| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Standard +0.3 This question involves standard exponential function techniques: using asymptote and point to find constants (routine substitution), stating range (direct observation), and solving exponential equations by substitution. The intersection requires solving a quadratic in e^(0.5x), which is a standard A-level technique. All parts are textbook-style with clear methods and no novel insight required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(A = 48\), \(B = 40\) | A1 |
| (b) | \(g(x) > 26\) or equivalent notation e.g. \(y > 26, g > 26, y \in (26, \infty)\) | B1 |
| (c) | \(\ln 400, \ln 4\) or equivalent forms e.g. \(\ln 400, \ln 4\) (not necessarily in this order, see note) | A1 |
**(a)** | $A = 48$, $B = 40$ | A1 | B1 $A = 48$. M1 Substitutes $x = 0$ and $y = 8$ into $f(x)$ and attempts to find the value of $B$. E.g. $8 = 48 - Be^{-0.5(0)} \Rightarrow 8 = 48 - B \Rightarrow B = \ldots$. |
**(b)** | $g(x) > 26$ or equivalent notation e.g. $y > 26, g > 26, y \in (26, \infty)$ | B1 | |
**(c)** | $\ln 400, \ln 4$ or equivalent forms e.g. $\ln 400, \ln 4$ (not necessarily in this order, see note) | A1 | M1 Sets their "$48 - 40e^{-0.5x} = 26 + e^{0.5x}$" and rearranges to produce a simplified equation of the form $e^{0.5x} + 40e^{-0.5x} - 22 = 0$. E.g. $48 - 40e^{-0.5x} = 26 + e^{0.5x} \Rightarrow e^{0.5x} + 40e^{-0.5x} - 22 = 0$. Correct quadratic equation. Look for $(e^{0.5x})^2 - 22e^{0.5x} + 40 = 0$ or $e^x - 22e^{0.5x} + 40 = 0$. M1 Full attempt to find the value of $x$. This involves solving a 3TQ in $e^{0.5x}$ followed by the use of lns. You may see different variables such as $t$. E.g. $t = e^{0.5x}$, $t^2 - 22t + 40 = 0$, $(t-20)(t-2) = 0 \Rightarrow t = 20, t = 2 \Rightarrow e^{0.5x} = 20 \Rightarrow x = 2\ln 20$, $e^{0.5x} = 2 \Rightarrow x = 2\ln 2$. A1 Correct answers only e.g. $\ln 400, \ln 4$ |
---10. The figure 4 shows the curves $\mathrm { f } ( x ) = A - B e ^ { - 0.5 x }$ and $\mathrm { g } ( x ) = 26 + e ^ { 0.5 x }$
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-14_718_1152_347_340}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Given that $\mathrm { f } ( x )$ passes through $( 0,8 )$ and has an horizontal asymptote $y = 48$\\
a. Find the values of $A$ and $B$ for $\mathrm { f } ( x )$\\
(3)\\
b. State the range of $\mathrm { g } ( x )$\\
(1)
The curves $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ meet at the points $C$ and $D$\\
c. Find the $x$-coordinates of the intersection points $C$ and $D$, in the form $\ln k$, where $k$ is an integer.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q10 [8]}}