| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 5 |
| 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 geometric problem requiring identification of an equilateral triangle (angle CAO = π/3), then applying sector and triangle area formulas. The algebra is straightforward with minimal steps, making it slightly easier than average for A-level. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks |
|---|---|
| Angle \(CAO = \frac{\pi}{3}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((\text{Sector } AOC) = \frac{1}{2}r^2 \times their\,\frac{\pi}{3}\) | M1 | SOI |
| \((\Delta ABC) = \frac{1}{2}(r)(2r)\sin\!\left(their\,\frac{\pi}{3}\right)\) or \(\frac{1}{2}(2r)(r)\frac{\sqrt{3}}{2}\) or \(\frac{1}{2}(r)(r)\sqrt{3}\) | M1 | For M1M1, \(their\,\frac{\pi}{3}\) must be of the form \(k\pi\) where \(0 < k < \frac{1}{2}\) |
| \((\Delta ABC) = \frac{1}{2}(r)(2r)\sin\!\left(\frac{\pi}{3}\right)\) or \(\frac{1}{2}(2r)(r)\frac{\sqrt{3}}{2}\) or \(\frac{1}{2}(r)(r)\sqrt{3}\) | A1 | All correct |
| \(r^2\!\left(\frac{\sqrt{3}}{2}\right) - \frac{1}{2}r^2\!\left(\frac{\pi}{3}\right)\) | A1 |
## Question 4:
**Part (i)**
Angle $CAO = \frac{\pi}{3}$ | **B1** |
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**Part (ii)**
$(\text{Sector } AOC) = \frac{1}{2}r^2 \times their\,\frac{\pi}{3}$ | **M1** | SOI
$(\Delta ABC) = \frac{1}{2}(r)(2r)\sin\!\left(their\,\frac{\pi}{3}\right)$ **or** $\frac{1}{2}(2r)(r)\frac{\sqrt{3}}{2}$ **or** $\frac{1}{2}(r)(r)\sqrt{3}$ | **M1** | For M1M1, $their\,\frac{\pi}{3}$ must be of the form $k\pi$ where $0 < k < \frac{1}{2}$
$(\Delta ABC) = \frac{1}{2}(r)(2r)\sin\!\left(\frac{\pi}{3}\right)$ **or** $\frac{1}{2}(2r)(r)\frac{\sqrt{3}}{2}$ **or** $\frac{1}{2}(r)(r)\sqrt{3}$ | **A1** | All correct
$r^2\!\left(\frac{\sqrt{3}}{2}\right) - \frac{1}{2}r^2\!\left(\frac{\pi}{3}\right)$ | **A1** |4\\
\includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-05_360_639_255_753}
The diagram shows a semicircle $A C B$ with centre $O$ and radius $r$. Arc $O C$ is part of a circle with centre $A$.\\
(i) Express angle $C A O$ in radians in terms of $\pi$.\\
(ii) Find the area of the shaded region in terms of $r , \pi$ and $\sqrt { } 3$, simplifying your answer.\\
\hfill \mbox{\textit{CAIE P1 2019 Q4 [5]}}