The question marker か can only be used in polite questions right? Then why plain volitional can use it?


今日は何をしようか。So is it a set rule that volitional form in plain (not using -ましょう)
can use the question marker か?




According to "A Dictionary of Basic Japanese Grammar", page 166, か can also be
used in plain form.  
Examples:  
これは本だ。This is a book.  
これは本か。 Is this a book?  
貴方は先生だ。You are a teacher.  
貴方は先生か。 Are you a teacher?  
In the examples above the だ drops and is replaced by か.

あの本は高い。That book is expensive.  
あの本は高いか。 Is that book expensive?  
私が話す。I speak.  
私が話すか。 Do I speak?  
In the examples above か is added at the end of the sentence.

