= Solution
{c}
Use homogeneous coordinates $x_0,\ldots,x_r$ on <projective space> and let
$$
U_i=X\cap D_+(x_i),\qquad 0\leq i\leq r.
$$
These $r+1$ <open subschemes> cover $X$. Each is a <closed subscheme> of the <affine scheme> $D_+(x_i)$, hence is an <affine scheme>.
Every nonempty finite intersection is affine as well. Indeed, inside $D_+(x_{i_0})$ the intersection with $D_+(x_{i_1}),\ldots,D_+(x_{i_p})$ is the <principal open subscheme> obtained by inverting
$$
\prod_{j=1}^p\frac{x_{i_j}}{x_{i_0}}.
$$
Intersecting it with the <closed subscheme> $X$ gives $U_{i_0}\cap\cdots\cap U_{i_p}$, again an <affine scheme>.
For any <quasi-coherent sheaf> $F$, <vanishing of quasi-coherent cohomology on an affine scheme> makes every such intersection acyclic for $F$. The <acyclic cover theorem> therefore identifies the <sheaf cohomology> of $F$ with the <Čech cohomology> of this cover. Its alternating <Čech cochain complex> has terms
$$
\check C^p(\mathcal U,F)=\prod_{0\leq i_0<\cdots<i_p\leq r}\Gamma(U_{i_0}\cap\cdots\cap U_{i_p},F),
$$
which vanish for $p>r$: one cannot choose $p+1$ distinct indices from $r+1$ indices. Consequently
$$
\boxed{H^k(X,F)=0\quad(k>r),\qquad N=\max\{1,r\}.}
$$
The choice $N=\max\{1,r\}$ meets the strict requirement $N>0$ even when $r=0$. It is independent of $F$, and applies to all <quasi-coherent sheaves>, including those which are not <coherent sheaves>. This is the <cohomological dimension bound from an affine cover>.
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