| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Integral bounds for series |
| Difficulty | Standard +0.8 This is a Further Pure 2 question requiring understanding of Riemann sums and integral approximations. Part (i) is straightforward geometric reasoning (2 marks), part (ii) requires drawing and analyzing a different rectangle configuration (3 marks), and part (iii) involves applying both inequalities to bound a sum and compute an approximation. While the concepts are accessible, the multi-step reasoning about upper/lower bounds and the FP2 context place it moderately above average difficulty. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08g Integration as limit of sum: Riemann sums |
| Answer | Marks |
|---|---|
| LHS = sum of areas of rectangles, area = \(1 \times \text{y-value from } x = 1 \text{ to } x = n\) | B1 |
| RHS = Area under curve from \(x = 0\) to \(n\) | B1 |
| Answer | Marks |
|---|---|
| Diagram showing areas of rectangles | B1 |
| Use sum of areas of rectangles | B1 |
| Explain/show area inequality with limits in integral clearly specified | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt integral as \(kx^{\frac{3}{5}}\) | M1 | |
| Limits gives 348.(1) and 352.(0) | A1 | Allow one correct |
| Get 350 | A1 | From two correct values only |
## (i)
LHS = sum of areas of rectangles, area = $1 \times \text{y-value from } x = 1 \text{ to } x = n$ | B1 |
RHS = Area under curve from $x = 0$ to $n$ | B1 |
## (ii)
Diagram showing areas of rectangles | B1 |
Use sum of areas of rectangles | B1 |
Explain/show area inequality with limits in integral clearly specified | B1 |
## (iii)
Attempt integral as $kx^{\frac{3}{5}}$ | M1 |
Limits gives 348.(1) and 352.(0) | A1 | Allow one correct
Get 350 | A1 | From two correct values only
---\includegraphics{figure_7}
The diagram shows the curve with equation $y = \sqrt{x}$, together with a set of $n$ rectangles of unit width.
\begin{enumerate}[label=(\roman*)]
\item By considering the areas of these rectangles, explain why
$$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} > \int_0^n \sqrt{x} dx.$$ [2]
\item By drawing another set of rectangles and considering their areas, show that
$$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n} < \int_1^{n+1} \sqrt{x} dx.$$ [3]
\item Hence find an approximation to $\sum_{n=1}^{100} \sqrt{n}$, giving your answer correct to 2 significant figures. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q7 [8]}}