Easiest way is to note the players perception skill (and other useful info) on a stat reference block and make the roll yourself.
This means you're making a roll for some reason however, which may get the players meta hackles twitching.
An option to avoid this that doesn't even involve rolling a dice (if you don't even want that to be seen) is pre-roll a load of d20's on a scrap of paper; cross each one off as you use it in turn for these "secret rolls".
Or you roll extra d20 random rolls all the time, for no reason at all. This has the added advantage of making the players more paranoid ;) (this is rather than the players making extra pointless rolls) you can speed up this with coloured dice with one colour per player and roll a bunch of them.
Hopefully however players can simply ignore these perception checks without the meta.
Related: ( How do I use Passive Perception to have some characters notice parts of the environment? )
Never forget "Never ask your players for a skill roll you don't want them to fail."
Perception is for observation, Investigation is for deduction.
Some of this answer will be observations on how Wizards has done it so far and some of this will be logic, and some of it will be mechanics.
First for the headline question. It depends. when to use Investigation, and when to use Perception is not entirely clear yet, and I'm hoping we'll get more guidance in the DMG late this year. The guidance from the rules is that the two skills mostly seem to differ in the methods by which they are found.
Depending on the exercise, either, or both of the skills may be used.
For passive checks, you're almost always looking at Perception being the skill of record. While all skills can be used passively, some skills make more sense than others. Perception is the hallmark passive skill, whereas investigation makes less sense as a passive skill.
If the character is alert to the possibility of hidden objects/traps, but not actively searching, he's using passive perception.
The guidance for this seems to be (though we can't confirm yet), that the DC for actively looking for something is regularly about 5 less than the DC for passively looking for something (or sometimes actively looking always reveals it).
So here's how I would play it. Traps can be noticed with perception passively (usually DC 15). If the PC is looking, ask them how they are looking. If it's observational, then use Perception. If it's deductive, use Investigation. When they are searching for items, again, either skill is appropriate. This is somewhat counter to how WOTC has written adventures so far. They are always written to use perception to notice traps. Passive with a higher DC and active with a lower.
Investigation also has broader uses such as when you are trying to track clues, or put something together. It's also a great "roll for a hint" kind of skill if your PCs get stuck and need some help figuring out what to do next.
Best Answer
Passive skills (especially perception, but anything really), are representative of a sort of "always on" ability. So, for instance, if you're walking in a forest and there is a goblin ambush, you see them if your passive perception beats their stealth score. However, if you're specifically looking for ambushes, you'll roll it actively.
So in the secret doors case, when you walk into a room, if your passive perception is high enough to spot the secret door, you notice it. If you're specifically looking for a secret door, then you roll an active check to try to notice (naturally, if you roll less than 10, you don't learn anything new).
Generally, the only time to roll an active perception check is if you are looking for something that your passive perception did not pick up. Naturally, in the course of play this happens fairly regularly. But the passive skills are designed to reflect a characters natural competence and should be used when they are sufficient, or there is no particular rush.