Using frequency to turn a circuit on
This is a fairly simple task for a DSP (Digital Signal Processor) such as the Microchip dsPIC family.  The DSPIC33FJ32GP102 for example, with 32KB of flash and 2KB RAM, costs $3.19 in single quantities in a simple to use DIP package.

The most efficient way to detect tones like this is to use the Goertzel algorithm.

Microchip provides a library for the dsPIC processors for detecting DTMF frequencies which I have successfully used in a project.  It uses the Goertzel algorithm to detect two simultaneous tones, selected from 16 frequencies spaced fairy close together (a few hundred Hz).  So your situation is even simpler since you only need to look for one tone (or you can use it as is for DTMF).  The full source code for the library is provided so you can modify it per your needs, including Goertzel.s.

As set up, the library uses a tolerance of ±1.5% to validate the frequencies (±1.5 Hz for the 100 Hz tone, ±15 Hz for 1 kHz,  and ±150 Hz for 10 kHz); and 40 ms minimum to detected a tone.  These parameters should be fairly easy to modify.

The 1.5% percent is would be necessary if you were to stick with DTMF tones, since they are fairly close together (hundreds of Hz), but if you were to use your propsed 100 Hz, 1kHz and 10kHz then ±10% should be doable.

Here's a project someone did to detect ambulance sirens using a dsPIC.  (It used FFT instead of Goertzel but still an interesting read.)

Note that if you don't already have a circuit for generating the tones, you can use a second dsPIC for that.  Microchip has a DTMF send library in addition to the DTMF receive which you could modify (either to use as is for DTMF, or modify for a single tone).