| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Range space basis and dimension |
| Difficulty | Challenging +1.2 This is a systematic linear algebra question requiring row reduction to find rank (dimension), basis vectors for range and null spaces, and solving a linear system. While it involves multiple parts and Further Maths content, each step follows standard algorithms taught in FP1 with no novel insights required. The techniques are mechanical: row-reduce to find rank, read off basis vectors, find null space via solving homogeneous system, and use the cartesian equation to find k. More routine than typical proof-based questions but harder than single-technique Core questions due to length and Further Maths context. |
| Spec | 4.03l Singular/non-singular matrices4.03s Consistent/inconsistent: systems of equations4.03t Plane intersection: geometric interpretation |
The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }$ is represented by the matrix $\mathbf { M }$, where
$$\mathbf { M } = \left( \begin{array} { r r r r }
2 & 1 & - 1 & 4 \\
3 & 4 & 6 & 1 \\
- 1 & 2 & 8 & - 7
\end{array} \right)$$
The range space of T is $R$. In any order,\\
(i) show that the dimension of $R$ is 2 ,\\
(ii) find a basis for $R$ and obtain a cartesian equation for $R$,\\
(iii) find a basis for the null space of T .
The vector $\left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)$ belongs to $R$. Find the value of $k$ and, with this value of $k$, find the general solution of
$$\mathbf { M x } = \left( \begin{array} { l }
8 \\
7 \\
k
\end{array} \right)$$
\hfill \mbox{\textit{CAIE FP1 2012 Q11 OR}}