Filing status

Not sure what you mean by "quiz paper". When you prepare your tax return and enter a close relative dependent that qualifies you to file as Head of Household, Turbotax normally asks you if you want to file as Head of Household. Not sure why it didn't do that if you answered all the questions correctly. Here are the requirements to file Head of Household:

1. You are not married or “considered unmarried” on the last day of the year. You are considered unmarried if you lived apart from your spouse the entire last 6 months of the year.
2. If "considered unmarried" you file a separate return from your spouse.
3. You paid more than half the cost of keeping up your home for the year.
4. You have a qualifying child (whether or not you claim the dependency exemption) or you claim a dependency exemption for a “qualifying person”  who is related to you and who lived with you in the home for more than half the year. An exception is a parent does not have to live with you to be a qualifying person.

If you believe you qualify but filed single by mistake and your tax return has already been accepted, you will have to prepare, print, sign and mail an amended return form 1040X to change the status. The form 1040X will be available in TurboTax later in February.

 DO NOT change your original tax return until you start the amended return. If you have already changed it from the way it was originally filed and accepted, you must change it back before you start the amended return. Wait until you receive the refund from your original return before beginning your amended return to make sure you enter the correct amount. Here is an FAQ to help you:
https://ttlc.intuit.com/questions/1894381-how-to-amend-change-or-correct-a-return-you-already-filed