| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| 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 a stationary point by differentiating a product (requiring product rule), then integrating the same function (requiring integration by parts twice). The product rule application is straightforward, but solving for the stationary point and performing repeated integration by parts with exponential functions represents solid A-level technique rather than routine drill, placing it moderately above average difficulty. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the correct product rule | M1 | |
| Obtain correct derivative in any form, e.g. \((2-2x)e^{\frac{1}{2}x} + \frac{1}{2}(2x-x^2)e^{\frac{1}{2}x}\) | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain \(x = \sqrt{5} - 1\) only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate by parts and reach \(a(2x-x^2)e^{\frac{1}{2}x} + b\int(2-2x)e^{\frac{1}{2}x}\,dx\) | M1* | |
| Obtain \(2e^{\frac{1}{2}x}(2x-x^2) - 2\int(2-2x)e^{\frac{1}{2}x}\,dx\), or equivalent | A1 | |
| Complete the integration correctly, obtaining \((12x - 2x^2 - 24)e^{\frac{1}{2}x}\), or equivalent | A1 | |
| Use limits \(x = 0\), \(x = 2\) correctly having integrated by parts twice | DM1 | |
| Obtain answer \(24 - 8e\), or exact simplified equivalent | A1 |
## Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the correct product rule | M1 | |
| Obtain correct derivative in any form, e.g. $(2-2x)e^{\frac{1}{2}x} + \frac{1}{2}(2x-x^2)e^{\frac{1}{2}x}$ | A1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain $x = \sqrt{5} - 1$ only | A1 | |
## Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate by parts and reach $a(2x-x^2)e^{\frac{1}{2}x} + b\int(2-2x)e^{\frac{1}{2}x}\,dx$ | M1* | |
| Obtain $2e^{\frac{1}{2}x}(2x-x^2) - 2\int(2-2x)e^{\frac{1}{2}x}\,dx$, or equivalent | A1 | |
| Complete the integration correctly, obtaining $(12x - 2x^2 - 24)e^{\frac{1}{2}x}$, or equivalent | A1 | |
| Use limits $x = 0$, $x = 2$ correctly having integrated by parts twice | DM1 | |
| Obtain answer $24 - 8e$, or exact simplified equivalent | A1 | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{84df6b9a-6118-44a2-9c18-512039ded4fd-3_416_677_258_733}
The diagram shows part of the curve $y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }$ and its maximum point $M$.\\
(i) Find the exact $x$-coordinate of $M$.\\
(ii) Find the exact value of the area of the shaded region bounded by the curve and the positive $x$-axis.
\hfill \mbox{\textit{CAIE P3 2016 Q7 [9]}}