| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Area of region bounded by circle and line |
| Difficulty | Standard +0.8 This is a multi-part question requiring understanding of circle geometry, radians, and sector areas. Part (a) uses the inscribed angle theorem (angle at center is twice angle at circumference), part (b) requires applying the cosine rule in triangle ACO, and part (c) involves calculating the difference between two sector areas. While each individual step uses standard techniques, the combination and the need to recognize geometric relationships makes this moderately challenging, above average difficulty. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Angle \(ACO = 0.7\) | B1 | Don't allow AWRT 0.7. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([R =]\ 1.53r\) | B1 | Allow AWRT \(1.53r\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sector \(OAB = \frac{1}{2}r^2 \times 2.8\ [= 1.4r^2]\) | B1 | |
| Sector \(CAB = \frac{1}{2}(\text{their } R)^2 \times 2 \times \text{their } 0.7\) | *M1 | |
| \(1.638r^2\) | A1 | Allow AWRT \(1.64r^2\). |
| \([2] \times \frac{1}{2}r^2 \sin(\pi - 1.4)\) OR \([2] \times \frac{1}{2}r \times \text{their } R \sin 0.7\) | *M1 | |
| \(2 \times 0.4927r^2\) | A1 | Allow AWRT \(0.98r^2\) to \(0.99r^2\). |
| \(1.4r^2 - (\text{their } 1.638r^2 - \text{their } 0.985r^2)\) | DM1 | |
| \(0.747r^2\) to \(0.748r^2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Finding any useful sector area of circle radius \(r\) | B1 | May be 'nested' in a segment. |
| Finding the area of sector CAB | *M1A1 | May be 'nested' in a segment. |
| Finding the area of one useful triangle | *M1 | May be 'nested' in a segment. |
| Finding the total area of useful triangles | A1 | May be 'nested' in a segment. |
| A correct plan for the shaded area | DM1 | |
| \(0.747r^2\) to \(0.748r^2\) | A1 |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Angle $ACO = 0.7$ | B1 | Don't allow AWRT 0.7. |
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[R =]\ 1.53r$ | B1 | Allow AWRT $1.53r$. |
## Question 10(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sector $OAB = \frac{1}{2}r^2 \times 2.8\ [= 1.4r^2]$ | B1 | |
| Sector $CAB = \frac{1}{2}(\text{their } R)^2 \times 2 \times \text{their } 0.7$ | *M1 | |
| $1.638r^2$ | A1 | Allow AWRT $1.64r^2$. |
| $[2] \times \frac{1}{2}r^2 \sin(\pi - 1.4)$ OR $[2] \times \frac{1}{2}r \times \text{their } R \sin 0.7$ | *M1 | |
| $2 \times 0.4927r^2$ | A1 | Allow AWRT $0.98r^2$ to $0.99r^2$. |
| $1.4r^2 - (\text{their } 1.638r^2 - \text{their } 0.985r^2)$ | DM1 | |
| $0.747r^2$ to $0.748r^2$ | A1 | |
**Alternative methods guidance:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Finding any useful sector area of circle radius $r$ | B1 | May be 'nested' in a segment. |
| Finding the area of sector CAB | *M1A1 | May be 'nested' in a segment. |
| Finding the area of one useful triangle | *M1 | May be 'nested' in a segment. |
| Finding the total area of useful triangles | A1 | May be 'nested' in a segment. |
| A correct plan for the shaded area | DM1 | |
| $0.747r^2$ to $0.748r^2$ | A1 | |10\\
\includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-12_552_582_255_778}
The diagram shows points $A , B$ and $C$ lying on a circle with centre $O$ and radius $r$. Angle $A O B$ is 2.8 radians. The shaded region is bounded by two arcs. The upper arc is part of the circle with centre $O$ and radius $r$. The lower arc is part of a circle with centre $C$ and radius $R$.
\begin{enumerate}[label=(\alph*)]
\item State the size of angle $A C O$ in radians.
\item Find $R$ in terms of $r$.
\item Find the area of the shaded region in terms of $r$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q10 [9]}}