  In this paper, a modified Eddington-inspired-Born-Infeld (EiBI) theory with a
pure trace term $g_{\mu\nu}R$ being added to the determinantal action is
analysed from a cosmological point of view. It corresponds to the most general
action constructed from a rank two tensor that contains up to first order terms
in curvature. This term can equally be seen as a conformal factor multiplying
the metric $g_{\mu\nu}$. This very interesting type of amendment has not been
considered within the Palatini formalism despite the large amount of works on
the Born-Infeld-inspired theory of gravity. This model can provide smooth
bouncing solutions which were not allowed in the EiBI model for the same EiBI
coupling. Most interestingly, for a radiation filled universe there are some
regions of the parameter space that can naturally lead to a de Sitter
inflationary stage without the need of any exotic matter field. Finally, in
this model we discover a new type of cosmic "quasi-sudden" singularity, where
the cosmic time derivative of the Hubble rate becomes very large but finite at
a finite cosmic time.
