| Exam Board | Edexcel |
|---|---|
| Module | CP1 (Core Pure 1) |
| Year | 2024 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Systems of differential equations |
| Type | Predict population extinction or event time |
| Difficulty | Challenging +1.2 This is a standard coupled differential equations problem requiring systematic elimination to form a second-order DE, solving using auxiliary equation methods (complex roots), finding particular solutions from initial conditions, and interpreting results. While it involves multiple steps and techniques, each step follows well-established procedures taught in Core Pure 1 with no novel insights required. The modeling context and interpretation parts (d-e) are straightforward applications. |
| Spec | 4.10h Coupled systems: simultaneous first order DEs |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{d^2x}{dt^2}=\frac{dx}{dt}+\frac{dy}{dt}=\frac{dx}{dt}+3y-2x\) | M1 | Differentiates first equation and substitutes for \(\frac{dy}{dt}\) from second equation |
| \(=\frac{dx}{dt}+3\left(\frac{dx}{dt}-x\right)-2x\) | M1 | Proceeds to equation in \(x\) and \(\frac{dx}{dt}\) only by substituting for \(y\) |
| \(\frac{d^2x}{dt^2}-4\frac{dx}{dt}+5x=0\) | A1* | Achieves printed answer with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(m^2-4m+5=0 \Rightarrow m=\ldots\) | M1 | Uses model to form and solve Auxiliary Equation |
| \(m=2\pm i\) | A1 | Correct roots of AE |
| \(x=e^{\alpha t}(A\cos\beta t+B\sin\beta t)\) | M1 | Uses model to form Complementary Function appropriate for their roots; accept complex index form |
| \(x=e^{2t}(A\cos t+B\sin t)\) | A1 | Correct General Solution; must include \(x\); accept complex index form |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(y=\frac{dx}{dt}-x=e^{2t}(B\cos t - A\sin t+2A\cos t+2B\sin t)-e^{2t}(A\cos t+B\sin t)\) | M1 | Uses model and answer to part (b) to give \(y\) in terms of \(t\); must involve product rule attempt; or repeat whole process of (a) and (b) on \(y\) |
| \(y=e^{2t}\big((A+B)\cos t+(B-A)\sin t\big)\) | A1 | Correct simplified equation; like terms gathered; accept complex index form |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(A=100,\ 275=A+B \Rightarrow B=175\) | M1 | Realises need to use initial conditions; must have consistent constants |
| \(x=y \Rightarrow 100\cos t+175\sin t=275\cos t+75\sin t \Rightarrow \tan t=\ldots\) | dM1 | Sets \(x=y\) and collects to reach \(\tan t=\ldots\); alternatively convert to sine or cosine |
| \(\tan t=1.75\) | A1 | Correct equation; accept equivalent trig approaches |
| \(T=24\tan^{-1}(1.75)=\ldots\) | M1 | Solves their \(\tan t=\ldots\) and multiplies by 24; must be working in radians |
| \(=25.24\) | A1 | Correct value; ignore units |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| E.g. Both populations become negative for some times which is not possible | B1 | Must have at least one correct general equation for \(x\) or \(y\); suggests suitable correct limitation — must focus on equations producing negative values; "gives negative values" is minimal acceptable answer; answers about unlimited growth without correct statement are B0 |
## Question 8:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{d^2x}{dt^2}=\frac{dx}{dt}+\frac{dy}{dt}=\frac{dx}{dt}+3y-2x$ | M1 | Differentiates first equation and substitutes for $\frac{dy}{dt}$ from second equation |
| $=\frac{dx}{dt}+3\left(\frac{dx}{dt}-x\right)-2x$ | M1 | Proceeds to equation in $x$ and $\frac{dx}{dt}$ only by substituting for $y$ |
| $\frac{d^2x}{dt^2}-4\frac{dx}{dt}+5x=0$ | A1* | Achieves printed answer with no errors |
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $m^2-4m+5=0 \Rightarrow m=\ldots$ | M1 | Uses model to form and solve Auxiliary Equation |
| $m=2\pm i$ | A1 | Correct roots of AE |
| $x=e^{\alpha t}(A\cos\beta t+B\sin\beta t)$ | M1 | Uses model to form Complementary Function appropriate for their roots; accept complex index form |
| $x=e^{2t}(A\cos t+B\sin t)$ | A1 | Correct General Solution; must include $x$; accept complex index form |
### Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $y=\frac{dx}{dt}-x=e^{2t}(B\cos t - A\sin t+2A\cos t+2B\sin t)-e^{2t}(A\cos t+B\sin t)$ | M1 | Uses model and answer to part (b) to give $y$ in terms of $t$; must involve product rule attempt; or repeat whole process of (a) and (b) on $y$ |
| $y=e^{2t}\big((A+B)\cos t+(B-A)\sin t\big)$ | A1 | Correct simplified equation; like terms gathered; accept complex index form |
### Part (d):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $A=100,\ 275=A+B \Rightarrow B=175$ | M1 | Realises need to use initial conditions; must have consistent constants |
| $x=y \Rightarrow 100\cos t+175\sin t=275\cos t+75\sin t \Rightarrow \tan t=\ldots$ | dM1 | Sets $x=y$ and collects to reach $\tan t=\ldots$; alternatively convert to sine or cosine |
| $\tan t=1.75$ | A1 | Correct equation; accept equivalent trig approaches |
| $T=24\tan^{-1}(1.75)=\ldots$ | M1 | Solves their $\tan t=\ldots$ and multiplies by 24; must be working in radians |
| $=25.24$ | A1 | Correct value; ignore units |
### Part (e):
| Working/Answer | Mark | Guidance |
|---|---|---|
| E.g. Both populations become negative for some times which is not possible | B1 | Must have at least one correct general equation for $x$ or $y$; suggests suitable correct limitation — must focus on equations producing negative values; "gives negative values" is minimal acceptable answer; answers about unlimited growth without correct statement are B0 |\begin{enumerate}
\item A scientist is studying the effect of introducing a population of type $A$ bacteria into a population of type $B$ bacteria.
\end{enumerate}
At time $t$ days, the number of type $A$ bacteria, $x$, and the number of type $B$ bacteria, $y$, are modelled by the differential equations
$$\begin{aligned}
& \frac { \mathrm { d } x } { \mathrm {~d} t } = x + y \\
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 3 y - 2 x
\end{aligned}$$
(a) Show that
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
(b) Determine a general solution for the number of type $A$ bacteria at time $t$ days.\\
(c) Determine a general solution for the number of type $B$ bacteria at time $t$ days.
The model predicts that, at time $T$ hours, the number of bacteria in the two populations will be equal.
Given that $x = 100$ and $y = 275$ when $t = 0$\\
(d) determine the value of $T$, giving your answer to 2 decimal places.\\
(e) Suggest a limitation of the model.
\hfill \mbox{\textit{Edexcel CP1 2024 Q8 [15]}}