Which is more efficient really depends on your setup and needs. The passive with an amp gives you a centralized control station - you can turn everything on/off from one spot. They're usually better for a permanent installation.
But if the cable lengths are extremely long or you need to be able to cannibalize one of the speakers for use in a different area (say a guest lecturer in a different room needs a speaker of his own), then at least one active speaker may be warranted.
A mixer is usually used to combine multiple inputs together into one output. Not to modulate different output volumes to multiple outputs. I don't think I've seen a mixer which supports (i.e. has separate volume controls for) more than two independent...