Q: Can I use a monitor larger than 27" on Mac Mini?

Some displays may work via Thunderbolt connection by adapter;

you may need to check to see what specification and then if the

numbers are what you can tolerate. If you actually have a 2011

Mac Mini or a different spec model near 2012, display support

will vary. To correctly identify unit by build year model and not

the year it was sold, is more direct. Also by serial number lookup.

• Thunderbolt ports and displays: Frequently asked questions (FAQ) - Apple Support

Thunderbolt Display 27-Inch - Everymac.com

Details such as the resolution may be rather low, so a larger display

(HDMI) could be unsuitable where the trade-off compromises quality.

• Connecting a displayport monitor to my Mac Mini (late 2012)

Mar 7, 2013 - I have a 27" ASUS PB278Q connected to a late 2012 mac mini using a mini ...

Displayport from and mac device should handle that resolution just fine. ... HDMI supports

resolutions above 1920*1080 just fine, just not on Mac

• Mac mini (Late 2012) - Technical Specifications - Apple Support

Aug 3, 2016 - Mac mini (Late 2012) - Technical Specifications. Size and Weight

... Support for 1080p resolution at up to 60Hz; Support for 3840-by-2160 ...

• Primary Display Capabilities of Macs - Macs By Capability ...

The primary display capabilities -- native resolution or optimum resolution -- of ... with a built-in display

and the maximum resolution supported for other systems as ..... i7" 2.6 (Late 2012) 1920x1200* 1 Mac

mini "Core i7" 2.3 (Late 2012/Server) . -- See chart from this page, below:

{Since the indefinite identification of your Mac Mini leaves a few questions - graphic support, ports;

the above are guidelines for unibody series Mid 2010 to Late 2014. I've a 'Late 2012 i7' quad server.}

General searches can start with an example such as this; modify 'build model' date:

https://www.google.com/#q=display+sizes+supported+by+Late+2012+Mac+Mini

So it looks up to  2560x1600 with Thunderbolt (at various resolutions depending on display.)

There are some other sources of question/answer discussion about displays here:

•Mac Software - Display Link forum:

http://www.displaylink.org/forum/forumdisplay.php?f=30

Hopefully this may be at least a guide of a sort. Some brands of digital TVs have issues.

Good luck & happy computing!

PS: you may be able to adapt some display resolution using

• SwitchResX: http://www.madrau.com/index.html - to suit...

Good luck & happy trails!