Throughout the paper \(K\) denotes an algebraically closed field and \(A\) is a \(K\)-algebra of finite representation type given by quivers. There are a representation-finite \(K\)-algebra \(A\) of global dimension dimension two, and indecomposable not faithful \(A\)-modules \(T\) and \(U\) with the following properties: (i) \(T\) (resp.\ \(U\)) is a summand of a 2-tilting (resp.\ 2-cotilting) module; (ii) \(\text{Ker\,Hom}_R(T,-)\cap T^{\perp_\infty}=0\) (resp.\ \(\text{Ker\,Hom}_R(-,U)\cap U^{\perp_\infty}= 0\)), but no proper submodule of \(T\) (resp.\ \(U\)) has this property (Theorem 4). There are a \(K\)-algebra \(A\) and a 2-cotilting \(A\)-module \(U\) with the following properties: (i) \(E(U)/U\) is semisimple; (ii) every simple \(A\)-module of injective dimension at most one is isomorphic to a summand of \(E(U)/U\); (iii) \(\text{Ext}_A^1(E(U)/U,E(U)/U)\neq 0\) (Theorem 5).