Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map\_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. Suppose that for any fibrant object $R$, the induced map 
$map\_{M}(B,R)\rightarrow map\_{M}(A,R)$ is a weak homotopy equivalence of simplicial sets. Do we have that $f$ is a weak equivalence ?



detecting weak equivalences in a simplicial model category
Yes. This is Proposition 9.7.1 in Hirschhorn's book. You don't even need $A$ to be cofibrant.


