| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Area of region bounded by circle and line |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring cosine rule to find an angle, then standard sector area formulas and arc length calculations. While it involves two circles and requires careful setup, the techniques are routine P1 content with no novel problem-solving insight needed—slightly easier than average. |
| 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 |
|---|---|---|
| \(\cos BAO = \frac{6}{8}\) or \(\frac{8^2 + 12^2 - 8^2}{2\times8\times12}\) | M1 | Or other correct method |
| \(BAO = 0.723\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Sector \(ABC = \frac{1}{2}\times12^2\times\) *their* \(0.7227\) | *M1 | Accept 52.1 |
| Triangle \(AOB = \frac{1}{2}\times8\times12\sin(\text{their } 0.7227)\) or \(\frac{1}{2}\times12\times\sqrt{28}\) | *M1 | or \(\frac{1}{2}\times8\times8\sin(\pi - 2\times\text{their } 0.7227)\). Expect 31.7 or 31.8 |
| Shaded area \(=\) *their* \(52.0 -\) *their* \(31.7 = 20.3\) | DM1, A1 | M1 dependent on both previous M marks |
| Answer | Marks | Guidance |
|---|---|---|
| Arc \(BC = 12\times\) *their* \(0.7227\) | *M1 | Expect 8.67 |
| Perimeter \(= 8 + 4 +\) *their* \(8.67 = 20.7\) | DM1, A1 |
## Question 9(a):
| $\cos BAO = \frac{6}{8}$ or $\frac{8^2 + 12^2 - 8^2}{2\times8\times12}$ | M1 | Or other correct method |
|---|---|---|
| $BAO = 0.723$ | A1 | |
## Question 9(b):
| Sector $ABC = \frac{1}{2}\times12^2\times$ *their* $0.7227$ | *M1 | Accept 52.1 |
|---|---|---|
| Triangle $AOB = \frac{1}{2}\times8\times12\sin(\text{their } 0.7227)$ or $\frac{1}{2}\times12\times\sqrt{28}$ | *M1 | or $\frac{1}{2}\times8\times8\sin(\pi - 2\times\text{their } 0.7227)$. Expect 31.7 or 31.8 |
| Shaded area $=$ *their* $52.0 -$ *their* $31.7 = 20.3$ | DM1, A1 | M1 dependent on both previous M marks |
## Question 9(c):
| Arc $BC = 12\times$ *their* $0.7227$ | *M1 | Expect 8.67 |
|---|---|---|
| Perimeter $= 8 + 4 +$ *their* $8.67 = 20.7$ | DM1, A1 | |
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\includegraphics[max width=\textwidth, alt={}, center]{3b44e558-f91d-4175-acda-eceb70dad82c-12_497_652_260_744}
In the diagram, arc $A B$ is part of a circle with centre $O$ and radius 8 cm . Arc $B C$ is part of a circle with centre $A$ and radius 12 cm , where $A O C$ is a straight line.
\begin{enumerate}[label=(\alph*)]
\item Find angle $B A O$ in radians.
\item Find the area of the shaded region.
\item Find the perimeter of the shaded region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q9 [9]}}