| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Applied context modeling |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine application. Part (a) uses the standard R cos(θ-α) = a cos θ + b sin θ technique with straightforward calculation. Parts (b)-(d) involve finding max/min values and substitution, all following predictable patterns. While multi-part with context, each step is algorithmic with no novel insight required—slightly easier than average due to the mechanical nature of the techniques. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(K = 500\) | B1 | Correct value for \(K\). Condone \(R = 500\) |
| \(\tan\alpha = \frac{480}{140} \Rightarrow \alpha = ...\) | M1 | Award for \(\tan\alpha = \pm\frac{480}{140}\), or \(\tan\alpha = \pm\frac{140}{480}\), or \(\sin\alpha = \pm\frac{480}{"500"}\), or \(\cos\alpha = \pm\frac{140}{"500"}\) leading to \(\alpha = ...\). Note \(\alpha = \) awrt 1.3 (rad) implies this mark. |
| \(\alpha = \) awrt \(73.74°\) or \(500\cos(\theta + 73.74)°\) | A1 | \(\alpha =\) awrt \(73.74\{°\}\) or correct expression \(500\cos(\theta+73.74)\{°\}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(R = 1000 + 500\cos(30t + 73.74)°\) or \(R = 1000 + 140\cos(30t)° - 480\sin(30t)°\) | B1ft | Correct equation in either form including the \(R=\) following through on their numerical \(K\) \((0 < K \leq 750)\) and their numerical \(\alpha\). Allow e.g. \(R = 1500 - \)"500"\(+\)"500"\(\cos(30t+\)"73.74"\()\{°\}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\{R_{\min} =\} 500\) | B1ft | 500 or follow through on (their \(A\) – their \(K\)) or \((1500 - 2\times\) their \(K\)) provided non-negative and less than 1500. Must be clear this is the answer to (b)(ii). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(t = 3.5 \Rightarrow R = \)"1000"\(+\)"500"\(\cos(30(3.5)+\)"73.74"\()° = ...\) | M1 | Substitutes \(t = 3.5\) into their model. Condone substitution of a value of \(t\) in range \(3 \leq t \leq 4.5\) |
| \(R = \) awrt \(500.1...\) so the model is reliable | A1 | \(R=\) awrt \(500.1...\) or \(500\) (not awrt) following substitution of \(t=3.5\), suggesting model is valid/reliable/appropriate/good. Or \(\cos(30(3.5)+73.74)\{°\}\approx -1\) suggesting model is valid. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sin(30t+70)° = -1 \Rightarrow 30t + 70 = 270 \Rightarrow 30t = ...\) | M1 | Realises \(\sin(30t+70)°=-1\), reaches \(30t+70=270\) or \(-90\) and attempts to find \(t\). Condone attempts using differentiation via \(\cos(30t+70)°=0 \Rightarrow 30t+70=270\). Must use 270 or \(-90\), not 90. |
| \(30t = 200\) or \(t = \frac{20}{3}\) | A1 | Correct value for \(30t\) or \(t\). Accept rounded/truncated to at least 3 s.f. |
| \(R = \)"1000"\(+\)"500"\(\cos\!\left(30\!\left(\frac{20}{3}\right)+\)"73.74"\(\right)°\) or \(R=\)"1000"\(+140\cos(\)"200"\()°-480\sin(\)"200"\()°\) | dM1 | Substitutes their value of \(t>0\) (or \(30t>0\)) from \(30t+70=270\) into model for \(R\) |
| \(R = 1032\) (or 1033) | A1 | Correct number of rabbits. Allow 1032 or 1033 but must be whole numbers, not just 1030. |
## Question 12:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $K = 500$ | B1 | Correct value for $K$. Condone $R = 500$ |
| $\tan\alpha = \frac{480}{140} \Rightarrow \alpha = ...$ | M1 | Award for $\tan\alpha = \pm\frac{480}{140}$, or $\tan\alpha = \pm\frac{140}{480}$, or $\sin\alpha = \pm\frac{480}{"500"}$, or $\cos\alpha = \pm\frac{140}{"500"}$ leading to $\alpha = ...$. Note $\alpha = $ awrt 1.3 (rad) implies this mark. |
