| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Logo and design problems |
| Difficulty | Standard +0.3 This is a straightforward application of standard arc length and sector area formulas with given values. Part (a) uses s=rθ directly, part (b) requires adding arc length to triangle sides (using Pythagoras), and part (c) combines sector area and triangle area. All steps are routine calculations with no problem-solving insight 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 |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Arc length \(BD = r\theta = 6 \times 3.5 = 21\) (cm) | M1A1 | Correct method for arc length \(BD\). May work in degrees: \(\frac{\theta}{360} \times 2\pi \times 6\) where \(\theta\) between 200 and 201°. A1: 21 cm |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\angle AEB = \frac{3}{2}\pi - 3.5\) (= awrt 1.2) | B1 | Accept awrt 1.2 radians or awrt 69.5°. May appear anywhere. |
| \(AB^2 = 9^2 + 6^2 - 2 \times 9 \times 6\cos(\angle AEB)\) (= 79.1...) | M1 | Correctly uses cosine rule with lengths 6, 9 and their \(\angle AEB\) (\(\neq 3.5\)) |
| \(P = 6 + 9 + \text{"21"} + \text{"AB"}\) | dM1 | Adds lengths 6, 9, their (a), and their \(AB\). Do not allow \(AB = 9\) unless rounded from 8.89... Dependent on previous M. |
| \(P = \) awrt 44.9 (cm) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Area triangle \(ABE = \frac{1}{2} \times 6 \times 9\sin\!\left(\frac{3}{2}\pi - 3.5\right)\) or \(\frac{1}{2} \times 9 \times 6\cos(3.5 - \pi)\) (\(\approx 25.3\)) | M1 | Attempts area of triangle \(ABE\) with correct lengths 6 and 9 and their \(\angle AEB\). Equivalent working in degrees acceptable. |
| Area sector \(BDE = \frac{1}{2} \times 6^2 \times 3.5\) (= awrt 63) | B1 | \(\frac{1}{2} \times 6^2 \times 3.5\) (or implied by awrt 63) |
| \(\frac{1}{2} \times 6 \times 9\sin\!\left(\frac{3}{2}\pi - 3.5\right) + \frac{1}{2} \times 6^2 \times 3.5 = \text{"25.3"} + \text{"63"} = \ldots\) | dM1 | Attempts to add area of sector to area of triangle. Dependent on previous B and M marks. |
| Area = awrt \(88.3\ \text{cm}^2\) | A1 |
# Question 8:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Arc length $BD = r\theta = 6 \times 3.5 = 21$ (cm) | M1A1 | Correct method for arc length $BD$. May work in degrees: $\frac{\theta}{360} \times 2\pi \times 6$ where $\theta$ between 200 and 201°. A1: 21 cm |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\angle AEB = \frac{3}{2}\pi - 3.5$ (= awrt 1.2) | B1 | Accept awrt 1.2 radians or awrt 69.5°. May appear anywhere. |
| $AB^2 = 9^2 + 6^2 - 2 \times 9 \times 6\cos(\angle AEB)$ (= 79.1...) | M1 | Correctly uses cosine rule with lengths 6, 9 and their $\angle AEB$ ($\neq 3.5$) |
| $P = 6 + 9 + \text{"21"} + \text{"AB"}$ | dM1 | Adds lengths 6, 9, their (a), and their $AB$. Do not allow $AB = 9$ unless rounded from 8.89... Dependent on previous M. |
| $P = $ awrt 44.9 (cm) | A1 | |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Area triangle $ABE = \frac{1}{2} \times 6 \times 9\sin\!\left(\frac{3}{2}\pi - 3.5\right)$ or $\frac{1}{2} \times 9 \times 6\cos(3.5 - \pi)$ ($\approx 25.3$) | M1 | Attempts area of triangle $ABE$ with correct lengths 6 and 9 and their $\angle AEB$. Equivalent working in degrees acceptable. |
| Area sector $BDE = \frac{1}{2} \times 6^2 \times 3.5$ (= awrt 63) | B1 | $\frac{1}{2} \times 6^2 \times 3.5$ (or implied by awrt 63) |
| $\frac{1}{2} \times 6 \times 9\sin\!\left(\frac{3}{2}\pi - 3.5\right) + \frac{1}{2} \times 6^2 \times 3.5 = \text{"25.3"} + \text{"63"} = \ldots$ | dM1 | Attempts to add area of sector to area of triangle. Dependent on previous B and M marks. |
| Area = awrt $88.3\ \text{cm}^2$ | A1 | |
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{de511cb3-35c7-4225-b459-a136b6304b78-20_547_463_269_735}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Diagram not drawn to scale
Figure 2 shows the design for a company logo. The design consists of a triangle $A B E$ joined to a sector $B C D E$ of a circle with radius 6 cm and centre $E$. The line $A E$ is perpendicular to the line $D E$ and the length of $A E$ is 9 cm . The size of angle $D E B$ is 3.5 radians, as shown in Figure 2.
\begin{enumerate}[label=(\alph*)]
\item Find the length of the arc BCD.
Find, to one decimal place,
\item the perimeter of the logo,
\item the area of the logo.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2019 Q8 [10]}}