| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape area |
| Difficulty | Standard +0.3 This is a multi-part question on arc length and sector area with compound shapes, requiring standard formulas and careful calculation. While it involves several steps and combining areas, it's a typical P1 application question with no novel insight required—slightly easier than average due to being straightforward formula application with clear geometric setup. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2.5\times\frac{4\pi}{3}+2.24\times\frac{5\pi}{6}\ [=10.47[2]+5.86[4]\) or \(\frac{10\pi}{3}+\frac{28\pi}{15}]\) | B1 | For either arc correct. Arc ARB could be AR+RB |
| M1 | For adding two (or three) arc lengths using different radii and angles and nothing else. SOI | |
| \(16.34\) or \(\frac{26\pi}{5}\) | A1 | AWRT. Condone 16.33 only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Area \(AOB=\frac{1}{2}\times2.5^2\sin\frac{2\pi}{3}\ [=2.706]\) ; Area \(APB=\frac{1}{2}\times2.24^2\sin\frac{5\pi}{6}\ [=1.254]\) | M1 | For either \(\triangle AOB\) or \(\triangle APB\) (\(AB=4.33\), \(h=1.25\), \(0.58\)) or any other valid method |
| [Difference \(=\)] \(1.45\) | A1 | AWRT. Condone 1.46 only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Area \(AOB=\frac{1}{2}\times2.5^2\times\frac{4\pi}{3}\ [=13.09]\); Area \(APB=\frac{1}{2}\times2.24^2\times\frac{5\pi}{6}\ [=6.57]\) | B1 | For either sector area correct |
| \(\frac{1}{2}\times2.5^2\times\frac{4\pi}{3}+\frac{1}{2}\times2.24^2\times\frac{5\pi}{6}+\text{"their }10(b)\text{"}\) \([=13.09+6.57+\text{"their }10(b)\text{"}]\) | M1 | Adding two sector areas from different sectors and 'their 10(b)' and nothing else. SOI |
| \(21.1\) | A1 | CAO. Condone slight inaccuracies in intermediate working if correct answer arrived at |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $2.5\times\frac{4\pi}{3}+2.24\times\frac{5\pi}{6}\ [=10.47[2]+5.86[4]$ or $\frac{10\pi}{3}+\frac{28\pi}{15}]$ | B1 | For either arc correct. Arc ARB could be AR+RB |
| | M1 | For adding two (or three) arc lengths using different radii and angles and nothing else. SOI |
| $16.34$ or $\frac{26\pi}{5}$ | A1 | AWRT. Condone 16.33 only |
---
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Area $AOB=\frac{1}{2}\times2.5^2\sin\frac{2\pi}{3}\ [=2.706]$ ; Area $APB=\frac{1}{2}\times2.24^2\sin\frac{5\pi}{6}\ [=1.254]$ | M1 | For either $\triangle AOB$ or $\triangle APB$ ($AB=4.33$, $h=1.25$, $0.58$) or any other valid method |
| [Difference $=$] $1.45$ | A1 | AWRT. Condone 1.46 only |
---
## Question 10(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Area $AOB=\frac{1}{2}\times2.5^2\times\frac{4\pi}{3}\ [=13.09]$; Area $APB=\frac{1}{2}\times2.24^2\times\frac{5\pi}{6}\ [=6.57]$ | B1 | For either sector area correct |
| $\frac{1}{2}\times2.5^2\times\frac{4\pi}{3}+\frac{1}{2}\times2.24^2\times\frac{5\pi}{6}+\text{"their }10(b)\text{"}$ $[=13.09+6.57+\text{"their }10(b)\text{"}]$ | M1 | Adding two sector areas from different sectors and 'their 10(b)' and nothing else. SOI |
| $21.1$ | A1 | CAO. Condone slight inaccuracies in intermediate working if correct answer arrived at |
---\begin{enumerate}[label=(\alph*)]
\item Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
\item Find the difference in area of the two triangles $A O B$ and $A P B$, giving your answer correct to 2 decimal places.
\item Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q10 [8]}}