| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2018 |
| Session | March |
| Marks | 7 |
| Topic | Sine and Cosine Rules |
| Type | Bearings and navigation |
| Difficulty | Standard +0.3 This is a standard sine/cosine rule application with bearings. Part (i) requires the cosine rule and bearing conversion (routine but multi-step), part (ii) uses basic trigonometry to find perpendicular distance, and part (iii) tests conceptual understanding. Slightly above average due to the ambiguous case consideration and multi-part nature, but all techniques are standard A-level fare. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\sin\theta}{\sin 25} = \frac{4.8}{2.2}\); \(\theta_1 = 67.2°\), so bearing is 067°; \(\theta_2 = 180° - 67°\); \(\theta_2 = 112.8°\) | M1, A1, M1, A1 | Attempt correct use of the sine rule to find at least one angle; Obtain 067°, or better; Attempt correct method for \(\theta_2\); Obtain 113°, or better; Must use 25°; 3 figure bearing required; 180° − their angle |
| Answer | Marks | Guidance |
|---|---|---|
| \(d = 4.8\sin 25° = 2.03\) km | M1, A1 | Attempt perpendicular distance; Obtain 2.03 km, or better; Allow all complete methods |
| Answer | Marks | Guidance |
|---|---|---|
| Coastline may not be straight between \(P\) and \(Q\) | E1 | Any sensible reason; Stations may not be exactly on coastline |
**(i)**
| $\frac{\sin\theta}{\sin 25} = \frac{4.8}{2.2}$; $\theta_1 = 67.2°$, so bearing is 067°; $\theta_2 = 180° - 67°$; $\theta_2 = 112.8°$ | M1, A1, M1, A1 | Attempt correct use of the sine rule to find at least one angle; Obtain 067°, or better; Attempt correct method for $\theta_2$; Obtain 113°, or better; Must use 25°; 3 figure bearing required; 180° − their angle |
**(ii)**
| $d = 4.8\sin 25° = 2.03$ km | M1, A1 | Attempt perpendicular distance; Obtain 2.03 km, or better; Allow all complete methods |
**(iii)**
| Coastline may not be straight between $P$ and $Q$ | E1 | Any sensible reason; Stations may not be exactly on coastline |7 Two lifeboat stations, $P$ and $Q$, are situated on the coastline with $Q$ being due south of $P$. A stationary ship is at sea, at a distance of 4.8 km from $P$ and a distance of 2.2 km from $Q$. The ship is on a bearing of $155 ^ { \circ }$ from $P$.\\
(i) Find any possible bearings of the ship from $Q$.\\
(ii) Find the shortest distance from the ship to the line $P Q$.\\
(iii) Give a reason why the actual distance from the ship to the coastline may be different to your answer to part (ii).
\hfill \mbox{\textit{OCR H240/01 2018 Q7 [7]}}