| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2004 |
| Session | June |
| Marks | 7 |
| 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 standard sector and triangle area formulas with given values. Students must calculate the area of triangle OCD minus the area of sector OAB, then find the perimeter by adding two line segments and an arc length. All formulas are directly applicable with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks |
|---|---|
| (i) Area of sector = \(\frac{1}{2}r^2 \times 0.8\) \((14.4)\) | Area of triangle = \(\frac{1}{2} \times 10^2 \times \sin 0.8\) \((35.9)\) → Shaded area = \(21.5\) |
| M1 M1 A1 [3] | Use of \(\frac{1}{2}r^2\theta\) with radians; Use of \(\frac{1}{2}ab\sin C\) or \(\frac{1}{2}bh\) with trig; Correct only |
| (ii) Arc length = \(6 \times 0.8\) \((4.8)\) | CD (by cos rule) or \(2 \times 10\sin 0.4\) \((7.8)\) → Perimeter = \(8 + 4.8 + 7.8 = 20.6\) |
| M1 M1 A1 A1 [4] | Use of \(s=r\theta\) with radians; Any correct method – allow if in (i); Correct only |
**(i)** Area of sector = $\frac{1}{2}r^2 \times 0.8$ $(14.4)$ | Area of triangle = $\frac{1}{2} \times 10^2 \times \sin 0.8$ $(35.9)$ → Shaded area = $21.5$
| M1 M1 A1 [3] | Use of $\frac{1}{2}r^2\theta$ with radians; Use of $\frac{1}{2}ab\sin C$ or $\frac{1}{2}bh$ with trig; Correct only |
**(ii)** Arc length = $6 \times 0.8$ $(4.8)$ | CD (by cos rule) or $2 \times 10\sin 0.4$ $(7.8)$ → Perimeter = $8 + 4.8 + 7.8 = 20.6$
| M1 M1 A1 A1 [4] | Use of $s=r\theta$ with radians; Any correct method – allow if in (i); Correct only |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-2_501_682_1302_735}
In the diagram, $O C D$ is an isosceles triangle with $O C = O D = 10 \mathrm {~cm}$ and angle $C O D = 0.8$ radians. The points $A$ and $B$, on $O C$ and $O D$ respectively, are joined by an arc of a circle with centre $O$ and radius 6 cm . Find\\
(i) the area of the shaded region,\\
(ii) the perimeter of the shaded region.
\hfill \mbox{\textit{CAIE P1 2004 Q5 [7]}}