| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 8 |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Challenging +1.2 This is a Further Maths question on systems of linear equations and plane configurations requiring matrix methods (determinant calculation, rank analysis). While it involves multiple steps and case analysis, the techniques are standard FM content: finding when det=0 for non-unique solutions, then checking consistency via row reduction. The conceptual demand is moderate—students must understand the geometric meaning of different rank configurations—but the execution follows well-practiced procedures without requiring novel insight. |
| Spec | 4.03s Consistent/inconsistent: systems of equations4.03t Plane intersection: geometric interpretation |
Three planes have equations
\begin{align}
4x - 5y + z &= 8\\
3x + 2y - kz &= 6\\
(k - 2)x + ky - 8z &= 6
\end{align}
where $k$ is a real constant.
The planes do not meet at a unique point.
\begin{enumerate}[label=(\alph*)]
\item Find the possible values of $k$. [3 marks]
\item For each value of $k$ found in part (a), identify the configuration of the given planes.
Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system. [5 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q10 [8]}}