| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2018 |
| Session | December |
| Marks | 6 |
| Topic | Linear transformations |
| Type | 3D transformation matrices |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on 3D transformations. Part (a) requires recognizing that a diagonal matrix with eigenvalues 1,1,-1 reflects in a coordinate plane (immediate from inspection). Part (b) uses the reflection property that the middle eigenvalue must equal -1, giving a simple quadratic. Part (c) asks for conceptual understanding that reflections are self-inverse. All parts are direct applications of standard theory with minimal computation or insight required. |
| Spec | 4.03f Linear transformations 3D: reflections and rotations about axes4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) The x-z plane | B1 | or \(y = 0\) |
| (b) \(\frac{2a - a^2}{3} = -1\) | B1 | |
| \(a^2 - 2a - 3 = 0 \Rightarrow a = -1, 3\) | M1 | BC. Rearranging the quadratic equation and solving. |
| \(a > 0 \Rightarrow a = 3\) | A1 | |
| (c) Any reflection is self-inverse... oe | B1 | eg "If you do a reflection twice it gets back to where it started" |
| ...\(\text{so } A^2 = AA^{-1} = I\) | B1 |
**(a)** The x-z plane | B1 | or $y = 0$
**(b)** $\frac{2a - a^2}{3} = -1$ | B1 |
$a^2 - 2a - 3 = 0 \Rightarrow a = -1, 3$ | M1 | BC. Rearranging the quadratic equation and solving.
$a > 0 \Rightarrow a = 3$ | A1 |
**(c)** Any reflection is self-inverse... oe | B1 | eg "If you do a reflection twice it gets back to where it started"
...$\text{so } A^2 = AA^{-1} = I$ | B1 |
---You are given that the matrix $\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}$, where $a$ is a positive constant, represents the transformation R which is a reflection in 3-D.
\begin{enumerate}[label=(\alph*)]
\item State the plane of reflection of R. [1]
\item Determine the value of $a$. [3]
\item With reference to R explain why $\mathbf{A}^2 = \mathbf{I}$, the $3\times 3$ identity matrix. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2018 Q4 [6]}}