  A combined analysis of the reactions $\pi^-p\rightarrow K^0\Lambda$ and $\eta
n$ is carried out with a chiral quark model. The data in the center-of-mass
(c.m.) energy range from threshold up to $W\simeq 1.8$ GeV are reasonably
described. For $\pi^-p\rightarrow K^0\Lambda$, it is found that $N(1535)S_{11}$
and $N(1650)S_{11}$ paly crucial roles near threshold. The $N(1650)S_{11}$
resonance contributes to the reaction through configuration mixing with
$N(1535)S_{11}$. The constructive interference between $N(1535)S_{11}$ and
$N(1650)S_{11}$ is responsible for the peak structure around threshold in the
total cross section. The $n$-pole, $u$- and $t$-channel backgrounds provide
significant contributions to the reaction as well. While, for the
$\pi^-p\rightarrow \eta n$ process, the "first peak" in the total cross section
is dominant by $N(1535)S_{11}$, which has a sizeable destructive interference
with $N(1650)S_{11}$. Around $P_\pi \simeq $ 1.0 GeV/c ($W\simeq 1.7$ GeV),
there seems to be a small bump structure in the total cross section, which
might be explained by the interference between the $u$-channel and
$N(1650)S_{11}$. The $N(1520)D_{13}$ resonance notably affects the angle
distributions of the cross sections, although less effects are seen in the
total cross section. The role of $P$-wave state $N(1720)P_{13}$ should be
further confirmed by future experiments. If $N(1720)P_{13}$ has a narrow width
of $\Gamma\simeq 120$ MeV as found in our previous work by a study of the
$\pi^0$ photoproduction processes, obvious evidence should be seen in the
$\pi^-p\rightarrow K^0\Lambda$ and $\eta n$ processes as well.
