| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper integrals with discontinuity |
| Difficulty | Challenging +1.2 This is a straightforward improper integral question from FP1. Part (a) requires identifying the discontinuity at x=0 (routine observation). Part (b) involves splitting the integrand, integrating x^{-0.5} and x^{-2.5} using standard power rule, then evaluating the limit as the lower bound approaches 0. While it's a Further Maths topic and requires understanding of improper integrals, the execution is mechanical with no conceptual surprises—harder than average A-level due to the FP1 content, but a standard textbook exercise within that syllabus. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands |
| Answer | Marks | Guidance |
|---|---|---|
| The integrand is undefined/discontinuous at \(x = 0\) (lower limit) | B1 | Must reference \(x=0\) being in interval |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \frac{x-4}{x^{1.5}}\,dx = \int (x^{-0.5} - 4x^{-1.5})\,dx\) | M1 | Split into two terms |
| \(= 2x^{0.5} + 8x^{-0.5} (+c)\) | A1 | Both terms correct |
| \(= \left[2x^{1/2} + \frac{8}{x^{1/2}}\right]_t^4\) as \(t \to 0^+\) | M1 | Use of limit |
| At \(x=4\): \(2(2) + \frac{8}{2} = 4 + 4 = 8\) | ||
| As \(t \to 0^+\): \(\frac{8}{t^{1/2}} \to \infty\) | A1 | Conclude integral does not have a finite value |
# Question 2:
## Part (a)
| The integrand is undefined/discontinuous at $x = 0$ (lower limit) | B1 | Must reference $x=0$ being in interval |
## Part (b)
| $\int \frac{x-4}{x^{1.5}}\,dx = \int (x^{-0.5} - 4x^{-1.5})\,dx$ | M1 | Split into two terms |
| $= 2x^{0.5} + 8x^{-0.5} (+c)$ | A1 | Both terms correct |
| $= \left[2x^{1/2} + \frac{8}{x^{1/2}}\right]_t^4$ as $t \to 0^+$ | M1 | Use of limit |
| At $x=4$: $2(2) + \frac{8}{2} = 4 + 4 = 8$ | | |
| As $t \to 0^+$: $\frac{8}{t^{1/2}} \to \infty$ | A1 | Conclude integral does not have a finite value |
---2
\begin{enumerate}[label=(\alph*)]
\item Explain why $\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x$ is an improper integral.
\item Either find the value of the integral $\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x$ or explain why it does not have a finite value.\\[0pt]
[4 marks]
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-04_1970_1712_737_150}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2015 Q2 [5]}}