| 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 area. The integration is non-routine, requiring strategic application of integration by parts with exponential and polynomial terms, and careful algebraic manipulation to reach the exact form required. More challenging than standard P3 questions but within expected scope. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 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 \(x = 2 \pm \sqrt{3}\) | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate by parts and reach \(k(1+x^2)e^{-\frac{1}{2}x} + l\int xe^{-\frac{1}{2}x}\,dx\) | *M1 | |
| Obtain \(-2(1+x^2)e^{-\frac{1}{2}x} + 4\int xe^{-\frac{1}{2}x}\,dx\), or equivalent | A1 | |
| Complete the integration and obtain \((-18 - 8x - 2x^2)e^{-\frac{1}{2}x}\), or equivalent | A1 | |
| Use limits \(x = 0\) and \(x = 2\) correctly, having fully integrated twice by parts | DM1 | |
| Obtain the given answer | A1 | |
| Total | 5 |
## Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| 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 $x = 2 \pm \sqrt{3}$ | A1 | |
| **Total** | **4** | |
## Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate by parts and reach $k(1+x^2)e^{-\frac{1}{2}x} + l\int xe^{-\frac{1}{2}x}\,dx$ | *M1 | |
| Obtain $-2(1+x^2)e^{-\frac{1}{2}x} + 4\int xe^{-\frac{1}{2}x}\,dx$, or equivalent | A1 | |
| Complete the integration and obtain $(-18 - 8x - 2x^2)e^{-\frac{1}{2}x}$, or equivalent | A1 | |
| Use limits $x = 0$ and $x = 2$ correctly, having fully integrated twice by parts | DM1 | |
| Obtain the given answer | A1 | |
| **Total** | **5** | |9\\
\includegraphics[max width=\textwidth, alt={}, center]{21878d10-7f16-4dbb-86ef-65a7ba5eeafb-16_446_956_260_593}
The diagram shows the curve $y = \left( 1 + x ^ { 2 } \right) \mathrm { e } ^ { - \frac { 1 } { 2 } x }$ for $x \geqslant 0$. The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x = 0$ and $x = 2$.\\
(i) Find the exact values of the $x$-coordinates of the stationary points of the curve.\\
(ii) Show that the exact value of the area of $R$ is $18 - \frac { 42 } { \mathrm { e } }$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q9 [9]}}