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From
[『スーツ』](https://ja.wikipedia.org/wiki/SUITS/%E3%82%B9%E3%83%BC%E3%83%84_\(%E6%97%A5%E6%9C%AC%E3%81%AE%E3%83%86%E3%83%AC%E3%83%93%E3%83%89%E3%83%A9%E3%83%9E\))

> A: ど…どうしたの？(Seeing B is unhappy: "What's going on with you?")
>
> B: どうしたの **か** って？人が死んだ。("You know what's wrong! Someone died!")

What I usually hear is that the part before a quotative って repeats what the
other party just said. What does か add to the tone of the rhetorical question?
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question
nuances particle-か rhetorical-questions
What does か do in a rhetorical question?
This sounds like incredulity.

> どうしたのかって？人が死んだ。

> Are you asking _what happened_? Someone died.

Without context, I'm not really sure what the original question really meant.
Were they asking "What happened?" or "What's going on with you?" If it's the
later, then

> You asking me, "what's going on with me?" Someone died.

か here is signaling a degree of incredulity that whoever asked the question in
the first place asked what they did.
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どうしたの and どうしたのか are the same except that the latter is usually regarded as
the literary style. The のか style is fairly bookish and stiff at the end of a
sentence, but it's less so in the middle of a sentence. Here, どうしたのって and
どうしたのかって are almost the same except that the latter may sound a little more
serious/formal.

  * 昨日は何食べたのって聞かれた。(informal)
  * 昨日は何を食べたの **か** (と)聞かれました。(formal)

(But is that really a rhetorical question? A typical rhetorical question is
something that implies "no". Something like 誰が分かるのか ("Who knows?") is a
rhetorical question because it implies "no one knows", but in this case どうしたの
is just an ordinary question and it does not imply "nothing happened" or
"nothing is wrong". Person B simply quoted it without changing the meaning of
the question itself.)

* * *

You probably know this, but if だ/です were used, it would add a nuance of
accusation/shock (see
[this](https://japanese.stackexchange.com/q/90333/5010)).

  * どうしたの **だ** って？
  * どうしたのか **だ** って？
  * どうしたの **です** って？
  * どうしたのか **です** って？
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answer