Include rejected paper in Masters application?
I would recommend putting it in your cv as a Manuscript in Preparation section where you list authors, title only. If you don’t do that, it’s something to bring up very briefly as either a bullet point of writing a resume, or in the cover letter. You may wish to spin this as “motivated to disseminate research” rather than to prove your aptitude for research, since publishing and doing research aren’t synonymous. Good luck