| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Matrix satisfying given equation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths matrix algebra question requiring routine computation of (A-I)², recognizing it equals a scalar multiple of I, then solving a simple equation by comparing matrices. The steps are mechanical with no conceptual difficulty beyond basic matrix operations, making it slightly easier than average even for FP1. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| 4(a) | \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) | B1 |
| Attempt at \((A - I)^2\) | M1 | with at most one numerical error |
| \((A - I)^2 = \begin{bmatrix} 0 & 4 \\ 3 & 0 \end{bmatrix}\begin{bmatrix} 0 & 4 \\ 3 & 0 \end{bmatrix} = 12I\) | A1 | 3 marks |
| 4(b) | \(A - B = \begin{bmatrix} 0 & 1 \\ 3 - p & 0 \end{bmatrix}\) | B1 |
| \((A - B)^2 = \begin{bmatrix} 3 - p & 0 \\ 0 & 3 - p \end{bmatrix}\) | M1A1 | M1 A0 if 3 entries correct |
| \(... = (A - I)^2\) for \(p = -9\) | A1F | ft wrong value of \(k\) |
| 4 marks | ||
| Total for Q4 | 7 marks |
4(a) | $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ | B1 | stated or used at any stage |
| Attempt at $(A - I)^2$ | M1 | with at most one numerical error |
| $(A - I)^2 = \begin{bmatrix} 0 & 4 \\ 3 & 0 \end{bmatrix}\begin{bmatrix} 0 & 4 \\ 3 & 0 \end{bmatrix} = 12I$ | A1 | 3 marks |
4(b) | $A - B = \begin{bmatrix} 0 & 1 \\ 3 - p & 0 \end{bmatrix}$ | B1 |
| $(A - B)^2 = \begin{bmatrix} 3 - p & 0 \\ 0 & 3 - p \end{bmatrix}$ | M1A1 | M1 A0 if 3 entries correct |
| $... = (A - I)^2$ for $p = -9$ | A1F | ft wrong value of $k$ |
| | | 4 marks |
| **Total for Q4** | **7 marks** |
---4 It is given that
$$\mathbf { A } = \left[ \begin{array} { l l }
1 & 4 \\
3 & 1
\end{array} \right]$$
and that $\mathbf { I }$ is the $2 \times 2$ identity matrix.
\begin{enumerate}[label=(\alph*)]
\item Show that $( \mathbf { A } - \mathbf { I } ) ^ { 2 } = k \mathbf { I }$ for some integer $k$.
\item Given further that
$$\mathbf { B } = \left[ \begin{array} { l l }
1 & 3 \\
p & 1
\end{array} \right]$$
find the integer $p$ such that
$$( \mathbf { A } - \mathbf { B } ) ^ { 2 } = ( \mathbf { A } - \mathbf { I } ) ^ { 2 }$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q4 [7]}}