| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Curve with minimum point |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard techniques: (i) differentiate using product rule, set to zero, and solve for the minimum point; (ii) integrate by parts to find the area. Both are routine applications of A-level methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08i Integration by parts |
| Answer | Marks |
|---|---|
| (i) Use correct product rule | M1 |
| Obtain correct derivative in any form | A1 |
| Equate derivative to zero and find non-zero \(x\) | M1 |
| Obtain \(x = \exp(-\frac{1}{4})\), or equivalent | A1 |
| Obtain \(y = -l(3e)\), or any ln-free equivalent | A1 |
| Answer | Marks |
|---|---|
| (ii) Integrate and read \(kx^4 \ln x + \int x^3 \cdot \frac{1}{x} dx\) | M1 |
| Obtain \(\frac{1}{4}x^4 \ln x - \frac{1}{4}\int x^3 \, dx\) | A1 |
| Obtain integral \(\frac{1}{4}x^4 \ln x - \frac{1}{16}x^4\), or equivalent | A1 |
| Use limits \(x = 1\) and \(x = 2\) correctly, having integrated twice | M1 |
| Obtain answer \(4 \ln 2 - \frac{15}{16}\), or exact equivalent | A1 |
**(i)** Use correct product rule | M1 |
Obtain correct derivative in any form | A1 |
Equate derivative to zero and find non-zero $x$ | M1 |
Obtain $x = \exp(-\frac{1}{4})$, or equivalent | A1 |
Obtain $y = -l(3e)$, or any ln-free equivalent | A1 |
[5 marks total]
**(ii)** Integrate and read $kx^4 \ln x + \int x^3 \cdot \frac{1}{x} dx$ | M1 |
Obtain $\frac{1}{4}x^4 \ln x - \frac{1}{4}\int x^3 \, dx$ | A1 |
Obtain integral $\frac{1}{4}x^4 \ln x - \frac{1}{16}x^4$, or equivalent | A1 |
Use limits $x = 1$ and $x = 2$ correctly, having integrated twice | M1 |
Obtain answer $4 \ln 2 - \frac{15}{16}$, or exact equivalent | A1 |
[5 marks total]9\\
\includegraphics[max width=\textwidth, alt={}, center]{bbc19395-6f88-4a7c-b5d4-59ced9ccdcf2-4_597_895_258_625}
The diagram shows the curve $y = x ^ { 3 } \ln x$ and its minimum point $M$.\\
(i) Find the exact coordinates of $M$.\\
(ii) Find the exact area of the shaded region bounded by the curve, the $x$-axis and the line $x = 2$.
\hfill \mbox{\textit{CAIE P3 2010 Q9 [10]}}