| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Tangent and sector - two tangents from external point |
| Difficulty | Standard +0.3 This is a straightforward application of standard circle theorems (tangent perpendicular to radius), basic trigonometry (tan for tangent length), and sector area formulas. Part (i) requires simple geometry, part (ii) is guided ('show that') and uses routine radian-based calculations. Slightly above average due to multiple steps and radian context, but all techniques are standard C2 material with no novel problem-solving required. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(AB = 7.8(0), 7.798\) to \(7.799\) seen | M1 for correct use of sine rule For long methods M1A1 for art \(7.8\) | 2 |
| area \(= 52.2\) to \(52.3\) | M1 for \([2\times][0.5 \times]\) their \(AB \times 11.4 \times\) \(\sin 36°\) | 2 |
| (ii) \(\tan 0.91 = ST/12.6\) \(ST = 12.6 \times \tan 0.91\) and completion \((16.208...)\) area \(OSTR = [2\times][0.5 \times]12.6 \times\) their\((16.2)\) nb \(204.....\) area of sector \(= 0.5 \times 12.6^2 \times 1.82\) \(=144.47...\) Logo \(= 59.6\) to \(60.0\) arc \(= 12.6 \times 1.82 [=22.9...]\) perimeter \(= 55.3\) to \(55.4\) | M1 E1 M1 M1 A1 A1 M1 A1 | Accept \(16.2\) if ST is explicit but for long methods with pa check that their explicit expression \(= 16.2\) oe using degrees soi by correct ans Accept \(144, 144.5\) oe using degrees |
| 8 |
(i) $AB = 7.8(0), 7.798$ to $7.799$ seen | M1 for correct use of sine rule For long methods M1A1 for art $7.8$ | 2
area $= 52.2$ to $52.3$ | M1 for $[2\times][0.5 \times]$ their $AB \times 11.4 \times$ $\sin 36°$ | 2
(ii) $\tan 0.91 = ST/12.6$ $ST = 12.6 \times \tan 0.91$ and completion $(16.208...)$ area $OSTR = [2\times][0.5 \times]12.6 \times$ their$(16.2)$ nb $204.....$ area of sector $= 0.5 \times 12.6^2 \times 1.82$ $=144.47...$ Logo $= 59.6$ to $60.0$ arc $= 12.6 \times 1.82 [=22.9...]$ perimeter $= 55.3$ to $55.4$ | M1 E1 M1 M1 A1 A1 M1 A1 | Accept $16.2$ if ST is explicit but for long methods with pa check that their explicit expression $= 16.2$ oe using degrees soi by correct ans Accept $144, 144.5$ oe using degrees | 4
| | 810 Arrowline Enterprises is considering two possible logos:
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-4_1123_1676_356_230}
\captionsetup{labelformat=empty}
\caption{Fig. 10.1}
\end{center}
\end{figure}
Fig. 10.2\\
(i) Fig. 10.1 shows the first logo ABCD . It is symmetrical about AC .
Find the length of AB and hence find the area of this logo.\\
(ii) Fig. 10.2 shows a circle with centre O and radius 12.6 cm . ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo.
Show that $\mathrm { ST } = 16.2 \mathrm {~cm}$ to 3 significant figures.\\
Find the area and perimeter of this logo.
\hfill \mbox{\textit{OCR MEI C2 2005 Q10 [12]}}