| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| 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 application of logarithmic differentiation and stationary point analysis. Students must differentiate using the chain rule for ln(2x+9) and standard derivative of ln x, set dy/dx = 0 to solve a simple equation, then use the second derivative test. All steps are routine A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Differentiate to obtain form \(\frac{k_1}{2x+9} - \frac{k_2}{x}\) | M1 | |
| Obtain correct \(\frac{6}{2x+9} - \frac{2}{x}\) | A1 | |
| Equate first derivative to zero and attempt solution to \(x = ...\) | M1 | Dependent on previous M1 |
| Obtain \(x = 9\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use appropriate method for determining nature of stationary point | M1 | Second derivative or gradient or value of \(y\) |
| Conclude minimum with no errors seen | A1 |
## Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate to obtain form $\frac{k_1}{2x+9} - \frac{k_2}{x}$ | M1 | |
| Obtain correct $\frac{6}{2x+9} - \frac{2}{x}$ | A1 | |
| Equate first derivative to zero and attempt solution to $x = ...$ | M1 | Dependent on previous M1 |
| Obtain $x = 9$ | A1 | |
## Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use appropriate method for determining nature of stationary point | M1 | Second derivative or gradient or value of $y$ |
| Conclude minimum with no errors seen | A1 | |2 A curve has equation $y = 3 \ln ( 2 x + 9 ) - 2 \ln x$.\\
(i) Find the $x$-coordinate of the stationary point.\\
(ii) Determine whether the stationary point is a maximum or minimum point.\\
\hfill \mbox{\textit{CAIE P2 2018 Q2 [6]}}