| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Show formula then optimise: cylinder/prism (single variable) |
| Difficulty | Standard +0.3 This is a standard optimization problem with guided steps. Part (a) is algebraic manipulation using V=l²h to eliminate l. Part (b) requires routine differentiation and solving dS/dh=0. Part (c) asks for second derivative test. All steps are clearly signposted with no novel insight required, making it slightly easier than average. |
| 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 | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(V=hl^2\Rightarrow 250000=hl^2\) or \(l=\frac{500}{\sqrt{h}}\) | B1 | Correct equation linking \(h\) and \(l\), seen or implied by substitution |
| \(S=l^2+4hl=\frac{``250000"}{h}+4h\times\frac{``500"}{\sqrt{h}}\) | M1 | Attempts surface area and substitutes to eliminate \(l\) |
| \(S=\frac{250000}{h}+2000\sqrt{h}\) | A1* | Achieves given result with no errors (starred — given result) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{\mathrm{d}S}{\mathrm{d}h}=-\frac{250000}{h^2}+2000\times\frac{1}{2}h^{-\frac{1}{2}}\) | M1A1 | M1: power decreased by one in at least one term; A1: correct derivative, need not be simplified |
| \(\frac{\mathrm{d}S}{\mathrm{d}h}=0\Rightarrow -\frac{250000}{h^2}+2000\times\frac{1}{2}h^{-\frac{1}{2}}=0\Rightarrow h^k=...\) | dM1 | Sets derivative to zero, attempts to solve as far as a power of \(h\); depends on first M |
| \(\Rightarrow h^{\frac{3}{2}}=250\Rightarrow h=...\) | ddM1 | Attempts to solve equation with fractional index; depends on both M marks |
| \(h=250^{\frac{2}{3}}\) | A1 | Or states \(k=\frac{2}{3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{\mathrm{d}^2S}{\mathrm{d}h^2}=\frac{500000}{h^3}-500h^{-\frac{3}{2}}\) | M1 | Attempts second derivative, reaching form \(\frac{A}{h^3}-Bh^{-\frac{3}{2}}\) with \(A,B>0\); first derivative test scores M0 |
| \(\left.\frac{\mathrm{d}^2S}{\mathrm{d}h^2}\right\ | _{h=39.7}=6>0\), hence gives the minimum value | A1 |
## Question 9(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $V=hl^2\Rightarrow 250000=hl^2$ or $l=\frac{500}{\sqrt{h}}$ | **B1** | Correct equation linking $h$ and $l$, seen or implied by substitution |
| $S=l^2+4hl=\frac{``250000"}{h}+4h\times\frac{``500"}{\sqrt{h}}$ | **M1** | Attempts surface area and substitutes to eliminate $l$ |
| $S=\frac{250000}{h}+2000\sqrt{h}$ | **A1*** | Achieves given result with no errors (starred — given result) |
---
## Question 9(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{\mathrm{d}S}{\mathrm{d}h}=-\frac{250000}{h^2}+2000\times\frac{1}{2}h^{-\frac{1}{2}}$ | **M1A1** | M1: power decreased by one in at least one term; A1: correct derivative, need not be simplified |
| $\frac{\mathrm{d}S}{\mathrm{d}h}=0\Rightarrow -\frac{250000}{h^2}+2000\times\frac{1}{2}h^{-\frac{1}{2}}=0\Rightarrow h^k=...$ | **dM1** | Sets derivative to zero, attempts to solve as far as a power of $h$; depends on first M |
| $\Rightarrow h^{\frac{3}{2}}=250\Rightarrow h=...$ | **ddM1** | Attempts to solve equation with fractional index; depends on both M marks |
| $h=250^{\frac{2}{3}}$ | **A1** | Or states $k=\frac{2}{3}$ |
---
## Question 9(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{\mathrm{d}^2S}{\mathrm{d}h^2}=\frac{500000}{h^3}-500h^{-\frac{3}{2}}$ | **M1** | Attempts second derivative, reaching form $\frac{A}{h^3}-Bh^{-\frac{3}{2}}$ with $A,B>0$; first derivative test scores M0 |
| $\left.\frac{\mathrm{d}^2S}{\mathrm{d}h^2}\right\|_{h=39.7}=6>0$, hence gives the minimum value | **A1** | Substitutes $h$ to get awrt 6, correct conclusion referring to sign ($>0$ or positive); allow if expression not fully simplified as long as clearly positive |
The image provided appears to be essentially a blank page, containing only:
- "PMT" in the top right corner
- A footer with Pearson Education Limited's registered company information
There is **no mark scheme content** visible on this page to extract. It appears to be a back/cover page of a mark scheme document.
If you have additional pages with actual mark scheme content, please share those and I'll be happy to extract and format the information for you.9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-30_469_863_251_593}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of a square based, open top box.\\
The height of the box is $h \mathrm {~cm}$, and the base edges each have length $l \mathrm {~cm}$.\\
Given that the volume of the box is $250000 \mathrm {~cm} ^ { 3 }$
\begin{enumerate}[label=(\alph*)]
\item show that the external surface area, $S \mathrm {~cm} ^ { 2 }$, of the box is given by
$$S = \frac { 250000 } { h } + 2000 \sqrt { h }$$
\item Use algebraic differentiation to show that $S$ has a stationary point when $h = 250 ^ { k }$ where $k$ is a rational constant to be found.
\item Justify by further differentiation that this value of $h$ gives the minimum external surface area of the box.
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-32_2647_1838_118_116}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q9 [10]}}