| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - polynomial/exponential products |
| Difficulty | Moderate -0.3 This is a straightforward application of the product rule to find a stationary point, followed by a standard second derivative test. The algebra is simple (linear factor times exponential), and both parts are routine A-level calculus procedures requiring no problem-solving insight, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks |
|---|---|
| (i) Use product rule | M1* |
| Obtain correct derivative in any form, e.g. \((x - 1)e^x\) | A1 |
| Equate derivative to zero and solve for \(x\) | M1* (dep) |
| Obtain \(x = 1\) | A1 |
| Obtain \(y = -e\) | A1 |
| [5] | |
| (ii) Carry out a method for determining the nature of a stationary point | M1 |
| Show that the point is a minimum point, with no errors seen | A1 |
| [2] |
**(i)** Use product rule | M1* |
Obtain correct derivative in any form, e.g. $(x - 1)e^x$ | A1 |
Equate derivative to zero and solve for $x$ | M1* (dep) |
Obtain $x = 1$ | A1 |
Obtain $y = -e$ | A1 |
| [5] |
**(ii)** Carry out a method for determining the nature of a stationary point | M1 |
Show that the point is a minimum point, with no errors seen | A1 |
| [2] |6 It is given that the curve $y = ( x - 2 ) \mathrm { e } ^ { x }$ has one stationary point.\\
(i) Find the exact coordinates of this point.\\
(ii) Determine whether this point is a maximum or a minimum point.
\hfill \mbox{\textit{CAIE P2 2008 Q6 [7]}}