| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.3 This is a standard Further Maths question on matrix transformations requiring plotting an image, calculating a determinant, interpreting its geometric meaning, and decomposing a matrix into two simpler transformations. While it involves multiple parts, each step follows routine procedures (matrix multiplication, determinant calculation, recognizing reflection/shear components) that are well-practiced in Further Pure. The decomposition requires some insight but the matrix structure clearly suggests reflection in y-axis combined with a shear, making this slightly easier than average for Further Maths content. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03h Determinant 2x2: calculation |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | ( − 1 0 ) ( 0 1 1 0 ) ( 0 − 1 − 1 0 ) |
| Answer | Marks |
|---|---|
| C′ (0, 1) plotted correctly | SC B1 For unlabelled |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (b) | (i) |
| [1] | 1.1 | |
| 9 | (b) | (ii) |
| orientation is reversed | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (c) | (i) |
| Answer | Marks |
|---|---|
| invariant line y-axis, mapping (−1, 0) to (−1, −2) | B1 |
| Answer | Marks |
|---|---|
| A1 | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | oe, e.g. mapping (1, 0) to (1, 2) |
| Answer | Marks |
|---|---|
| Shear | M1 |
| invariant line y-axis, mapping (1, 0) to (1, −2) | A1 |
| then reflection in y-axis | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (c) | (ii) |
| Answer | Marks |
|---|---|
| 2 1 0 1 − 2 1 | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2a | ( 1 0 ) |
| Answer | Marks |
|---|---|
| 0 1 −2 1 −2 1 | Must match the order |
Question 9:
9 | (a) | M1
A1
[2] | 1.1
1.1 | ( − 1 0 ) ( 0 1 1 0 ) ( 0 − 1 − 1 0 )
=
− 2 1 0 0 1 1 0 − 2 − 1 1
A′ (−1, −2), B′ (−1, −1),
C′ (0, 1) plotted correctly | SC B1 For unlabelled
diagram with no
working
9 | (b) | (i) | d e t M = − 1 1 − 0 ( − 2 ) = − 1 | B1
[1] | 1.1
9 | (b) | (ii) | area is preserved
orientation is reversed | B1
B1
[2] | 1.1
1.1
9 | (c) | (i) | Reflection in y axis
then shear
invariant line y-axis, mapping (−1, 0) to (−1, −2) | B1
M1
A1 | 3.1a
3.1a
1.1 | oe, e.g. mapping (1, 0) to (1, 2)
Alternatively
Shear | M1
invariant line y-axis, mapping (1, 0) to (1, −2) | A1
then reflection in y-axis | B1
[3]
9 | (c) | (ii) | ( − 1 0 )
Reflection:
0 1
( 1 0 )
Shear:
2 1
( 1 0 ) ( − 1 0 ) ( − 1 0 )
=
2 1 0 1 − 2 1 | B1
B1
B1
[3] | 1.1
1.1
2.2a | ( 1 0 )
Or shear first
− 2 1
− 1 0
then reflection
0 1
(−1 0)( 1 0) (−1 0)
=
0 1 −2 1 −2 1 | Must match the order
described in ci for final
mark9 The transformation Too the plane has associated matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l l } - 1 & 0 \\ - 2 & 1 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item On the grid in the Printed Answer Booklet, plot the image $\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }$ of the unit square OABC under the transformation T.
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the value of $\operatorname { det } \mathbf { M }$.
\item Explain the significance of the value of $\operatorname { det } \mathbf { M }$ in relation to the image $\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }$.
\end{enumerate}\item T is equivalent to a sequence of two transformations of the plane.
\begin{enumerate}[label=(\roman*)]
\item Specify fully two transformations equivalent to T .
\item Use matrices to verify your answer.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2021 Q9 [11]}}