| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Radians, Arc Length and Sector Area |
| Type | Sector with attached triangle |
| Difficulty | Moderate -0.3 This is a straightforward application of standard sector and triangle formulas with minimal problem-solving required. Part (a) uses the sector area formula directly, part (b) requires finding triangle area using ½absinC (all values given), and part (c) applies arc length formula plus Pythagoras or cosine rule. All necessary information is provided explicitly, making this slightly easier than average for C2. |
| Spec | 1.05c 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 |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | Usually answered in radians: Uses Area \(ZYW = \frac{1}{2} \times 5^2 \times (angle) = 12.5 \times 0.7 = 8.75\) o.e. (cm²) | M1 A1 |
| (2) | ||
| (b) | Area of triangle \(XYZ = \frac{1}{2} \times 7 \times 5 \times \sin Y = (11.273)\) (cm²) | M1 |
| Area of whole flag = "8.75" + "11.273" = 20.02 (cm²) | M1, A1 | M1: for adding two numerical areas – triangle and sector (not dependent on previous M marks). A1: for 20.02 (do not need to see units) (Allow answers which round to 20.02 e.g. do not allow 20.07 or allow 20.05) |
| (3) | ||
| (c) | \((XZ)^2 = 7^2 + 5^2 - 2 \times 7 \times 5 \cos(\pi - 0.7)\) Or \((XZ)^2 = (7 + 5\cos 0.7)^2 + (5\sin 0.7)^2\) | M1, M1, ddM1 |
| Use of arc length formula \(s = 50\) (= 3.5) | ||
| Total perimeter = 12 + "3.5" + "11.293" = 26.79 cm | A1 | A1: 26.79 – allow awrt |
| (4) | ||
| (9 marks) |
| **Part** | **Answer/Working** | **Marks** | **Notes** |
|---|---|---|---|
| (a) | **Usually answered in radians:** Uses Area $ZYW = \frac{1}{2} \times 5^2 \times (angle) = 12.5 \times 0.7 = 8.75$ o.e. (cm²) | M1 A1 | M1: uses $A = 12.5 \times 0$ with $\theta$ in radians or completely correct work in degrees. (If angle given as 0.7 π and formula not quoted correctly do not give this mark). A1: 8.75 or $\frac{35}{4}$ or equivalent (do not need to see units) |
| | | **(2)** | |
| (b) | Area of triangle $XYZ = \frac{1}{2} \times 7 \times 5 \times \sin Y = (11.273)$ (cm²) | M1 | M1: for use of $A = \frac{1}{2} \times 7 \times 5 \times \sin Y$ (where $Y = 0.7$ or attempt at $(\pi - 0.7)$ they give same answer) Do not need to see 11.273 (Do not allow use of 0.7 or $\pi - 0.7$ instead of their respective sines). This may arise from use of $A = \frac{1}{2} \times a \times b \times \sin C$ formula or from $A = \frac{1}{2} \times b \times h$ with $h$ found by correct method so either $A = \frac{1}{2} \times 7 \times (5\sin Y)$ or $A = \frac{1}{2} \times 5 \times (7\sin Y)$ |
| | Area of whole flag = "8.75" + "11.273" = 20.02 (cm²) | M1, A1 | M1: for adding two numerical areas – triangle and sector (not dependent on previous M marks). A1: for 20.02 (do not need to see units) (Allow answers which round to 20.02 e.g. do not allow 20.07 or allow 20.05) |
| | | **(3)** | |
| (c) | $(XZ)^2 = 7^2 + 5^2 - 2 \times 7 \times 5 \cos(\pi - 0.7)$ Or $(XZ)^2 = (7 + 5\cos 0.7)^2 + (5\sin 0.7)^2$ | M1, M1, ddM1 | M1: Uses cosine rule with correct angle (allow 2.4) or uses right angle triangle with correct sides. (do not need see $XZ = 11.293$) This may calculated in part (b). M1: Uses arc length formula $s = 50$ (= 3.5). ddM1: (Needs to have earned both previous M marks) Adds 7 + 5 + their arc length + their $XZ$. This mark should not be awarded if they use their answer for $XZ^2$ instead of $XZ$. |
| | Use of arc length formula $s = 50$ (= 3.5) | | |
| | Total perimeter = 12 + "3.5" + "11.293" = 26.79 cm | A1 | A1: 26.79 – allow awrt |
| | | **(4)** | |
| | | **(9 marks)** | |
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8daf56fa-bfce-454e-bbb8-fecd8170d77e-10_310_716_214_621}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Not to scale
Figure 2 shows a flag $X Y W Z X$.
The flag consists of a triangle $X Y Z$ joined to a sector $Z Y W$ of a circle with radius 5 cm and centre $Y$.
The angle of the sector, angle $Z Y W$, is 0.7 radians.
The points $X , Y$ and $W$ lie on a straight line with $X Y = 7 \mathrm {~cm}$ and $Y W = 5 \mathrm {~cm}$.
Find
\begin{enumerate}[label=(\alph*)]
\item the area of the sector $Z Y W$ in $\mathrm { cm } ^ { 2 }$,
\item the area of the flag, in $\mathrm { cm } ^ { 2 }$, to 2 decimal places,
\item the length of the perimeter, $X Y W Z X$, of the flag, in cm to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2018 Q4 [9]}}