  We consider the Cauchy problem of systems of quasilinear wave equations in
2-dimensional space. We assume that the propagation speeds are distinct and
that the nonlinearities contain quadratic and cubic terms of the first and
second order derivatives of the solution. We know that if the all quadratic and
cubic terms of nonlinearities satisfy $Strong$ $Null$-$condition$, then there
exists a global solution for sufficiently small initial data. In this paper, we
study about the lifespan of the smooth solution, when the cubic terms in the
quasi-linear nonlinearities do not satisfy the Strong null-condition. In the
proof of our claim, we use the $ghost$ $weight$ energy method and the
$L^{\infty}$-$L^{\infty}$ estimates of the solution, which is slightly
improved.
