| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Sector with attached triangle |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring standard arc length formula (s=rθ), basic trigonometry in triangles, and area calculations. Part (a) is direct application of arc length formula, part (b) uses simple trigonometry with given lengths, and part (c) combines sector area with triangle areas. All steps are routine with clear scaffolding and no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(OC \times 2.3 = 27.6\) | M1 | Uses \(l = r\theta\) with \(l = 27.6\) and \(\theta = 2.3\) correctly substituted |
| \(OC = \frac{27.6}{2.3} = 12\) m | A1* | Achieves expression for \(OC\) before proceeding to \(OC = 12\)(m) with no errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \((2AOB =)\ \pi - 2.3\) | M1 | Attempts to subtract 2.3 from \(\pi\) |
| \(\frac{\pi - 2.3}{2} \Rightarrow 0.421\) rad | A1* | Achieves 0.421 (rad) with no errors seen |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Area \(OCDE = \frac{1}{2} \times 12^2 \times 2.3\) | M1 | Attempts to use \(A = \frac{1}{2}r^2\theta\) with \(r=12\) and \(\theta = 2.3\) |
| \(= 165.6\) m² (accept awrt 166) | A1 | awrt 166 (may be implied by later work) |
| \((OB =)\ \frac{35 - 27.6}{2} + 12 = 15.7\) m | B1 | Correct expression or value for length \(OB\) or \(OF\) |
| Area of \(OAB = \frac{1}{2} \times 15.7 \times 7.5 \times \sin 0.421\ (= 24.0...\) m²\()\) | M1 | Attempts area of at least one congruent triangle using \(OB\) from \(\frac{35-27.6}{2}+12\), \(OA = 7.5\), \(\theta = 0.421\) in \(\frac{1}{2} \times OA \times OB \times \sin C\) |
| Total area \(= ``165.6" + 2 \times ``24.1"\) | dM1 | Combines appropriate areas; dependent on previous method marks and B mark |
| \(=\) awrt \(214\) m² | A1 | awrt 214 (m²); condone lack of units |
## Question 8:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $OC \times 2.3 = 27.6$ | M1 | Uses $l = r\theta$ with $l = 27.6$ and $\theta = 2.3$ correctly substituted |
| $OC = \frac{27.6}{2.3} = 12$ m | A1* | Achieves expression for $OC$ before proceeding to $OC = 12$(m) with no errors seen |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $(2AOB =)\ \pi - 2.3$ | M1 | Attempts to subtract 2.3 from $\pi$ |
| $\frac{\pi - 2.3}{2} \Rightarrow 0.421$ rad | A1* | Achieves 0.421 (rad) with no errors seen |
### Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Area $OCDE = \frac{1}{2} \times 12^2 \times 2.3$ | M1 | Attempts to use $A = \frac{1}{2}r^2\theta$ with $r=12$ and $\theta = 2.3$ |
| $= 165.6$ m² (accept awrt 166) | A1 | awrt 166 (may be implied by later work) |
| $(OB =)\ \frac{35 - 27.6}{2} + 12 = 15.7$ m | B1 | Correct expression or value for length $OB$ or $OF$ |
| Area of $OAB = \frac{1}{2} \times 15.7 \times 7.5 \times \sin 0.421\ (= 24.0...$ m²$)$ | M1 | Attempts area of at least one congruent triangle using $OB$ from $\frac{35-27.6}{2}+12$, $OA = 7.5$, $\theta = 0.421$ in $\frac{1}{2} \times OA \times OB \times \sin C$ |
| Total area $= ``165.6" + 2 \times ``24.1"$ | dM1 | Combines appropriate areas; dependent on previous method marks and B mark |
| $=$ awrt $214$ m² | A1 | awrt 214 (m²); condone lack of units |
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-18_505_1301_257_572}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the plan view of a stage.\\
The plan view shows two congruent triangles $A B O$ and $G F O$ joined to a sector $O C D E O$ of a circle, centre $O$, where
\begin{itemize}
\item angle $C O E = 2.3$ radians
\item arc length $C D E = 27.6 \mathrm {~m}$
\item $A O G$ is a straight line of length 15 m
\begin{enumerate}[label=(\alph*)]
\item Show that $O C = 12 \mathrm {~m}$.
\item Show that the size of angle $A O B$ is 0.421 radians correct to 3 decimal places.
\end{itemize}
Given that the total length of the front of the stage, $B C D E F$, is 35 m ,
\item find the total area of the stage, giving your answer to the nearest square metre.
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 2023 Q8 [10]}}