(a) 求使方程组

有唯一解的 $k$ 的取值集合，并从几何角度解释这种情况。

(b) 已知 $k=\frac{7}{2}$，证明 1(a) 中的方程组无解，并从几何角度解释这种情况。

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。

先用第三行减去第二行的两倍。新的一行只有一个非零项，沿这一行展开可直接找出使行列式为零的参数。

Subtracting twice the second row from the third, the coefficient determinant becomes

Thus the system has a unique solution for $\boxed{k\in\mathbb R\setminus\{\frac72\}}$. In this case the three planes intersect at a single point.

后两式可以确定 $x$ 和 $y+2z$，把它们代回第一式检查。再用各个法向量判断两两相交的直线方向。

Subtracting twice the second equation from the third gives $-8x=-1$, so $x=\frac18$ and the second equation gives $y+2z=1$. The left-hand side of the first equation is then

The system is inconsistent. The normals are pairwise non-parallel, so each pair of planes intersects in a line. All three normals are perpendicular to $(0,-2,1)$, so these three distinct intersection lines are parallel, forming the edges of a triangular prism.