| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Natural logarithm equation solving |
| Difficulty | Standard +0.0 This is a standard C3 multi-part question covering routine logarithm and exponential techniques: solving ln equations, sketching log curves, finding inverse functions, and function composition. All parts follow textbook procedures with no novel problem-solving required, making it typical average difficulty for A-level. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.06d Natural logarithm: ln(x) function and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(4 - \ln 3x = 0\), \(\ln 3x = 4\), \(x = \frac{1}{3}e^4\) | M1 A1 | |
| (b) Graph showing curve with asymptote at x-axis, passing through \((\frac{1}{3}e^4, 0)\) | B2 | |
| (c) \(y = 4 - \ln 3x\) | M1 | |
| \(\ln 3x = 4 - y\) | M1 | |
| \(x = \frac{1}{3}e^{4-y}\) | M1 | |
| \(\therefore f^{-1}(x) = \frac{1}{3}e^{4-x}\) | A1 | |
| (d) \(fg(x) = 4 - \ln 3e^{2-x}\) | M1 | |
| \(= 4 - (\ln 3 + \ln e^{2-x})\) | M1 | |
| \(= 4 - \ln 3 - (2-x)\) | M1 | |
| \(= x + 2 - \ln 3\) | A1 | [a=2, b=3] |
**(a)** $4 - \ln 3x = 0$, $\ln 3x = 4$, $x = \frac{1}{3}e^4$ | M1 A1 |
**(b)** Graph showing curve with asymptote at x-axis, passing through $(\frac{1}{3}e^4, 0)$ | B2 |
**(c)** $y = 4 - \ln 3x$ | M1 |
$\ln 3x = 4 - y$ | M1 |
$x = \frac{1}{3}e^{4-y}$ | M1 |
$\therefore f^{-1}(x) = \frac{1}{3}e^{4-x}$ | A1 |
**(d)** $fg(x) = 4 - \ln 3e^{2-x}$ | M1 |
$= 4 - (\ln 3 + \ln e^{2-x})$ | M1 |
$= 4 - \ln 3 - (2-x)$ | M1 |
$= x + 2 - \ln 3$ | A1 | [a=2, b=3] | (10 marks)
---6. The function f is defined by
$$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0 .$$
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\mathrm { f } ( x ) = 0$.
\item Sketch the curve $y = \mathrm { f } ( x )$.
\item Find an expression for the inverse function, $\mathrm { f } ^ { - 1 } ( x )$.
The function $g$ is defined by
$$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
\item Show that
$$\operatorname { fg } ( x ) = x + a - \ln b$$
where $a$ and $b$ are integers to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q6 [10]}}