| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Optimise geometric shape surface area/volume |
| Difficulty | Moderate -0.3 This is a standard optimization problem requiring volume constraint manipulation, differentiation of a simple rational function, and finding a minimum using second derivative test. All steps are routine C2 techniques with no novel insight required, making it slightly easier than average but still requiring multiple connected steps. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks |
|---|---|
| 7 | i |
| Answer | Marks |
|---|---|
| iii | h = 120/x2 |
| Answer | Marks |
|---|---|
| A = 145.9 to 146 | B1 |
| Answer | Marks |
|---|---|
| A1 | at least one interim step shown |
| Answer | Marks |
|---|---|
| Dependent on previous M1 | 3 |
Question 7:
7 | i
ii
iii | h = 120/x2
A = 2x2 + 4xh o.e.
completion to given answer
A′ = 4x − 480/x2 o.e.
A′′ = 4 + 960 / x3
use of A′ = 0
x = 3120 or 4.9(3..)
Test using A′ or A′′ to confirm
minimum
Substitution of their x in A
A = 145.9 to 146 | B1
M1
A1
2
2
M1
A1
T1
M1
A1 | at least one interim step shown
1 for kx-2 o.e. included
ft their A′ only if kx-2 seen ; 1 if one
error
Dependent on previous M1 | 3
4
5\includegraphics{figure_7}
Fig. 10 shows a solid cuboid with square base of side $x$ cm and height $h$ cm. Its volume is $120$ cm$^3$.
\begin{enumerate}[label=(\roman*)]
\item Find $h$ in terms of $x$. Hence show that the surface area, $A$ cm$^2$, of the cuboid is given by
$$A = 2x^2 + \frac{480}{x}.$$ [3]
\item Find $\frac{dA}{dx}$ and $\frac{d^2A}{dx^2}$. [4]
\item Hence find the value of $x$ which gives the minimum surface area. Find also the value of the surface area in this case. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 Q7 [12]}}