| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring identification of a shear transformation and then solving a matrix equation MQ = P by finding Q = M·P^(-1). While it involves matrix inversion and geometric interpretation, these are standard FP1 techniques with no novel problem-solving required. Slightly above average difficulty due to being Further Maths content, but routine for that level. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Shear | B1 | Must be a shear, otherwise 0/2 |
| \(x\)-axis invariant and e.g. \((0,1)\to(3,1)\) | B1 | For invariant only allow parallel to or along \(x\)-axis, in \(x\) direction; for image allow \(0.322°\), \(18.4°\), \(\tan^{-1}(1/3)\) or complement, ignore scale factor if all OK otherwise. Column vectors for coordinates OK |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| *Either:* \(\begin{pmatrix}1&3\\0&1\end{pmatrix}\begin{pmatrix}a&b\\c&d\end{pmatrix}=\begin{pmatrix}-3&-1\\-1&0\end{pmatrix}\) or \(PQ=M\) | B1 | Obtain correct matrix equation |
| \(\begin{pmatrix}0&-1\\-1&0\end{pmatrix}\) | M1 | Sensible attempt at multiplication of a pair of \(2\times2\) matrices |
| A2 | Obtain correct answer, A1 for 3 elements correct | |
| Reflection in \(y=-x\) | B2 | Must be describing \(Q\). If they say \(M=QP\Rightarrow P^{-1}M=Q\), they can only get (possibly) M1 B2 if they obtain "correct" matrix for \(Q\) |
| [6] | ||
| *Or:* \(\begin{pmatrix}0&-1\\-1&0\end{pmatrix}\) | B2 | Diagram showing at least unit square and image under M, coordinates shown |
| B2 | State reflection in \(y=-x\) | |
| B2 | Correct matrix |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Shear | B1 | Must be a shear, otherwise 0/2 |
| $x$-axis invariant and e.g. $(0,1)\to(3,1)$ | B1 | For invariant only allow parallel to or along $x$-axis, in $x$ direction; for image allow $0.322°$, $18.4°$, $\tan^{-1}(1/3)$ or complement, ignore scale factor if all OK otherwise. Column vectors for coordinates OK |
| **[2]** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| *Either:* $\begin{pmatrix}1&3\\0&1\end{pmatrix}\begin{pmatrix}a&b\\c&d\end{pmatrix}=\begin{pmatrix}-3&-1\\-1&0\end{pmatrix}$ or $PQ=M$ | B1 | Obtain correct matrix equation |
| $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$ | M1 | Sensible attempt at multiplication of a pair of $2\times2$ matrices |
| | A2 | Obtain correct answer, A1 for 3 elements correct |
| Reflection in $y=-x$ | B2 | Must be describing $Q$. If they say $M=QP\Rightarrow P^{-1}M=Q$, they can only get (possibly) M1 B2 if they obtain "correct" matrix for $Q$ |
| **[6]** | | |
| *Or:* $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$ | B2 | Diagram showing at least unit square and image under **M**, coordinates shown |
| | B2 | State reflection in $y=-x$ |
| | B2 | Correct matrix |
---7 The matrix $\left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)$ represents a transformation P .\\
(i) Describe fully the transformation P .
The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1 \\ - 1 & 0 \end{array} \right)$.\\
(ii) Given that $\mathbf { M }$ represents transformation Q followed by transformation P , find the matrix that represents the transformation Q and describe fully the transformation Q .\\
\hfill \mbox{\textit{OCR FP1 2016 Q7 [8]}}