Defeating first two in quiz, disappearing outside (11)
I believe the answer is:
vanquishing
'defeating' is the definition.
('vanquish' can be a synonym of 'defeat')
'first two in quiz disappearing outside' is the wordplay.
'first two in quiz' becomes 'qu' (I can't justify this - if you can you should believe this answer much more).
'disappearing' becomes 'vanishing' ('vanish' can be a synonym of 'disappear').
'outside' is an insertion indicator.
'qu' put into 'vanishing' is 'VANQUISHING'.
Can you help me to learn more?
(Other definitions for vanquishing that I've seen before include "Trouncing" , "Conquering" .)