usage - In what situation can I use ~かい (for interrogative question)?

One day, I asked my japanese friend how I could invite some friends to eat. He said
"一緒に食べに行くかい？"

I know that we could also say:

行きましょうか（行こうか）？
行きませんか（行かない）？

行きますか（行くの）？

but I got interested in why he used "kai".
I'd like to know others situations that japanese people use "kai"

Answer

As opposed to 「か」, which is open-ended and can have any sort of answer, 「かい」 is expected to have an answer in the affirmative or negative only, that is, yes or no, with subsequent explanation optional.

Example:

　誰か来たのか
　誰か来たのかい

　誰が来たのか

× 誰が来たのかい