| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape perimeter |
| Difficulty | Standard +0.3 This is a standard compound shape problem requiring arc length and sector area formulas with radians. While it involves multiple steps and algebraic manipulation with surds, it's a routine application of well-practiced techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Recognise that at least one of angles \(A\), \(B\), \(C\) is \(\frac{\pi}{3}\) | B1 | SOI; allow \(60°\). |
| One arc \(6 \times their\ \frac{\pi}{3}\) leading to two arcs \(2 \times 6 \times their\ \frac{\pi}{3}\) | M1 | SOI e.g. may see \(2\pi\) or \(4\pi\). Use of correct formula for length of arc and multiply by 2. |
| Perimeter \(= 6 + 4\pi\) | A1 | Must be exact value. |
| Alternative: Calculate circumference of whole circle \(= 12\pi\) | B1 | |
| One arc \(\frac{1}{6}\times 12\pi\) leading to two arcs \(2\times\frac{1}{6}\times 12\pi\) | M1 | SOI e.g. may see \(2\pi\) or \(4\pi\). |
| Perimeter \(= 6 + 4\pi\) | A1 | Must be exact value. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sector \(= \frac{1}{2}\times 6^2 \times their\!\left(\frac{\pi}{3}\right)\) | M1 | Use of correct formula for area of sector. SOI e.g. may see \(6\pi\) or \(12\pi\). |
| \(\frac{1}{2}\times(6^2)\times their\!\left(\frac{\pi}{3}\right) - \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right) + 6\pi \left[= 6\pi - 9\sqrt{3} + 6\pi\right]\) | M1 A1 | M1 for attempt at strategy with values substituted: area of segment + area of sector. A1 if correct (unsimplified). |
| Area \(= 12\pi - 9\sqrt{3}\) | A1 | Must be simplified exact value. |
| Alternative 1: \(2\times\!\left(\frac{1}{2}\times 6^2\times their\!\left(\frac{\pi}{3}\right)\right) - \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right)\) | M1 A1 | M1 strategy: \(2\times\textbf{sector} - \textbf{triangle}\). A1 if correct (unsimplified). |
| Area \(= 12\pi - 9\sqrt{3}\) | A1 | Must be simplified exact value. |
| Alternative 2: \(2\times\!\left(\frac{1}{2}\times 6^2\times their\!\left(\frac{\pi}{3}\right) - \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right)\right) + \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right) \left[= 12\pi - 18\sqrt{3} + 9\sqrt{3}\right]\) | M1 A1 | M1 strategy: \(2\times\textbf{segment} + \textbf{triangle}\). A1 if correct (unsimplified). |
| Area \(\left[= 6\pi - 9\sqrt{3} + 6\pi\right] = 12\pi - 9\sqrt{3}\) | A1 | Must be simplified exact value. |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Recognise that at least one of angles $A$, $B$, $C$ is $\frac{\pi}{3}$ | B1 | SOI; allow $60°$. |
| One arc $6 \times their\ \frac{\pi}{3}$ leading to two arcs $2 \times 6 \times their\ \frac{\pi}{3}$ | M1 | SOI e.g. may see $2\pi$ or $4\pi$. Use of correct formula for length of arc and multiply by 2. |
| Perimeter $= 6 + 4\pi$ | A1 | Must be exact value. |
| **Alternative:** Calculate circumference of whole circle $= 12\pi$ | B1 | |
| One arc $\frac{1}{6}\times 12\pi$ leading to two arcs $2\times\frac{1}{6}\times 12\pi$ | M1 | SOI e.g. may see $2\pi$ or $4\pi$. |
| Perimeter $= 6 + 4\pi$ | A1 | Must be exact value. |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sector $= \frac{1}{2}\times 6^2 \times their\!\left(\frac{\pi}{3}\right)$ | M1 | Use of correct formula for area of sector. SOI e.g. may see $6\pi$ or $12\pi$. |
| $\frac{1}{2}\times(6^2)\times their\!\left(\frac{\pi}{3}\right) - \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right) + 6\pi \left[= 6\pi - 9\sqrt{3} + 6\pi\right]$ | M1 A1 | M1 for attempt at strategy with values substituted: **area of segment + area of sector**. A1 if correct (unsimplified). |
| Area $= 12\pi - 9\sqrt{3}$ | A1 | Must be simplified exact value. |
| **Alternative 1:** $2\times\!\left(\frac{1}{2}\times 6^2\times their\!\left(\frac{\pi}{3}\right)\right) - \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right)$ | M1 A1 | M1 strategy: $2\times\textbf{sector} - \textbf{triangle}$. A1 if correct (unsimplified). |
| Area $= 12\pi - 9\sqrt{3}$ | A1 | Must be simplified exact value. |
| **Alternative 2:** $2\times\!\left(\frac{1}{2}\times 6^2\times their\!\left(\frac{\pi}{3}\right) - \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right)\right) + \frac{1}{2}\times(6^2)\times\sin\!\left(their\!\left(\frac{\pi}{3}\right)\right) \left[= 12\pi - 18\sqrt{3} + 9\sqrt{3}\right]$ | M1 A1 | M1 strategy: $2\times\textbf{segment} + \textbf{triangle}$. A1 if correct (unsimplified). |
| Area $\left[= 6\pi - 9\sqrt{3} + 6\pi\right] = 12\pi - 9\sqrt{3}$ | A1 | Must be simplified exact value. |\begin{enumerate}[label=(\alph*)]
\item Find the perimeter of the plate, giving your answer in terms of $\pi$.
\item Find the area of the plate, giving your answer in terms of $\pi$ and $\sqrt { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q6 [7]}}