| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area of sector/segment problems |
| Difficulty | Standard +0.3 This is a straightforward geometric problem requiring area and perimeter calculations for sectors. Part (i) involves adding three sector areas using the standard formula A = ½r²θ, which is direct substitution. Part (ii) requires recognizing that arc lengths cancel when computing the perimeter. Both parts are 'show that' questions with given answers, requiring only verification rather than problem-solving insight. This is slightly easier than average due to its routine nature and guided structure. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (i) Sector \(OCD = \frac{1}{2}(2r)^2\theta\ (= 2r^2\theta)\) | B1 | \(2r^2\theta\) seen somewhere |
| Sector(s) \(OAB/OEF = (2)\frac{1}{2}r^2(\pi - \theta)\) | B1 | Accept with/without factor (2); AG www |
| Total \(= r^2(\pi + \theta)\) | B1 | |
| [3] | ||
| (ii) Arc \(CD = 2r\theta\) | B1 | Accept with/without factor (2) |
| Arc(s) \(AB/EF\ \ 2r(\pi - \theta)\) | B1 | |
| Straight edges \(= 4r\) | B1 | Must be simplified |
| Total \(2\pi r + 4r\) (which is independent of \(\theta\)) | B1 | |
| [4] |
## Question 4:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(i)** Sector $OCD = \frac{1}{2}(2r)^2\theta\ (= 2r^2\theta)$ | B1 | $2r^2\theta$ seen somewhere |
| Sector(s) $OAB/OEF = (2)\frac{1}{2}r^2(\pi - \theta)$ | B1 | Accept with/without factor (2); **AG** www |
| Total $= r^2(\pi + \theta)$ | B1 | |
| **[3]** | | |
| **(ii)** Arc $CD = 2r\theta$ | B1 | Accept with/without factor (2) |
| Arc(s) $AB/EF\ \ 2r(\pi - \theta)$ | B1 | |
| Straight edges $= 4r$ | B1 | Must be simplified |
| Total $2\pi r + 4r$ (which is independent of $\theta$) | B1 | |
| **[4]** | | |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-2_606_579_895_785}
The diagram shows a metal plate $O A B C D E F$ consisting of 3 sectors, each with centre $O$. The radius of sector $C O D$ is $2 r$ and angle $C O D$ is $\theta$ radians. The radius of each of the sectors $B O A$ and $F O E$ is $r$, and $A O E D$ and $C B O F$ are straight lines.\\
(i) Show that the area of the metal plate is $r ^ { 2 } ( \pi + \theta )$.\\
(ii) Show that the perimeter of the metal plate is independent of $\theta$.
\hfill \mbox{\textit{CAIE P1 2015 Q4 [7]}}