Fe- and Cu-doped ZnO of nominal compositions Zn0.95Fe0.05O and Zn0.94Fe0.05Cu0.01O were synthesized by a wet chemical route. X-ray diffraction analysis of the samples annealed at 575 K showed that they are single phase without any secondary phases. DC magnetization measurements of Cu co-doped samples (Zn0.94Fe0.05Cu0.01O) as a function of field at room temperature showed ferromagnetic signature while the samples without Cu co-doping (Zn0.95Fe0.05O) are paramagnetic in nature. On increasing the temperature of annealing from 575 K to 1,075 K an impurity phase emerges in both the samples, which has been identified as a variant of ZnFe2O4. Both the samples heated at and above 1,075 K are found to be paramagnetic at room temperature. These observations, the absence of room temperature ferromagnetism in Zn0.95Fe0.05O and the disappearance of ferromagnetism in Zn0.94Fe0.05Cu0.01O on raising the temperature of annealing clearly rules out the likelihood of room temperature ferromagnetism arising from the impurity phases like gamma-Fe2O3 and/or ZnFe2O4 that might have been formed during the synthesis. Our results strongly suggest that room temperature ferromagnetism in Zn0.94Fe0.05Cu0.01O can be attributed to the formation of a secondary phase of Cu-doped ZnFe2O4.