| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Integral bounds for series |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring understanding of integral bounds for series, but the approach is guided through scaffolded parts. Part (i) asks for explanation of a given diagram, part (ii) extends the pattern, and part (iii) applies the method to a larger sum. While it requires geometric insight about upper/lower Riemann sums and careful index manipulation, the structure is supportive and the techniques are standard for FP2 level. |
| Spec | 1.08g Integration as limit of sum: Riemann sums4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| (i) LHS is the total area of the four rectangles; RHS is the corresponding area under the curve, which is clearly greater | B1 | For identifying rectangle areas (not heights) |
| \(-\) | B1 | For correct explanation |
| \(-\) | 2 | |
| (ii) \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} > \int_1^4 \frac{1}{x+2} \, dx\) | M1 | For attempt at relevant new inequality |
| \(-\) | A1 | For correct statement |
| \(-\) | 2 | |
| (iii) Sum is the area of \(999\) rectangles; Bounds are \(\int_1^{999} \frac{1}{x+2} \, dx\) and \(\int_2^{1000} \frac{1}{x+1} \, dx\) | M1 | For considering the sum as an area again |
| \(-\) | M1 | For stating either integral as a bound |
| So lower bound is \([\ln(x+2)]_1^{999} = \ln(500.5)\) | A1 | For showing the given value correctly |
| and upper bound is \([\ln(x+1)]_2^{1000} = \ln(1000)\) | A1 | Ditto |
| \(-\) | 4 |
(i) LHS is the total area of the four rectangles; RHS is the corresponding area under the curve, which is clearly greater | B1 | For identifying rectangle areas (not heights) |
$-$ | B1 | For correct explanation |
$-$ | **2** | |
(ii) $\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} > \int_1^4 \frac{1}{x+2} \, dx$ | M1 | For attempt at relevant new inequality |
$-$ | A1 | For correct statement |
$-$ | **2** | |
(iii) Sum is the area of $999$ rectangles; Bounds are $\int_1^{999} \frac{1}{x+2} \, dx$ and $\int_2^{1000} \frac{1}{x+1} \, dx$ | M1 | For considering the sum as an area again |
$-$ | M1 | For stating either integral as a bound |
So lower bound is $[\ln(x+2)]_1^{999} = \ln(500.5)$ | A1 | For showing the given value correctly |
and upper bound is $[\ln(x+1)]_2^{1000} = \ln(1000)$ | A1 | Ditto |
$-$ | **4** | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_444_999_1258_539}
The diagram shows the curve $y = \frac { 1 } { x + 1 }$ together with four rectangles of unit width.\\
(i) Explain how the diagram shows that
$$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } < \int _ { 0 } ^ { 4 } \frac { 1 } { x + 1 } \mathrm {~d} x$$
The curve $y = \frac { 1 } { x + 2 }$ passes through the top left-hand corner of each of the four rectangles shown.\\
(ii) By considering the rectangles in relation to this curve, write down a second inequality involving $\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 }$ and a definite integral.\\
(iii) By considering a suitable range of integration and corresponding rectangles, show that
$$\ln ( 500.5 ) < \sum _ { r = 2 } ^ { 1000 } \frac { 1 } { r } < \ln ( 1000 ) .$$
\hfill \mbox{\textit{OCR FP2 Q5 [8]}}