| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Year | 2024 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Find invariant points |
| Difficulty | Challenging +1.8 This question requires students to decompose a composite transformation matrix into reflection and rotation components, work backwards to find the reflection line parameter k, then find invariant points by solving (T-I)x=0. It demands strong understanding of transformation matrices, matrix multiplication, and the specific forms of reflection/rotation matrices. The multi-step nature, need to work backwards from a composite transformation, and requirement to handle both parts systematically makes this significantly harder than standard transformation questions, though the actual calculations are manageable for Further Maths students. |
| Spec | 4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products4.03g Invariant points and lines |
\begin{enumerate}
\item A point $P$ is reflected in the line $y = k x$, where $k$ is a constant. It is then rotated anticlockwise about $O$ through an acute angle $\theta$, where $\cos \theta = 0 \cdot 8$. The resulting transformation matrix is given by $T$, where
\end{enumerate}
$$T = \frac { 1 } { 85 } \left[ \begin{array} { r r }
- 84 & - 13 \\
- 13 & 84
\end{array} \right]$$
(a) Determine the value of $k$.\\
Find the invariant points of $T$.\\
\hfill \mbox{\textit{WJEC Further Unit 1 2024 Q8 [12]}}