| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 This is a standard C3 product rule question with routine differentiation of exponential and trigonometric functions, followed by solving dy/dx=0 numerically and applying the second derivative test. While it requires multiple steps and careful algebra, all techniques are core syllabus material with no novel insight required, making it slightly easier than average. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n 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 |
|---|---|---|
| (a) \(\frac{dy}{dx} = 3e^{3x} \times \cos 2x + e^{3x} \times (-2 \sin 2x) = e^{3x}(3 \cos 2x - 2 \sin 2x)\) | M1 A1 | |
| (b) \(\frac{d^2y}{dx^2} = 3e^{3x} \times (3 \cos 2x - 2 \sin 2x) + e^{3x}(-6 \sin 2x - 4 \cos 2x)\) | M1 A1 | |
| \(= e^{3x}(5 \cos 2x - 12 \sin 2x)\) | A1 | |
| (c) SP: \(e^{3x}(3 \cos 2x - 2 \sin 2x) = 0\) | M1 | |
| \(3 \cos 2x = 2 \sin 2x\) | M1 | |
| \(\tan 2x = \frac{3}{2}\) | M1 | |
| \(2x = 0.98279\), \(x = 0.491\) (3sf) | M1 A1 | |
| (d) when \(x = 0.491\), \(\frac{d^2y}{dx^2} = -31.5\), \(\frac{d^2y}{dx^2} < 0\) \(\therefore\) maximum | M1 A1 | (11 marks) |
**(a)** $\frac{dy}{dx} = 3e^{3x} \times \cos 2x + e^{3x} \times (-2 \sin 2x) = e^{3x}(3 \cos 2x - 2 \sin 2x)$ | M1 A1
**(b)** $\frac{d^2y}{dx^2} = 3e^{3x} \times (3 \cos 2x - 2 \sin 2x) + e^{3x}(-6 \sin 2x - 4 \cos 2x)$ | M1 A1
$= e^{3x}(5 \cos 2x - 12 \sin 2x)$ | A1
**(c)** SP: $e^{3x}(3 \cos 2x - 2 \sin 2x) = 0$ | M1
$3 \cos 2x = 2 \sin 2x$ | M1
$\tan 2x = \frac{3}{2}$ | M1
$2x = 0.98279$, $x = 0.491$ (3sf) | M1 A1
**(d)** when $x = 0.491$, $\frac{d^2y}{dx^2} = -31.5$, $\frac{d^2y}{dx^2} < 0$ $\therefore$ maximum | M1 A1 | (11 marks)6. A curve has the equation $y = \mathrm { e } ^ { 3 x } \cos 2 x$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { 3 x } ( 5 \cos 2 x - 12 \sin 2 x )$.
The curve has a stationary point in the interval $[ 0,1 ]$.
\item Find the $x$-coordinate of the stationary point to 3 significant figures.
\item Determine whether the stationary point is a maximum or minimum point and justify your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q6 [11]}}