| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Session | Specimen |
| Marks | 8 |
| Topic | Vectors Introduction & 2D |
| Type | River crossing: reach point directly opposite (find angle and/or time) |
| Difficulty | Moderate -0.3 This is a standard relative velocity problem requiring vector resolution and basic trigonometry. Part (i)(a) is immediate (perpendicular to bank), part (i)(b) requires setting up sin θ = 5/8 for shortest path, and part (ii) involves straightforward time calculations. While it's a multi-part question worth 8 marks total, the techniques are routine for Further Maths students and require no novel insight—slightly easier than average due to its textbook nature. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.10h Vectors in kinematics: uniform acceleration in vector form3.02e Two-dimensional constant acceleration: with vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(90°\) | B1 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\alpha = 38.7°\) and \(\beta\) (to the bank) \(= 51.3°\) | M1, M1, M1, A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(t_1 = \frac{100}{5} = 20\) sec | B1 | |
| \(t_2 = \frac{100}{5\sin 51.32°} = 25.6\) sec | M1, A1 | 3 |
### (i)(a)
$90°$ | B1 | **1**
### (i)(b)
Shortest route $\Rightarrow \alpha$ is a maximum ($\alpha$ is the angle between resultant path and bank)
$\sin\alpha = \frac{5\sin\theta}{8}$
$\alpha_{\max} \Rightarrow \theta = 90°$ ($\theta$ is the angle between headed path and resultant path)
$\alpha = 38.7°$ and $\beta$ (to the bank) $= 51.3°$ | M1, M1, M1, A1 | **4**
### (ii)
$t_1 = \frac{100}{5} = 20$ sec | B1 |
$t_2 = \frac{100}{5\sin 51.32°} = 25.6$ sec | M1, A1 | **3**A girl can paddle her canoe at $5 \text{ m s}^{-1}$ in still water. She wishes to cross a river which is $100 \text{ m}$ wide and flowing at $8 \text{ m s}^{-1}$.
\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item Write down the angle to the river bank at which the boat must head, in order to cross the river in the least possible time. [1]
\item Find the acute angle to the river bank at which the boat must head, in order to cross the river by the shortest route. [4]
\end{enumerate}
\item Calculate the times taken for each of the two cases in part (i). [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 Q5 [8]}}