| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Stationary points then area/volume |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: (i) finding stationary points using product rule differentiation (routine calculation leading to a given answer), and (ii) integration by parts applied twice to a polynomial-exponential product. Both parts are textbook exercises with no novel insight required, making this slightly easier than average for A-level. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | ||
| Use product rule to find first derivative | M1 | |
| Obtain \(2xe^{2-x} - x^2e^{2-x}\) | A1 | |
| Confirm \(x = 2\) at M | A1 | [3] |
| (ii) | ||
| Attempt integration by parts and reach \(\pm x^2e^{2-x} \pm \int 2xe^{2-x}\,dx\) | *M1 | |
| Obtain \(-x^2e^{2-x} + \int 2xe^{2-x}\,dx\) | A1 | |
| Attempt integration by parts and reach \(\pm x^2e^{2-x} \pm 2xe^{2-x} \pm 2e^{2-x}\) | *M1 | |
| Obtain \(-x^2e^{2-x} - 2xe^{2-x} - 2e^{2-x}\) | A1 | |
| Use limits 0 and 2 having integrated twice | M1 dep *M | |
| Obtain \(2e^2 - 10\) | A1 | [6] |
**(i)** |
Use product rule to find first derivative | M1 |
Obtain $2xe^{2-x} - x^2e^{2-x}$ | A1 |
Confirm $x = 2$ at M | A1 | [3]
**(ii)** |
Attempt integration by parts and reach $\pm x^2e^{2-x} \pm \int 2xe^{2-x}\,dx$ | *M1 |
Obtain $-x^2e^{2-x} + \int 2xe^{2-x}\,dx$ | A1 |
Attempt integration by parts and reach $\pm x^2e^{2-x} \pm 2xe^{2-x} \pm 2e^{2-x}$ | *M1 |
Obtain $-x^2e^{2-x} - 2xe^{2-x} - 2e^{2-x}$ | A1 |
Use limits 0 and 2 having integrated twice | M1 dep *M |
Obtain $2e^2 - 10$ | A1 | [6]9\\
\includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_399_696_255_721}
The diagram shows the curve $y = x ^ { 2 } \mathrm { e } ^ { 2 - x }$ and its maximum point $M$.\\
(i) Show that the $x$-coordinate of $M$ is 2 .\\
(ii) Find the exact value of $\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm { e } ^ { 2 - x } \mathrm {~d} x$.
\hfill \mbox{\textit{CAIE P3 2015 Q9 [9]}}