| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Integration with differentiation context |
| Difficulty | Standard +0.8 This question requires product rule differentiation to find the maximum (setting derivative to zero), then integration by parts twice for the area calculation. The integration of x²e^(-x/2) is non-routine and requires careful application of integration by parts multiple times, making it moderately challenging but still within standard P3 scope. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use product or quotient rule | M1* | |
| Obtain first derivative \(2xe^{-\frac{1}{2}x} - \frac{1}{2}x^2e^{-\frac{1}{2}x}\) or equivalent | A1 | |
| Equate derivative to zero and solve for non-zero \(x\) | M1(dep*) | |
| Obtain answer \(x = 4\) | A1 | Total: 4 marks |
| (ii) Integrate by parts once, obtaining \(kx^2e^{-\frac{1}{2}x} + \int \left[xe^{-\frac{1}{2}x}\right]dx\), where \(kl \neq 0\) | M1 | |
| Obtain integral \(-2x^2e^{-\frac{1}{2}x} + 4\int xe^{-\frac{1}{2}x}dx\), or any unsimplified equivalent | A1 | |
| Complete the integration, obtaining \(-2(x^2 + 4x + 8)e^{-\frac{1}{2}x}\) or equivalent | A1 | |
| Having integrated by parts twice, use limits \(x = 0\) and \(x = 1\) in the complete integral | M1 | |
| Obtain simplified answer \(16 – 26e^{-\frac{1}{2}}\) or equivalent | A1 | Total: 5 marks |
**(i)** Use product or quotient rule | M1* | |
Obtain first derivative $2xe^{-\frac{1}{2}x} - \frac{1}{2}x^2e^{-\frac{1}{2}x}$ or equivalent | A1 | |
Equate derivative to zero and solve for non-zero $x$ | M1(dep*) | |
Obtain answer $x = 4$ | A1 | **Total: 4 marks** |
**(ii)** Integrate by parts once, obtaining $kx^2e^{-\frac{1}{2}x} + \int \left[xe^{-\frac{1}{2}x}\right]dx$, where $kl \neq 0$ | M1 | |
Obtain integral $-2x^2e^{-\frac{1}{2}x} + 4\int xe^{-\frac{1}{2}x}dx$, or any unsimplified equivalent | A1 | |
Complete the integration, obtaining $-2(x^2 + 4x + 8)e^{-\frac{1}{2}x}$ or equivalent | A1 | |
Having integrated by parts twice, use limits $x = 0$ and $x = 1$ in the complete integral | M1 | |
Obtain simplified answer $16 – 26e^{-\frac{1}{2}}$ or equivalent | A1 | **Total: 5 marks** |7\\
\includegraphics[max width=\textwidth, alt={}, center]{8c533469-393c-4e4c-a6ec-eab1303741e7-3_480_901_973_621}
The diagram shows the curve $y = x ^ { 2 } e ^ { - \frac { 1 } { 2 } x }$.\\
(i) Find the $x$-coordinate of $M$, the maximum point of the curve.\\
(ii) Find the area of the shaded region enclosed by the curve, the $x$-axis and the line $x = 1$, giving your answer in terms of e.
\hfill \mbox{\textit{CAIE P3 2004 Q7 [9]}}