You are right, Fortunately their questions mostly relies on first order logic consistency and equivalence. Thank you

@EmilJeřábek I need to solve this problem in PhD entrance exam in the next week, I will have only 3 minutes to solve it that's why I was looking for a deterministic way.

Yes, sounds I need a trick more than an algorithm! The examples I mentioned in the question was the problem of the last year.

I'll have PhD entrance exam in the next week and they ask the completeness of a set of formulas problem every year. sounds I should give it up!

@JoelDavidHamkins that is right.

You mentioned the Rice theorem. it says any non-trivial attribute of algorithms is undecidable. Undecidable means both semi-decidable which equals to recursively enumerable sets and non recursively enumerable sets. Semi decidable problems have an algorithm. I would like to know if this problem is semi-decidable is there any partial algorithm (partial function) to solve the problem?

@CarlMummert despite of being decidable or semi-decidable what is the process of determining the completeness of a set?

How?! any inconsistent set of formulas is complete while it has no model