= Solution
Take a <transitive model> $M\models\mathsf{ZFC}+2^{\aleph_0}=\aleph_2$. In $M$, choose a <bijection> $e:\omega_2\to\mathcal P(\omega)$ and encode its graph by a set $A\subseteq\omega_2$, using a fixed bijection between $\omega_2\times\omega$ and $\omega_2$. The <relative constructible universe> $L(A)$ can decode $e$, and therefore contains every <real number> of $M$; being an inner model of $M$, it has no additional reals.
Models with the same reals have the same <first uncountable ordinal>, because their reals code exactly the same countable well-orders. If $L(A)$ satisfied the <Continuum hypothesis>, its bijection between $\omega_1$ and the reals would also belong to $M$, contradicting $M\models2^{\aleph_0}=\aleph_2$. This is the construction in <relative constructible universe can violate the continuum hypothesis>, and it gives
$$
\boxed{A\subseteq\omega_2\quad\text{and}\quad L(A)\models\neg\mathsf{CH}.}
$$
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