| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Write down transformation matrix |
| Difficulty | Standard +0.3 This is a standard FP1 matrices question covering routine transformations, matrix multiplication, and a straightforward induction proof. Parts (i)-(iii) are basic recall and computation (6 marks total). Part (iv) is a textbook induction on matrix powers with a simple algebraic step in the inductive case. While induction is FP1 content, this particular proof follows a completely standard template with no tricky algebra or insight required. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
\begin{enumerate}[label=(\roman*)]
\item Write down the matrix $\mathbf{C}$ which represents a stretch, scale factor $2$, in the $x$-direction. [2]
\item The matrix $\mathbf{D}$ is given by $\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$. Describe fully the geometrical transformation represented by $\mathbf{D}$. [2]
\item The matrix $\mathbf{M}$ represents the combined effect of the transformation represented by $\mathbf{C}$ followed by the transformation represented by $\mathbf{D}$. Show that
$$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
\item Prove by induction that $\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}$, for all positive integers $n$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 Q9 [12]}}