| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Compound shape area |
| Difficulty | Moderate -0.3 This is a multi-part question involving standard sector area formula, cosine rule for finding an angle, and combining areas of basic shapes. While it requires careful organization across three parts and involves both triangles and a sector, each individual step uses routine A-level techniques without requiring novel insight or particularly challenging problem-solving. |
| Spec | 1.03c Straight line models: in variety of contexts1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2} \times 6^2 \times 1.3 = \ldots\) | M1 | Attempts \(\frac{1}{2}r^2\theta\) with \(r=6\), \(\theta=1.3\); may use degree equivalent ~74° or 75° |
| \(= 23.4 \text{ (m}^2\text{)}\) | A1 | Allow equivalent values e.g. \(\frac{117}{5}\), \(23\frac{2}{5}\); correct answer only scores both marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(12.2^2 = 6^2 + 10.8^2 - 2 \times 6 \times 10.8\cos(ABE)\) | M1 | Correct numerical application of cosine rule for required angle |
| \(\cos(ABE) = \frac{6^2 + 10.8^2 - 12.2^2}{2 \times 6 \times 10.8} \left(= \frac{19}{648}\right)\) | A1 | |
| \(ABE = 1.54\) | A1 | For awrt 1.54 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Area \(ABE = \frac{1}{2} \times 10.8 \times 6\sin(ABE)\) | M1 | Correct method for area \(ABE\) |
| Area \(BCD = \frac{1}{2} \times 6\cos(\pi - 1.3 - \text{"1.54"}) \times 6\sin(\pi - 1.3 - \text{"1.54"})\) | M1 | Correct method for area \(BCD\) including the \(\frac{1}{2}\); do NOT allow mixing of degrees and radians |
| Total area \(= 60.9\text{m}^2\) | A1 | Allow awrt 60.9; condone lack of units |
## Question 5(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2} \times 6^2 \times 1.3 = \ldots$ | M1 | Attempts $\frac{1}{2}r^2\theta$ with $r=6$, $\theta=1.3$; may use degree equivalent ~74° or 75° |
| $= 23.4 \text{ (m}^2\text{)}$ | A1 | Allow equivalent values e.g. $\frac{117}{5}$, $23\frac{2}{5}$; correct answer only scores both marks |
---
## Question 5(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $12.2^2 = 6^2 + 10.8^2 - 2 \times 6 \times 10.8\cos(ABE)$ | M1 | Correct numerical application of cosine rule for required angle |
| $\cos(ABE) = \frac{6^2 + 10.8^2 - 12.2^2}{2 \times 6 \times 10.8} \left(= \frac{19}{648}\right)$ | A1 | |
| $ABE = 1.54$ | A1 | For awrt 1.54 |
---
## Question 5(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Area $ABE = \frac{1}{2} \times 10.8 \times 6\sin(ABE)$ | M1 | Correct method for area $ABE$ |
| Area $BCD = \frac{1}{2} \times 6\cos(\pi - 1.3 - \text{"1.54"}) \times 6\sin(\pi - 1.3 - \text{"1.54"})$ | M1 | Correct method for area $BCD$ including the $\frac{1}{2}$; do NOT allow mixing of degrees and radians |
| Total area $= 60.9\text{m}^2$ | A1 | Allow awrt 60.9; condone lack of units |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7e2b7c81-e678-4078-964b-8b78e3b63f43-10_529_1403_255_267}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the plan view of a garden.\\
The shape of the garden $A B C D E A$ consists of a triangle $A B E$ and a right-angled triangle $B C D$ joined to a sector $B D E$ of a circle with radius 6 m and centre $B$.
The points $A , B$ and $C$ lie on a straight line with $A B = 10.8 \mathrm {~m}$\\
Angle $B C D = \frac { \pi } { 2 }$ radians, angle $E B D = 1.3$ radians and $A E = 12.2 \mathrm {~m}$
\begin{enumerate}[label=(\alph*)]
\item Find the area of the sector $B D E$, giving your answer in $\mathrm { m } ^ { 2 }$
\item Find the size of angle $A B E$, giving your answer in radians to 2 decimal places.
\item Find the area of the garden, giving your answer in $\mathrm { m } ^ { 2 }$ to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P1 2024 Q5 [7]}}