| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| 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 C2 content: triangle area formula, cosine rule, arc length, and sector area. Part (a) is routine substitution into area = ½ab sin C. Part (b) applies cosine rule directly. Part (c) requires recognizing that arc AD has radius 8cm and calculating arc length and shaded area using sector formulas. All steps are standard textbook exercises with no novel problem-solving 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 |
|---|---|---|
| Correct exponential shape passing through \((0,1)\) | B1 | y-intercept = 1 stated or marked |
| Increasing curve | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\log 9 = \log 15\) | M1 | |
| \(x = \frac{\log 15}{\log 9} = 1.23\) | A1 | 3 s.f. |
| Answer | Marks |
|---|---|
| \(f(x) = 9^{-x}\) | B1 |
# Question 4:
**(a)**
Correct exponential shape passing through $(0,1)$ | B1 | y-intercept = 1 stated or marked
Increasing curve | B1 |
**(b)**
$x\log 9 = \log 15$ | M1 |
$x = \frac{\log 15}{\log 9} = 1.23$ | A1 | 3 s.f.
**(c)**
$f(x) = 9^{-x}$ | B1 |
---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]{48c5470e-6489-4b25-98a6-1b4e101ab01c-004_417_883_1436_557}
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 Q4}}