| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and tangent/normal |
| Difficulty | Standard +0.3 This is a standard two-part question combining routine differentiation (product rule with ln x and x^(1/2)) and a volumes of revolution integral. Part (i) requires straightforward application of the product rule and finding a tangent equation. Part (ii) involves setting up and evaluating ∫πy² dx with integration by parts, which is a standard P3/C4 technique. While it requires multiple steps and careful algebra, both parts follow well-practiced procedures without requiring novel insight or particularly challenging manipulation. |
| Spec | 1.07m Tangents and normals: gradient and equations1.08i Integration by parts4.08e Mean value of function: using integral |
| Answer | Marks | Guidance |
|---|---|---|
| Use correct product rule | M1 | |
| Obtain derivative in any correct form, e.g. \(\frac{\ln x}{2\sqrt{x}} + \frac{\sqrt{x}}{x}\) | A1 | |
| Carry out a complete method to form an equation of the tangent at \(x = 1\) | M1 | |
| Obtain answer \(y = x - 1\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply that the indefinite integral for the volume is \(\pi\int x(\ln x)^2\,dx\) | B1 | |
| Integrate by parts and reach \(ax^2(\ln x)^2 + b\int x^2 \cdot \frac{\ln x}{x}\,dx\) | M1* | |
| Obtain \(\frac{1}{2}x^2(\ln x)^2 - \int x\ln x\,dx\), or unsimplified equivalent | A1 | |
| Attempt second integration by parts reaching \(cx^2\ln x + d\int x^2 \cdot \frac{1}{x}\,dx\) | M1(dep*) | |
| Complete the integration correctly, obtaining \(\frac{1}{2}x^2(\ln x)^2 - \frac{1}{2}x^2\ln x + \frac{1}{4}x^2\) | A1 | |
| Substitute limits \(x = 1\) and \(x = e\), having integrated twice | M1(dep*) | |
| Obtain answer \(\frac{1}{4}\pi(e^2-1)\), or exact equivalent | A1 | [7] |
**(i)**
| Use correct product rule | M1 | |
| Obtain derivative in any correct form, e.g. $\frac{\ln x}{2\sqrt{x}} + \frac{\sqrt{x}}{x}$ | A1 | |
| Carry out a complete method to form an equation of the tangent at $x = 1$ | M1 | |
| Obtain answer $y = x - 1$ | A1 | [4] |
**(ii)**
| State or imply that the indefinite integral for the volume is $\pi\int x(\ln x)^2\,dx$ | B1 | |
| Integrate by parts and reach $ax^2(\ln x)^2 + b\int x^2 \cdot \frac{\ln x}{x}\,dx$ | M1* | |
| Obtain $\frac{1}{2}x^2(\ln x)^2 - \int x\ln x\,dx$, or unsimplified equivalent | A1 | |
| Attempt second integration by parts reaching $cx^2\ln x + d\int x^2 \cdot \frac{1}{x}\,dx$ | M1(dep*) | |
| Complete the integration correctly, obtaining $\frac{1}{2}x^2(\ln x)^2 - \frac{1}{2}x^2\ln x + \frac{1}{4}x^2$ | A1 | |
| Substitute limits $x = 1$ and $x = e$, having integrated twice | M1(dep*) | |
| Obtain answer $\frac{1}{4}\pi(e^2-1)$, or exact equivalent | A1 | [7] |
**Note:** [If $\pi$ omitted, or $2\pi$ or $\pi/2$ used, give B0 and then follow through.]
[Integration using parts $x\ln x$ and $\ln x$ is also viable.]
---9\\
\includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-3_421_767_1567_689}
The diagram shows the curve $y = x ^ { \frac { 1 } { 2 } } \ln x$. The shaded region between the curve, the $x$-axis and the line $x = \mathrm { e }$ is denoted by $R$.\\
(i) Find the equation of the tangent to the curve at the point where $x = 1$, giving your answer in the form $y = m x + c$.\\
(ii) Find by integration the volume of the solid obtained when the region $R$ is rotated completely about the $x$-axis. Give your answer in terms of $\pi$ and e.
\hfill \mbox{\textit{CAIE P3 2012 Q9 [11]}}