| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Tangent and sector - single tangent line |
| Difficulty | Standard +0.3 This is a straightforward application of Pythagoras' theorem, arc length, and sector area formulas. Students must find angle POQ using right-angled triangle OPT, then apply standard formulas. While it requires multiple steps, each is routine and the problem structure is typical for this topic, making it slightly easier than average. |
| Spec | 1.03f Circle properties: angles, chords, tangents1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Pythagoras \(\rightarrow OT = 13, QT = 8\) cm | B1 | For \(QT\) in either part. |
| Angle \(POQ = \tan^{-1}(12/5) = 1.176\) | M1 | Could use \(\sin^{-1}\) or \(\cos^{-1}\) – in (i) or (ii) |
| \(S = r\theta \rightarrow 5.88\) | M1 | For \(5 \times\) angle in rads or equivalent in ° |
| \(\rightarrow\) Perimeter \(= 25.9\) cm | A1 | [4] |
| (ii) Area of sector \(= \frac{1}{2}r^2\theta\) used | M1 | Correct formula used. |
| Area of triangle \(OPT = \frac{1}{2} \times 12 \times 5\) | B1 | Anywhere |
| Shaded area \(= 30 - 12.5 \times 1.176\) | A1 | co |
| \(\rightarrow 15.3\) cm² | [3] |
**(i)** Pythagoras $\rightarrow OT = 13, QT = 8$ cm | B1 | For $QT$ in either part.
Angle $POQ = \tan^{-1}(12/5) = 1.176$ | M1 | Could use $\sin^{-1}$ or $\cos^{-1}$ – in (i) or (ii)
$S = r\theta \rightarrow 5.88$ | M1 | For $5 \times$ angle in rads or equivalent in °
$\rightarrow$ Perimeter $= 25.9$ cm | A1 | [4]
**(ii)** Area of sector $= \frac{1}{2}r^2\theta$ used | M1 | Correct formula used.
Area of triangle $OPT = \frac{1}{2} \times 12 \times 5$ | B1 | Anywhere
Shaded area $= 30 - 12.5 \times 1.176$ | A1 | co
$\rightarrow 15.3$ cm² | [3]5\\
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-2_543_883_1274_630}
The diagram shows a circle with centre $O$ and radius 5 cm . The point $P$ lies on the circle, $P T$ is a tangent to the circle and $P T = 12 \mathrm {~cm}$. The line $O T$ cuts the circle at the point $Q$.\\
(i) Find the perimeter of the shaded region.\\
(ii) Find the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2008 Q5 [7]}}