| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Moderate -0.3 This is a straightforward C2 question testing basic transformations, sketching exponentials, and solving exponential equations using logarithms. Part (c) requires equating 4^(-x) = 3×4^x and applying logs, which is a standard technique, though the algebraic manipulation (getting 1 = 3×4^(2x)) requires care. Slightly easier than average due to being a routine multi-part question with clear methods. |
| 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 |
|---|---|---|
| Stretch(I) in \(y\)-direction(II) scale factor 3(III) | M1 | OE Need (I) and either (II) or (III) |
| [All correct] | A1 (2 marks) | Need (I) and (II) and (III); [>1 transformation scores 0/2] |
| Answer | Marks | Guidance |
|---|---|---|
| Shape with correct asymptotic behaviour in 2nd quadrant, below point of intersection with \(y\)-axis | B1 | Shape with indication of correct asymptotic behaviour in 2nd quadrant below pt of intersection with \(y\)-axis |
| Only intersection is with \(y\)-axis, intercept is 3 stated/indicated | B1 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(3 \times 4^x = 4^{-x}\) | M1 | OE equation in \(x\) |
| \(\log 3 + \log 4^x = \log 4^{-x}\) | m1 | Log Law 1 (or Law 2 applied to \(\frac{4^x}{4^{-x}} = 3\) or \(\frac{1}{3}\) OE) used correctly; or correct rearrangement to \(4^{2x} = \frac{1}{3}\); OE simplified e.g. \(16^x = 3^{-1}\) or \(4^x = (1/\sqrt{3})\) |
| \(\log 3 + x\log 4 = -x\log 4\) | m1 | Log Law 3 applied correctly twice (dependent on both M1 & m1); or a correct method using logs to solve an equation of form \(a^{kx} = b\), \(b > 0\) (dependent on M1 and valid method to \(a^{kx}\)) |
| \(x = \frac{-\log 3}{2\log 4} \left(= \frac{-\log 3}{\log 16}\right)\) | A1 | Correct expression for \(x\) or for \(-x\); e.g. \(x = \frac{1}{2}\log_4\left(\frac{1}{3}\right)\); PI by correct 3sf value or better |
| \(x = -0.396\) (to 3sf) | A1 (5 marks) | If logs not used explicitly then max of M1m1m0 |
# Question 7:
## Part 7(a):
Stretch(I) in $y$-direction(II) scale factor 3(III) | M1 | OE Need (I) and either (II) or (III)
[All correct] | A1 (2 marks) | Need (I) and (II) and (III); [>1 transformation scores 0/2]
## Part 7(b):
Shape with correct asymptotic behaviour in 2nd quadrant, below point of intersection with $y$-axis | B1 | Shape with indication of correct asymptotic behaviour in 2nd quadrant below pt of intersection with $y$-axis
Only intersection is with $y$-axis, intercept is 3 stated/indicated | B1 (2 marks) |
## Part 7(c):
$3 \times 4^x = 4^{-x}$ | M1 | OE equation in $x$
$\log 3 + \log 4^x = \log 4^{-x}$ | m1 | Log Law 1 (or Law 2 applied to $\frac{4^x}{4^{-x}} = 3$ or $\frac{1}{3}$ OE) used correctly; or correct rearrangement to $4^{2x} = \frac{1}{3}$; OE simplified e.g. $16^x = 3^{-1}$ or $4^x = (1/\sqrt{3})$
$\log 3 + x\log 4 = -x\log 4$ | m1 | Log Law 3 applied correctly twice (dependent on both M1 & m1); or a correct method using logs to solve an equation of form $a^{kx} = b$, $b > 0$ (dependent on M1 and valid method to $a^{kx}$)
$x = \frac{-\log 3}{2\log 4} \left(= \frac{-\log 3}{\log 16}\right)$ | A1 | Correct expression for $x$ or for $-x$; e.g. $x = \frac{1}{2}\log_4\left(\frac{1}{3}\right)$; PI by correct 3sf value or better
$x = -0.396$ (to 3sf) | A1 (5 marks) | If logs not used explicitly then max of M1m1m0
---7
\begin{enumerate}[label=(\alph*)]
\item Describe a geometrical transformation that maps the graph of $y = 4 ^ { x }$ onto the graph of $y = 3 \times 4 ^ { x }$.
\item Sketch the curve with equation $y = 3 \times 4 ^ { x }$, indicating the value of the intercept on the $y$-axis.
\item The curve with equation $y = 4 ^ { - x }$ intersects the curve $y = 3 \times 4 ^ { x }$ at the point $P$. Use logarithms to find the $x$-coordinate of $P$, giving your answer to three significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2013 Q7 [9]}}