Here's some advice from Super Moderator MoonlitSonata.
Apple security questions.
Got it sorted!
From:
https://discussions.apple.com/docs/DOC-4551
Another user had success doing the following:
I got some help from an apple assistant on the phone. It is kind of round about way to get in.
Here is what he said to do and it is working for me...
a. on the device that is asking you for the security questions go to "settings", > "store" >
tap the Apple ID and choose view"Apple ID" and sign in.
b. Tap on payment information and add a credit/debit card of your preference then select
"done", in the upper right corner
c. sign out and back into iTunes on the device by going to "settings"> "store" > tap the
Apple ID and choose "sign-out" > Tap "sign -in" > "use existing Apple ID" and you
should be asked to verify your security code for the credit /debit card and NOT the
security questions.
d. At this time you can remove the card by going back in to edit the payment info and
selecting "none" as the card type then saving the changes by selecting "done". You
should now be able to use your iTunes store credit without answering the security
questions.