| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Real-world modelling (tides, daylight, etc.) |
| Difficulty | Moderate -0.3 This is a straightforward application of a given sinusoidal model requiring substitution at t=6.5 for part (a), then solving a simple trigonometric equation 5 + 2sin(30t)° = 3.8 for part (b). The context is routine, the algebra is minimal, and the inverse sine calculation is standard, making this slightly easier than average despite being multi-part. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(D = 5+2\sin(30\times6.5)^\circ \approx 4.48\,\text{m}\) | B1 | Must include units; allow \(D=4.482\ldots=4.5\,\text{m}\); allow correct answer without working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3.8=5+2\sin(30t)^\circ \Rightarrow \sin(30t)^\circ = -0.6\) | M1 | Using \(D=3.8\) and proceeding to \(\sin(30t)^\circ=k\), \( |
| \(\sin(30t)^\circ = -0.6\) | A1 | May be implied by any correct answer for \(t\), e.g. \(t=7.2\); calculation must be in degrees |
| \(t=10.77\) (finding first value of \(t\) for \(\sin(30t)^\circ=k\) after \(t=8.5\)) | dM1 | Note \(\sin(30t)^\circ=-0.6 \Rightarrow 30t=323.1^\circ \Rightarrow t\approx10.8\); award if \(30t=\text{inv}\sin(\text{their}\,-0.6)\) gives first value where \(30t>255\) |
| \(10\text{:}46\,\text{a.m.}\) or \(10\text{:}47\,\text{a.m.}\) | A1 | Allow 12-hour or 24-hour clock notation; do NOT allow 646 minutes or 10 hours 46 minutes |
# Question 8:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $D = 5+2\sin(30\times6.5)^\circ \approx 4.48\,\text{m}$ | B1 | Must include units; allow $D=4.482\ldots=4.5\,\text{m}$; allow correct answer without working |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3.8=5+2\sin(30t)^\circ \Rightarrow \sin(30t)^\circ = -0.6$ | M1 | Using $D=3.8$ and proceeding to $\sin(30t)^\circ=k$, $|k|\leq1$ |
| $\sin(30t)^\circ = -0.6$ | A1 | May be implied by any correct answer for $t$, e.g. $t=7.2$; calculation must be in degrees |
| $t=10.77$ (finding first value of $t$ for $\sin(30t)^\circ=k$ after $t=8.5$) | dM1 | Note $\sin(30t)^\circ=-0.6 \Rightarrow 30t=323.1^\circ \Rightarrow t\approx10.8$; award if $30t=\text{inv}\sin(\text{their}\,-0.6)$ gives first value where $30t>255$ |
| $10\text{:}46\,\text{a.m.}$ or $10\text{:}47\,\text{a.m.}$ | A1 | Allow 12-hour or 24-hour clock notation; do NOT allow 646 minutes or 10 hours 46 minutes |
---\begin{enumerate}
\item The depth of water, $D$ metres, in a harbour on a particular day is modelled by the formula
\end{enumerate}
$$D = 5 + 2 \sin ( 30 t ) ^ { \circ } \quad 0 \leqslant t < 24$$
where $t$ is the number of hours after midnight.
A boat enters the harbour at 6:30 am and it takes 2 hours to load its cargo. The boat requires the depth of water to be at least 3.8 metres before it can leave the harbour.\\
(a) Find the depth of the water in the harbour when the boat enters the harbour.\\
(b) Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.)
\hfill \mbox{\textit{Edexcel Paper 1 2018 Q8 [5]}}