Standard +0.3 This is a straightforward application of the determinant condition for unique solutions (det ≠ 0). Students need to form the coefficient matrix, calculate a 3×3 determinant (routine FP1 skill), set it equal to zero, and solve for k. It's slightly easier than average because it's a direct method application with no conceptual subtlety—just mechanical execution of a standard technique.
5 By using the determinant of an appropriate matrix, or otherwise, find the value of \(k\) for which the simultaneous equations
$$\begin{aligned}
2 x - y + z & = 7 \\
3 y + z & = 4 \\
x + k y + k z & = 5
\end{aligned}$$
do not have a unique solution for \(x , y\) and \(z\).
5 By using the determinant of an appropriate matrix, or otherwise, find the value of $k$ for which the simultaneous equations
$$\begin{aligned}
2 x - y + z & = 7 \\
3 y + z & = 4 \\
x + k y + k z & = 5
\end{aligned}$$
do not have a unique solution for $x , y$ and $z$.
\hfill \mbox{\textit{OCR FP1 2009 Q5 [5]}}