| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Trig equation from real-world model |
| Difficulty | Moderate -0.3 This is a straightforward application of trigonometric functions in a modeling context. Parts (a)-(d) require only basic understanding of cosine properties (range, period) with no equation solving beyond cos(30t)° = 1. Part (e) involves solving a simple trig equation. Part (f) asks for a qualitative modification (damping factor) rather than rigorous mathematics. Slightly easier than average due to the routine nature of all parts. |
| Spec | 1.02z Models in context: use functions in modelling1.05a Sine, cosine, tangent: definitions for all arguments1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(6000\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2000\) | B1f | ft their \((a) - 4000\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Oscillates or Goes up and down oe. Fluctuates. Moves in a cycle | B1 | Ignore all else. NOT "Increases for 1st 6 months then decreases" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(30t = 360\). Time to return to initial size \(= 12\) months | M1, A1 | May be implied by answer. Allow \(t=12\), or \(t=12\) months, or just 12 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(4500 = 5000 - 1000\cos(30t)°\) | M1 | Substitute \(P=4500\). May be implied by next line |
| \(\cos(30t)° = 0.5\) | A1 | Correct rearrangement |
| \(30t = 60\) or \(300\) (both) | M1 | Attempt \(30t = \cos^{-1}(\text{their } 0.5)\), giving \(\alpha\) and \(360-\alpha\). Condone \(30t = \frac{\pi}{3},\ \frac{5\pi}{3}\) |
| 2nd time \(P=4500\) is when \(t=10\) | A1 | or after 10 months. Allow \(t=10\) months, or just 10. SC: if not gained 1st M1A1, correct answer with no/inadequate working and/or T&I: \(t=10\) stated: B2; \(t=10\) embedded: B1B0 |
| Alternative: \(30t=60\) or \(-60\) (\(t=2\) or \(-2\)). 2nd time \(P=4500\) is when \(t=-2+12=10\) | M1, A1 | \(30t=60\ (t=2)\). End of 1st cycle at \(t=12\); 2nd time \(P=4500\) is when \(t=12-2=10\) |
| \(30t=60\ (t=2)\); \(6-2=4;\ t=6+4=10\) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| eg \(P = 5000 - 1000e^{-t}\cos(30t)°\); \(P=5000-1000e^{-kt}\cos(30t)°\ (k>0)\). Answers in words must be equivalent to one of these | B1 | or other good answers, eg \(P=5000-(1000\cos(30t)°)^{1/t}\); \(P=5000-\frac{1000}{t}\cos(30t)°\ (t>0)\) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $6000$ | **B1** | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $2000$ | **B1f** | ft their $(a) - 4000$ |
---
## Question 4(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Oscillates or Goes up and down oe. Fluctuates. Moves in a cycle | **B1** | Ignore all else. NOT "Increases for 1st 6 months then decreases" |
---
## Question 4(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $30t = 360$. Time to return to initial size $= 12$ months | **M1, A1** | May be implied by answer. Allow $t=12$, or $t=12$ months, or just 12 |
---
## Question 4(e):
| Answer | Mark | Guidance |
|--------|------|----------|
| $4500 = 5000 - 1000\cos(30t)°$ | **M1** | Substitute $P=4500$. May be implied by next line |
| $\cos(30t)° = 0.5$ | **A1** | Correct rearrangement |
| $30t = 60$ or $300$ (both) | **M1** | Attempt $30t = \cos^{-1}(\text{their } 0.5)$, giving $\alpha$ and $360-\alpha$. Condone $30t = \frac{\pi}{3},\ \frac{5\pi}{3}$ |
| 2nd time $P=4500$ is when $t=10$ | **A1** | or after 10 months. Allow $t=10$ months, or just 10. SC: if not gained 1st M1A1, correct answer with no/inadequate working and/or T&I: $t=10$ stated: B2; $t=10$ embedded: B1B0 |
| **Alternative:** $30t=60$ or $-60$ ($t=2$ or $-2$). 2nd time $P=4500$ is when $t=-2+12=10$ | **M1, A1** | $30t=60\ (t=2)$. End of 1st cycle at $t=12$; 2nd time $P=4500$ is when $t=12-2=10$ |
| $30t=60\ (t=2)$; $6-2=4;\ t=6+4=10$ | **M1, A1** | |
---
## Question 4(f):
| Answer | Mark | Guidance |
|--------|------|----------|
| eg $P = 5000 - 1000e^{-t}\cos(30t)°$; $P=5000-1000e^{-kt}\cos(30t)°\ (k>0)$. Answers in words must be equivalent to one of these | **B1** | or other good answers, eg $P=5000-(1000\cos(30t)°)^{1/t}$; $P=5000-\frac{1000}{t}\cos(30t)°\ (t>0)$ |
---4 The size, $P$, of a population of a certain species of insect at time $t$ months is modelled by the following formula.\\
$P = 5000 - 1000 \cos ( 30 t ) ^ { \circ }$
\begin{enumerate}[label=(\alph*)]
\item Write down the maximum size of the population.
\item Write down the difference between the largest and smallest values of $P$.
\item Without giving any numerical values, describe briefly the behaviour of the population over time.
\item Find the time taken for the population to return to its initial size for the first time.
\item Determine the time on the second occasion when $P = 4500$.
A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and $P$ converges to 5000 .
\item Suggest a change to the model that will take account of this observation.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2021 Q4 [10]}}