| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Standard +0.3 This is a straightforward separable variables differential equation requiring standard separation, integration (including a simple trigonometric integral), and application of initial conditions. While it involves multiple parts and some algebraic manipulation with fractional powers, it follows a completely standard template with no conceptual surprises or novel problem-solving required. Slightly above average difficulty due to the N^(3/2) term and multi-part nature, but still routine for P3 level. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Separate variables correctly | B1 | \(\frac{dN}{N^{\frac{3}{2}}}=(k\cos 0.02t)\,dt\). Allow without integral signs |
| Obtain term \(-\frac{2}{\sqrt{N}}\) | B1 | OE. Ignore position of \(k\) |
| Obtain term \(50k\sin 0.02t\) | B1 | OE. Ignore position of \(k\) |
| Use \(t=0\), \(N=100\) to evaluate a constant, or as limits, in a solution containing terms \(\frac{a}{\sqrt{N}}\) and \(b\sin 0.02t\), where \(ab\neq 0\) | M1 | e.g. \(c=-0.2\) or \(c=\frac{-0.2}{k}\) |
| Obtain correct solution in any form, e.g. \(-\frac{2}{\sqrt{N}}=50k\sin 0.02t - 0.2\) | A1 | OE ISW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the substitution \(N=625\) and \(t=50\) in expression of appropriate form to evaluate \(k\) | M1 | Expression must contain \(a+bk\sin 0.02t\), \((\sqrt{N})^{\pm n}\), where \(n=-1,1,3\) or \(5\) and \(a\) and \(b\) are constants \(ab\neq 0\) or \((a+bk\sin 0.02t)^{\pm 2}\) and \((N)^{\pm n}\). Allow with \(k\) replaced by \(\frac{1}{k}\), error due to \(k(N^{-3/2})\) when separating variables in 8(a). If invert term by term when 3 terms shown then M0 |
| Obtain \(k=0.00285[2148]\) | A1 | Must evaluate \(\sin 1\). Degrees \(k=0.138\). M1 A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Rearrange and obtain \(N=4(0.2-0.142(607)\sin 0.02t)^{-2}\). Substitution for \(k\) required | M1 | Anything of the form \(N=c(d-ek\sin 0.02t)^{-2}\), where \(c\), \(d\) and \(e\) are constants \(cde\neq 0\) and value of \(k\) substituted. Allow with \(k\) replaced by \(1/k\), error due to \(k(N^{-3/2})\) when separating variables in 8(a). OE ISW |
| Accept answers between 1209 and 1215 | A1 | ISW. Substitute \(\sin 0.02t=1\) or \(t=50\sin^{-1}1\) or \(78.5\) or \(25\pi\). Answer with no working (rubric) 0/2. SC \(N=\ldots\) not seen but correct numerical answer B1 1/2 |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Separate variables correctly | B1 | $\frac{dN}{N^{\frac{3}{2}}}=(k\cos 0.02t)\,dt$. Allow without integral signs |
| Obtain term $-\frac{2}{\sqrt{N}}$ | B1 | OE. Ignore position of $k$ |
| Obtain term $50k\sin 0.02t$ | B1 | OE. Ignore position of $k$ |
| Use $t=0$, $N=100$ to evaluate a constant, or as limits, in a solution containing terms $\frac{a}{\sqrt{N}}$ and $b\sin 0.02t$, where $ab\neq 0$ | M1 | e.g. $c=-0.2$ or $c=\frac{-0.2}{k}$ |
| Obtain correct solution in any form, e.g. $-\frac{2}{\sqrt{N}}=50k\sin 0.02t - 0.2$ | A1 | OE ISW |
---
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the substitution $N=625$ and $t=50$ in expression of appropriate form to evaluate $k$ | M1 | Expression must contain $a+bk\sin 0.02t$, $(\sqrt{N})^{\pm n}$, where $n=-1,1,3$ or $5$ and $a$ and $b$ are constants $ab\neq 0$ or $(a+bk\sin 0.02t)^{\pm 2}$ and $(N)^{\pm n}$. Allow with $k$ replaced by $\frac{1}{k}$, error due to $k(N^{-3/2})$ when separating variables in 8(a). If invert term by term when 3 terms shown then M0 |
| Obtain $k=0.00285[2148]$ | A1 | Must evaluate $\sin 1$. Degrees $k=0.138$. M1 A0 |
---
## Question 8(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rearrange and obtain $N=4(0.2-0.142(607)\sin 0.02t)^{-2}$. Substitution for $k$ required | M1 | Anything of the form $N=c(d-ek\sin 0.02t)^{-2}$, where $c$, $d$ and $e$ are constants $cde\neq 0$ and value of $k$ substituted. Allow with $k$ replaced by $1/k$, error due to $k(N^{-3/2})$ when separating variables in 8(a). OE ISW |
| Accept answers between 1209 and 1215 | A1 | ISW. Substitute $\sin 0.02t=1$ or $t=50\sin^{-1}1$ or $78.5$ or $25\pi$. Answer with no working (rubric) 0/2. **SC** $N=\ldots$ not seen but correct numerical answer **B1** 1/2 |
---8 At time $t$ days after the start of observations, the number of insects in a population is $N$. The variation in the number of insects is modelled by a differential equation of the form $\frac { \mathrm { d } N } { \mathrm {~d} t } = k N ^ { \frac { 3 } { 2 } } \cos 0.02 t$, where $k$ is a constant and $N$ is a continuous variable. It is given that when $t = 0 , N = 100$.
\begin{enumerate}[label=(\alph*)]
\item Solve the differential equation, obtaining a relation between $N , k$ and $t$.
\item Given also that $N = 625$ when $t = 50$, find the value of $k$.
\item Obtain an expression for $N$ in terms of $t$, and find the greatest value of $N$ predicted by this model.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q8 [9]}}