Which of these is the grammatically correct form:
Where are you going to apply?
Where are you applying?
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Both of them are correct.
We can use either one of them to ask a question about the future.
Some grammars call #1 the "going to" future, and suggest that we use it when talking about something that has been planned or arranged (we have taken some step to make something happen).
They may call #2 the present continuous and say that we use this tense to talk about the future when we have only an intention to do something (but we have not taken any steps to arrange the future action).
In reality, they are both examples of the present continuous, and the factors that describe when we use them are subtle and complex. Short and simple explanations don't work very well here.
Quoting from a book I own: The present continuous indicates that we have a firm intention or have made a definite decision to do something, although this may not already be arranged. So both could be used.
Keep in mind that the present continous can't be used to talk about:
In a formal style you could also use will.