| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - polynomial/exponential products |
| Difficulty | Standard +0.3 This is a standard C3 question on product rule differentiation and integration of exponential functions. Part (ii) requires routine application of product rule to find stationary points, while other parts involve straightforward substitution and transformation reasoning. Slightly easier than average due to the guided multi-part structure and standard techniques throughout. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07q Product and quotient rules: differentiation1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(g(x) = 3f(\frac{1}{2}x) = 3(1-\frac{1}{2}x)e^x\) | B1 | o.e.; mark final answer |
| Through \((2,0)\) and \((0,3)\) | B1 | condone errors in writing coordinates (e.g. \((0,2)\)) |
| Reasonable shape | B1dep | dependent on previous B1 |
| TP at \((1, \frac{3e}{2})\) or \((1, 4.1)\) (or better) | B1 | Must be evidence that \(x=1\), \(y=4.1\) is indeed the TP — appearing in a table of values is not enough on its own |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(6 \times \frac{1}{4}(e^2-3) \left[= \frac{3(e^2-3)}{2}\right]\) | B1 | o.e. mark final answer |
# Question 1:
## Part (iv)
| Answer | Mark | Guidance |
|--------|------|----------|
| $g(x) = 3f(\frac{1}{2}x) = 3(1-\frac{1}{2}x)e^x$ | B1 | o.e.; mark final answer |
| Through $(2,0)$ and $(0,3)$ | B1 | condone errors in writing coordinates (e.g. $(0,2)$) |
| Reasonable shape | B1dep | dependent on previous B1 |
| TP at $(1, \frac{3e}{2})$ or $(1, 4.1)$ (or better) | B1 | Must be evidence that $x=1$, $y=4.1$ is indeed the TP — appearing in a table of values is not enough on its own |
**[4]**
## Part (v)
| Answer | Mark | Guidance |
|--------|------|----------|
| $6 \times \frac{1}{4}(e^2-3) \left[= \frac{3(e^2-3)}{2}\right]$ | B1 | o.e. mark final answer |
**[1]**
---1 Fig. 8 shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = ( 1 - x ) \mathrm { e } ^ { 2 x }$, with its turning point P .
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{75eebbfb-7bfa-4382-a6d7-1c5a7f3f419a-1_722_817_450_642}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}
(i) Write down the coordinates of the intercepts of $y = \mathrm { f } ( x )$ with the $x$ - and $y$-axes.\\
(ii) Find the exact coordinates of the turning point P .\\
(iii) Show that the exact area of the region enclosed by the curve and the $x$ - and $y$-axes is $\frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 } - 3 \right)$. The function $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$.\\
(iv) Express $\mathrm { g } ( x )$ in terms of $x$.
Sketch the curve $y = \mathrm { g } ( x )$ on the copy of Fig. 8, indicating the coordinates of its intercepts with the $x$ - and $y$-axes and of its turning point.\\
(v) Write down the exact area of the region enclosed by the curve $y = \mathrm { g } ( x )$ and the $x$ - and $y$-axes.
\hfill \mbox{\textit{OCR MEI C3 Q1 [18]}}