| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions in differential equations |
| Difficulty | Challenging +1.2 This is a structured two-part question combining partial fractions (a standard A-level technique) with separable differential equations. Part (i) is routine decomposition with a repeated linear factor. Part (ii) requires separation, integration using the partial fractions result, and applying initial conditions—all standard procedures for P3/Further Pure. The algebra is somewhat involved but follows predictable steps without requiring novel insight. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply the form \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{10-x}\) | B1 | Form \(\frac{Dx+E}{x^2} + \frac{C}{10-x}\) acceptable, leads to \(D=1, E=10, C=1\) |
| Use any relevant method to determine a constant | M1 | If \(A\) or \(B\) (\(D\) or \(E\)) omitted from form of fractions, give B0M1A0A0 in (i) |
| Obtain one of the values \(A = 1,\ B = 10,\ C = 1\) | A1 | |
| Obtain the remaining two values | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Separate variables and attempt integration of both sides | M1 | Separation of form \(\frac{a\,dx}{x^2(10-x)} = b\,dt\) essential for M1. F.t. is on \(A, B, C\) |
| Obtain terms \(\ln x,\ -10/x,\ -\ln(10-x)\), or equivalent | A1\(\sqrt{}\) + A1\(\sqrt{}\) + A1\(\sqrt{}\) | If \(A\) or \(B\) (\(D\) or \(E\)) omitted: M1A1\(\sqrt{}\)A1\(\sqrt{}\)M1A0 in (ii) |
| Evaluate constant or use limits \(x = 1,\ t = 0\) with solution containing 3 of the terms \(k\ln x,\ l/x,\ m\ln(10-x)\) and \(t\), or equivalent | M1 | |
| Obtain any correct expression for \(t\), e.g. \(t = \ln\!\left(\dfrac{9x}{10-x}\right) - \dfrac{10}{x} + 10\) | A1 |
## Question 8:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply the form $\frac{A}{x} + \frac{B}{x^2} + \frac{C}{10-x}$ | B1 | Form $\frac{Dx+E}{x^2} + \frac{C}{10-x}$ acceptable, leads to $D=1, E=10, C=1$ |
| Use any relevant method to determine a constant | M1 | If $A$ or $B$ ($D$ or $E$) omitted from form of fractions, give B0M1A0A0 in (i) |
| Obtain one of the values $A = 1,\ B = 10,\ C = 1$ | A1 | |
| Obtain the remaining two values | A1 | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Separate variables and attempt integration of both sides | M1 | Separation of form $\frac{a\,dx}{x^2(10-x)} = b\,dt$ essential for M1. F.t. is on $A, B, C$ |
| Obtain terms $\ln x,\ -10/x,\ -\ln(10-x)$, or equivalent | A1$\sqrt{}$ + A1$\sqrt{}$ + A1$\sqrt{}$ | If $A$ or $B$ ($D$ or $E$) omitted: M1A1$\sqrt{}$A1$\sqrt{}$M1A0 in (ii) |
| Evaluate constant or use limits $x = 1,\ t = 0$ with solution containing 3 of the terms $k\ln x,\ l/x,\ m\ln(10-x)$ and $t$, or equivalent | M1 | |
| Obtain any correct expression for $t$, e.g. $t = \ln\!\left(\dfrac{9x}{10-x}\right) - \dfrac{10}{x} + 10$ | A1 | |8 (i) Express $\frac { 100 } { x ^ { 2 } ( 10 - x ) }$ in partial fractions.\\
(ii) Given that $x = 1$ when $t = 0$, solve the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 100 } x ^ { 2 } ( 10 - x )$$
obtaining an expression for $t$ in terms of $x$.
\hfill \mbox{\textit{CAIE P3 2009 Q8 [10]}}