Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$

Given a hypersurface $i\text{\hspace{0.17em}⁣}:X\hookrightarrow {\tilde{P}}^n$ in a weighted projective space, we compute the intersection form on the second cohomology $H^2(X,\mathbb{Z}{)}^{\otimes n-1}\to \mathbb{Z}$ for the purpose of identifying Fano manifolds obtained from smoothing singular Fanos. In the process, we describe the integer cohomology groups $H^k(X,\mathbb{Z})$ for $k<n$ and give an explicit formula for the pullback map $i^{\ast }$.