| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Area transformation under matrices |
| Difficulty | Moderate -0.3 This is a straightforward application of the area transformation formula (area scales by |det(M)|). Part (i) is routine matrix inversion using the 2×2 formula, and part (ii) requires calculating a determinant and multiplying by the original area from the diagram. Both are standard textbook exercises requiring recall of formulas rather than problem-solving insight. |
| Spec | 4.03i Determinant: area scale factor and orientation4.03n Inverse 2x2 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| \(\det(\mathbf{M}) = 8 + 2p\) | M1 | Finding determinant |
| \(\mathbf{M}^{-1} = \frac{1}{8+2p}\begin{pmatrix}1 & 2 \\ -p & 8\end{pmatrix}\) | A1 | Correct inverse |
| Answer | Marks | Guidance |
|---|---|---|
| Area of triangle from graph \(= \frac{1}{2}(3)(3) = 4.5\) (reading vertices e.g. \((1,0),(4,-1),(4,5)\)) | M1 | |
| \( | \det\mathbf{M} | = |
| Area of image \(= 14 \times 4.5 = 63\) | A1 |
## Question 1:
**(i)**
$\det(\mathbf{M}) = 8 + 2p$ | M1 | Finding determinant
$\mathbf{M}^{-1} = \frac{1}{8+2p}\begin{pmatrix}1 & 2 \\ -p & 8\end{pmatrix}$ | A1 | Correct inverse
**(ii)**
Area of triangle from graph $= \frac{1}{2}(3)(3) = 4.5$ (reading vertices e.g. $(1,0),(4,-1),(4,5)$) | M1 |
$|\det\mathbf{M}| = |8(1)-(-2)(3)| = 14$ | M1 |
Area of image $= 14 \times 4.5 = 63$ | A1 |
---1 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { c c } 8 & - 2 \\ p & 1 \end{array} \right)$, where $p \neq - 4$.\\
(i) Find the inverse of $\mathbf { M }$ in terms of $p$.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{578345cb-e7a1-41fd-abf8-a977912965e8-2_1086_885_584_587}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
The triangle shown in Fig. 1 undergoes the transformation represented by the matrix $\left( \begin{array} { c c } 8 & - 2 \\ 3 & 1 \end{array} \right)$. Find the area of the image of the triangle following this transformation.
\hfill \mbox{\textit{OCR MEI FP1 2016 Q1 [4]}}