Both GQ Mus and V1974 Cyg were observed to turnoff in X-rays by ROSAT. The turnoff of V1974 Cyg was also observed with EUVE. GQ Mus was observed near the beginning of its outburst with EXOSAT and then 7 years later by ROSAT in the all-sky survey. Later ROSAT PSPC observations showed that its X-ray intensity was slowly declining with time and it was not detected in the last pointing that occurred in August 1993. We observed GQ Mus with IUE over the entire active phase of its outburst and found a change in the slope of the UV continuum around the time that the X-rays turned off. V1974 Cyg was observed by ROSAT throughout its entire active phase in X-rays which lasted about 18 months. V1974 Cyg was detected in the EUVE all-sky survey, but not in pointed observations that occurred in August 1993 (and June and November 1994). We use the measured times of the active phases to determine important properties of these two novae. For example, for V1974 Cyg we predict that more than 10−5 M ⊙ of helium rich material was left on the white dwarf when it returned to quiescence. For GQ Mus, ~ 10−4 M ⊙ was left on the white dwarf. These results imply that a significant amount of the helium seen in nova ejecta was produced in outbursts prior to the one that was just observed. They also imply that the mechanism which mixes core material into the ejecta must be efficient.