The coolest and dimmest space objects emit radiation of longer wavelengths than those we can see. a detector for which type of electromagnetic radiation is most suitable to study cool dim stars?

Infrared detectors.

Explanation:

1) Remember the relation between the wavelength and the frequency: they are inversely related. The longer the wavelength the higher the frequency.

2) Infrared radiaton is the electromagnetic radiation with longer wavelength than the visible light and is next to it in the spectrum.

3) Infrared detectors use the properties of some materials to detect the infrared light (waves) either by their effect in the temperature or by capturing the photons emitted by the object analyzed.

4) The effects that are most used to analyze the radiaton are 1) that the change of temperature induces changes in resistance (thermal detectors) and 2) the photons excite the atoms (photodetectors).

Maybe you should get some from your teacher and learn more so you can get your grades up and maybe study a bit more