| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Moderate -0.3 This is a multi-part question covering standard C2 techniques: trapezium rule application, transformation of exponential graphs, and solving exponential equations using logarithms. All parts are routine textbook exercises requiring direct application of learned methods with no novel problem-solving, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06g Equations with exponentials: solve a^x = b1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(h = 0.5\); Integral \(= h/2\{\ldots\}\) | B1, M1 | PI; OE summing of areas of the four traps. |
| \(\{\ldots\} = f(0) + 2[f(\frac{1}{2}) + f(1) + f(\frac{3}{2})] + f(2) = 1 + 2[\sqrt{6} + 6 + 6\sqrt{6}] + 36 = 1 + 2[2.449\ldots + 6 + 14.6969\ldots] + 36 = 37 + 2 \times 23.146\ldots = 83.292\ldots\) | A1, A1 (4 marks) | Condone 1 numerical slip. Accept 3sf values if not exact.; CAO; must be 20.8 |
| (a)(ii) Relevant trapazia drawn on a copy of given graph; {Approximation is an}overestimate | M1, A1 (2 marks) | Accept single trapezium with its sloping side above the curve; Dep. on 4 trapazia with each of their upper vertices lying on the curve |
| (b)(i) Stretch (I) in \(x\)-direction (II); (scale factor \(\frac{1}{3}\)) (III) | M1, A1 (2 marks) | Need (I) and one of (II, III); M0 if more than one transformation |
| (ii) \(6^{3x} = 84\); \(\log_{10} 6^y = \log_{10} 84\) | M1, M1 | PI; Take logs of both sides of \(a^t = b\), PI by 'correct' value(s) later or \(3x = \log_6 84\) |
| \(3x \log_{10} 6 = \log_{10} 84\) | m1 | Use of \(\log 6^y = 3x\log 6\) OE or \(3x = \log_6 84\) seen |
| \(x = \frac{\lg 84}{3\lg 6} = 0.82429\ldots = 0.824\) (to 3dp) | A1 (4 marks) | Must see that logs have been used before any of the last 3 marks are awarded in (b)(ii). Condone > 3dp |
| (c) \(f(x) = 6^{x-1} - 2\) | B2,1 (2 marks) | B1 for either \(6^{x-1} + 2\) or for \(6^{x+1} - 2\) |
**(a)(i)** $h = 0.5$; Integral $= h/2\{\ldots\}$ | B1, M1 | PI; OE summing of areas of the four traps.
$\{\ldots\} = f(0) + 2[f(\frac{1}{2}) + f(1) + f(\frac{3}{2})] + f(2) = 1 + 2[\sqrt{6} + 6 + 6\sqrt{6}] + 36 = 1 + 2[2.449\ldots + 6 + 14.6969\ldots] + 36 = 37 + 2 \times 23.146\ldots = 83.292\ldots$ | A1, A1 (4 marks) | Condone 1 numerical slip. Accept 3sf values if not exact.; CAO; must be 20.8
**(a)(ii)** Relevant trapazia drawn on a copy of given graph; {Approximation is an}overestimate | M1, A1 (2 marks) | Accept single trapezium with its sloping side above the curve; Dep. on 4 trapazia with each of their upper vertices lying on the curve
**(b)(i)** Stretch (I) in $x$-direction (II); (scale factor $\frac{1}{3}$) (III) | M1, A1 (2 marks) | Need (I) and one of (II, III); M0 if more than one transformation
**(ii)** $6^{3x} = 84$; $\log_{10} 6^y = \log_{10} 84$ | M1, M1 | PI; Take logs of both sides of $a^t = b$, PI by 'correct' value(s) later or $3x = \log_6 84$
$3x \log_{10} 6 = \log_{10} 84$ | m1 | Use of $\log 6^y = 3x\log 6$ OE or $3x = \log_6 84$ seen
$x = \frac{\lg 84}{3\lg 6} = 0.82429\ldots = 0.824$ (to 3dp) | A1 (4 marks) | Must see that logs have been used before any of the last 3 marks are awarded in (b)(ii). Condone > 3dp
**(c)** $f(x) = 6^{x-1} - 2$ | B2,1 (2 marks) | B1 for either $6^{x-1} + 2$ or for $6^{x+1} - 2$
**Total for Q8: 14 marks**
---8 The diagram shows a sketch of the curve with equation $y = 6 ^ { x }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-5_403_506_370_769}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with five ordinates (four strips) to find an approximate value for $\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x$, giving your answer to three significant figures.
\item Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of $\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Describe a single geometrical transformation that maps the graph of $y = 6 ^ { x }$ onto the graph of $y = 6 ^ { 3 x }$.
\item The line $y = 84$ intersects the curve $y = 6 ^ { 3 x }$ at the point $A$. By using logarithms, find the $x$-coordinate of $A$, giving your answer to three decimal places.\\
(4 marks)
\end{enumerate}\item The graph of $y = 6 ^ { x }$ is translated by $\left[ \begin{array} { c } 1 \\ - 2 \end{array} \right]$ to give the graph of the curve with equation $y = \mathrm { f } ( x )$. Write down an expression for $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2008 Q8 [14]}}