Question 23

The determinant of the coefficient matrix $\begin{bmatrix} 1 & k \\ k & 1 \end{bmatrix}$ is $\Delta = 1 - k^2$.

• If $k \ne \pm 1$: $\Delta \ne 0$, so the system has a unique solution for infinitely many real values of $k$ ((C) is false).
• If $k = 1$: the equations become $x + y = 1$ and $x + y = -1$, which are parallel and inconsistent (no solution).
• If $k = -1$: the equations become $x - y = 1$ and $-x + y = -1$, which are identical (infinitely many solutions).

Thus, there is exactly one value of $k$ ($k = 1$) with no solution (A), and exactly one value of $k$ ($k = -1$) with an infinite number of solutions (D).