  Let $g$ and $c$ be any integers satisfying $g\geq3$ and $0\leq c\leq
\lfloor\frac{g-1}{2}\rfloor$. It is known that there exists a polarized K3
surface $(X,H)$ such that $X$ is a K3 surface of Picard number 2, and $H$ is a
very ample line bundle on $X$ of sectional genus $g$ and Clifford index $c$, by
Johnsen and Knutsen([J-K] and [Kn]). In this paper, we give a necessary and
sufficient condition for a Lazarsfeld-Mukai bundles of rank 2 associated with a
smooth curve $C$ belonging to the linear system $|H|$ and a base point free
pencil on $C$ not to be $H$-slope stable.
