| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area of sector/segment problems |
| Difficulty | Moderate -0.5 This is a straightforward geometry problem requiring basic trigonometry (cosine rule) and sector area formulas. Part (i) is pure calculation with a given answer to verify, and part (ii) involves standard sector area subtraction. The multi-step nature and geometric setup add slight complexity, but all techniques are routine for P1 level with no novel insight required. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos A = \frac{8}{10} \rightarrow A = 0.6435\) | B1 | AG Allow other valid methods e.g. \(\sin A = \frac{6}{10}\) |
| Answer | Marks |
|---|---|
| Area \(\triangle ABC = \frac{1}{2} \times 16 \times 6\) or \(\frac{1}{2} \times 10 \times 16 \sin 0.6435 = 48\) | (M1A1) |
| Area 1 sector \(\frac{1}{2} \times 10^2 \times 0.6435\) | M1 |
| Shaded area \(= 2 \times their \text{ sector} - their \triangle ABC\) | M1 |
| Answer | Marks |
|---|---|
| \(\triangle BDE = 12\), \(\triangle BDC = 30\) | (B1 B1) |
| Sector \(= 32.18\) | M1 |
| \(2\times\text{segment} + \triangle BDE\) | M1 |
| \(= 16.4\) | A1 |
## Question 5(i):
$\cos A = \frac{8}{10} \rightarrow A = 0.6435$ | B1 | AG Allow other valid methods e.g. $\sin A = \frac{6}{10}$
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## Question 5(ii):
**EITHER:**
Area $\triangle ABC = \frac{1}{2} \times 16 \times 6$ or $\frac{1}{2} \times 10 \times 16 \sin 0.6435 = 48$ | (M1A1) |
Area 1 sector $\frac{1}{2} \times 10^2 \times 0.6435$ | M1 |
Shaded area $= 2 \times their \text{ sector} - their \triangle ABC$ | M1 |
**OR:**
$\triangle BDE = 12$, $\triangle BDC = 30$ | (B1 B1) |
Sector $= 32.18$ | M1 |
$2\times\text{segment} + \triangle BDE$ | M1 |
$= 16.4$ | A1 |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-08_446_844_260_648}
The diagram shows an isosceles triangle $A B C$ in which $A C = 16 \mathrm {~cm}$ and $A B = B C = 10 \mathrm {~cm}$. The circular arcs $B E$ and $B D$ have centres at $A$ and $C$ respectively, where $D$ and $E$ lie on $A C$.\\
(i) Show that angle $B A C = 0.6435$ radians, correct to 4 decimal places.\\
(ii) Find the area of the shaded region.\\
\hfill \mbox{\textit{CAIE P1 2017 Q5 [6]}}