| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find tangent line equation |
| Difficulty | Standard +0.2 This is a straightforward C3 question testing standard techniques: evaluating a function at given points, differentiating ln and exponential functions, and finding a tangent line equation. All parts are routine applications of core methods with no problem-solving insight required, making it slightly easier than the typical A-level question. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations |
\includegraphics{figure_1}
Figure 1 shows a sketch of the curve with equation $y = \text{f}(x)$, where
$$\text{f}(x) = 10 + \ln(3x) - \frac{1}{2}e^x, \quad 0.1 \leq x \leq 3.3.$$
Given that f(k) = 0,
\begin{enumerate}[label=(\alph*)]
\item show, by calculation, that $3.1 < k < 3.2$. [2]
\item Find f'(x). [3]
\end{enumerate}
The tangent to the graph at $x = 1$ intersects the $y$-axis at the point $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item \begin{enumerate}[label=(\roman*)]
\item Find an equation of this tangent.
\item Find the exact $y$-coordinate of $P$, giving your answer in the form $a + \ln b$. [5]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q6 [10]}}