| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Stationary points then area/volume |
| Difficulty | Standard +0.8 This question requires finding stationary points by differentiating a product (requiring product rule), then integrating by parts twice to find an area. Part (ii) is particularly demanding as it requires repeated integration by parts with an exponential function and careful algebraic manipulation to reach the exact form. The 5 marks and 'show that' format indicate substantial working, making this harder than a routine integration by parts question but not exceptionally difficult for P3 level. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| 9(i) | Use correct product or quotient rule | M1 |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and obtain a 3 term quadratic equation in x | M1 | |
| Obtain answers x=2± 3 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(ii) | Integrate by parts and reach k(1+x2)e−1 2 x +l∫xe−1 2 x dx | *M1 |
| Obtain−2(1+x2)e−1 2 x +4∫xe−1 2 x dx, or equivalent | A1 | |
| Complete the integration and obtain(−18−8x−2x2)e−1 2 x, or equivalent | A1 | |
| Use limits x = 0 and x = 2 correctly, having fully integrated twice by parts | DM1 | |
| Obtain the given answer | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(i) ---
9(i) | Use correct product or quotient rule | M1
Obtain correct derivative in any form | A1
Equate derivative to zero and obtain a 3 term quadratic equation in x | M1
Obtain answers x=2± 3 | A1
4
--- 9(ii) ---
9(ii) | Integrate by parts and reach k(1+x2)e−1 2 x +l∫xe−1 2 x dx | *M1
Obtain−2(1+x2)e−1 2 x +4∫xe−1 2 x dx, or equivalent | A1
Complete the integration and obtain(−18−8x−2x2)e−1 2 x, or equivalent | A1
Use limits x = 0 and x = 2 correctly, having fully integrated twice by parts | DM1
Obtain the given answer | A1
5
Question | Answer | Marks\includegraphics{figure_9}
The diagram shows the curve $y = (1 + x^2)\text{e}^{-\frac{3x}{4}}$ for $x \geqslant 0$. The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x = 0$ and $x = 2$.
\begin{enumerate}[label=(\roman*)]
\item Find the exact values of the $x$-coordinates of the stationary points of the curve. [4]
\item Show that the exact value of the area of $R$ is $18 - \frac{42}{\text{e}}$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q9 [9]}}