| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Metal plate perimeter and area |
| Difficulty | Standard +0.3 This is a straightforward application of circle geometry and radians. Part (i) uses inverse trig (arcsin) with a given answer to verify, parts (ii) and (iii) require standard arc length and sector area formulas. The symmetry simplifies calculations, and all techniques are routine for A-level Pure Maths, making it slightly easier than average. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\sin \frac{1}{2}\theta = \frac{6}{10}\) | M1 | Use of trig with/without radians |
| Angle \(DOE = 1.287\) radians. | A1 | co – answer given. |
| [2] | ||
| (ii) \(P = 12 + 12 + 2 × 10 ×\) angle \(BOD\) | M1 | Use of \(s = r\theta\) for arc length. |
| Angle \(BOD = (\pi - 1.287)\) | M1 | Correct angle |
| \(\to 61.1\) | A1 | co |
| [3] | ||
| (iii) Sector \(DOE = \frac{1}{2} × 10^2 × 1.287\) | M1 | Correct formula used with radians. |
| Triangle \(DOE = \frac{1}{2} × 10^2 × \sin 1.287\) | M1 | Correct formula used with radians. |
| Area \(= \pi × 10^2 - (2\) sectors \(- 2\) triangles) (or \(48 + 48 + 2×\frac{1}{2}×10^2×(\pi - 1.287)\)) | M1 | M1 |
| \(\to 281\) or \(282\) | A1 | co |
| [3] |
(i) $\sin \frac{1}{2}\theta = \frac{6}{10}$ | M1 | Use of trig with/without radians
Angle $DOE = 1.287$ radians. | A1 | co – answer given.
| [2] |
(ii) $P = 12 + 12 + 2 × 10 ×$ angle $BOD$ | M1 | Use of $s = r\theta$ for arc length.
Angle $BOD = (\pi - 1.287)$ | M1 | Correct angle
$\to 61.1$ | A1 | co
| [3] |
(iii) Sector $DOE = \frac{1}{2} × 10^2 × 1.287$ | M1 | Correct formula used with radians.
Triangle $DOE = \frac{1}{2} × 10^2 × \sin 1.287$ | M1 | Correct formula used with radians.
Area $= \pi × 10^2 - (2$ sectors $- 2$ triangles) (or $48 + 48 + 2×\frac{1}{2}×10^2×(\pi - 1.287)$) | M1 | M1 |
$\to 281$ or $282$ | A1 | co
| [3] |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_744_675_255_735}
The diagram shows a metal plate $A B C D E F$ which has been made by removing the two shaded regions from a circle of radius 10 cm and centre $O$. The parallel edges $A B$ and $E D$ are both of length 12 cm .\\
(i) Show that angle $D O E$ is 1.287 radians, correct to 4 significant figures.\\
(ii) Find the perimeter of the metal plate.\\
(iii) Find the area of the metal plate.
\hfill \mbox{\textit{CAIE P1 2010 Q7 [8]}}