= Solution
\b[Interpretation of the removed locus.] Over $\mathbb Z$, the notation $[1:0:0]$ specifies a section $\operatorname{Spec}\mathbb Z\to\mathbb P^2_{\mathbb Z}$, with image $V_+(x_1,x_2)\cong\operatorname{Spec}\mathbb Z$. We remove this entire <closed subscheme>. This is the interpretation forced by the asserted <open cover> $X=U_1\cup U_2$. Removing one closed point in a particular fiber would not give that <open cover>. The native PDF uses the same notation as the transcription.
Put $a=x_0/x_1$, $b=x_2/x_1$ on $U_1$. On $U_2$ put $c=x_0/x_2$, $d=x_1/x_2$. On their intersection $c=a/b$ and $d=b^{-1}$, giving <coordinate rings>
$$
\Gamma(U_1,\mathcal O_X)=\mathbb Z[a,b],\qquad
\Gamma(U_2,\mathcal O_X)=\mathbb Z[a/b,b^{-1}],\qquad
\Gamma(U_1\cap U_2,\mathcal O_X)=L:=\mathbb Z[a,b,b^{-1}].
$$
The <Čech cochain complex> for the <structure sheaf>, using increasing distinct indices, is
$$
0\longrightarrow\mathbb Z[a,b]\oplus\mathbb Z[a/b,b^{-1}]
\xrightarrow{\delta}L\longrightarrow0,\qquad
\delta(f,g)=g-f.
$$
Thus $\check H^0$ is the intersection of the two subrings of $L$, while $\check H^1$ is the <quotient module> of $L$ by their additive sum.
The <Laurent polynomial ring> $L$ is a free <abelian group> on the <monomials> $a^ib^j$ with $i\geq0$, $j\in\mathbb Z$. The first subring has the <basis> $j\geq0$. The second has the <basis>
$$
c^id^k=a^ib^{-i-k},\qquad i,k\geq0,
$$
so its allowed exponents satisfy $j\leq-i$. A <monomial> belongs to both precisely when $i=j=0$. The <monomials> belonging to neither satisfy $i\geq2$ and $-i<j<0$. Since the subgroups are spanned by subsets of the same <basis>, their sum introduces no other relations. Therefore
$$
\boxed{\begin{aligned}
\check H^0(\{U_1,U_2\},\mathcal O_X)&=\mathbb Z,\\
\check H^1(\{U_1,U_2\},\mathcal O_X)&=\bigoplus_{i\geq2}\ \bigoplus_{k=1}^{i-1}\mathbb Z\,[a^ib^{-k}]\\
&=\bigoplus_{m,n\geq1}\mathbb Z\!\left[\frac{x_0^{m+n}}{x_1^mx_2^n}\right],\\
\check H^q(\{U_1,U_2\},\mathcal O_X)&=0\qquad(q\geq2).
\end{aligned}}
$$
Here the square brackets denote <cohomology classes>, and the <direct sums> are as <abelian groups>. In particular, $[a^2b^{-1}]=[x_0^2/(x_1x_2)]$ is a nonzero <cohomology class>. This <scheme> is a <separated scheme>, being an <open subscheme> of <projective space> over $\mathbb Z$, and is a <quasi-compact scheme> by (d). Both charts and their intersection are <affine schemes>, so this <Čech cohomology> equals <sheaf cohomology>. The nonzero first <sheaf cohomology> of the <quasi-coherent> <structure sheaf> rules out an <affine scheme>:
$$
\boxed{X\text{ is not affine}.}
$$
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