| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Specific function transformation description |
| Difficulty | Moderate -0.3 Part (a) is a standard recall question about horizontal stretch transformations. Parts (b-d) involve routine sketching and solving exponential equations by equating and rearranging. The algebraic manipulation in parts (c) and (d) requires completing the square or substitution but follows standard A-level techniques without requiring novel insight. Slightly easier than average due to the straightforward nature of the transformations and algebraic steps. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Stretch in the \(x\)-direction | B1 | |
| Stretch scale factor \(\frac{1}{2}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Graph: general shape of exponential, less steep than given graph for positive \(x\)] | B1 | General shape of exponential graph less steep than the given graph for positive \(x\) (note red graph is printed) |
| [Horizontal asymptote above \(x\)-axis, \(y\)-axis intersection above that of given graph] | B1 | Horizontal asymptote above the \(x\)-axis and intersection with \(y\)-axis must be above that for the given graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Graphs intersect when \(e^{2x} = k + e^x\), so \(e^{2x} - e^x - k = 0\) | M1 | Attempts to solve simultaneously. Allow \(k = -\frac{1}{4}\) substituted |
| Discriminant \((-1)^2 - 4(-k)\) | M1 | Uses discriminant of the equation |
| Is negative for \(k < -\frac{1}{4}\), so no real roots and no points of intersection | E1 | Must state no real roots or no points of intersection |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| When \(k=2\): \(e^{2x} - e^x - 2 = 0\), gives \(e^x = -1,\ 2\) | M1 | Evaluates \(e^x\) from their quadratic and attempts to use natural logs |
| So \(x = \ln 2\) as \(e^x = -1\) is not possible | A1 | Must state that \(\ln 2\) is a root and that there are no others. Allow SC1 for substituting \(x = \ln 2\) in both equations and concluding it must be a root |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Stretch in the $x$-direction | B1 | |
| Stretch scale factor $\frac{1}{2}$ | B1 | |
---
## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Graph: general shape of exponential, less steep than given graph for positive $x$] | B1 | General shape of exponential graph less steep than the given graph for positive $x$ (note red graph is printed) |
| [Horizontal asymptote above $x$-axis, $y$-axis intersection above that of given graph] | B1 | Horizontal asymptote above the $x$-axis and intersection with $y$-axis must be above that for the given graph |
---
## Question 9(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Graphs intersect when $e^{2x} = k + e^x$, so $e^{2x} - e^x - k = 0$ | M1 | Attempts to solve simultaneously. Allow $k = -\frac{1}{4}$ substituted |
| Discriminant $(-1)^2 - 4(-k)$ | M1 | Uses discriminant of the equation |
| Is negative for $k < -\frac{1}{4}$, so no real roots and no points of intersection | E1 | Must state no real roots or no points of intersection |
---
## Question 9(d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| When $k=2$: $e^{2x} - e^x - 2 = 0$, gives $e^x = -1,\ 2$ | M1 | Evaluates $e^x$ from their quadratic and attempts to use natural logs |
| So $x = \ln 2$ as $e^x = -1$ is not possible | A1 | Must state that $\ln 2$ is a root and that there are no others. Allow SC1 for substituting $x = \ln 2$ in both equations and concluding it must be a root |
---9 The graph shows the function $\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-6_595_732_322_242}
\begin{enumerate}[label=(\alph*)]
\item Describe the transformation of the graph of $y = e ^ { x }$ that gives the graph of $y = e ^ { 2 x }$.
A second function is defined by $\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }$.
\item A copy of the graph of $\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }$ is given in the Printed Answer Booklet.
Add a sketch of the graph of $\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }$ in a case where $k$ is a positive constant.
\item Show that the two graphs do not intersect for values of $k$ less than $- \frac { 1 } { 4 }$.
\item In the case where $k = 2$, show that the only point of intersection occurs when $x = \ln 2$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 1 2023 Q9 [9]}}