  We show, for a field ${\mathbb F}$ of prime characteristic $p>0$, that the
truncated filiform Lie algebra ${\frak m}_0(p)$ admits a family ${\frak
m}_0^\lambda(p)$ of restricted Lie algebra structures parameterized by elements
$\lambda\in {\mathbb F}^p$. We compute the ordinary cohomology groups
$H^q({\frak m}_0^\lambda(p))$ and restricted cohomology groups $H^q_*({\frak
m}_0^\lambda(p))$ for $q=1, 2$, and we give explicit descriptions of bases for
these cohomology spaces. We apply our results to restricted one-dimensional
central Extensions of the algebras ${\frak m}_0^\lambda(p)$.
