| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2021 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Decompose matrix into transformation sequence |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring matrix algebra (determinants, matrix multiplication), understanding of geometric transformations, and decomposition of a transformation into components. While systematic, it demands conceptual understanding of orientation, scale factors, and the relationship between matrix operations and geometric transformations—significantly harder than standard A-level single maths questions. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation4.03n Inverse 2x2 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | (i) |
| Answer | Marks |
|---|---|
| Always negative, so reverses orientation | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1a |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | (ii) |
| Area scale factor = | B1 | |
| [1] | 1.1 | not − λ2 |
| 6 | (b) | (i) |
| Answer | Marks |
|---|---|
| ⇒M 2−λ2 I=0 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.2 | |
| 2.2a | correct matrix multiplication for M2 | |
| 6 | (b) | (ii) |
| enlargement [about O] scale factor | B1 | |
| [1] | 1.1 | |
| 6 | (c) | (i) |
| Answer | Marks |
|---|---|
| 0 −1 0 −1 0 1 | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | (1 2 ) (1 0 )(a b) |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (c) | (ii) |
| with invariant line Ox mapping (0, 1) to (2, 1) | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | oe |
Question 6:
6 | (a) | (i) | detM=−1−(λ+1)(λ−1)
=−λ2
Always negative, so reverses orientation | M1
A1
A1
[3] | 1.1a
1.1
2.2a
6 | (a) | (ii) | λ2
Area scale factor = | B1
[1] | 1.1 | not − λ2
6 | (b) | (i) | 1 λ+1 2 λ2 0
M 2 = = =λ2 I
λ−1 −1 0 λ2
⇒M 2−λ2 I=0 | M1
A1
[2] | 1.2
2.2a | correct matrix multiplication for M2
6 | (b) | (ii) | λ2
enlargement [about O] scale factor | B1
[1] | 1.1
6 | (c) | (i) | (1 0 )
reflection in Ox has matrix
0 −1
(1 0 )−1(1 2 )
matrix for S =
0 −1 0 −1
(1 0 )−1(1 2 ) (1 2)
= =
0 −1 0 −1 0 1 | B1
M1
A1
[3] | 1.1
3.1a
1.1 | (1 2 ) (1 0 )(a b)
or =
0 −1 0 −1 c d
⇒a=1, b=2, c=0, d =1
6 | (c) | (ii) | S is a shear
with invariant line Ox mapping (0, 1) to (2, 1) | M1
A1
[2] | 1.1
1.1 | oe6 A transformation T of the plane has associated matrix $\mathbf { M } = \left( \begin{array} { c c } 1 & \lambda + 1 \\ \lambda - 1 & - 1 \end{array} \right)$, where $\lambda$ is a non-zero\\
constant.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that T reverses orientation.
\item State, in terms of $\lambda$, the area scale factor of T .
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that $\mathbf { M } ^ { 2 } - \lambda ^ { 2 } \mathbf { I } = \mathbf { 0 }$.
\item Hence specify the transformation equivalent to two applications of T .
\end{enumerate}\item In the case where $\lambda = 1 , \mathrm {~T}$ is equivalent to a transformation S followed by a reflection in the $x$-axis.
\begin{enumerate}[label=(\roman*)]
\item Determine the matrix associated with S .
\item Hence describe the transformation S .
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2021 Q6 [12]}}