| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2004 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - logarithmic functions |
| Difficulty | Moderate -0.3 This is a straightforward stationary point question requiring the product rule to differentiate x²ln(x), solving dy/dx = 0 to get x = e^(-1/2), and using the second derivative test. It's slightly easier than average as it follows a standard procedure with no conceptual surprises, though it does require competent handling of logarithmic differentiation. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the product rule to obtain first derivative (must involve 2 terms) | M1 | |
| Obtain derivative \(2x\ln x + x^2 \cdot \frac{1}{x}\), or equivalent | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(x = e^{-0.5}\) or \(\frac{1}{\sqrt{e}}\), or equivalent (e.g. 0.61) | A1 | Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Determine nature of stationary point using correct second derivative \((3 + 2\ln x)\) or correct first derivative or equation of curve (3 \(y\)-values, central one \(y(\exp(-0.5))\)) | M1 | |
| Show point is a minimum completely correctly | A1 | Total: 2 |
## Question 5(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the product rule to obtain first derivative (must involve 2 terms) | M1 | |
| Obtain derivative $2x\ln x + x^2 \cdot \frac{1}{x}$, or equivalent | A1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain answer $x = e^{-0.5}$ or $\frac{1}{\sqrt{e}}$, or equivalent (e.g. 0.61) | A1 | **Total: 4** |
## Question 5(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Determine nature of stationary point using correct second derivative $(3 + 2\ln x)$ or correct first derivative or equation of curve (3 $y$-values, central one $y(\exp(-0.5))$) | M1 | |
| Show point is a minimum completely correctly | A1 | **Total: 2** |
---5 The curve with equation $y = x ^ { 2 } \ln x$, where $x > 0$, has one stationary point.\\
(i) Find the $x$-coordinate of this point, giving your answer in terms of e .\\
(ii) Determine whether this point is a maximum or a minimum point.
\hfill \mbox{\textit{CAIE P2 2004 Q5 [6]}}