| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Commutativity of transformations |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on matrix transformations requiring routine calculations (matrix multiplication to check commutativity) and standard recognition of transformations (reflection and stretch). The invariant line part is also a standard textbook exercise. While it's Further Maths content, the techniques are mechanical with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03g Invariant points and lines |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | DR |
| Answer | Marks |
|---|---|
| so not commutative | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1a |
| Answer | Marks |
|---|---|
| 2.2a | both MN and NM calculated |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (b) | (i) |
| [1] | 1.1 | |
| 6 | (b) | (ii) |
| scale factor 2 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (c) | order of transformations matters |
| [1] | 2.2a | must refer to transformations not just matrices |
| 6 | (d) | 2 0 𝑥 2𝑥 2 0 𝑥 2𝑥 |
| Answer | Marks |
|---|---|
| invariant | M1 |
| Answer | Marks |
|---|---|
| [2] | 2.1 |
| 2.2a | Allow SC1 for correct geometrical argument, e.g. stretch in |
Question 6:
6 | (a) | DR
0 1 0 1 2 0 0 1
MN = (cid:4672) (cid:4673) or (cid:4672) (cid:4673)(cid:4672) (cid:4673)= (cid:4672) (cid:4673)
2 0 1 0 0 1 2 0
0 2 2 0 0 1 0 2
NM = (cid:4672) (cid:4673) or (cid:4672) (cid:4673)(cid:4672) (cid:4673)= (cid:4672) (cid:4673)
1 0 0 1 1 0 1 0
so not commutative | M1
A1*
A1
[3] | 1.1a
1.1
2.2a | both MN and NM calculated
both correct
dep A1*
6 | (b) | (i) | M is reflection in y = x | B1
[1] | 1.1
6 | (b) | (ii) | N is stretch parallel to the x-axis
scale factor 2 | M1
A1
[2] | 1.1
1.1
6 | (c) | order of transformations matters | B1
[1] | 2.2a | must refer to transformations not just matrices
6 | (d) | 2 0 𝑥 2𝑥 2 0 𝑥 2𝑥
(cid:4672) (cid:4673)(cid:4672) (cid:4673) =(cid:3436) (cid:3440) or (cid:4672) (cid:4673)(cid:4672) (cid:4673) = (cid:4672) (cid:4673)
0 1 𝑦 𝑦 0 1 𝑚𝑥+𝑐 𝑚𝑥+𝑐
y-coordinate unchanged so lines parallel to x-axis
invariant | M1
A1
[2] | 2.1
2.2a | Allow SC1 for correct geometrical argument, e.g. stretch in
x-direction leaves y-coordinates unchanged
www. Must refer to invariant lines.6 The matrices $\mathbf { M }$ and $\mathbf { N }$ are $\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)$ and $\left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)$ respectively.
\begin{enumerate}[label=(\alph*)]
\item In this question you must show detailed reasoning.
Determine whether $\mathbf { M }$ and $\mathbf { N }$ commute under matrix multiplication.
\item Specify the transformation of the plane associated with each of the following matrices.
\begin{enumerate}[label=(\roman*)]
\item M
\item N
\end{enumerate}\item State the significance of the result in part (a) for the transformations associated with $\mathbf { M }$ and $\mathbf { N }$. [1]
\item Use an algebraic method to show that all lines parallel to the $x$-axis are invariant lines of the transformation associated with N.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2023 Q6 [9]}}