| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Specific function transformation description |
| Difficulty | Moderate -0.3 Part (a) is a routine transformation question requiring identification of horizontal stretch and translation from y=e^x to y=e^(2x-5). Part (b) involves standard calculus techniques (finding gradient, normal equation, intercepts, triangle area) but requires careful algebraic manipulation to reach the given form. The multi-step nature and algebraic complexity elevate it slightly above pure routine, but all techniques are standard C3 material with no novel insight required. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Stretch [Parallel to] \(x\)[-axis] | M1 | I and II or III |
| [SF] \(0.5\) | A1 | \(I + II + III\) |
| Translation \(\begin{bmatrix} k \\ 0 \end{bmatrix}\) | M1 | or (2nd) Stretch [parallel to] \(y\)[-axis] |
| \(\begin{bmatrix} 2.5 \\ 0 \end{bmatrix}\) | A1 | SF \(e^{-5}\); for the '2 stretch' method, if the '\(y\)' direction stretch is first, marks can only be earned if there is a second stretch in '\(x\)' direction |
| OR Translation \(\begin{bmatrix} k \\ 0 \end{bmatrix}\) | (M1) | The stretches can be in either order |
| \(\begin{bmatrix} 5 \\ 0 \end{bmatrix}\) | (A1) | |
| then Stretch [Parallel to] \(x\)[-axis] [SF] \(0.5\) | (M1)(A1) | Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = 2e^{2x-5}\) | B1 | |
| Grad normal \(= -\dfrac{1}{\text{their gradient}}\) | B1F | Condone expression in terms of \(x\) |
| \(y - e^{-1} = -\dfrac{e}{2}(x-2)\) | B1 | Must be exact values; (At \(A\), \(y=0\)) |
| \(x = 2 + \dfrac{2}{e^2}\) | M1 | Attempt to find at least one intercept from 'their' normal, subst \(x=0\) or \(y=0\) in any straight line equation |
| \(y = e + \dfrac{1}{e} = \dfrac{e^2+1}{e}\) | A1 | Both \(x\) and \(y\) values correct |
| Area \(= 0.5 \times \dfrac{(e^2+1)}{e} \times \dfrac{2(1+e^2)}{e^2}\) | ||
| \(= \dfrac{(e^2+1)^2}{e^3}\) | A1 | Total: 6 |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Stretch [Parallel to] $x$[-axis] | M1 | I and II or III |
| [SF] $0.5$ | A1 | $I + II + III$ |
| Translation $\begin{bmatrix} k \\ 0 \end{bmatrix}$ | M1 | **or** (2nd) Stretch [parallel to] $y$[-axis] |
| $\begin{bmatrix} 2.5 \\ 0 \end{bmatrix}$ | A1 | SF $e^{-5}$; for the '2 stretch' method, if the '$y$' direction stretch is first, marks **can only** be earned if there is a **second** stretch in '$x$' direction |
| **OR** Translation $\begin{bmatrix} k \\ 0 \end{bmatrix}$ | (M1) | The stretches can be in either order |
| $\begin{bmatrix} 5 \\ 0 \end{bmatrix}$ | (A1) | |
| **then** Stretch [Parallel to] $x$[-axis] [SF] $0.5$ | (M1)(A1) | **Total: 4** |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 2e^{2x-5}$ | B1 | |
| Grad normal $= -\dfrac{1}{\text{their gradient}}$ | B1F | Condone expression in terms of $x$ |
| $y - e^{-1} = -\dfrac{e}{2}(x-2)$ | B1 | Must be exact values; (At $A$, $y=0$) |
| $x = 2 + \dfrac{2}{e^2}$ | M1 | Attempt to find at least one intercept from 'their' normal, subst $x=0$ **or** $y=0$ in any straight line equation |
| $y = e + \dfrac{1}{e} = \dfrac{e^2+1}{e}$ | A1 | Both $x$ **and** $y$ values correct |
| Area $= 0.5 \times \dfrac{(e^2+1)}{e} \times \dfrac{2(1+e^2)}{e^2}$ | | |
| $= \dfrac{(e^2+1)^2}{e^3}$ | A1 | **Total: 6** |
---4
\begin{enumerate}[label=(\alph*)]
\item Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { e } ^ { x }$ onto the graph of $y = \mathrm { e } ^ { 2 x - 5 }$.
\item The normal to the curve $y = \mathrm { e } ^ { 2 x - 5 }$ at the point $P \left( 2 , \mathrm { e } ^ { - 1 } \right)$ intersects the $x$-axis at the point $A$ and the $y$-axis at the point $B$.
Show that the area of the triangle $O A B$ is $\frac { \left( \mathrm { e } ^ { 2 } + 1 \right) ^ { m } } { \mathrm { e } ^ { n } }$, where $m$ and $n$ are integers.\\[0pt]
[6 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2016 Q4 [10]}}