!