Operating system bootability sucks bigtime! Whether it's due to the Windows programmers or Linux programmers or the BIOS firmware programmers of various companies. I don't understand why these people can't come to an agreement of how to detect bootloaders and boot in a simple way.
If you have Windows installed on one disk and Linux installed on another disk, if you reinstall Windows, you won't be able to boot into Linux. In-fact, even when you press F11, GRUB won't be shown in the boot options. There are two things you can do in such a situation:
1. Point BIOS to the Linux bootloader
Simply restart the computer, go to BIOS, and depending on where in the boot options you find the option to specify the Linux bootloader, simply select it and set it as the first boot option instead of the Windows bootloader.
2. Do a boot repair
If the BIOS does not show you any option for the Linux bootloader, boot from a bootable Linux pen drive and run boot repair. In the start menu, simply start typing "boot", and you'll see the boot repair application. At least in Linux Mint, it's pre-installed.
You might see an error message, but you could ignore that and go to BIOS like I mentioned in the first point, and set Linux GRUB as the first boot option. Everything will work fine then.
A slightly more involved method of doing it I've mentioned here, and there are some more options here.
