| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - mixed transcendental products |
| Difficulty | Standard +0.8 This question requires product rule differentiation of an exponential-trigonometric product, then solving a transcendental equation (likely requiring tan inverse) to find the stationary point, followed by second derivative test or sign analysis. While the differentiation is standard A-level technique, solving the resulting equation and completing both parts requires solid algebraic manipulation and multiple connected steps, placing it moderately above average difficulty. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule | M1 | |
| Obtain derivative in any correct form e.g. \(2e^{2x}(\sin x + 3\cos x) + e^{2x}(\cos x - 3\sin x)\) | A1 | |
| Equate derivative to zero and obtain an equation in one trigonometric ratio | M1 | |
| Obtain \(x = 1.43\) only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to determine the nature of the stationary point e.g. \(x=1.42,\ y'=0.06e^{2.84}>0\); \(x=1.44,\ y'=-0.07e^{2.88}<0\) | M1 | |
| Show that it is a maximum point | A1 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule | M1 | |
| Obtain derivative in any correct form e.g. $2e^{2x}(\sin x + 3\cos x) + e^{2x}(\cos x - 3\sin x)$ | A1 | |
| Equate derivative to zero and obtain an equation in one trigonometric ratio | M1 | |
| Obtain $x = 1.43$ only | A1 | |
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## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to determine the nature of the stationary point e.g. $x=1.42,\ y'=0.06e^{2.84}>0$; $x=1.44,\ y'=-0.07e^{2.88}<0$ | M1 | |
| Show that it is a maximum point | A1 | |
---4 The curve with equation $y = \mathrm { e } ^ { 2 x } ( \sin x + 3 \cos x )$ has a stationary point in the interval $0 \leqslant x \leqslant \pi$.
\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinate of this point, giving your answer correct to 2 decimal places.
\item Determine whether the stationary point is a maximum or a minimum.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q4 [6]}}