| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | November |
| Marks | 10 |
| 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 question requiring standard integration techniques (∫cos(at)dt), substitution of initial conditions, and finding a minimum using calculus. All steps are routine for A-level, though the multi-part structure and trigonometric integration place it slightly above average difficulty. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02h Square roots: of complex numbers |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Separate variables correctly and attempt integration of both sides | M1* | |
| Obtain term \(\ln N\), or equivalent | A1 | |
| Obtain term \(-\frac{k}{0.02}\sin(0.02t)\), or equivalent | A1 | |
| Use \(t = 0, N = 125\) to evaluate a constant, or as limits, in a solution containing terms of the form \(a\ln N\) and \(b\sin(0.02t)\), or equivalent | M1 | |
| Obtain any correct form of solution, e.g. \(\ln N = 50e\sin(0.02t) + \ln 125\) | A1 | [5] |
| (ii) Substituting \(N = 166\) and \(t = 30\), evaluate \(k\) | M1(dep*) | |
| Obtain \(k = 0.0100479\)...(accept \(k = 0.01\)) | A1 | [2] |
| (iii) Rearrange and obtain \(N = 125\exp(0.502\sin(0.02t))\), or equivalent | B1 | |
| Set \(\sin(0.02t) = -1\) in the expression for \(N\), or equivalent | M1 | |
| Obtain least value \(75.6\) (accept answers in the interval \([75, 76]\)) | A1 | [3] |
**(i)** Separate variables correctly and attempt integration of both sides | M1* |
Obtain term $\ln N$, or equivalent | A1 |
Obtain term $-\frac{k}{0.02}\sin(0.02t)$, or equivalent | A1 |
Use $t = 0, N = 125$ to evaluate a constant, or as limits, in a solution containing terms of the form $a\ln N$ and $b\sin(0.02t)$, or equivalent | M1 |
Obtain any correct form of solution, e.g. $\ln N = 50e\sin(0.02t) + \ln 125$ | A1 | [5]
**(ii)** Substituting $N = 166$ and $t = 30$, evaluate $k$ | M1(dep*) |
Obtain $k = 0.0100479$...(accept $k = 0.01$) | A1 | [2]
**(iii)** Rearrange and obtain $N = 125\exp(0.502\sin(0.02t))$, or equivalent | B1 |
Set $\sin(0.02t) = -1$ in the expression for $N$, or equivalent | M1 |
Obtain least value $75.6$ (accept answers in the interval $[75, 76]$) | A1 | [3]
[For the B1, accept 0.5 following $k = 0.01$, and allow 4.8 or better for ln 125.]7 The number of insects in a population $t$ days after the start of observations is denoted by $N$. The variation in the number of insects is modelled by a differential equation of the form
$$\frac { \mathrm { d } N } { \mathrm {~d} t } = k N \cos ( 0.02 t )$$
where $k$ is a constant and $N$ is taken to be a continuous variable. It is given that $N = 125$ when $t = 0$.\\
(i) Solve the differential equation, obtaining a relation between $N , k$ and $t$.\\
(ii) Given also that $N = 166$ when $t = 30$, find the value of $k$.\\
(iii) Obtain an expression for $N$ in terms of $t$, and find the least value of $N$ predicted by this model.
\hfill \mbox{\textit{CAIE P3 2007 Q7 [10]}}