| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Describe reflection from matrix |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing basic matrix transformations. Part (a) involves routine matrix multiplication, recognizing a standard reflection (M is reflection in y=-x), and connecting M² to applying the transformation twice. Part (b) requires knowing that reflections have determinant -1 and solving a simple linear equation. While it's Further Maths content, the techniques are mechanical with no problem-solving insight required, making it slightly easier than average. |
| 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 orientation |
| Answer | Marks | Guidance |
|---|---|---|
| 8 | (a) | (i) |
| Answer | Marks | Guidance |
|---|---|---|
| 0 1 | B1 | |
| [1] | 1.1 | |
| 8 | (a) | (ii) |
| [1] | 1.1 | |
| 8 | (a) | (iii) |
| identity transformation. | B1 | |
| [1] | 2.2a | |
| 8 | (b) | det N = (k + 1)(k + 2) |
| Answer | Marks |
|---|---|
| So no roots ⇒ can never represent a reflection. | B1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
Question 8:
8 | (a) | (i) | 1 0
M2 =
0 1 | B1
[1] | 1.1
8 | (a) | (ii) | Reflection in y = −x | B1
[1] | 1.1
8 | (a) | (iii) | Two reflections in y = −x are equivalent to the
identity transformation. | B1
[1] | 2.2a
8 | (b) | det N = (k + 1)(k + 2)
If N represents a reflection then det N = −1
So k2 + 3k + 3 = 0
discriminant = 9 – 12 = −3 < 0
So no roots ⇒ can never represent a reflection. | B1
B1
M1
A1
[4] | 1.1
2.1
1.1
2.2a8
\begin{enumerate}[label=(\alph*)]
\item The matrix $\mathbf { M }$ is $\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$.
\begin{enumerate}[label=(\roman*)]
\item Find $\mathbf { M } ^ { 2 }$.
\item Write down the transformation represented by $\mathbf { M }$.
\item Hence state the geometrical significance of the result of part (i).
\end{enumerate}\item The matrix $\mathbf { N }$ is $\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)$, where $k$ is a constant.
Using determinants, investigate whether $\mathbf { N }$ can represent a reflection.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2020 Q8 [7]}}