| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| 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 integrals question requiring standard techniques: integrate, apply limits, and evaluate as upper limit approaches infinity. Part (a) is routine (converges), part (b) diverges due to the constant term (standard recognition), and part (c) expands to show divergence. All three parts follow textbook patterns with no novel insight required, making this slightly easier than average for Further Maths. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks |
|---|---|
| \(\int x^{-3}\,dx = kx^{-2}\ (+c)\) | M1 |
| \(x^{-n} \to 0\) as \(x \to \infty\) | M1 |
| Improper integral has value 1 | A1 (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| No value as \(x\) term tends to \(\infty\) | B1 (1 mark) | OE |
| Answer | Marks |
|---|---|
| \(\int x^{-2}\,dx = kx^{-1}\ (+c)\) | M1 |
| \(x^{-1} \to 0\) as \(x \to \infty\) | m1 |
| Improper integral has value 5 | A1 (3 marks) |
## Question 4:
**Part (a)**
$\int x^{-3}\,dx = kx^{-2}\ (+c)$ | M1 |
$x^{-n} \to 0$ as $x \to \infty$ | M1 |
Improper integral has value 1 | A1 (3 marks) |
**Part (b)**
No value as $x$ term tends to $\infty$ | B1 (1 mark) | OE
**Part (c)**
$\int x^{-2}\,dx = kx^{-1}\ (+c)$ | M1 |
$x^{-1} \to 0$ as $x \to \infty$ | m1 |
Improper integral has value 5 | A1 (3 marks) |
---4 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=(\alph*)]
\item $\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } \mathrm {~d} x$;\\
(3 marks)
\item $\quad \int _ { 2 } ^ { \infty } \left( 8 x ^ { - 3 } + 1 \right) \mathrm { d } x$;
\item $\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2005 Q4 [7]}}