| 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 expression (requiring integration by parts twice). The differentiation is moderately complex, and the integration by parts must be applied carefully with the exponential-polynomial product. While systematic, it demands solid technique and careful algebra across multiple steps, placing it 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:
### Part (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 | |
### Part (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]{f4614578-f5f6-4283-8185-8b5598ad91d5-3_416_679_258_731}
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]}}