You Do Need power plug adapter while traveling from Cocos (Keeling) Islands to Balearic Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Balearic Islands are C and F and in Cocos (Keeling) Islands is I. This is how they look like.

Cocos (Keeling) Islands

In Cocos (Keeling) Islands, outlets of type I are in use.

Type I

Balearic Islands

In Balearic Islands, outlets of type C and F are in use.

Type C

Type F