| $\alpha = $ awrt $73.74°$ or $500\cos(\theta + 73.74)°$ | A1 | $\alpha =$ awrt $73.74\{°\}$ or correct expression $500\cos(\theta+73.74)\{°\}$ |
### Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = 1000 + 500\cos(30t + 73.74)°$ or $R = 1000 + 140\cos(30t)° - 480\sin(30t)°$ | B1ft | Correct equation in either form including the $R=$ following through on their numerical $K$ $(0 < K \leq 750)$ and their numerical $\alpha$. Allow e.g. $R = 1500 - $"500"$+$"500"$\cos(30t+$"73.74"$)\{°\}$ |
### Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\{R_{\min} =\} 500$ | B1ft | 500 or follow through on (their $A$ – their $K$) or $(1500 - 2\times$ their $K$) provided non-negative and less than 1500. Must be clear this is the answer to (b)(ii). |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $t = 3.5 \Rightarrow R = $"1000"$+$"500"$\cos(30(3.5)+$"73.74"$)° = ...$ | M1 | Substitutes $t = 3.5$ into their model. Condone substitution of a value of $t$ in range $3 \leq t \leq 4.5$ |
| $R = $ awrt $500.1...$ so the model is reliable | A1 | $R=$ awrt $500.1...$ **or** $500$ (not awrt) following substitution of $t=3.5$, suggesting model is valid/reliable/appropriate/good. Or $\cos(30(3.5)+73.74)\{°\}\approx -1$ suggesting model is valid. |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sin(30t+70)° = -1 \Rightarrow 30t + 70 = 270 \Rightarrow 30t = ...$ | M1 | Realises $\sin(30t+70)°=-1$, reaches $30t+70=270$ or $-90$ and attempts to find $t$. Condone attempts using differentiation via $\cos(30t+70)°=0 \Rightarrow 30t+70=270$. Must use 270 or $-90$, **not** 90. |
| $30t = 200$ or $t = \frac{20}{3}$ | A1 | Correct value for $30t$ or $t$. Accept rounded/truncated to at least 3 s.f. |
| $R = $"1000"$+$"500"$\cos\!\left(30\!\left(\frac{20}{3}\right)+$"73.74"$\right)°$ or $R=$"1000"$+140\cos($"200"$)°-480\sin($"200"$)°$ | dM1 | Substitutes their value of $t>0$ (or $30t>0$) from $30t+70=270$ into model for $R$ |
| $R = 1032$ (or 1033) | A1 | Correct number of rabbits. Allow 1032 or 1033 but must be whole numbers, not just 1030. |
---\begin{enumerate}
\item (a) Express $140 \cos \theta - 480 \sin \theta$ in the form $K \cos ( \theta + \alpha )$\\
where $K > 0$ and $0 < \alpha < 90 ^ { \circ }$\\
State the value of $K$ and give the value of $\alpha$, in degrees, to 2 decimal places.
\end{enumerate}
A scientist studies the number of rabbits and the number of foxes in a wood for one year.
The number of rabbits, $R$, is modelled by the equation
$$R = A + 140 \cos ( 30 t ) ^ { \circ } - 480 \sin ( 30 t ) ^ { \circ }$$
where $t$ months is the time after the start of the year and $A$ is a constant.\\
Given that, during the year, the maximum number of rabbits in the wood is 1500\\
(b) (i) find a complete equation for this model.\\
(ii) Hence write down the minimum number of rabbits in the wood during the year according to the model.
The actual number of rabbits in the wood is at its minimum value in the middle of April.\\
(c) Use this information to comment on the model for the number of rabbits.
The number of foxes, $F$, in the wood during the same year is modelled by the equation
$$F = 100 + 70 \sin ( 30 t + 70 ) ^ { \circ }$$
The number of foxes is at its minimum value after $T$ months.\\
(d) Find, according to the models, the number of rabbits in the wood at time $T$ months.
\hfill \mbox{\textit{Edexcel Paper 1 2024 Q12 [11]}}