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 outputs. They may have lots of different output ports so you can plug in a variety of different cables, but they're usually all hooked up to the single master volume slider. When we needed different signal volume sent to different speakers, we usually connected one of the outputs from the primary mixer into a second mixer or an amp.