| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Natural logarithm equation solving |
| Difficulty | Moderate -0.8 Both parts are routine C3 logarithm exercises requiring standard techniques: (a) take ln of both sides and rearrange, (b) use log laws to combine and solve a linear equation. No problem-solving insight needed, just direct application of learned procedures with straightforward algebra. |
| Spec | 1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(e^{3x-9} = 8 \Rightarrow 3x - 9 = \ln 8\) | M1 | Takes ln of both sides and uses power law |
| \(x = \frac{\ln 8 + 9}{3}\) | A1 | Correct unsimplified answer or equivalent such as \(\frac{\ln 8e^9}{3}\), \(3 + \ln(\sqrt[3]{8})\), \(\frac{\log 8}{3 \log e} + 3\) |
| \(= \ln 2 + 3\) | A1 | cso; accept \(\ln 2e^3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\ln\left(\frac{2y+5}{4-y}\right) = 2\) | M1 | Uses correct method to combine two ln terms into single ln term; condone slips on signs/coefficients but not on \(e^2\) |
| \(\left(\frac{2y+5}{4-y}\right) = e^2\) | M1 | Attempt to undo ln to get equation in \(y\); must be awarded after attempt to combine ln terms |
| \(2y + 5 = e^2(4-y) \Rightarrow 2y + e^2y = 4e^2 - 5\) | dM1 | Dependent on both previous M marks; making \(y\) subject, both \(y\) terms collected and factorised |
| \(y = \frac{4e^2 - 5}{2 + e^2}\) | A1 | Or equivalent such as \(y = 4 - \frac{13}{2+e^2}\); ISW after correct answer |
## Question 2:
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $e^{3x-9} = 8 \Rightarrow 3x - 9 = \ln 8$ | M1 | Takes ln of both sides and uses power law |
| $x = \frac{\ln 8 + 9}{3}$ | A1 | Correct unsimplified answer or equivalent such as $\frac{\ln 8e^9}{3}$, $3 + \ln(\sqrt[3]{8})$, $\frac{\log 8}{3 \log e} + 3$ |
| $= \ln 2 + 3$ | A1 | cso; accept $\ln 2e^3$ |
### Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\ln\left(\frac{2y+5}{4-y}\right) = 2$ | M1 | Uses correct method to combine two ln terms into single ln term; condone slips on signs/coefficients but not on $e^2$ |
| $\left(\frac{2y+5}{4-y}\right) = e^2$ | M1 | Attempt to undo ln to get equation in $y$; must be awarded after attempt to combine ln terms |
| $2y + 5 = e^2(4-y) \Rightarrow 2y + e^2y = 4e^2 - 5$ | dM1 | Dependent on both previous M marks; making $y$ subject, both $y$ terms collected and factorised |
| $y = \frac{4e^2 - 5}{2 + e^2}$ | A1 | Or equivalent such as $y = 4 - \frac{13}{2+e^2}$; ISW after correct answer |
---2. Find the exact solutions, in their simplest form, to the equations
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { e } ^ { 3 x - 9 } = 8$
\item $\ln ( 2 y + 5 ) = 2 + \ln ( 4 - y )$\\\includegraphics[max width=\textwidth, alt={}, center]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-05_37_1813_0_6}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2017 Q2 [7]}}