| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Sequential triangle calculations (basic) |
| Difficulty | Moderate -0.8 This is a straightforward three-part question testing standard sine/cosine rule applications with given values. Part (a) uses the basic area formula (1/2)ab sin C, part (b) applies the cosine rule directly, and part (c) uses the sine rule. All steps are routine with no problem-solving or insight required, making it easier than average but not trivial due to multi-step nature. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{[Area]} = \frac{1}{2} \times 7.4 \times 5.26 \times \sin 63°\) | M1 | |
| \(= 17.3(407\ldots) \text{ [m}^2\text{]}\) | A1 | Accept any value from 17.3 to 17.341. 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(BC^2 = 5.26^2 + 7.4^2 - 2 \times 5.26 \times 7.4 \cos 63\) | M1 | RHS of cosine rule used |
| \(\ldots = 27.66(76) + 54.76 - 35.34(22\ldots)\) | m1 | Correct order of evaluation |
| \(\Rightarrow BC = \sqrt{47.08(5\ldots)} = 6.861(8\ldots)\) | ||
| \(BC = 6.86 \text{ [m] to 3sf}\) | A1 | AG. Cand. must show a 4th sf in either \(\sqrt{47.08(5\ldots)}\) or \(6.861(8)\) before giving the printed answer 6.86. 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\sin B}{\sin 63} = \frac{5.26}{BC}\) | M1 | Sine rule involving 'sin B' [If valid cosine rule used to find cos B, no marks awarded until stage of converting to sin B] |
| \(\sin B = 0.68 \text{ to 2sf}\) | A1 | If not 0.68, accept AWRT any value from 0.682 to 0.684 inclusive. 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin B = 0.68 \text{ to 2sf}\) | (M1), (A1) | (6.86...) could be c's ans (b). If not 0.68, accept AWRT any value from 0.682 to 0.684 inclusive |
**3(a)**
$\text{[Area]} = \frac{1}{2} \times 7.4 \times 5.26 \times \sin 63°$ | M1 |
$= 17.3(407\ldots) \text{ [m}^2\text{]}$ | A1 | Accept any value from 17.3 to 17.341. 2 marks total
**3(b)**
$BC^2 = 5.26^2 + 7.4^2 - 2 \times 5.26 \times 7.4 \cos 63$ | M1 | RHS of cosine rule used
$\ldots = 27.66(76) + 54.76 - 35.34(22\ldots)$ | m1 | Correct order of evaluation
$\Rightarrow BC = \sqrt{47.08(5\ldots)} = 6.861(8\ldots)$ |
$BC = 6.86 \text{ [m] to 3sf}$ | A1 | AG. Cand. must show a 4th sf in either $\sqrt{47.08(5\ldots)}$ or $6.861(8)$ before giving the printed answer 6.86. 3 marks total
**3(c)**
$\frac{\sin B}{\sin 63} = \frac{5.26}{BC}$ | M1 | Sine rule involving 'sin B' [If valid cosine rule used to find cos B, no marks awarded until stage of converting to sin B]
$\sin B = 0.68 \text{ to 2sf}$ | A1 | If not 0.68, accept AWRT any value from 0.682 to 0.684 inclusive. 2 marks total
**ALTn**
$\frac{1}{2} \times 7.4 \times (6.86\ldots) \times \sin B = \text{c's ans (a)}$
$\sin B = 0.68 \text{ to 2sf}$ | (M1), (A1) | (6.86...) could be c's ans (b). If not 0.68, accept AWRT any value from 0.682 to 0.684 inclusive
**Total for Q3: 7 marks**
---3 The diagram shows a triangle $A B C$.
\begin{tikzpicture}[scale=0.7]
% Coordinates
\coordinate[label=above:{$A$}] (A) at (4,7);
\coordinate[label=below left:{$B$}] (B) at (0,0);
\coordinate[label=right:{$C$}] (C) at (8,2);
% Triangle
\draw (A) -- (B) -- (C) -- cycle;
% Angle arc at A (63°)
\pic[draw, angle radius=8mm, angle eccentricity=1.45, "$63^\circ$" font=\small]
{angle = B--A--C};
% Side labels
\node[above left] at ($(A)!0.5!(B)$) {$7.4\,\mathrm{m}$};
\node[above right] at ($(A)!0.5!(C)$) {$5.26\,\mathrm{m}$};
\end{tikzpicture}
The size of angle $A$ is $63 ^ { \circ }$, and the lengths of $A B$ and $A C$ are 7.4 m and 5.26 m respectively.
\begin{enumerate}[label=(\alph*)]
\item Calculate the area of triangle $A B C$, giving your answer in $\mathrm { m } ^ { 2 }$ to three significant figures.
\item Show that the length of $B C$ is 6.86 m , correct to three significant figures.
\item Find the value of $\sin \boldsymbol { B }$ to two significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2009 Q3 [7]}}