第一句to ask是不定式作定语，修饰question. 不定式与被修饰名词之间为逻辑上的动宾关系。

但第二句的不定式并不是定语。which room to live in是带疑问词的不定式，是疑问分句的简略形式。还原为名词性从句则为：which room I am to live in 或者which room I should live in. 也就是说，don't know的宾语并不是which room, 而是which room to live in整个带疑问词的不定式短语。

I don't know which room to live in. 带疑问词的不定式作宾语。

= I don't know which room I should live in. 间接疑问句作宾语。

= I don't know which room I am to live in. 间接疑问句作宾语。