It rained the whole week except on Sunday.
In this question is the word 'whole' an adverb? Or since the word 'week' is a noun, will it become a determiner?
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Since whole modifies the noun week, it is an adjective (but see below about determiners). Here are other sentences with an adjective modifying week:
It rained all of last week except on Sunday.
It rained twice during that unpleasant week.
Adverbs modify anything other than a noun: a verb, an adjective, a preposition, another adverb, or a whole clause or sentence.
A determiner is an adjective that specifies something about which instance or instances the noun refers to: a new instance not previously talked about (as indicated by a), an instance previously talked about (as indicated by the), the number of instances, the concept in general rather than any specific instance, etc. In your sentence, whole specifies how much of the instance is referred to, so it's usually considered a determiner.
Some people don't count determiners as adjectives. I don't think it matters as long as you understand the difference between specifying which instance of the noun (as in "last week", "the whole week", and "that unpleasant week") and providing additional information about the instance ("that unpleasant week"). If you're asking in preparation for taking an exam, though, you should find out whether the author of the exam counts determiners as adjectives or not. If you're not sure, then bet that they consider them separate parts of speech.
Yes it can be an adverb - just not in your example sentence.
An adverb is a word or phrase that modifies or qualifies an adjective, verb, or other adverb. In your example, it is modifying the noun "week".
For it to be an adverb, there would need to be another adjective or verb used, for example: