| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Shaded region with arc |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard applications of triangle area formula, cosine rule, arc length, and sector area. Part (a) uses Area = ½ab sin C (routine), part (b) applies cosine rule directly, and part (c) requires recognizing that arc AD has radius 8cm and calculating arc length and shaded area by subtraction. All steps are standard C2 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.05b 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 | Marks | Guidance |
| Area of triangle \(= \frac{1}{2}(12)(8)\sin\theta\) | M1 | Use of \(\frac{1}{2}ab\sin C\) or full equivalent |
| \(\sin\theta = \frac{20}{48}\ [=0.41(666\ldots)]\) | A1 | OE (giving 0.412 to 0.42) |
| \(\Rightarrow \theta = 0.4297(7\ldots) = 0.430\) to 3sf | A1 | AG (need to see \(>\)3sf value) |
| Total (a) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\{AB^2 =\} 8^2 + 12^2 - 2\times8\times12\times\cos\theta\) | M1 | |
| \(= 64 + 144 - 174.5\ldots\) | m1 | Accept 33 to 34 inclusive if three values not separate |
| \(\Rightarrow AB = 5.78\ldots = 5.8\) cm to 2sf | A1 | If not 2sf condone 5.78 to 5.79 inclusive. Condone \(\pm\) |
| Total (b) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Arc \(AD = 8\theta\); \(= 3.44\ldots = 3.4\) cm to 2sf | M1; A1 | If not 2sf condone 3.438 to 3.44 inclusive |
| Total (c)(i) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Area of sector \(= \frac{1}{2}r^2\theta\) | M1 | Stated or used [or 13.7(6..) seen] |
| Shaded area \(=\) Area of triangle \(-\) sector area | M1 | Difference of areas |
| Shaded area \(= 20 - 0.5\times8^2\times\theta = 6.2\) cm² to 2sf | A1 | Condone 6.24 to 6.2472 |
| Total (c)(ii) | 3 |
## Question 4(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area of triangle $= \frac{1}{2}(12)(8)\sin\theta$ | M1 | Use of $\frac{1}{2}ab\sin C$ or full equivalent |
| $\sin\theta = \frac{20}{48}\ [=0.41(666\ldots)]$ | A1 | OE (giving 0.412 to 0.42) |
| $\Rightarrow \theta = 0.4297(7\ldots) = 0.430$ to 3sf | A1 | AG (need to see $>$3sf value) |
| **Total (a)** | **3** | |
## Question 4(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\{AB^2 =\} 8^2 + 12^2 - 2\times8\times12\times\cos\theta$ | M1 | |
| $= 64 + 144 - 174.5\ldots$ | m1 | Accept 33 to 34 inclusive if three values not separate |
| $\Rightarrow AB = 5.78\ldots = 5.8$ cm to 2sf | A1 | If not 2sf condone 5.78 to 5.79 inclusive. Condone $\pm$ |
| **Total (b)** | **3** | |
## Question 4(c)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Arc $AD = 8\theta$; $= 3.44\ldots = 3.4$ cm to 2sf | M1; A1 | If not 2sf condone 3.438 to 3.44 inclusive |
| **Total (c)(i)** | **2** | |
## Question 4(c)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area of sector $= \frac{1}{2}r^2\theta$ | M1 | Stated or used [or 13.7(6..) seen] |
| Shaded area $=$ Area of triangle $-$ sector area | M1 | Difference of areas |
| Shaded area $= 20 - 0.5\times8^2\times\theta = 6.2$ cm² to 2sf | A1 | Condone 6.24 to 6.2472 |
| **Total (c)(ii)** | **3** | |
---4 The triangle $A B C$, shown in the diagram, is such that $A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}$ and angle $A C B = \theta$ radians.
The area of triangle $A B C = 20 \mathrm {~cm} ^ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\theta = 0.430$ correct to three significant figures.
\item Use the cosine rule to calculate the length of $A B$, giving your answer to two significant figures.
\item The point $D$ lies on $C B$ such that $A D$ is an arc of a circle centre $C$ and radius 8 cm . The region bounded by the arc $A D$ and the straight lines $D B$ and $A B$ is shaded in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-3_424_894_1434_555}
Calculate, to two significant figures:
\begin{enumerate}[label=(\roman*)]
\item the length of the $\operatorname { arc } A D$;
\item the area of the shaded region.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q4 [11]}}