| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Suggest and prove formula |
| Difficulty | Standard +0.8 This is a standard Further Maths induction question requiring matrix multiplication and pattern recognition. While it involves multiple steps (computing M^4, conjecturing the general form, then proving by induction), the techniques are routine for FP1 students and the pattern (3^n in top-left, 3^n-1 in bottom-left) is straightforward to spot from the given example. The induction proof itself is mechanical once the conjecture is made. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Matrix multiplication attempted | M1 | Attempt at matrix multiplication |
| Obtain \(\mathbf{M}^2\) correctly | A1 | |
| Obtain given answer correctly | A1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}3^n & 0 \\ 3^n-1 & 1\end{pmatrix}\) | B1, B1 | 3 elements correct; 4th element correct |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show result true for \(n=1\) or \(2\) | B1 | |
| Attempt to find \(\mathbf{M}^k\mathbf{M}\) or vice versa | M1 | |
| \(\begin{pmatrix}3^{k+1}&0\\3^{k+1}-1&1\end{pmatrix}\) | A1 | Obtain correct answer |
| Complete statement of induction conclusion | B1 | Must have first 3 marks |
| [4] |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Matrix multiplication attempted | M1 | Attempt at matrix multiplication |
| Obtain $\mathbf{M}^2$ correctly | A1 | |
| Obtain given answer correctly | A1 | |
| **[3]** | | |
---
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}3^n & 0 \\ 3^n-1 & 1\end{pmatrix}$ | B1, B1 | 3 elements correct; 4th element correct |
| **[2]** | | |
---
## Question 7(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show result true for $n=1$ or $2$ | B1 | |
| Attempt to find $\mathbf{M}^k\mathbf{M}$ or vice versa | M1 | |
| $\begin{pmatrix}3^{k+1}&0\\3^{k+1}-1&1\end{pmatrix}$ | A1 | Obtain correct answer |
| Complete statement of induction conclusion | B1 | Must have first 3 marks |
| **[4]** | | |
---7 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 2 & 1 \end{array} \right)$.\\
(i) Show that $\mathbf { M } ^ { 4 } = \left( \begin{array} { l l } 81 & 0 \\ 80 & 1 \end{array} \right)$.\\
(ii) Hence suggest a suitable form for the matrix $\mathbf { M } ^ { n }$, where $n$ is a positive integer.\\
(iii) Use induction to prove that your answer to part (ii) is correct.
\hfill \mbox{\textit{OCR FP1 2012 Q7 [9]}}