| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Infinite series convergence and sum |
| Difficulty | Standard +0.3 This is a straightforward improper integral question requiring students to recognize why integrals are improper (discontinuity at x=0) and evaluate them using limits. Part (a) tests conceptual understanding, while part (b) involves routine integration with power rule and limit evaluation. One integral converges, one diverges—a standard FP1 exercise with minimal problem-solving demand beyond applying the definition of improper integrals. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | \(x^{-\frac{1}{2}} \to \infty\) as \(x \to 0\) | E1 |
| 5(b)(i) | \(\int x^{-\frac{1}{2}} dx = 2x^{\frac{1}{2}} (+c)\) | M1A1 |
| \(\int_0^{\frac{1}{6}} x^{-\frac{1}{2}} dx = \frac{1}{2}\) | A1F | 3 marks |
| 5(b)(ii) | \(\int x^{-\frac{2}{3}} dx = -4x^{-\frac{1}{4}} (+c)\) | M1A1 |
| \(x^{-\frac{1}{4}} \to \infty\) as \(x \to 0\), so no value | E1F | 3 marks |
| Total for Q5 | 7 marks |
5(a) | $x^{-\frac{1}{2}} \to \infty$ as $x \to 0$ | E1 | 1 mark | Condone "$x^{-\frac{1}{2}}$ has no value at $x = 0$" |
5(b)(i) | $\int x^{-\frac{1}{2}} dx = 2x^{\frac{1}{2}} (+c)$ | M1A1 | M1 for correct power of $x$ |
| $\int_0^{\frac{1}{6}} x^{-\frac{1}{2}} dx = \frac{1}{2}$ | A1F | 3 marks | ft wrong coefficient of $x^{\frac{1}{2}}$ |
5(b)(ii) | $\int x^{-\frac{2}{3}} dx = -4x^{-\frac{1}{4}} (+c)$ | M1A1 | M1 for correct power of $x$ |
| $x^{-\frac{1}{4}} \to \infty$ as $x \to 0$, so no value | E1F | 3 marks | ft wrong coefficient of $x^{-\frac{1}{4}}$ |
| **Total for Q5** | **7 marks** |
---5
\begin{enumerate}[label=(\alph*)]
\item Explain why $\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x$ is an improper integral.
\item For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
\begin{enumerate}[label=(\roman*)]
\item $\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x$;
\item $\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q5 [7]}}