| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 11 |
| Topic | 3x3 Matrices |
| Type | Geometric interpretation of systems |
| Difficulty | Standard +0.8 This is a Further Maths question on matrices and systems of equations requiring multiple techniques: finding when a matrix is singular (determinant ≠ 0), solving a system with parameter p (likely using inverse or row reduction), and analyzing consistency conditions for a singular system. Part (c)(ii) requires geometric interpretation of a degenerate case. While each component is standard FM material, the multi-part structure with parameters and the geometric interpretation elevate it above routine exercises to moderately challenging. |
| Spec | 4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03r Solve simultaneous equations: using inverse matrix4.03s Consistent/inconsistent: systems of equations |
$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of $k$ for which the matrix $\mathbf{M}$ has an inverse.
[2]
\item Find, in terms of $p$, the coordinates of the point where the following planes intersect
\begin{align}
2x - y + z &= p \\
3x - 6y + 4z &= 1 \\
3x + 2y - z &= 0
\end{align}
[5]
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $q$ for which the set of simultaneous equations
\begin{align}
2x - y + z &= 1 \\
3x - 5y + 4z &= q \\
3x + 2y - z &= 0
\end{align}
can be solved.
\item For this value of $q$, interpret the solution of the set of simultaneous equations geometrically.
[4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q12 [11]}}