| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard differentiation of exponentials, substitution to find constants, and basic coordinate geometry (finding normal equation and triangle area). While it has multiple steps (10 marks total), each step uses routine C3 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07m Tangents and normals: gradient and equations |
The curve $C$ with equation $y = p + qe^x$, where $p$ and $q$ are constants, passes through the point $(0, 2)$. At the point $P(\ln 2, p + 2q)$ on $C$, the gradient is $5$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $p$ and the value of $q$. [5]
\end{enumerate}
The normal to $C$ at $P$ crosses the $x$-axis at $L$ and the $y$-axis at $M$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the area of $\triangle OLM$, where $O$ is the origin, is approximately $53.8$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q17 [10]}}