| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Sequence of transformations order |
| Difficulty | Standard +0.3 This is a standard C3 transformations question requiring identification of graph endpoints, describing transformations (horizontal shift and vertical stretch), sketching the result, and finding dx/dy using implicit differentiation. All techniques are routine for this module, though part (d) requires careful algebraic manipulation of inverse trig functions. Slightly above average due to the multi-step nature and the less common dx/dy calculation. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P(-1,\pi)\) | B1 | Condone \((-1, 180°)\) |
| \(Q(1,0)\) | B1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Translate | E1 | |
| \(\begin{bmatrix}1\\0\end{bmatrix}\) | B1 | or equivalent in words |
| Stretch SF 2 \(\parallel\) \(y\)-axis | M1 | Stretch + one other correct |
| Full description correct | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Correct shape in 1st quadrant | B1 | |
| \(2\pi\) and 2 marked correctly | B1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{y}{2}=\cos^{-1}(x-1)\) | M1 | |
| \(\cos\left(\frac{y}{2}\right)=x-1\) | ||
| \(x=\cos\left(\frac{y}{2}\right)+1\) | A1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(-\frac{1}{2}\sin\left(\frac{y}{2}\right)\) | M1 A1 | |
| At \(y=2\), \(\frac{dx}{dy}=-\frac{1}{2}\sin 1\) | A1 | 3 |
# Question 8:
## Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $P(-1,\pi)$ | B1 | Condone $(-1, 180°)$ |
| $Q(1,0)$ | B1 | 2 | |
## Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Translate | E1 | |
| $\begin{bmatrix}1\\0\end{bmatrix}$ | B1 | or equivalent in words |
| Stretch SF 2 $\parallel$ $y$-axis | M1 | Stretch + one other correct |
| Full description correct | A1 | 4 | all correct |
## Part (c):
| Working | Mark | Guidance |
|---------|------|----------|
| Correct shape in 1st quadrant | B1 | |
| $2\pi$ and 2 marked correctly | B1 | 2 | |
## Part (d)(i):
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{y}{2}=\cos^{-1}(x-1)$ | M1 | |
| $\cos\left(\frac{y}{2}\right)=x-1$ | | |
| $x=\cos\left(\frac{y}{2}\right)+1$ | A1 | 2 | |
## Part (d)(ii):
| Working | Mark | Guidance |
|---------|------|----------|
| $-\frac{1}{2}\sin\left(\frac{y}{2}\right)$ | M1 A1 | | $k\sin(...)$; $\frac{dx}{dy}$ correct |
| At $y=2$, $\frac{dx}{dy}=-\frac{1}{2}\sin 1$ | A1 | 3 | Condone AWRT $-0.42$ |
---8 The sketch shows the graph of $y = \cos ^ { - 1 } x$.\\
\includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-5_593_686_383_683}
\begin{enumerate}[label=(\alph*)]
\item Write down the coordinates of $P$ and $Q$, the end points of the graph.
\item Describe a sequence of two geometrical transformations that maps the graph of $y = \cos ^ { - 1 } x$ onto the graph of $y = 2 \cos ^ { - 1 } ( x - 1 )$.
\item Sketch the graph of $y = 2 \cos ^ { - 1 } ( x - 1 )$.
\item \begin{enumerate}[label=(\roman*)]
\item Write the equation $y = 2 \cos ^ { - 1 } ( x - 1 )$ in the form $x = \mathrm { f } ( y )$.
\item Hence find the value of $\frac { \mathrm { d } x } { \mathrm {~d} y }$ when $y = 2$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2009 Q8 [13]}}