| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Area of region bounded by circle and line |
| Difficulty | Standard +0.8 This question requires understanding of rhombus properties, sector areas, and coordinate geometry with radians. Students must find two sector areas, subtract them, work with non-standard angles (1.2 radians), and apply the cosine rule or similar techniques to find PQ. The multi-step nature and integration of several topics (geometry, trigonometry, circles) makes this moderately challenging, though the techniques themselves are standard A-level material. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{2} \times 5^2 \times 1.2\) | B1 | |
| \(\frac{1}{2} \times 5^2 \times \sin 1.2\) | B1 | |
| \(2\left[\frac{1}{2} \times 5^2 \times 1.2 - \frac{1}{2} \times 5^2 \times \sin 1.2\right]\) | M1 | Subtraction and multiplication by 2 |
| \(6.70\) | A1 | Accept 6.7 or anything rounding to 6.70 |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(5\cos 0.6\) | M1 | |
| \(5 - \text{"} 5\cos 0.6\text{"}\) | M1 | Subtraction from 5 |
| \(10(1 - \cos 0.6)\) | M1 | Multiplication by 2 |
| \(1.75\) | A1 | |
| [4] |
## Question 8:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{2} \times 5^2 \times 1.2$ | B1 | |
| $\frac{1}{2} \times 5^2 \times \sin 1.2$ | B1 | |
| $2\left[\frac{1}{2} \times 5^2 \times 1.2 - \frac{1}{2} \times 5^2 \times \sin 1.2\right]$ | M1 | Subtraction and multiplication by 2 |
| $6.70$ | A1 | Accept 6.7 or anything rounding to 6.70 |
| **[4]** | | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $5\cos 0.6$ | M1 | |
| $5 - \text{"} 5\cos 0.6\text{"}$ | M1 | Subtraction from 5 |
| $10(1 - \cos 0.6)$ | M1 | Multiplication by 2 |
| $1.75$ | A1 | |
| **[4]** | | |
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_600_883_1731_630}
The diagram shows a rhombus $A B C D$. Points $P$ and $Q$ lie on the diagonal $A C$ such that $B P D$ is an arc of a circle with centre $C$ and $B Q D$ is an arc of a circle with centre $A$. Each side of the rhombus has length 5 cm and angle $B A D = 1.2$ radians.\\
(i) Find the area of the shaded region $B P D Q$.\\
(ii) Find the length of $P Q$.
\hfill \mbox{\textit{CAIE P1 2010 Q8 [8]}}