What does ikimasen ka mean?

Answer 1

=> 'Ikimasen' can be polite form of (1) 行く /i ku/ [to go/to continue] in negative conjugation.

It also can be the same form of (2) 生きる /i ki ru/ [to live].

=> The 'ka' takes it to interrogative (question form). So we have negative interrogative, which is very commonly used in asking someone politely/formally to do something. Same case could happen in English but less frequently.

In case (1) it would mean 'won't (you) come/go to...?' or 'please come/go to...' or 'let's go to...' or 'shall we go to...?' as the most common usage; and it would mean 'won't/doesn't go' in general.

Example: Uchi ni ikimasen ka? = 'Won't you come by my place?'

As you see it can be a simple question, or a polite request or suggestion in form of question in English as well.

In case (2) it wouldn't make much sense for it to be used in suggestive or requesting interrogative like in case (1) but it's not impossible or meaningless; it could mean 'won't you live...' or 'please live ...' or 'shall we/let's live...' in special cases. It would make much more sense for it to mean 'won't/doesn't live' in general, regardless of grammatical persons.

Example: Chotto akaruku ikimasen ka? = 'Why don't (we) live a bit more cheerfully?' [suggestive question]

Example 2: 'Ano otoko WA akuruku ikimasen ka' = 'Doesn't he live cheerfully?' [simple question]

Answer 2

Ikimasu could mean several things, and without seeing the kanji I cannot know to which 'ikimasu' you are referring. One of the more common meanings is "go/will go."