| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points coordinates |
| Difficulty | Standard +0.8 This requires applying the product rule to differentiate e^(2x)cos(x), setting the derivative to zero, solving a transcendental equation involving exponentials and trigonometry (tan x = -2), then substituting back to find both coordinates. The multi-step process and the need to handle the transcendental equation make this moderately challenging, though the techniques themselves are standard C3 material. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07q Product and quotient rules: differentiation |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = 2e^{2x}\cos x - e^{2x}\sin x\) | M1 product rule used |
| \(\frac{dy}{dx} = e^{2x}(2\cos x - \sin x)\) | A1 cao – mark final answer |
| \(\frac{dy}{dx} = 0\) | M1 their derivative = 0 |
| \(2\cos x - \sin x = 0\) | M1 |
| Answer | Marks |
|---|---|
| \(2 = \frac{\sin x}{\cos x} = \tan x\) | M1 \(\sin x / \cos x = \tan x\) used |
| \(x = 1.11\) | A1 \(1.1\) or \(0.35\) or better, or \(\arctan 2\), not \(63.4°\) but condone answer given in both degrees and radians here |
| \(y = 4.09\) | A1 cao consistent with their derivatives (e.g. \(2e^{2x} - e^{2x}\tan x\) is A0 or \(\sin^2 x + \cos^2 x = 1\) used) \(1.1071487\ldots, 0.352416\ldots,\) penalise incorrect rounding, no choice |
# Question 1
$y = e^{2x}\cos x$
$\frac{dy}{dx} = 2e^{2x}\cos x - e^{2x}\sin x$ | M1 product rule used
$\frac{dy}{dx} = e^{2x}(2\cos x - \sin x)$ | A1 cao – mark final answer
$\frac{dy}{dx} = 0$ | M1 their derivative = 0
$2\cos x - \sin x = 0$ | M1
$2\cos x = \sin x$
$2 = \frac{\sin x}{\cos x} = \tan x$ | M1 $\sin x / \cos x = \tan x$ used
$x = 1.11$ | A1 $1.1$ or $0.35$ or better, or $\arctan 2$, not $63.4°$ but condone answer given in both degrees and radians here
$y = 4.09$ | A1 cao consistent with their derivatives (e.g. $2e^{2x} - e^{2x}\tan x$ is A0 or $\sin^2 x + \cos^2 x = 1$ used) $1.1071487\ldots, 0.352416\ldots,$ penalise incorrect rounding, no choice
[6]1 Fig. 1 shows part of the curve $y = \mathrm { e } ^ { 2 x } \cos x$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{01bdea17-c698-44ae-a45a-7da4de631de4-1_669_1032_459_538}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Find the coordinates of the turning point P .
\hfill \mbox{\textit{OCR MEI C3 Q1 [6]}}