| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Find intersection of exponential curves |
| Difficulty | Standard +0.3 This is a straightforward C3 exponential question requiring basic substitution (x=0 for part a) and algebraic manipulation to find intersection points. Part (b) involves solving e^(x+2) = 3 + 2e^x, which rearranges to a linear equation in e^x, then substituting back—standard techniques with 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 |
|---|---|
| \(A(0,5), B(0, e^2)\) | B2 |
| \(\therefore AB = e^2 - 5\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(3 + 2e^x = e^{x+2} = e^x e^2\) | M1 | |
| \(3 = e^x(e^2 - 2)\) | ||
| \(e^x = \frac{3}{e^2-2}\), \(x = \ln\frac{3}{e^2-2}\) | M1 A1 | |
| \(\therefore y = e^x e^x = e^{2x} \times \frac{3}{e^2-2} = \frac{3e^2}{e^2-2}\) | M1 A1 | (8) |
**(a)**
$A(0,5), B(0, e^2)$ | B2 |
$\therefore AB = e^2 - 5$ | B1 |
**(b)**
$3 + 2e^x = e^{x+2} = e^x e^2$ | M1 |
$3 = e^x(e^2 - 2)$ | |
$e^x = \frac{3}{e^2-2}$, $x = \ln\frac{3}{e^2-2}$ | M1 A1 |
$\therefore y = e^x e^x = e^{2x} \times \frac{3}{e^2-2} = \frac{3e^2}{e^2-2}$ | M1 A1 | (8)
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c8b85e00-4549-4219-a75d-85f67ccb8e16-2_638_675_644_445}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curves $y = 3 + 2 \mathrm { e } ^ { x }$ and $y = \mathrm { e } ^ { x + 2 }$ which cross the $y$-axis at the points $A$ and $B$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the exact length $A B$.
The two curves intersect at the point $C$.
\item Find an expression for the $x$-coordinate of $C$ and show that the $y$-coordinate of $C$ is $\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q2 [8]}}