| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Infinite series convergence and sum |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question combining telescoping series (standard FP1 technique) with complex number algebra requiring systematic algebraic manipulation to find square roots, then solving a quartic via substitution, and finally geometric interpretation on an Argand diagram. While each technique is within FP1 scope, the combination of multiple topics, the algebraic complexity of finding exact complex square roots, and the geometric locus work make this moderately challenging for Further Maths students. |
| Spec | 4.02h Square roots: of complex numbers4.02i Quadratic equations: with complex roots4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct denominator | M1 | Use correct denominator |
| Correct answer | A1 | Obtain given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Express as differences using (i) | M1 | Express terms as differences using (i) |
| At least first 3 terms | M1 | Do this for at least \(1^{\text{st}}\) 3 terms |
| First 3 terms correct | A1 | First 3 terms all correct |
| Last 3 terms correct | A1 | Last 3 terms all correct (in terms of \(n\) or \(r\)) |
| Pairs cancelling | M1 | Show pairs cancelling |
| \(1 + \frac{1}{3} - \frac{1}{2n-1} - \frac{1}{2n+1}\) | A1 | Obtain correct answer, a.e.f. (in terms of \(n\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{4}{3}\) | B1ft | Given answer deduced correctly, ft their (ii) |
## Question 9:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct denominator | M1 | Use correct denominator |
| Correct answer | A1 | Obtain given answer correctly |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Express as differences using (i) | M1 | Express terms as differences using (i) |
| At least first 3 terms | M1 | Do this for at least $1^{\text{st}}$ 3 terms |
| First 3 terms correct | A1 | First 3 terms all correct |
| Last 3 terms correct | A1 | Last 3 terms all correct (in terms of $n$ or $r$) |
| Pairs cancelling | M1 | Show pairs cancelling |
| $1 + \frac{1}{3} - \frac{1}{2n-1} - \frac{1}{2n+1}$ | A1 | Obtain correct answer, a.e.f. (in terms of $n$) |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{4}{3}$ | B1ft | Given answer deduced correctly, ft their (ii) |
**Total: 9 marks**
---9 (i) Show that $\frac { 1 } { 2 r - 3 } - \frac { 1 } { 2 r + 1 } = \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$.\\
(ii) Hence find an expression, in terms of $n$, for
$$\sum _ { r = 2 } ^ { n } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$$
(iii) Show that $\sum _ { r = 2 } ^ { \infty } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 } = \frac { 4 } { 3 }$.\\
(i) Use an algebraic method to find the square roots of the complex number $2 + \mathrm { i } \sqrt { 5 }$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are exact real numbers.\\
(ii) Hence find, in the form $x + \mathrm { i } y$ where $x$ and $y$ are exact real numbers, the roots of the equation
$$z ^ { 4 } - 4 z ^ { 2 } + 9 = 0$$
(iii) Show, on an Argand diagram, the roots of the equation in part (ii).\\
(iv) Given that $\alpha$ is the root of the equation in part (ii) such that $0 < \arg \alpha < \frac { 1 } { 2 } \pi$, sketch on the same Argand diagram the locus given by $| z - \alpha | = | z |$.
\hfill \mbox{\textit{OCR FP1 2009 Q9 [9]}}