  Bianchi type I cosmological model in (n+1)-dimensional gravity with several
forms is considered. When the electric non-composite brane ansatz is adopted,
the Wheeler-DeWitt (WDW) equation for the model, written in a conformally
covariant form, is analyzed. Under certain restrictions, asymptotic solutions
to the WDW equation near the singularity are found, which reduce the problem to
the so-called quantum billiard on the (n-1)-dimensional Lobachevsky space
H^{n-1}. Two examples of quantum billiards are considered: a 2-dimensional
quantum billiard for a 4D model with three 2-forms and a 9D quantum billiard
for an 11D model with 120 4-forms which mimics SM2-brane sector of D=11
supergravity. For certain solutions, vanishing of the wave function at the
singularity is proved.
