| Exam Board | WJEC |
|---|---|
| Module | Unit 1 (Unit 1) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Algebraic side lengths |
| Difficulty | Standard +0.3 This is a straightforward application of the cosine rule to set up a quadratic equation, followed by using the sine rule. The problem requires standard techniques with clear setup (given angle, side relationship), making it slightly easier than average but still requiring multi-step algebraic manipulation. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Choice of variable (x) for \(AB \Rightarrow AC = x + 2\) | B1 | AO3 |
| \((x + 2)^2 = x^2 + 12^2 – 2 \times x \times 12 \times \frac{2}{3}\) | M1 | AO3 |
| \(x^2 + 4x + 4 = x^2 + 144 – 16x\) | A1 | AO3 |
| \(20x = 140 \Rightarrow x = 7\) | A1 | AO3 |
| \(AB = 7\), \(AC = 9\) | A1 | AO3 (Amend proof for candidates who choose \(AC = x\)) |
| (b) \(\sin \angle ABC = \frac{\sqrt{5}}{3}\) | B1 | AO1 |
| \(\frac{\sin \angle BAC}{12} = \frac{\sin \angle ABC}{9}\) | M1 | AO1 (f.t. candidate's derived values for \(AC\) and \(\sin \angle ABC\)) |
| \(\sin \angle BAC = \frac{4\sqrt{5}}{9}\) | A1 | AO1 (c.a.o.) |
**(a)** Choice of variable (x) for $AB \Rightarrow AC = x + 2$ | B1 | AO3
$(x + 2)^2 = x^2 + 12^2 – 2 \times x \times 12 \times \frac{2}{3}$ | M1 | AO3
$x^2 + 4x + 4 = x^2 + 144 – 16x$ | A1 | AO3
$20x = 140 \Rightarrow x = 7$ | A1 | AO3
$AB = 7$, $AC = 9$ | A1 | AO3 (Amend proof for candidates who choose $AC = x$)
**(b)** $\sin \angle ABC = \frac{\sqrt{5}}{3}$ | B1 | AO1
$\frac{\sin \angle BAC}{12} = \frac{\sin \angle ABC}{9}$ | M1 | AO1 (f.t. candidate's derived values for $AC$ and $\sin \angle ABC$)
$\sin \angle BAC = \frac{4\sqrt{5}}{9}$ | A1 | AO1 (c.a.o.)
**Total: [7]**
---In triangle $ABC$, $BC = 12$ cm and $\cos ABC = \frac{2}{3}$.
The length of $AC$ is 2 cm greater than the length of $AB$.
\begin{enumerate}[label=(\alph*)]
\item Find the lengths of $AB$ and $AC$. [4]
\item Find the exact value of $\sin BAC$. Give your answer in its simplest form. [3]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 1 Q13 [7]}}