I think Apple has a problem here with recovery mode, at least on newer iMacs. I have a new 5K iMac with wireless, rechargeable keyboard and mouse.
1. You can't use startup keys with wireless keyboards because they don't connect to the system early enough in the boot process.
2. But you can if you wait until the startup chime and then press the keys.
3. But you can't because Apple removed the startup chime and the recommended workaround (sudo nvram BootAudio=%01) doesn't work, at least on my brand new iMac.
4. OK, you really can because the rechargeable keyboard does work early enough in the boot process if you plug in the charging cable.
5. But there is no Recovery Partition on my brand new iMac (doesn't show up in the boot manager; doesn't show up in disk utility). Why? I don't know, but apparently Recovery Mode boots by going over the internet to Apple's servers. So now I am in Recovery Mode.
6. But now my mouse doesn't work in Recovery Mode. I can use the arrow keys to move up and down among the 4 options in Recovery Mode, but I can't select any of them with the Enter or Return keys. Recovery Mode is useless.
7. Summary: Because of a series of uncoordinated changes made by Apple, I can't use Recovery Mode on a brand new 5K iMac.
Suggestions for further work-arounds appreciated (other than the obvious option of buying a USB mouse and keyboard).
Apple, please think this through and fix it.