| Exam Board | WJEC |
|---|---|
| Module | Unit 3 (Unit 3) |
| Year | 2024 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with linear factors – decompose and integrate (definite) |
| Difficulty | Standard +0.3 This is a standard A-level partial fractions question with routine integration and sign analysis. Part (a) is textbook partial fractions decomposition with three linear factors. Part (b) involves straightforward integration of logarithmic terms and algebraic manipulation. Part (c) requires basic sign checking and recognizing that f(x) has a vertical asymptote (discontinuity at x=2.5) rather than a root, which is a common exam trap but requires minimal insight. All techniques are standard C3/C4 material with no novel problem-solving required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08d Evaluate definite integrals: between limits1.09a Sign change methods: locate roots |
The function $f$ is given by
$$f(x) = \frac{25x + 32}{(2x - 5)(x + 1)(x + 2)}.$$
\begin{enumerate}[label=(\alph*)]
\item Express $f(x)$ in terms of partial fractions. [4]
\item Show that $\int_1^2 f(x) dx = -\ln P$, where $P$ is an integer whose value is to be found. [5]
\item Show that the sign of $f(x)$ changes in the interval $x = 2$ to $x = 3$. Explain why the change of sign method fails to locate a root of the equation $f(x) = 0$ in this case. [2]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 3 2024 Q1 [11]}}