| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with infinite upper limit (power/logarithm functions) |
| Difficulty | Standard +0.3 This is a straightforward improper integrals question requiring basic integration of a power function and evaluation of limits. Part (a) is routine integration, while part (b) tests understanding of convergence at both zero and infinity—standard FP1 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| 2(a) \(\int 2x^{-3} dx = -x^{-2} (+c)\) | M1A1 | M1 for correct index |
| 2(a)(cont) \(\int_p^q 2x^{-3} dx = p^{-2} - q^{-2}\) | A1F | OE; ft wrong coefficient of \(x^{-2}\) |
| 2(b)(i) As \(p \to 0, p^{-2} \to \infty\), so no value | B1 | |
| 2(b)(ii) As \(q \to \infty, q^{-2} \to 0\), so value is \(\frac{1}{4}\) | M1A1F | ft wrong coefficient of \(x^{-2}\) or reversal of limits |
**2(a)** $\int 2x^{-3} dx = -x^{-2} (+c)$ | M1A1 | M1 for correct index | 3 marks
**2(a)(cont)** $\int_p^q 2x^{-3} dx = p^{-2} - q^{-2}$ | A1F | OE; ft wrong coefficient of $x^{-2}$
**2(b)(i)** As $p \to 0, p^{-2} \to \infty$, so no value | B1 | | 3 marks
**2(b)(ii)** As $q \to \infty, q^{-2} \to 0$, so value is $\frac{1}{4}$ | M1A1F | ft wrong coefficient of $x^{-2}$ or reversal of limits
**Total: 6 marks**
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\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $p$ and $q$, the value of the integral $\int _ { p } ^ { q } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x$.
\item Show that only one of the following improper integrals has a finite value, and find that value:
\begin{enumerate}[label=(\roman*)]
\item $\int _ { 0 } ^ { 2 } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x$;
\item $\int _ { 2 } ^ { \infty } \frac { 2 } { x ^ { 3 } } \mathrm {~d} x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q2 [6]}}