The correct option is B Sinecure
The word for the phrase 'An office or post with no work but high status' is Option B, Sinecure. Option A, Honorary is given as an honor to someone who has not done a course of study. Option C, Gratis means something that is free or costs nothing. Option D, Ex-officio means holding a position due to power or influence and not by election. .Thus, options A, C and D are incorrect and the correct word for the phrase is Option B, Sinecure.