| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Sequence of transformations order |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 topics: function transformations (routine identification of stretch and translation), numerical integration using mid-ordinate rule (straightforward application), exact integration of exponentials (standard technique), and area between curves (standard but requires careful setup). All parts are textbook exercises requiring competent technique but no novel insight, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Stretch parallel to \(x\)-axis | B1 | |
| Scale factor \(\frac{1}{2}\) | B1 | |
| Translate \(\begin{pmatrix}0\\3\end{pmatrix}\) | B1, B1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | \(y\) | |
| 2.25 | 93.017 | |
| 2.75 | 247.692 | |
| 3.25 | 668.142 | |
| 3.75 | 1811.042 | |
| Use of mid-ordinate rule | M1 | |
| Correct \(x\) values | A1 | |
| 3 correct \(y\) values (2 sf) | A1 | |
| Area \(= 0.5 \times 2819.893 = 1410\) | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(A = \int e^{2x} + 3 \; dx\) | M1 | + attempt to integrate |
| \(= \left[\frac{1}{2}e^{2x} + 3x\right]\) | A1 | correct |
| \(\left(\frac{1}{2}e^8 + 12\right) - \left(\frac{1}{2}e^4 + 6\right)\) | m1 | substitute 2, 4 into their \(\int\) |
| \(= \frac{1}{2}\left(e^8 - e^4\right) + 6\) | A1 | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x_1 = 2\), \(y_1 = e^4 + 3\) (57.6); \(x_2 = 4\), \(y_2 = e^8 + 3\) (2980) | M1, A1 | attempt at \(y(2)\) or \(y(4)\); both correct |
| Area of \(A + B = 2\left(e^8 - e^4\right) + 2\left(e^8 + 3\right)\) | M1 | attempt to find correct area |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \frac{7}{2}e^8 - \frac{3}{2}e^4\) | A1 | 4 |
## Question 8:
### Part (a):
Stretch parallel to $x$-axis | B1 |
Scale factor $\frac{1}{2}$ | B1 |
Translate $\begin{pmatrix}0\\3\end{pmatrix}$ | B1, B1 | 4
**Total: 4 marks**
### Part (b):
| $x$ | $y$ |
|------|------|
| 2.25 | 93.017 |
| 2.75 | 247.692 |
| 3.25 | 668.142 |
| 3.75 | 1811.042 |
Use of mid-ordinate rule | M1 |
Correct $x$ values | A1 |
3 correct $y$ values (2 sf) | A1 |
Area $= 0.5 \times 2819.893 = 1410$ | A1 | 4 | CAO
**Total: 4 marks**
### Part (c):
$A = \int e^{2x} + 3 \; dx$ | M1 | + attempt to integrate
$= \left[\frac{1}{2}e^{2x} + 3x\right]$ | A1 | correct
$\left(\frac{1}{2}e^8 + 12\right) - \left(\frac{1}{2}e^4 + 6\right)$ | m1 | substitute 2, 4 into their $\int$
$= \frac{1}{2}\left(e^8 - e^4\right) + 6$ | A1 | 4 | $\left(\frac{1}{2}e^4\left(e^4-1\right)+6\right)$
**Total: 4 marks**
### Part (d):
$x_1 = 2$, $y_1 = e^4 + 3$ (57.6); $x_2 = 4$, $y_2 = e^8 + 3$ (2980) | M1, A1 | attempt at $y(2)$ or $y(4)$; both correct
Area of $A + B = 2\left(e^8 - e^4\right) + 2\left(e^8 + 3\right)$ | M1 | attempt to find correct area
Area $B = 4e^8 - 2e^4 + 6 - \frac{1}{2}e^8 + \frac{1}{2}e^4 - 6$
$= \frac{7}{2}e^8 - \frac{3}{2}e^4$ | A1 | 4
**Total: 4 marks**
**Question 8 Total: 16 marks**
**Paper Total: 75 marks**8 The diagram shows part of the graph of $y = \mathrm { e } ^ { 2 x } + 3$.\\
\includegraphics[max width=\textwidth, alt={}, center]{d5b78fa6-ea3c-497b-94d8-1d5f61288aa5-4_833_1034_1027_513}
\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 $y = \mathrm { e } ^ { 2 x } + 3$.
\item Use the mid-ordinate rule with four strips of equal width to find an estimate for the area of the shaded region $A$, giving your answer to three significant figures.
\item Find the exact value of the area of the shaded region $A$.
\item The region $B$ is indicated on the diagram. Find the area of the region $B$, giving your answer in the form $p \mathrm { e } ^ { 8 } + q \mathrm { e } ^ { 4 }$, where $p$ and $q$ are numbers to be determined.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2005 Q8 [16]}}