1. Given the sets
$$
A=\{x \mid 1 \leqslant x \leqslant 2\}, B=\left\{x \mid x^{2}-a x+4 \geqslant 0\right\} \text {. }
$$

If $A \subseteq B$, then the range of real number $a$ is
$$
-1, a \leqslant 4 \text {. }
$$

It is easy to see that the inequality $x^{2}-a x+4 \geqslant 0$ holds for all $x$ in the interval $[1,2]$.
Therefore, $a \leqslant 4$.