search cathay's press releases. every order of aircraft includes an order for an engine. there are usually 3 options for every aircraft. the a340-300 only has the cfm56 because that was the only option for that aircraft. usually the engine of choice is the rr rb211-524g/h for the 747s (except for the 747-8f and the 747-400er), and the rest of the fleet is a variation of the rr trent (either the 500, 700 or the 800 series). i believe that the 777-300ers have the ge90.
you'll retain more of this information if you search the press releases yourself
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