| Exam Board | OCR |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Find general invariant lines |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on linear transformations requiring standard techniques: applying a matrix to vertices, finding invariant points by solving (M-I)x=0, and interpreting the determinant. While it's Further Maths content, the methods are routine and well-practiced, making it slightly easier than average overall but appropriately challenging for AS Further Pure. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(A'\) is \(\left(-\frac{3}{5},\frac{4}{5}\right)\), \(B'\) is \(\left(\frac{1}{5},\frac{7}{5}\right)\) and \(C'\) is \(\left(\frac{4}{5},\frac{3}{5}\right)\) | B1 | |
| So \(-\frac{3}{5}x+\frac{4}{5}y=x\) and \(\frac{4}{5}x+\frac{3}{5}y=y\) | M1 | At least one seen; M1 \(\Rightarrow\) midpoint is \(\left(\frac{1}{5},\frac{2}{5}\right)\) |
| Both of which lead to \(y=2x\) | E1 | Must conclude from both equations; E1 \(\Rightarrow y=2x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\mathbf{M}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\\y\end{pmatrix}\) | M1 | Seen or implied; OR M1 \(A'\left(\frac{-3}{5},\frac{4}{5}\right)\) \(A(1,0)\) |
| So \(-\frac{3}{5}x+\frac{4}{5}y=x\) and \(\frac{4}{5}x+\frac{3}{5}y=y\) | M1 | At least one seen; M1 \(\Rightarrow\) midpoint is \(\left(\frac{1}{5},\frac{2}{5}\right)\) |
| Both of which lead to \(y=2x\) | E1 | Must conclude from both equations; E1 \(\Rightarrow y=2x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\det\mathbf{M}=-1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Area remains the same | B1 | |
| Orientation of the image has changed | B1 | Accept 'sense', 'order of labelling' |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $A'$ is $\left(-\frac{3}{5},\frac{4}{5}\right)$, $B'$ is $\left(\frac{1}{5},\frac{7}{5}\right)$ and $C'$ is $\left(\frac{4}{5},\frac{3}{5}\right)$ | B1 | |
| So $-\frac{3}{5}x+\frac{4}{5}y=x$ and $\frac{4}{5}x+\frac{3}{5}y=y$ | M1 | At least one seen; **M1** $\Rightarrow$ midpoint is $\left(\frac{1}{5},\frac{2}{5}\right)$ |
| Both of which lead to $y=2x$ | E1 | Must conclude from both equations; **E1** $\Rightarrow y=2x$ |
---
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\mathbf{M}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\\y\end{pmatrix}$ | M1 | Seen or implied; **OR M1** $A'\left(\frac{-3}{5},\frac{4}{5}\right)$ $A(1,0)$ |
| So $-\frac{3}{5}x+\frac{4}{5}y=x$ and $\frac{4}{5}x+\frac{3}{5}y=y$ | M1 | At least one seen; **M1** $\Rightarrow$ midpoint is $\left(\frac{1}{5},\frac{2}{5}\right)$ |
| Both of which lead to $y=2x$ | E1 | Must conclude from both equations; **E1** $\Rightarrow y=2x$ |
---
## Question 5(iii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\det\mathbf{M}=-1$ | B1 | |
---
## Question 5(iii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Area remains the same | B1 | |
| Orientation of the image has changed | B1 | Accept 'sense', 'order of labelling' |
---5 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)$.
\begin{enumerate}[label=(\roman*)]
\item The diagram in the Printed Answer Booklet shows the unit square $O A B C$. The image of the unit square under the transformation represented by $\mathbf { M }$ is $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$. Draw and clearly label $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$.
\item Find the equation of the line of invariant points of this transformation.
\item (a) Find the determinant of $\mathbf { M }$.\\
(b) Describe briefly how this value relates to the transformation represented by $\mathbf { M }$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core AS Q5 [9]}}