| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Curve with minimum point |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard techniques: finding x-intercepts (trivial), finding stationary points using quotient rule (routine), and integration by parts with ln x (textbook exercise). While it requires multiple steps, each component is a standard A-level technique with no novel insight required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) State \(x\)-coordinate of \(A\) is 1 | B1 | Total: 1 |
| (ii) Use product or quotient rule | M1 | [SR: if quotient rule misused with reversed numerator or \(x^2\) instead of \(x^4\) in denominator, award M0A0 but allow following M1A1A1.] |
| Obtain derivative in any correct form e.g. \(-\frac{2\ln x}{x^3} + \frac{1}{x} \cdot \frac{1}{x^2}\) | A1 | |
| Equate derivative to zero and solve for \(\ln x\) | M1 | |
| Obtain \(x = e^{\frac{1}{2}}\) or equivalent (accept 1.65) | A1 | |
| Obtain \(y = \frac{1}{2e}\) or exact equivalent not involving ln | A1 | Total: 5 |
| (iii) Attempt integration by parts, going the correct way | M1 | [If \(u = \ln x\) is used, apply an analogous scheme to the result of the substitution.] |
| Obtain \(-\frac{\ln x}{x} + \int \frac{1}{x} \cdot \frac{1}{x}\,dx\) or equivalent | A1 | |
| Obtain indefinite integral \(-\frac{\ln x}{x} - \frac{1}{x}\) | A1 | |
| Use \(x\)-coordinate of \(A\) and \(e\) as limits, having integrated twice | M1 | |
| Obtain exact answer \(1 - \frac{2}{e}\), or equivalent | A1 | Total: 5 |
## Question 10:
| Answer/Working | Mark | Guidance |
|---|---|---|
| **(i)** State $x$-coordinate of $A$ is 1 | B1 | **Total: 1** |
| **(ii)** Use product or quotient rule | M1 | [SR: if quotient rule misused with reversed numerator or $x^2$ instead of $x^4$ in denominator, award M0A0 but allow following M1A1A1.] |
| Obtain derivative in any correct form e.g. $-\frac{2\ln x}{x^3} + \frac{1}{x} \cdot \frac{1}{x^2}$ | A1 | |
| Equate derivative to zero and solve for $\ln x$ | M1 | |
| Obtain $x = e^{\frac{1}{2}}$ or equivalent (accept 1.65) | A1 | |
| Obtain $y = \frac{1}{2e}$ or exact equivalent not involving ln | A1 | **Total: 5** |
| **(iii)** Attempt integration by parts, going the correct way | M1 | [If $u = \ln x$ is used, apply an analogous scheme to the result of the substitution.] |
| Obtain $-\frac{\ln x}{x} + \int \frac{1}{x} \cdot \frac{1}{x}\,dx$ or equivalent | A1 | |
| Obtain indefinite integral $-\frac{\ln x}{x} - \frac{1}{x}$ | A1 | |
| Use $x$-coordinate of $A$ and $e$ as limits, having integrated twice | M1 | |
| Obtain exact answer $1 - \frac{2}{e}$, or equivalent | A1 | **Total: 5** |
---10\\
\includegraphics[max width=\textwidth, alt={}, center]{2718ebbb-29e3-46f7-8d8d-ec7d526483f8-3_458_920_1144_609}
The diagram shows the curve $y = \frac { \ln x } { x ^ { 2 } }$ and its maximum point $M$. The curve cuts the $x$-axis at $A$.\\
(i) Write down the $x$-coordinate of $A$.\\
(ii) Find the exact coordinates of $M$.\\
(iii) Use integration by parts to find the exact area of the shaded region enclosed by the curve, the $x$-axis and the line $x = \mathrm { e }$.
\hfill \mbox{\textit{CAIE P3 2004 Q10 [11]}}