Based on $J/psi$ events collected with the BESIII detector, with corresponding Monte Carlo samples, the tracking efficiency and its systematic uncertainty are studied using a control sample of $J/psi to p overline p pi^+ pi^-$. Validation methods and
different factors influencing the tracking efficiency are presented in detail. The tracking efficiency and its systematic uncertainty for protons and pions with the transverse momentum and polar angle dependence are also discussed.
A search for the rare decays of $J/psi to D_s^-rho^+ + c.c.$ and $J/psi to overline{D}{}^0overline{K}{}^{ast0} + c.c.$ is performed with a data sample of 225.3 million $J/psi$ events collected with the BESIII detector. No evident signal is observed.
Upper limits on the branching fractions are determined to be $mathcal{B}(J/psi to D_s^-rho^+ + c.c.) < 1.3 times 10^{-5}$ and $mathcal{B}(J/psi to overline{D}{}^0overline{K}{}^{ast0} + c.c.) < 2.5 times 10^{-6}$ at the $90%$ confidence level.
