1. Do you remember the day when we met first?

1-1. Do you remember when we met first?

2. Do you remember the day on which when we dmet first?

(Can we use #1-1?)

3. Tell me the reason why you were late.
4. Tell me why you were late.
(We can delete 'the reason' in Sentence 3. Then we get #4. In #4, what is the part of speech of 'why'? In #3, 'why' was a relative adverb.' In #4, is 'why' a relative adverb or 'an interrogative adverb'?)

5. This is the reason why you were late.
6. This is why you were late.
(In #5, 'why' is a relative adverb. In #6, is 'why' a relative adverb or an interrogative adverb?)

1. Do you remember the day when we first met? <~~Word order matters!

1-1. Do you remember when we met first? not OK

2. Do you remember the day on which when we met first? not OK

Any of these can be used correctly if you put "first" before "met."
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3. Tell me the reason why you were late.
4. Tell me why you were late.
(We can delete 'the reason' in Sentence 3. Then we get #4. Yes. You can use either 3 or 4, but the redundancy makes 4 the better phrasing.

In 4 and 6, "why" is an interrogative adverb that is used to introduce the indirect question. I wouldn't use 3 or 5 because of the redundancy.

1. Yes, I remember the day we first met. To answer this question, you can rely on your personal memory. It's a simple yes or no answer based on your recollection.

1-1. Yes, "Do you remember when we met first?" is a grammatically correct variation of the first question. The word order is different, but the meaning remains the same.

2. Yes, you can use the sentence "Do you remember the day on which when we met first?" However, it sounds a bit awkward and unnecessarily wordy. It's more common to use simpler and clearer phrasing such as "Do you remember the day we first met?"

Regarding your question if we could use "#1-1," it seems like you intended to refer to the first variation, which is "Do you remember when we met first?" Yes, you can use this variation to ask about the memory of the first meeting.

3. In sentence 3, "Tell me the reason why you were late," the word "why" is a relative adverb. It introduces and relates to the reason for being late, connecting it to the main clause.

4. In sentence 4, "Tell me why you were late," the word "why" is an interrogative adverb. It functions as an adverb by modifying the verb "tell." It is used to ask for the reason behind someone being late.

5. In sentence 5, "This is the reason why you were late," the word "why" is still a relative adverb. It introduces and relates to the reason, connecting it to the main clause.

6. In sentence 6, "This is why you were late," the word "why" is an interrogative adverb. It functions as an adverb by modifying the verb "is." It is used to ask for or state the cause or reason for someone being late.

So, to summarize, in sentences 3, 5, and 6, "why" functions as a relative adverb. In sentences 4 and 6, "why" functions as an interrogative adverb.