| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| 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 topics: writing transformation matrices, describing transformations, composing transformations, and matrix induction. Parts (i)-(iii) are straightforward recall and calculation (6 marks total). Part (iv) is a textbook induction proof with a simple algebraic step, requiring no novel insight. While Further Maths content is inherently harder, this represents a very standard example of its type, placing it slightly above average overall. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| \(\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}\) | B1B1 | Each column correct |
| Answer | Marks | Guidance |
|---|---|---|
| Shear, e.g. \((0,1)\) transforms to \((3,1)\) | B1B1 | One example or sensible explanation |
| Answer | Marks | Guidance |
|---|---|---|
| \(M = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}\) | M1 | Attempt to find \(DC\) (not \(CD\)) |
| A1 | Obtain given answer |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | Explicit check for \(n = 1\) or \(n = 2\) | |
| \(M^k = \begin{pmatrix} 2^k & 3(2^k - 1) \\ 0 & 1 \end{pmatrix}\) | M1 | Induction hypothesis that result is true for \(M^k\) |
| M1 | Attempt to multiply \(MM^k\) or vice versa | |
| A1 | Element \(3(2^{k+1}-1)\) derived correctly | |
| A1 | All other elements correct | |
| A1 | Explicit statement of induction conclusion |
### (i)
$\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ | B1B1 | Each column correct
**Subtotal: 2 marks**
### (ii)
Shear, e.g. $(0,1)$ transforms to $(3,1)$ | B1B1 | One example or sensible explanation
**Subtotal: 2 marks**
### (iii)
$M = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}$ | M1 | Attempt to find $DC$ (not $CD$)
| A1 | Obtain given answer
**Subtotal: 2 marks**
### (iv)
| B1 | Explicit check for $n = 1$ or $n = 2$
$M^k = \begin{pmatrix} 2^k & 3(2^k - 1) \\ 0 & 1 \end{pmatrix}$ | M1 | Induction hypothesis that result is true for $M^k$
| M1 | Attempt to multiply $MM^k$ or vice versa
| A1 | Element $3(2^{k+1}-1)$ derived correctly
| A1 | All other elements correct
| A1 | Explicit statement of induction conclusion
**Subtotal: 6 marks**
**Total: 12 marks**\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 2005 Q9 [12]}}