Please advise me on the correct usage of the words sleep and slept.
How was your night?
Did you slept well?
Is this sentence correct?
Did is the past of do. Slept is the past/past participle of sleep. When there are two verbs in a sentence, (in this case, do and sleep), using past tense for both verbs is incorrect.
Since you have started your sentence with a past tense, i.e. Did, you have to ask
How was your night? Did you sleep well?
Or, if you want to further condense the above sentence,
Did you sleep well last night?
How was your night? Slept well?
In the above, slept indicates the past action(verb).
No, it's not correct.
What I suggest to my students is to observe the patterns that arise because did can never exist with the past tense form (in your example slept) of any verb in a sentence.
Look at this pattern for all verbs using your verb.
Q: Did you sleep well?
A1: Yes, I slept well.
A2: No, I didn't sleep well.
Notice that the presence of did eliminates the need for the past tense form.
Notice that only when being affirmative (i.e. saying, "yes") and not using did should you use the past tense form.
Notice also, that you could answer affirmatively with did, in which case you again would not use the past tense form.
A3: I did sleep well.
English, generally, does not like doubling. If a word has a quality (like being past tense, as did is, and signaling the past tense as did does) then it accomplishes that task for the whole sentence (generally).