| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Optimization with constraints |
| Difficulty | Moderate -0.3 This is a straightforward optimization problem requiring differentiation of a simple function (power and reciprocal terms), solving a cubic equation that factors nicely, and applying the second derivative test. While it involves multiple steps across three parts, each technique is standard C2 material with no novel insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
On a journey, the average speed of a car is $v$ m s$^{-1}$. For $v \geq 5$, the cost per kilometre, $C$ pence, of the journey is modelled by
$$C = \frac{160}{v} + \frac{v^2}{100}.$$
Using this model,
\begin{enumerate}[label=(\alph*)]
\item show, by calculus, that there is a value of $v$ for which $C$ has a stationary value, and find this value of $v$. [5]
\item Justify that this value of $v$ gives a minimum value of $C$. [2]
\item Find the minimum value of $C$ and hence find the minimum cost of a 250 km car journey. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q38 [10]}}