| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Method of differences with given identity |
| Difficulty | Standard +0.3 This is a guided method of differences question where the partial fraction decomposition is given. Part (i) requires writing out terms and cancelling (standard telescoping series technique), and part (ii) is a direct application of the limit. While it involves Further Maths content, the question is highly structured with minimal problem-solving required, making it easier than average overall. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Multiply out AB, e.g. element (1,1): \(1(-5) + (-2)(8) + k(1) = -5 - 16 + k = k - 21\) | M1 | Attempt to multiply matrices |
| Compare with \(k - n\): so \(n = 21\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(AB = (k-n)I \Rightarrow A^{-1} = \frac{1}{k-n}B\) | M1 | |
| \(A^{-1} = \frac{1}{k-21}\begin{pmatrix}-5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5\end{pmatrix}\) | A1 A1 | B1 for scalar, A1 for matrix |
| Condition: \(k \neq 21\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Write as matrix equation \(A\mathbf{x} = \mathbf{b}\) where \(\mathbf{b} = (1, 12, 3)^T\) | M1 | |
| \(\mathbf{x} = A^{-1}\mathbf{b}\) using their inverse with appropriate \(k\) value | M1 | |
| Identify \(k = 1\) from equations (coefficient of \(z\) in first equation) | B1 | |
| \(\frac{1}{1-21}B\begin{pmatrix}1\\12\\3\end{pmatrix} = \frac{1}{-20}B\begin{pmatrix}1\\12\\3\end{pmatrix}\) | M1 | |
| \(x = 3, y = 1, z = 2\) | A1 A1 A1 |
## Question 10 (June 2007 paper):
**(i) Find the value of n**
| Working/Answer | Mark | Guidance |
|---|---|---|
| Multiply out AB, e.g. element (1,1): $1(-5) + (-2)(8) + k(1) = -5 - 16 + k = k - 21$ | M1 | Attempt to multiply matrices |
| Compare with $k - n$: so $n = 21$ | A1 | cao |
**(ii) Write down A⁻¹ and state condition on k**
| Working/Answer | Mark | Guidance |
|---|---|---|
| $AB = (k-n)I \Rightarrow A^{-1} = \frac{1}{k-n}B$ | M1 | |
| $A^{-1} = \frac{1}{k-21}\begin{pmatrix}-5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5\end{pmatrix}$ | A1 A1 | B1 for scalar, A1 for matrix |
| Condition: $k \neq 21$ | B1 | |
**(iii) Solve the simultaneous equations**
| Working/Answer | Mark | Guidance |
|---|---|---|
| Write as matrix equation $A\mathbf{x} = \mathbf{b}$ where $\mathbf{b} = (1, 12, 3)^T$ | M1 | |
| $\mathbf{x} = A^{-1}\mathbf{b}$ using their inverse with appropriate $k$ value | M1 | |
| Identify $k = 1$ from equations (coefficient of $z$ in first equation) | B1 | |
| $\frac{1}{1-21}B\begin{pmatrix}1\\12\\3\end{pmatrix} = \frac{1}{-20}B\begin{pmatrix}1\\12\\3\end{pmatrix}$ | M1 | |
| $x = 3, y = 1, z = 2$ | A1 A1 A1 | |
---10 (i) You are given that
$$\frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 }$$
Use the method of differences to show that
$$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
(ii) Hence find the sum of the infinite series
$$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$
RECOGNISING ACHIEVEMENT
\section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS}
\section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education}
\section*{MEI STRUCTURED MATHEMATICS}
Further Concepts For Advanced Mathematics (FP1)\\
Wednesday 18 JANUARY 2006 Afternoon ..... 1 hour 30 minutes\\
Additional materials:\\
8 page answer booklet\\
Graph paper\\
MEI Examination Formulae and Tables (MF2)
TIME 1 hour 30 minutes
\begin{itemize}
\item Write your name, centre number and candidate number in the spaces provided on the answer booklet.
\item Answer all the questions.
\item You are permitted to use a graphical calculator in this paper.
\item Final answers should be given to a degree of accuracy appropriate to the context.
\end{itemize}
\begin{itemize}
\item The number of marks is given in brackets [ ] at the end of each question or part question.
\item You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
\item The total number of marks for this paper is 72.
\end{itemize}
\hfill \mbox{\textit{OCR MEI FP1 Q10}}