| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Quadrilateral with diagonal |
| Difficulty | Standard +0.3 This is a standard multi-part trigonometry question requiring systematic application of cosine rule, sine rule, and area formulas across two triangles. While it has multiple steps (3 parts, likely 7-8 marks total), each step follows directly from the previous with no novel insight required—part (a) is guided ('show that'), part (b) uses standard sine/cosine rule techniques, and part (c) is straightforward area calculation. This is slightly easier than average because the structure is transparent and it's a textbook-style exercise testing technique rather than problem-solving. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(BD^2 = 6^2 + 9^2 - (2 \times 6 \times 9 \times \cos 60)\) | M1 A1 | |
| \(BD^2 = 36 + 81 - 54 = 63\) | M1 A1 | |
| \(BD = \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}\) cm | M1 A1 | |
| (b) \((3\sqrt{7})^2 = 3^2 + 8^2 - (2 \times 3 \times 8 \times \cos C)\) | M1 | |
| \(\cos C = \frac{9 + 64 - 63}{48} = \frac{10}{48} = \frac{5}{24}\) | M1 A1 | |
| \(\angle BCD = 78.0°\) (1dp) | M1 A1 | |
| (c) \(= \left(\frac{1}{2} \times 6 \times 9 \times \sin 60\right) + \left(\frac{1}{2} \times 3 \times 8 \times \sin 77.975\right)\) | M2 | |
| \(= 35.1\) cm² (3sf) | A1 | (10 marks) |
**(a)** $BD^2 = 6^2 + 9^2 - (2 \times 6 \times 9 \times \cos 60)$ | M1 A1 |
$BD^2 = 36 + 81 - 54 = 63$ | M1 A1 |
$BD = \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$ cm | M1 A1 |
**(b)** $(3\sqrt{7})^2 = 3^2 + 8^2 - (2 \times 3 \times 8 \times \cos C)$ | M1 |
$\cos C = \frac{9 + 64 - 63}{48} = \frac{10}{48} = \frac{5}{24}$ | M1 A1 |
$\angle BCD = 78.0°$ (1dp) | M1 A1 |
**(c)** $= \left(\frac{1}{2} \times 6 \times 9 \times \sin 60\right) + \left(\frac{1}{2} \times 3 \times 8 \times \sin 77.975\right)$ | M2 |
$= 35.1$ cm² (3sf) | A1 | **(10 marks)**8.
Figure 2
Figure 2 shows the quadrilateral $A B C D$ in which $A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm} , C D = 8 \mathrm {~cm}$, $A D = 9 \mathrm {~cm}$ and $\angle B A D = 60 ^ { \circ }$.
\begin{enumerate}[label=(\alph*)]
\item Using the cosine rule, show that $B D = 3 \sqrt { 7 } \mathrm {~cm}$.
\item Find the size of $\angle B C D$ in degrees.
\item Find the area of quadrilateral $A B C D$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [10]}}