1. Given $a \geqslant 2$, and
$$
\begin{array}{l}
A=\{x \mid-2 \leqslant x \leqslant a\}, \\
B=\{y \mid y=2 x+3, x \in A\}, \\
C=\left\{t \mid t=x^{2}, x \in A\right\} .
\end{array}
$$

If $C \subseteq B$, then the range of values for $a$ is
$$
-1 .\left[\frac{1}{2}, 3\right] \text {. }
$$

Notice that, $B=[-1,2a+3]$, to make $C \subseteq B$, it is only necessary that the maximum element of $C$ is in $B$. Therefore,
$$
\left\{\begin{array}{l}
(-2)^{2} \leqslant 2a+3, \\
a^{2} \leqslant 2a+3
\end{array} \Rightarrow \frac{1}{2} \leqslant a \leqslant 3 .\right.
$$