一条直线的方程为 $y=3x-2k$，一条曲线的方程为 $y=x^2-kx+2$，其中 $k$ 为常数。

证明：对 $k$ 的所有取值，直线与曲线都有交点。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

联立方程得到关于 $x$ 的二次式。把判别式配成完全平方，就能检查所有 $k$。

The intersection equation is

Its discriminant is $(k+3)^2-4(2+2k)=(k-1)^2\ge0$ for every real $k$. There is at least one real root for every $k$, and thus the graphs always meet.

也可以从两式中含 $k$ 的项找共同交点。选一个 $x$，使两个 $k$ 项完全相同。

At $x=2$, the line gives $y=6-2k$, and the curve gives $y=2^2-2k+2=6-2k$. The point $(2,6-2k)$ is therefore on both graphs for every real $k$.