1. $k \in \mathbf{R}$ the real roots $x$ of the equation $x^{4}-2 k x^{2}+k^{2}+2 k-3=$ 0 should satisfy ( ).
( A ) $-1 \leqslant x \leqslant 1$
( B ) $0 \leqslant x \leqslant \sqrt{2}$
( C) $-\sqrt{2} \leqslant x \leqslant \sqrt{2}$
(D) $-\sqrt{2} \leqslant x \leqslant 0$
-、1.( C).
Transform the original equation into an equation about $k$
$$
\begin{array}{l}
k^{2}+2\left(1-x^{2}\right) k+x^{4}-3=0, \\
\because k \in \mathbf{R}, \therefore \Delta \geqslant 0, \text { i.e., } \\
4\left(1-x^{2}\right)^{2}-4\left(x^{4}-3\right) \geqslant 0 .
\end{array}
$$

Solving this yields $-\sqrt{2} \leqslant x \leqslant \sqrt{2}$.