| 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 perimeter or area of 2D region |
| Difficulty | Moderate -0.3 This is a standard C2 optimization problem with guided steps. Part (i) requires setting up a perimeter equation (straightforward), part (ii) involves algebraic manipulation with the answer given, and part (iii) uses basic differentiation to find a maximum. The constraint is simple, the algebra is routine, and the question structure heavily scaffolds the solution. Slightly easier than average due to the extensive guidance provided. |
| Spec | 1.02z Models in context: use functions in modelling1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(2x+2y+\pi x = 10 \Rightarrow y = 5-x-\frac{1}{2}\pi x\) | M1, A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(A = 2xy + \frac{1}{2}\pi x^2\) | M1 | Attempt at correct elements |
| \(= 2x\left(5-x-\frac{1}{2}\pi x\right)+\frac{1}{2}\pi x^2\) | M1, A1 | Substitute for \(y\) |
| \(= 10x-2x^2-\frac{1}{2}\pi x^2\) | E1 | Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dA}{dx} = 10-4x-\pi x\) | M1, A1 | |
| \(= 0\) when \(x = \frac{10}{4+\pi}\) | M1, A1 | |
| Then \(y = 5 - \frac{10}{4+\pi} - \frac{\pi}{2}\times\frac{10}{4+\pi} = 5 - \frac{10+5\pi}{4+\pi}\) | M1 | |
| \(= \frac{5(4+\pi)-10-5\pi}{4+\pi} = \frac{10}{4+\pi}\) | ||
| Thus \(y = x\) | E1 | Total: 6 |
## Question 12(i):
$2x+2y+\pi x = 10 \Rightarrow y = 5-x-\frac{1}{2}\pi x$ | M1, A1 | **Total: 2**
## Question 12(ii):
$A = 2xy + \frac{1}{2}\pi x^2$ | M1 | Attempt at correct elements
$= 2x\left(5-x-\frac{1}{2}\pi x\right)+\frac{1}{2}\pi x^2$ | M1, A1 | Substitute for $y$
$= 10x-2x^2-\frac{1}{2}\pi x^2$ | E1 | **Total: 4**
## Question 12(iii):
$\frac{dA}{dx} = 10-4x-\pi x$ | M1, A1 |
$= 0$ when $x = \frac{10}{4+\pi}$ | M1, A1 |
Then $y = 5 - \frac{10}{4+\pi} - \frac{\pi}{2}\times\frac{10}{4+\pi} = 5 - \frac{10+5\pi}{4+\pi}$ | M1 |
$= \frac{5(4+\pi)-10-5\pi}{4+\pi} = \frac{10}{4+\pi}$ | |
Thus $y = x$ | E1 | **Total: 6**12 Fig. 12 shows a window. The base and sides are parts of a rectangle with dimensions $2 x$ metres horizontally by $y$ metres vertically. The top is a semicircle of radius $x$ metres. The perimeter of the window is 10 metres.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{73d1c02b-1b7b-426d-a171-c762597cfed4-4_428_433_1638_766}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{center}
\end{figure}
(i) Express $y$ as a function of $x$.\\
(ii) Find the total area, $A \mathrm {~m} ^ { 2 }$, in terms of $x$ and $y$. Use your answer to part (i) to show that this simplifies to
$$A = 10 x - 2 x ^ { 2 } - \frac { 1 } { 2 } \pi x ^ { 2 }$$
(iii) Prove that for the maximum value of $A$, $y = x$ exactly.\\
\section*{MEI STRUCTURED MATHEMATICS }
\section*{CONCEPTS FOR ADVANCED MATHEMATICS, C2}
\section*{Practice Paper C2-B \\
Insert sheet for question 11}
11 Speed-time graph with the first two points plotted.\\
\includegraphics[max width=\textwidth, alt={}, center]{73d1c02b-1b7b-426d-a171-c762597cfed4-5_768_1772_1389_205}
\hfill \mbox{\textit{OCR MEI C2 Q12 [12]}}