| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area of sector/segment problems |
| Difficulty | Moderate -0.5 This is a straightforward application of standard circle sector formulas (arc length, sector area) combined with basic right-angled triangle trigonometry. Part (i) requires identifying perimeter components (arc AB + OC + CB + OA), and part (ii) involves adding sector area and triangle area with given numerical values. No novel insight or complex problem-solving required, just routine formula application, making it slightly easier than average. |
| Spec | 1.03h Parametric equations: in modelling contexts1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Arc \(AB = r\theta\) | M1 | |
| \(OC = r\sin\theta\) or \(BC = r\cos\theta\) | M1 | oe eg \(BC = r\sin\frac{\theta}{\tan\theta}\) etc |
| \(r(1 + \theta + \cos\theta + \sin\theta)\) correctly derived | A1 | [3] \(OC\) & \(BC\) reversed loses M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Sector \(OAB = \frac{1}{2} \times 10^2 \times \frac{\pi}{5}\) \((= 31.42)\) | M1 | oe \(\Delta\) in terms of \(\pi\) and \(10\) |
| \(\Delta OCB = \frac{1}{2}\left(10\cos\frac{\pi}{5}\right)\left(10\sin\frac{\pi}{5}\right)\) \((= 23.78)\) | M1 | Allow \(OC\) & \(BC\) reversed (ie max 4/6) |
| Total area \(= 55.2\) | A1 | [3] |
## Question 5:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Arc $AB = r\theta$ | M1 | |
| $OC = r\sin\theta$ or $BC = r\cos\theta$ | M1 | oe eg $BC = r\sin\frac{\theta}{\tan\theta}$ etc |
| $r(1 + \theta + \cos\theta + \sin\theta)$ correctly derived | A1 | [3] $OC$ & $BC$ reversed loses M1A1 |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Sector $OAB = \frac{1}{2} \times 10^2 \times \frac{\pi}{5}$ $(= 31.42)$ | M1 | oe $\Delta$ in terms of $\pi$ and $10$ |
| $\Delta OCB = \frac{1}{2}\left(10\cos\frac{\pi}{5}\right)\left(10\sin\frac{\pi}{5}\right)$ $(= 23.78)$ | M1 | Allow $OC$ & $BC$ reversed (ie max 4/6) |
| Total area $= 55.2$ | A1 | [3] |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-2_512_903_1302_621}
The diagram represents a metal plate $O A B C$, consisting of a sector $O A B$ of a circle with centre $O$ and radius $r$, together with a triangle $O C B$ which is right-angled at $C$. Angle $A O B = \theta$ radians and $O C$ is perpendicular to $O A$.\\
(i) Find an expression in terms of $r$ and $\theta$ for the perimeter of the plate.\\
(ii) For the case where $r = 10$ and $\theta = \frac { 1 } { 5 } \pi$, find the area of the plate.
\hfill \mbox{\textit{CAIE P1 2011 Q5 [6]}}