| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2007 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - mixed transcendental products |
| Difficulty | Standard +0.3 This is a straightforward application of the product rule to find stationary points of e^(-x)sin(x), requiring students to solve e^(-x)(cos x - sin x) = 0, leading to tan x = 1, then use the second derivative test. While it involves transcendental functions, the technique is standard and the algebra is clean, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct product or quotient rule | M1 | |
| Obtain derivative in any correct form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(x = \frac{1}{4}\pi\) or \(0.785\) with no errors seen | A1 | [4] |
| (ii) Use an appropriate method for determining the nature of a stationary point | M1 | |
| Show the point is a maximum point with no errors seen | A1 | [2] |
**(i)** Use correct product or quotient rule | M1 |
Obtain derivative in any correct form | A1 |
Equate derivative to zero and solve for $x$ | M1 |
Obtain answer $x = \frac{1}{4}\pi$ or $0.785$ with no errors seen | A1 | [4]
**(ii)** Use an appropriate method for determining the nature of a stationary point | M1 |
Show the point is a maximum point with no errors seen | A1 | [2]
[SR: for the answer $45°$ deduct final A1 in part (i), and deduct A1 in part (ii) if this value in degrees is used in the exponential.]4 The curve with equation $y = \mathrm { e } ^ { - x } \sin x$ has one stationary point for which $0 \leqslant x \leqslant \pi$.\\
(i) Find the $x$-coordinate of this point.\\
(ii) Determine whether this point is a maximum or a minimum point.
\hfill \mbox{\textit{CAIE P3 2007 Q4 [6]}}