| Answer | Marks | Guidance |
|---|---|---|
| \(w^2 = 7.9^2 + 10.5^2 - 2(7.9)(10.5)\cos(30 + 35)\) | M1 | Use of cosine rule |
| \(w = 10.1\) | A1 | 10.12658... |
| \(\frac{\sin\theta}{7.9} = \frac{\sin(30 + 35)}{10.12658...}\) or \(\frac{\sin\alpha}{10.5} = \frac{\sin(30 + 35)}{10.12658...}\) | M1 | Use of sine rule with their \(w\) |
| \(\theta = 45.0\) or \(\alpha = 70.0\) | A1 | 44.99406... (\(\beta = 15.005...\)) or (\(\gamma = 74.994...\)) |
| Bearing = \(330 - \theta = 285\) or \(180 + 35 + \alpha = 285\) | A1 | 285.0059... |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Shortest distance = \(d = 1500\sin(44.99406... - 25)\) | M1 | \(1500\sin\beta\) |
| \(d = 513\) (3 sf) | A1 | 512.8841... |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = \frac{1500\cos(44.99406... - 25)}{10.12658...}\) | M1 | Use of \(s = ut\) with their \(w\) and \(\theta\) |
| \(= 139\) (3 sf) | A1 | 139.1972... (\(\beta\) consistent with (ii)) |
| [2] |
**(i)**
$w^2 = 7.9^2 + 10.5^2 - 2(7.9)(10.5)\cos(30 + 35)$ | M1 | Use of cosine rule
$w = 10.1$ | A1 | 10.12658...
$\frac{\sin\theta}{7.9} = \frac{\sin(30 + 35)}{10.12658...}$ or $\frac{\sin\alpha}{10.5} = \frac{\sin(30 + 35)}{10.12658...}$ | M1 | Use of sine rule with their $w$
$\theta = 45.0$ or $\alpha = 70.0$ | A1 | 44.99406... ($\beta = 15.005...$) or ($\gamma = 74.994...$)
Bearing = $330 - \theta = 285$ or $180 + 35 + \alpha = 285$ | A1 | 285.0059...
| [5] |
**(ii)**
Shortest distance = $d = 1500\sin(44.99406... - 25)$ | M1 | $1500\sin\beta$
$d = 513$ (3 sf) | A1 | 512.8841...
| [2] |
**(iii)**
$t = \frac{1500\cos(44.99406... - 25)}{10.12658...}$ | M1 | Use of $s = ut$ with their $w$ and $\theta$
$= 139$ (3 sf) | A1 | 139.1972... ($\beta$ consistent with (ii))
| [2] |\includegraphics{figure_2}
Boat $A$ is travelling with constant speed 7.9 m s$^{-1}$ on a course with bearing 035°. Boat $B$ is travelling with constant speed 10.5 m s$^{-1}$ on a course with bearing 330°. At one instant, the boats are 1500 m apart with $B$ on a bearing of 125° from $A$ (see diagram).
\begin{enumerate}[label=(\roman*)]
\item Find the magnitude and the bearing of the velocity of $B$ relative to $A$. [5]
\item Find the shortest distance between $A$ and $B$ in the subsequent motion. [2]
\item Find the time taken from the instant when $A$ and $B$ are 1500 m apart to the instant when $A$ and $B$ are at the point of closest approach. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR M4 2016 Q2 [9]}}