| Date | None Specimen | Marks available | 4 | Reference code | SPNone.2.hl.TZ0.7 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Determine | Question number | 7 | Adapted from | N/A |
Question
A ship, S, is 10 km north of a motorboat, M, at 12.00pm. The ship is travelling northeast with a constant velocity of \(20{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\). The motorboat wishes to intercept the ship and it moves with a constant velocity of \(30{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\) in a direction \(\theta \) degrees east of north. In order for the interception to take place, determine
the value of \(\theta \).
the time at which the interception occurs, correct to the nearest minute.
Markscheme
let the interception occur at the point P, t hrs after 12:00
then, SP = 20t and MP = 30t A1
using the sine rule,
\(\frac{{{\text{SP}}}}{{{\text{MP}}}} = \frac{2}{3} = \frac{{\sin \theta }}{{\sin 135}}\) M1A1
whence \(\theta = 28.1\) A1
[4 marks]
using the sine rule again,
\(\frac{{{\text{MP}}}}{{{\text{MS}}}} = \frac{{\sin 135}}{{\sin (45 - 28.1255 \ldots )}}\) M1A1
\(30t = 10 \times \frac{{\sin 135}}{{\sin 16.8745 \ldots }}\) M1
\(t = 0.81199 \ldots \) A1
the interception occurs at 12:49 A1
[5 marks]
