| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.3 This is a straightforward separable variables question requiring standard integration techniques (separating variables, integrating cos and 1/√M, applying initial conditions). The multi-part structure and need to find a minimum adds slight complexity beyond the most routine examples, but all steps follow standard procedures taught in P3 with no novel insight required. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Separate variables correctly and integrate one side | B1 | |
| Obtain term \(2\sqrt{M}\), or equivalent | B1 | |
| Obtain term \(50k\sin(0.02t)\), or equivalent | B1 | |
| Evaluate a constant of integration, or use limits \(M = 100, t = 0\) in a solution with terms of the form \(a\sqrt{M}\) and \(b\sin(0.02t)\) | M1* | |
| Obtain correct solution in any form, e.g. \(2\sqrt{M} = 50k\sin(0.02t) + 20\) | A1 | 5 |
| (ii) Use values \(M = 196, t = 50\) and calculate \(k\) | M1(dep*) | |
| Obtain answer \(k = 0.190\) | A1 | 2 |
| (iii) State an expression for \(M\) in terms of \(t\), e.g. \(M = (4.75\sin(0.02t) + 10)^2\) | M1(dep*) | |
| State that the least possible number of micro-organisms is 28 or 27.5 or 27.6 (27.5625) | A1 | 2 |
**(i)** Separate variables correctly and integrate one side | B1 |
Obtain term $2\sqrt{M}$, or equivalent | B1 |
Obtain term $50k\sin(0.02t)$, or equivalent | B1 |
Evaluate a constant of integration, or use limits $M = 100, t = 0$ in a solution with terms of the form $a\sqrt{M}$ and $b\sin(0.02t)$ | M1* |
Obtain correct solution in any form, e.g. $2\sqrt{M} = 50k\sin(0.02t) + 20$ | A1 | 5 |
**(ii)** Use values $M = 196, t = 50$ and calculate $k$ | M1(dep*) |
Obtain answer $k = 0.190$ | A1 | 2 |
**(iii)** State an expression for $M$ in terms of $t$, e.g. $M = (4.75\sin(0.02t) + 10)^2$ | M1(dep*) |
State that the least possible number of micro-organisms is 28 or 27.5 or 27.6 (27.5625) | A1 | 2 |7 The number of micro-organisms in a population at time $t$ is denoted by $M$. At any time the variation in $M$ is assumed to satisfy the differential equation
$$\frac { \mathrm { d } M } { \mathrm {~d} t } = k ( \sqrt { } M ) \cos ( 0.02 t )$$
where $k$ is a constant and $M$ is taken to be a continuous variable. It is given that when $t = 0 , M = 100$.\\
(i) Solve the differential equation, obtaining a relation between $M , k$ and $t$.\\
(ii) Given also that $M = 196$ when $t = 50$, find the value of $k$.\\
(iii) Obtain an expression for $M$ in terms of $t$ and find the least possible number of micro-organisms.
\hfill \mbox{\textit{CAIE P3 2015 Q7 [9]}}