| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Shaded region between arcs |
| Difficulty | Standard +0.3 This is a straightforward application of arc length and sector area formulas with radians. Students must find two arc lengths (with radii 20 and 15) and two straight line segments, then calculate the difference between two sector areas. All values are given directly, requiring only substitution into standard formulas with no geometric insight or problem-solving beyond recognizing the structure. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Perimeter \(= (15 \times 1.8) + (20 \times 1.8) + 5 + 5 = 73\text{cm}\) | M1 | Use \(r\theta\) at least once |
| A1 | Obtain at least one of 27cm or 36cm | |
| A1 | Obtain 73 |
| Answer | Marks | Guidance |
|---|---|---|
| Area \(= \left(\frac{1}{2} \times 20^2 \times 1.8\right) - \left(\frac{1}{2} \times 15^2 \times 1.8\right) = 157.5\text{cm}^2\) | M1 | Attempt area of sector using \(kr^2\theta\) |
| M1 | Find difference between attempts at two sectors | |
| A1 | Obtain 157.5 / 158 |
**(i)**
Perimeter $= (15 \times 1.8) + (20 \times 1.8) + 5 + 5 = 73\text{cm}$ | M1 | Use $r\theta$ at least once
| A1 | Obtain at least one of 27cm or 36cm
| A1 | Obtain 73
**(ii)**
Area $= \left(\frac{1}{2} \times 20^2 \times 1.8\right) - \left(\frac{1}{2} \times 15^2 \times 1.8\right) = 157.5\text{cm}^2$ | M1 | Attempt area of sector using $kr^2\theta$
| M1 | Find difference between attempts at two sectors
| A1 | Obtain 157.5 / 158
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{58680cd3-8744-42ee-83d4-35056592b2d0-2_647_797_1323_680}
The diagram shows a sector $O A B$ of a circle with centre $O$. The angle $A O B$ is 1.8 radians. The points $C$ and $D$ lie on $O A$ and $O B$ respectively. It is given that $O A = O B = 20 \mathrm {~cm}$ and $O C = O D = 15 \mathrm {~cm}$. The shaded region is bounded by the arcs $A B$ and $C D$ and by the lines $C A$ and $D B$.\\
(i) Find the perimeter of the shaded region.\\
(ii) Find the area of the shaded region.
\hfill \mbox{\textit{OCR C2 2006 Q4 [6]}}