| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - logarithmic functions |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on a standard transcendental function. Part (i) requires simple observation (ln x = 0 when x = 1). Part (ii) involves routine differentiation using quotient rule and solving f'(x) = 0, yielding x = e². Part (iii) is a guided integration by parts with a specific result to verify. All techniques are standard P3/C3 material 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 |
|---|---|
| State coordinates \((1, 0)\) | B1 |
| [1] |
| Answer | Marks |
|---|---|
| Use correct quotient or product rule | M1 |
| Obtain derivative in any correct form | A1 |
| Equate derivative to zero and solve for \(x\) | M1 |
| Obtain \(x = e^2\) correctly | A1 |
| [4] |
| Answer | Marks |
|---|---|
| Attempt integration by parts reaching \(a\sqrt{x}\ln x + a\int\sqrt{x} \cdot \frac{1}{x}dx\) | M1* |
| Obtain \(2\sqrt{x}\ln x - 2\int\frac{1}{\sqrt{x}}dx\) | A1 |
| Integrate and obtain \(2\sqrt{x}\ln x - 4\sqrt{x}\) | A1 |
| Use limits \(x = 1\) and \(x = 4\) correctly, having integrated twice | M1(dep*) |
| Justify the given answer | A1 |
| [5] |
**(i)**
| State coordinates $(1, 0)$ | B1 |
| [1] |
**(ii)**
| Use correct quotient or product rule | M1 |
| Obtain derivative in any correct form | A1 |
| Equate derivative to zero and solve for $x$ | M1 |
| Obtain $x = e^2$ correctly | A1 |
| [4] |
**(iii)**
| Attempt integration by parts reaching $a\sqrt{x}\ln x + a\int\sqrt{x} \cdot \frac{1}{x}dx$ | M1* |
| Obtain $2\sqrt{x}\ln x - 2\int\frac{1}{\sqrt{x}}dx$ | A1 |
| Integrate and obtain $2\sqrt{x}\ln x - 4\sqrt{x}$ | A1 |
| Use limits $x = 1$ and $x = 4$ correctly, having integrated twice | M1(dep*) |
| Justify the given answer | A1 |
| [5] |9\\
\includegraphics[max width=\textwidth, alt={}, center]{8d134c65-af23-4508-acef-49b6ab49e374-3_504_910_625_614}
The diagram shows the curve $y = \frac { \ln x } { \sqrt { } x }$ and its maximum point $M$. The curve cuts the $x$-axis at the point $A$.\\
(i) State the coordinates of $A$.\\
(ii) Find the exact value of the $x$-coordinate of $M$.\\
(iii) Using integration by parts, show that the area of the shaded region bounded by the curve, the $x$-axis and the line $x = 4$ is equal to $8 \ln 2 - 4$.
\hfill \mbox{\textit{CAIE P3 2009 Q9 [10]}}