| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Optimization with constraints |
| Difficulty | Moderate -0.3 This is a standard C2 optimization problem requiring differentiation of a sum of power functions, solving a cubic equation that factors nicely, and applying the second derivative test. While it involves multiple steps (10 marks total), each technique is routine for C2 level with no novel problem-solving required—slightly easier than average due to the straightforward algebraic manipulation. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative |
| Answer | Marks |
|---|---|
| \(\frac{dC}{dv} = -160v^{-2} + \frac{2v}{100}\) | M1 A1 |
| \(-160v^{-2} + \frac{2v}{100} = 0\) | M1 |
| \(v^3 = 8\,000\) resulting in \(v = 20\) | M1 A1 |
| (5 marks) |
| Answer | Marks |
|---|---|
| \(\frac{d^2C}{dv^2} = 320v^{-3} + \frac{1}{50}\) | M1 |
| \(> 0\), therefore minimum | A1 |
| (2 marks) |
| Answer | Marks |
|---|---|
| \(v = 20\): \(C = \frac{160}{20} + \frac{400}{100} = 12\) | B1ft |
| Cost \(= 250 \times 12 = £360\) | M1 A1 |
| (3 marks) | |
| (10 marks total) |
## (a)
$\frac{dC}{dv} = -160v^{-2} + \frac{2v}{100}$ | M1 A1 |
$-160v^{-2} + \frac{2v}{100} = 0$ | M1 |
$v^3 = 8\,000$ resulting in $v = 20$ | M1 A1 |
| (5 marks) |
## (b)
$\frac{d^2C}{dv^2} = 320v^{-3} + \frac{1}{50}$ | M1 |
$> 0$, therefore minimum | A1 |
| (2 marks) |
## (c)
$v = 20$: $C = \frac{160}{20} + \frac{400}{100} = 12$ | B1ft |
Cost $= 250 \times 12 = £360$ | M1 A1 |
| (3 marks) |
| **(10 marks total)** |
---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 Q7 [10]}}