| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear transformations |
| Type | Matrix powers and repeated transformations |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring matrix multiplication, pattern recognition, proof by induction, and geometric interpretation. While the matrix calculations are straightforward, formulating the general form M^n and executing a complete induction proof elevates this above standard A-level. The geometric interpretation of a shear transformation adds another layer. This is moderately challenging for FP1 but accessible to well-prepared students. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| \(M^2 = \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}\), \(M^3 = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}\) | B1 | Correct \(M^2\) seen |
| M1 | Convincing attempt at matrix multiplication for \(M^3\) | |
| A1, 3 | Obtain correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(M^n = \begin{pmatrix} 1 & 2n \\ 0 & 1 \end{pmatrix}\) | B1ft, 1 | State correct form, consistent with (i) |
| Answer | Marks |
|---|---|
| M1 | Correct attempt to multiply \(M\) & \(M^k\) or v.v. |
| A1 | Obtain element \(2(k+1)\) |
| A1 | Clear statement of induction step, from correct working |
| B1, 4 | Clear statement of induction conclusion, following their working |
| Answer | Marks |
|---|---|
| B1 | Shear |
| DB1 | x-axis invariant |
| DB1, 3 | e.g. \((1,1) \to (2i, 1)\) or equivalent using scale factor or angles |
**Part (i)**
$M^2 = \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}$, $M^3 = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}$ | B1 | Correct $M^2$ seen
| M1 | Convincing attempt at matrix multiplication for $M^3$
| A1, 3 | Obtain correct answer
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**Part (ii)**
$M^n = \begin{pmatrix} 1 & 2n \\ 0 & 1 \end{pmatrix}$ | B1ft, 1 | State correct form, consistent with (i)
---
**Part (iii)**
| M1 | Correct attempt to multiply $M$ & $M^k$ or v.v.
| A1 | Obtain element $2(k+1)$
| A1 | Clear statement of induction step, from correct working
| B1, 4 | Clear statement of induction conclusion, following their working
---
**Part (iv)**
| B1 | Shear
| DB1 | x-axis invariant
| DB1, 3 | e.g. $(1,1) \to (2i, 1)$ or equivalent using scale factor or angles10 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)$.\\
(i) Find $\mathbf { M } ^ { 2 }$ and $\mathbf { M } ^ { 3 }$.\\
(ii) Hence suggest a suitable form for the matrix $\mathbf { M } ^ { n }$.\\
(iii) Use induction to prove that your answer to part (ii) is correct.\\
(iv) Describe fully the single geometrical transformation represented by $\mathbf { M } ^ { 10 }$.
\hfill \mbox{\textit{OCR FP1 2010 Q10 [11]}}