| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions for differential equations |
| Difficulty | Standard +0.8 This is a substantial C4 question combining partial fractions with a separable differential equation. Part (a) is routine but involves a repeated linear factor. Part (b) requires separation of variables, integration using the partial fractions result, applying initial conditions, and algebraic manipulation to reach the specified form—more demanding than typical C4 questions but within standard syllabus scope. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
8
\begin{enumerate}[label=(\alph*)]
\item Express $\frac { 1 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ in the form $\frac { A } { 3 - 2 x } + \frac { B } { 1 - x } + \frac { C } { ( 1 - x ) ^ { 2 } }$.\\
(4 marks)
\item Solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \sqrt { y } } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$$
where $y = 0$ when $x = 0$, expressing your answer in the form
$$y ^ { p } = q \ln [ \mathrm { f } ( x ) ] + \frac { x } { 1 - x }$$
where $p$ and $q$ are constants.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2011 Q8 [13]}}