| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Chord and sector relationship |
| Difficulty | Moderate -0.3 This is a straightforward application of standard circle geometry formulas in radians. Part (i) uses the cosine rule or isosceles triangle properties, (ii) combines arc length with the chord, and (iii) subtracts segment area from sector area. All techniques are routine for P1 level with no problem-solving insight required, making it slightly easier than average but not trivial due to the multi-step nature. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) cosine rule or \(2 \times r × \sin \frac{1}{2}(2.4)\) | M1 A1 | Any complete valid method. co [2] |
| \(\to 14.9\) cm | ||
| (ii) Perimeter = (i) + \(r\theta\) | M1 | Uses \(s = r\theta\) with 2.4, or \(\pi - 2.4\), or \(2\pi - 2.4\) |
| \(\theta = 2\pi - 2.4\), | B1 | |
| \(\to 46.0\) cm | A1♦ | [3] |
| (iii) Area = Sector + triangle | M1 M1 | Uses \(\frac{1}{2}r^2\theta\). Uses any valid method. co |
| \(\frac{1}{2}×8^2(2\pi - 2.4) + \frac{1}{2}×8^2×\sin 2.4\) | A1 | |
| \(124.3 + 21.6 \to 146\) cm² | [3] |
(i) cosine rule or $2 \times r × \sin \frac{1}{2}(2.4)$ | M1 A1 | Any complete valid method. co [2]
$\to 14.9$ cm | |
(ii) Perimeter = (i) + $r\theta$ | M1 | Uses $s = r\theta$ with 2.4, or $\pi - 2.4$, or $2\pi - 2.4$
$\theta = 2\pi - 2.4$, | B1 |
$\to 46.0$ cm | A1♦ | [3] | Anywhere in parts (ii) or (iii). Adds 31.1 to (i) for ♦.
(iii) Area = Sector + triangle | M1 M1 | Uses $\frac{1}{2}r^2\theta$. Uses any valid method. co
$\frac{1}{2}×8^2(2\pi - 2.4) + \frac{1}{2}×8^2×\sin 2.4$ | A1 |
$124.3 + 21.6 \to 146$ cm² | [3] |\includegraphics{figure_6}
The diagram shows a metal plate made by removing a segment from a circle with centre $O$ and radius $8$ cm. The line $AB$ is a chord of the circle and angle $AOB = 2.4$ radians. Find
\begin{enumerate}[label=(\roman*)]
\item the length of $AB$, [2]
\item the perimeter of the plate, [3]
\item the area of the plate. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q6 [8]}}