| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area of sector/segment problems |
| Difficulty | Standard +0.3 This is a standard sector/segment problem requiring area and perimeter formulas for sectors with given radii. Part (i) involves setting up an equation relating two areas using the sector area formula, while part (ii) requires equating perimeters involving arc lengths. Both parts are routine applications of circle geometry formulas with straightforward algebraic manipulation, slightly above average due to the two-part structure and algebraic work required. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| (i) sector areas are \(\frac{1}{2} \times 11^2 \alpha, \frac{1}{2} \times 5^2 \alpha\) | B1 | Sight of \(11^2, 5^2\) |
| \(k = \frac{\frac{1}{2} \times 11^2 \alpha - \frac{1}{2} \times 5^2 \alpha}{\frac{1}{2} \times 5^2 \alpha}\) | M1 | Or \(\frac{11^2 - 5^2}{5^2}\) |
| \(k = \frac{96}{25}\) or 3.84 | A1 | |
| [3] | ||
| (ii) perimeter shaded region\(= 11\alpha + 5\alpha + 6 + 6 = 16\alpha + 12\) | B1 | |
| perimeter unshaded region \(= 5\alpha + 5 + 5 = 5\alpha + 10\) | B1 | |
| \(16\alpha + 12 = 2(5\alpha + 10)\) | M1 | |
| \(\alpha = \frac{4}{3}\) or 1.33 | A1 | |
| [4] |
**(i)** sector areas are $\frac{1}{2} \times 11^2 \alpha, \frac{1}{2} \times 5^2 \alpha$ | B1 | Sight of $11^2, 5^2$
$k = \frac{\frac{1}{2} \times 11^2 \alpha - \frac{1}{2} \times 5^2 \alpha}{\frac{1}{2} \times 5^2 \alpha}$ | M1 | Or $\frac{11^2 - 5^2}{5^2}$
$k = \frac{96}{25}$ or 3.84 | A1 |
| | [3]
**(ii)** perimeter shaded region$= 11\alpha + 5\alpha + 6 + 6 = 16\alpha + 12$ | B1 |
perimeter unshaded region $= 5\alpha + 5 + 5 = 5\alpha + 10$ | B1 |
$16\alpha + 12 = 2(5\alpha + 10)$ | M1 |
$\alpha = \frac{4}{3}$ or 1.33 | A1 |
| | [4]6\\
\includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-3_463_621_255_762}
The diagram shows sector $O A B$ with centre $O$ and radius 11 cm . Angle $A O B = \alpha$ radians. Points $C$ and $D$ lie on $O A$ and $O B$ respectively. Arc $C D$ has centre $O$ and radius 5 cm .\\
(i) The area of the shaded region $A B D C$ is equal to $k$ times the area of the unshaded region $O C D$. Find $k$.\\
(ii) The perimeter of the shaded region $A B D C$ is equal to twice the perimeter of the unshaded region $O C D$. Find the exact value of $\alpha$.
\hfill \mbox{\textit{CAIE P1 2013 Q6 [7]}}