I think the question setter could possibly be accused of being economical with the truth here. I'm not sure there was ever a game called battledore.
Badminton is commonly said (e.g. here on Wikipedia) to have evolved from a traditional game called battledore and shuttlecock. The battledore was the racquet, and the shuttlecock (as anyone who has ever played or watched badminton will know) is the thing you hit.
Battledore and shuttlecock was not a team game, or in any real sense a competitive one; players simply stood around in a group and hit the shuttlecock to each other (using their battledores), trying to keep it in the air for as long as possible.
Badminton evolved from this traditional game in British India. According to Wikipedia, no one knows how it came to be associated with the Duke of Beaufort's Gloucestershire pile, but in 1860 a London toy dealer named Isaac Spratt published a booklet entitled Badminton Battledore – A New Game. No copy is known to have survived.
So battledore and shuttlecock became Badminton battledore, which was abbreviated to badminton. As far as I can tell, it was never known simply as battledore.
The setter no doubt felt that to mention a shuttlecock would be too much of a giveaway. No excuse, in my book; find another way of wording it!
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