| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a standard C4 calculus question involving differentiation of exponentials, finding a normal line equation, and integration to find an area. Part (a) requires routine differentiation and normal line calculation (4 marks), part (b) is a simple verification (1 mark), and part (c) involves straightforward integration of an exponential function and finding the area of a trapezium (7 marks). While it has multiple parts and requires careful setup of the area calculation, all techniques are standard C4 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
\includegraphics{figure_3}
The curve $C$ with equation $y = 2e^x + 5$ meets the $y$-axis at the point $M$, as shown in Fig. 3.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the normal to $C$ at $M$ in the form $ax + by = c$, where $a$, $b$ and $c$ are integers. [4]
\end{enumerate}
This normal to $C$ at $M$ crosses the $x$-axis at the point $N(n, 0)$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that $n = 14$. [1]
\end{enumerate}
The point $P(\ln 4, 13)$ lies on $C$. The finite region $R$ is bounded by $C$, the axes and the line $PN$, as shown in Fig. 3.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the area of $R$, giving your answers in the form $p + q \ln 2$, where $p$ and $q$ are integers to be found. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q7 [12]}}