| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Segment area calculation |
| Difficulty | Standard +0.3 This is a straightforward application of standard arc length and sector area formulas with given radius and angle in radians. Part (i) requires arc length plus chord length (using cosine rule), and part (ii) requires sector area minus triangle area, then forming a ratio. All techniques are routine for P1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05c 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) \(s = r\theta\) | M1 | Used with major or minor arc |
| Angle of major arc \(= 2\pi - 2.2 = (4.083)\) | B1 | Could be gained in (ii). |
| Perimeter \(= 12 + 24.5 = 36.5\) or \(12\pi - 1.2\) (or full circle – minor arc B1) | A1 | co |
| [3] | ||
| (ii) Area of major sector \(= \frac{1}{2}r^2\theta = (73.49)\) | M1 | Used with major/minor sector. |
| Area of triangle \(= \frac{1}{2} \cdot 6^2 \sin 2.2 = (14.55)\) | M1 | Correct formula or method. \((2\pi - 2.2)/\sin 2.2\) gets M1M1 co |
| Ratio \(= 5.05 : 1\) (Allow \(5.03 \rightarrow 5.06\)) | A1 | |
| [3] |
(i) $s = r\theta$ | M1 | Used with major or minor arc |
Angle of major arc $= 2\pi - 2.2 = (4.083)$ | B1 | Could be gained in (ii). |
Perimeter $= 12 + 24.5 = 36.5$ or $12\pi - 1.2$ (or full circle – minor arc B1) | A1 | co |
| | [3] |
(ii) Area of major sector $= \frac{1}{2}r^2\theta = (73.49)$ | M1 | Used with major/minor sector. |
Area of triangle $= \frac{1}{2} \cdot 6^2 \sin 2.2 = (14.55)$ | M1 | Correct formula or method. $(2\pi - 2.2)/\sin 2.2$ gets M1M1 co |
Ratio $= 5.05 : 1$ (Allow $5.03 \rightarrow 5.06$) | A1 | |
| | [3] |3\\
\includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-2_485_623_790_760}
The diagram shows part of a circle with centre $O$ and radius 6 cm . The chord $A B$ is such that angle $A O B = 2.2$ radians. Calculate\\
(i) the perimeter of the shaded region,\\
(ii) the ratio of the area of the shaded region to the area of the triangle $A O B$, giving your answer in the form $k : 1$.
\hfill \mbox{\textit{CAIE P1 2014 Q3 [6]}}