| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Integration inequality bounds |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring understanding of Riemann sums and integral bounds, but follows a highly structured format with clear visual guidance. Parts (i) and (ii) are essentially guided explanations of standard upper/lower sum concepts, while part (iii) requires straightforward integration of √x. The conceptual insight (rectangles vs area under curve) is provided by the diagram and question structure, making this more of a careful execution exercise than a problem requiring novel reasoning. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08g Integration as limit of sum: Riemann sums |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos 3\theta = \cos(-3\theta)\) OR \(\cos\theta = \cos(-\theta)\) for all \(\theta\) | M1 | For a correct procedure for symmetry related to the equation OR to \(\cos 3\theta\) |
| \(\Rightarrow\) equation is unchanged, so symmetrical about \(\theta = 0\) | A1 | For correct explanation relating to equation AG |
| \(r = 0 \Rightarrow \cos 3\theta = -1\) | M1 | For obtaining equation for tangents |
| \(\Rightarrow \theta = \pm 1/3 \pi, \pi\) | A1 | A1 for any 2 values |
| A1 for all, no extras (ignore values outside range) | ||
| \(r^{1/4} (1+\cos 3\theta)^2 (d\theta)\) | (limits may be \([0, 1/4\pi]\) at any stage) | |
| \(= 1/2 \int_0^{1/\pi} 1 + 2\cos 3\theta + \cos^2 3\theta d\theta\) | M1 | For multiplying out, giving at least 2 terms |
| \(= 1/2 \int_0^{1/\pi} 1 + 2\cos 3\theta + 1/2(1 + \cos 6\theta) d\theta\) | M1 | For integration to \(A\theta + B\sin 3\theta + C\sin 6\theta\) AEF |
| \(= 1/2[\theta + 2/3\sin 3\theta + 1/2(\theta + 1/12\sin 6\theta)]_0^{1/\pi}\) | M1 | For completing integration and substituting (*dep) |
| their limits into terms in \(\cos 3\theta\) AEF | ||
| \(= 1/4\pi\) | A1 | For correct area www |
$\cos 3\theta = \cos(-3\theta)$ OR $\cos\theta = \cos(-\theta)$ for all $\theta$ | M1 | For a correct procedure for symmetry related to the equation OR to $\cos 3\theta$
$\Rightarrow$ equation is unchanged, so symmetrical about $\theta = 0$ | A1 | For correct explanation relating to equation AG
$r = 0 \Rightarrow \cos 3\theta = -1$ | M1 | For obtaining equation for tangents
$\Rightarrow \theta = \pm 1/3 \pi, \pi$ | A1 | A1 for any 2 values
| | A1 for all, no extras (ignore values outside range)
$r^{1/4} (1+\cos 3\theta)^2 (d\theta)$ | | (limits may be $[0, 1/4\pi]$ at any stage)
$= 1/2 \int_0^{1/\pi} 1 + 2\cos 3\theta + \cos^2 3\theta d\theta$ | M1 | For multiplying out, giving at least 2 terms
$= 1/2 \int_0^{1/\pi} 1 + 2\cos 3\theta + 1/2(1 + \cos 6\theta) d\theta$ | M1 | For integration to $A\theta + B\sin 3\theta + C\sin 6\theta$ AEF
$= 1/2[\theta + 2/3\sin 3\theta + 1/2(\theta + 1/12\sin 6\theta)]_0^{1/\pi}$ | M1 | For completing integration and substituting (*dep)
| | their limits into terms in $\cos 3\theta$ AEF
$= 1/4\pi$ | A1 | For correct area www7\\
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429}
The diagram shows the curve with equation $y = \sqrt { x }$. A set of $N$ rectangles of unit width is drawn, starting at $x = 1$ and ending at $x = N + 1$, where $N$ is an integer (see diagram).\\
(i) By considering the areas of these rectangles, explain why
$$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
(ii) By considering the areas of another set of rectangles, explain why
$$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
(iii) Hence find, in terms of $N$, limits between which $\sum _ { r = 1 } ^ { N } \sqrt { r }$ lies.
\section*{Jan 2006}
\hfill \mbox{\textit{OCR FP2 Q7 [9]}}