| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Topic | Projectiles |
| Type | Two moving objects interception (non-projectile) |
| Difficulty | Standard +0.8 This is a relative velocity/interception problem requiring vector resolution, solving simultaneous equations with trigonometry, and optimization for minimum speed. While the individual techniques are A-level standard, the multi-step nature, need to set up the geometry correctly, and the minimization in part (ii) requiring calculus or geometric insight elevates it above routine questions. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.10h Vectors in kinematics: uniform acceleration in vector form |
**(i)(a)**
$\sin\theta = \frac{600\sin 30°}{800}$ — M1A1
$\Rightarrow \theta = 22° \Rightarrow$ Bearing is $322°$ — A1 [3]
**(b)** Relative speed $= \frac{800\sin 127.97...°}{\sin 30°} = 1261.235...\text{ km h}^{-1}$ — M1A1
Time $= \frac{200}{1261.235...} \times 60 = 9.51$ minutes ($\Rightarrow$ Intercept at 1209 and 30 sec.) — M1A1 [4]
**(ii)** Steering perpendicular to relative path $\Rightarrow \frac{V}{600} = \sin 30°$ — M1
$\Rightarrow V = 300$ — A1
Speed is $300\text{ km h}^{-1}$ [2]
**[Total: 9]**2 At 1200 hours an aircraft, $A$, sets out to intercept a second aircraft, $B$, which is 200 km away on a bearing of $300 ^ { \circ }$ and is flying due east at $600 \mathrm {~km} \mathrm {~h} ^ { - 1 }$. Both aircraft are at the same altitude and continue to fly horizontally.
\begin{enumerate}[label=(\roman*)]
\item (a) Find the bearing on which $A$ should fly when travelling at $800 \mathrm {~km} \mathrm {~h} ^ { - 1 }$.\\
(b) Find the time at which $A$ intercepts $B$ in this case.
\item Find the least steady speed at which $A$ can fly to intercept $B$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2010 Q2 [9]}}