Saying what is the purpose of # inquiry is really formal and would only be used in writing in answer to an authority or in a special case like dealing with, say, HR in an unclear situation (I doubt you would even use it in talking to the police) - and you would want to replace the # with "this" our "your" or "the" because otherwise there's just something missing here.
You asked specifically about a very formal conversation but in the parts of the world I know, even a very formal (spoken) conversation does not always mean to use intensely formal wording. In many situations, using that phrase in a conversation, however formal, would just come across as awkwardly stilted language. Too little information in your question to be more specific, though.
Added in reaction to a comment of yours, although stretching the scope of the question but I think it is warranted here:
Wanting to sound "cold and formal" seems socially off in most situations in most parts of the world where English is used as a first language. If you really need to get somebody off your back unambigously, omitting all expression of courtesy should be quite more than enough (and could be seen as a sign of weakness or lesser education, depending on the situation and other things). Where it does not suffice, and you need or really want to tell a person to stop or go away, you will need to directly tell them so (if the situation is such that you can legally and acceptably do that).
In the latter case, you may again be better off using a basic amount of politeness in some situations (the directness of the statement doing the work here) unless a rude tone is required and may be being used already. But as others have already mentioned, that would depend on context (specifically situation, social class - not necessarily low here in all thinkable situations - etc).