| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Find inverse transformation matrix |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring knowledge of 3D transformation matrices and matrix algebra. Part (a) involves finding A = BR^(-1) using the standard rotation matrix, then showing terms cancel to eliminate θ—a multi-step process requiring careful matrix multiplication. Part (b) requires geometric interpretation of the resulting matrix. While systematic rather than requiring deep insight, it's above average difficulty due to the 3D context, matrix inverse manipulation, and being from Further Maths content. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03f Linear transformations 3D: reflections and rotations about axes |
| Answer | Marks |
|---|---|
| 7(a) | Finds the correct |
| Answer | Marks | Guidance |
|---|---|---|
| PI by correct A | AO2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| PI by correct A | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| independent of θ | AO2.1 | R1 |
| Answer | Marks |
|---|---|
| 7(b) | States fully correct |
| Answer | Marks | Guidance |
|---|---|---|
| plane. | AO3.2a | E1 |
| Total | 4 | |
| Q | Marking Instructions | AO |
Question 7:
--- 7(a) ---
7(a) | Finds the correct
matrix for R-1
PI by correct A | AO2.2a | B1 | cosθ sinθ 0
R-1 = −sinθ cosθ 0
0 0 1
A=BR-1
−cosθ sinθ 0 cosθ sinθ 0
A= sinθ cosθ 0 −sinθ cosθ 0
0 0 1 0 0 1
−1 0 0
A= 0 1 0
0 0 1
Ais independent ofθ.
Appropriate method to
find A, such as post
multiplyingBby R-1
PI by correct A | AO1.1a | M1
Completes a rigorous
argument to show the
required result,
including finding the
correct matrix forA.
Must include
conclusion that A is
independent of θ | AO2.1 | R1
--- 7(b) ---
7(b) | States fully correct
(single) geometrical
description.
Eg Reflection in y/ z
plane. | AO3.2a | E1 | Reflection in x =0 plane.
Total | 4
Q | Marking Instructions | AO | Marks | Typical SolutionThree non-singular square matrices, $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{R}$ are such that
$$\mathbf{AR} = \mathbf{B}$$
The matrix $\mathbf{R}$ represents a rotation about the $z$-axis through an angle $\theta$ and
$$\mathbf{B} = \begin{pmatrix} -\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathbf{A}$ is independent of the value of $\theta$.
[3 marks]
\item Give a full description of the single transformation represented by the matrix $\mathbf{A}$.
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 2019 Q7 [4]}}