Passive Perception is always on
At all times all creatures, including PCs, are using their passive Perception to notice things. That can be walls, doors, treasure chests, or invisible beholders.
Passive Perception serves as the minimum Perception that a creature has at any time.
You can use the Search action to make an active Perception check
You can make any kind of check using your normal action, but you can also use the Search action to try and roll better than your passive Perception. Even if you roll a 1, your passive Perception is still on.
However, this won't really help with seeing an invisible beholder, nor locating an anti-magic field.
Invisible creatures cannot be seen, but they can be located
An invisible creature is invisible, you can't see them. You can't just roll high on Perception and see the invisible. But they are probably still making noise, interacting with the environment, and doing things that let you know their location. Knowing that, you can either attack with disadvantage, or dispel the invisibility.
If the beholder hides, then they can act stealthily to become even less detectable. However, a canny player can use the Search action to attempt to locate the beholder. But that still won't make the beholder visible.
Locating the antimagic cone requires magic detection
Perception is a mundane sense; you can't use it to see magic. You need to use spells or abilities that let you detect magic.
However, your DM may be able to help you out.
Your DM may allow you to use passive Arcana or make an Arcana check to detect the magic if you are lucky. Or they could rule that since the cone comes from the eye, a PC can figure out roughly where the cone is. These seem like pretty reasonable rulings to me.