We present the results of a programme of scanning and mapping observations of astronomical masers and Jupiter designed to characterise the performance of the Mopra Radio Telescope at frequencies between 16-50 GHz using the 12-mm and 7-mm receivers. We use these observations to determine the telescope beam size, beam shape and overall telescope beam efficiency as a function of frequency. We find that the beam size is well fit by $lambda$/$D$ over the frequency range with a correlation coefficient of ~90%. We determine the telescope main beam efficiencies are between ~48-64% for the 12-mm receiver and reasonably flat at ~50% for the 7-mm receiver. Beam maps of strong H$_2$O (22 GHz) and SiO masers (43 GHz) provide a means to examine the radial beam pattern of the telescope. At both frequencies the radial beam pattern reveals the presence of three components, a central `core, which is well fit by a Gaussian and constitutes the telescopes main beam, and inner and outer error beams. At both frequencies the inner and outer error beams extend out to approximately 2 and 3.4 times the full-width half maximum of the main beam respectively. Sources with angular sizes a factor of two or more larger than the telescope main beam will couple to the main and error beams, and therefore the power contributed by the error beams needs to be considered. From measurements of the radial beam power pattern we estimate the amount of power contained in the inner and outer error beams is of order one-fifth at 22 GHz rising slightly to one-third at 43 GHz.
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