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We develop a device model for a resonant detector of electromagnetic radiation with a frequency in the terahertz (THz) range modulated by megahertz (MHz) or gigahertz (GHz) signals based on a micromachined high-electron mobility transistor (HEMT) with a metallized nanostring (NS) or metallic carbon nanotube (CNT) as mechanically the floating gate and analyze the detector operation. The device model describes both the NS/CNT mechanical motion and plasma effects in the HEMT two-dimensional electron channel. Using this model, we calculate the output gate alternating current and the detector responsivity as functions of the carrier (in the THz range) and modulation frequencies, which are in the THz and MHz (or GHz range), respectively. It is shown that the THz detector responsivity exhibits sharp and high maxima under the conditions of both mechanical and plasma resonances.