Thank you for your help.
1. Monday is the day when I am the busiest.
2. Monday is the day when I am busiest.
3. Monday is the day I am the busiest.
4. Monday is when I am the busiest.
5. Monday is the day that I am the busiest.
6. Monday is that I am the busiest.
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Q1: Are they all grammatical?
Q2: Do we have to put 'the' before 'busiest' in this sentence?
Q3:In Sentence 4, 'the day' is omitted. In this case, what is the part of speech of 'when'? An interrogative adverb or a relative adverb?
1 - 5 are fine. 6 is not.
Adding the word "the" seems to just provide emphasis, but doesn't change the meaning.
In 4, "when" is the interrogative that is introducing the indirect question. It's not a relative-anything.
Interrogatives: when, where, who, whether, how ... http://grammar.ccc.commnet.edu/grammar/marks/question.htm#indirect and http://www.perfect-english-grammar.com/indirect-questions.html
Relative pronouns introduce relative clauses -- http://www.chompchomp.com/terms/relativeclause.htm -- and some relative pronouns also serve as interrogatives.
A1: No, they are not all grammatical.
A2: No, you do not have to put 'the' before 'busiest' in this sentence. It would be grammatically correct to say "Monday is the day when I am busiest" without the word 'the'.
A3: In sentence 4, 'when' is used as a relative adverb. It is introducing a relative clause that modifies the noun phrase "the day".
Q1: Are they all grammatical?
A1: Out of the six sentences provided, sentences 1, 2, 3, and 4 are grammatically correct. Sentences 5 and 6, however, are not grammatical.
Q2: Do we have to put 'the' before 'busiest' in this sentence?
A2: In sentences 1, 2, 3, 4, and 5, it is grammatically correct to include 'the' before 'busiest.' This is because 'the' is used to specify that it is the particular day (Monday) in the context that is the busiest. However, in sentence 6, 'the' should not be included.
Q3: In Sentence 4, 'the day' is omitted. In this case, what is the part of speech of 'when'? An interrogative adverb or a relative adverb?
A3: In sentence 4, where 'the day' is omitted, the word 'when' can be classified as a relative adverb. It is used to introduce a relative clause that provides additional information about the time (Monday) being referred to.