| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| 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 application of improper integral evaluation using the p-test for convergence. Students must integrate power functions and apply limits, recognizing that x^(-3/4) diverges while x^(-5/4) converges. Part (c) combines the results linearly. This is standard FP1 material requiring careful execution but no novel insight, making it slightly easier than average. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int x^{-\frac{3}{4}}\,dx = 4x^{\frac{1}{4}}\ (+c)\) | M1A1 | M1 if index correct |
| This tends to \(\infty\) as \(x\to\infty\), so no value | A1F | ft wrong coefficient |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int x^{-\frac{5}{4}}\,dx = -4x^{-\frac{1}{4}}\ (+c)\) | M1A1 | M1 if index correct |
| \(\int_1^{\infty} x^{-\frac{5}{4}}\,dx = 0-(-4) = 4\) | A1F | ft wrong coefficient |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Subtracting \(4\) leaves \(\infty\), so no value | B1F | ft if \(c\) has 'no value' in (a) but has a finite answer in (b) |
| Total | 7 |
## Question 8(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int x^{-\frac{3}{4}}\,dx = 4x^{\frac{1}{4}}\ (+c)$ | M1A1 | M1 if index correct |
| This tends to $\infty$ as $x\to\infty$, so no value | A1F | ft wrong coefficient |
| **Total** | **3** | |
## Question 8(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int x^{-\frac{5}{4}}\,dx = -4x^{-\frac{1}{4}}\ (+c)$ | M1A1 | M1 if index correct |
| $\int_1^{\infty} x^{-\frac{5}{4}}\,dx = 0-(-4) = 4$ | A1F | ft wrong coefficient |
| **Total** | **3** | |
## Question 8(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Subtracting $4$ leaves $\infty$, so no value | B1F | ft if $c$ has 'no value' in (a) but has a finite answer in (b) |
| **Total** | **7** | |
---8 For each of the following improper integrals, find the value of the integral or explain why it does not have a value:
\begin{enumerate}[label=(\alph*)]
\item $\int _ { 1 } ^ { \infty } x ^ { - \frac { 3 } { 4 } } \mathrm {~d} x$;
\item $\int _ { 1 } ^ { \infty } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x$;
\item $\quad \int _ { 1 } ^ { \infty } \left( x ^ { - \frac { 3 } { 4 } } - x ^ { - \frac { 5 } { 4 } } \right) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q8 [7]}}