| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Moderate -0.3 This C2 question tests standard transformations and exponential equations. Part (a) requires recognizing that (1/3)^x = 3^(-x), a reflection in the y-axis. Part (b) is routine sketching with y-intercepts. Part (c) involves solving 3^(-x) = 2(3^x) algebraically, leading to 3^(2x) = 1/2, which is straightforward logarithm work. The question is slightly easier than average due to its structured guidance and standard techniques, though the algebraic manipulation in part (c) provides some substance for C2 level. |
| 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 |
|---|---|---|
| (a) Reflection in the \(y\)-axis | B1 | |
| (b) \(y = (\frac{1}{3})^x\) and \(y = 2(3^x)\) with points \((0,2)\) and \((0,1)\) marked | B3 | |
| (c) \((\frac{1}{3})^y = 2(3^y)\) | M1 | |
| \(1 = 2 \times (3^y)^2\) | M1 | |
| \(3^{2y} = \frac{1}{2}\), \(2y = \frac{\lg \frac{1}{2}}{\lg 3}\) | M1 | |
| \(x = \frac{\lg \frac{1}{2}}{2 \lg 3} = -0.32\) | A1 | |
| \(3^x = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{1}{\sqrt{2}}\) | M1 | |
| \(y = 2(3^x) = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}\) | A1 | (9 marks) |
**(a)** Reflection in the $y$-axis | B1 |
**(b)** $y = (\frac{1}{3})^x$ and $y = 2(3^x)$ with points $(0,2)$ and $(0,1)$ marked | B3 |
**(c)** $(\frac{1}{3})^y = 2(3^y)$ | M1 |
$1 = 2 \times (3^y)^2$ | M1 |
$3^{2y} = \frac{1}{2}$, $2y = \frac{\lg \frac{1}{2}}{\lg 3}$ | M1 |
$x = \frac{\lg \frac{1}{2}}{2 \lg 3} = -0.32$ | A1 |
$3^x = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{1}{\sqrt{2}}$ | M1 |
$y = 2(3^x) = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}$ | A1 | (9 marks)\begin{enumerate}[label=(\alph*)]
\item Describe fully a single transformation that maps the graph of $y = 3^x$ onto the graph of $y = (\frac{1}{3})^x$. [1]
\item Sketch on the same diagram the curves $y = (\frac{1}{3})^x$ and $y = 2(3^x)$, showing the coordinates of any points where each curve crosses the coordinate axes. [3]
\end{enumerate}
The curves $y = (\frac{1}{3})^x$ and $y = 2(3^x)$ intersect at the point $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the $x$-coordinate of $P$ to 2 decimal places and show that the $y$-coordinate of $P$ is $\sqrt{2}$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q5 [9]}}