| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Constraint derivation verification |
| Difficulty | Easy -1.2 This is a straightforward linear programming question requiring routine translation of word problems into inequalities and basic algebraic manipulation. Parts (a)-(d) involve simple constraint formulation with guided steps ('show that'), while (e)-(f) are standard graphical LP solution methods covered extensively in D1. No novel problem-solving or complex reasoning required—purely procedural application of textbook techniques. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x + 2y \leq 12\) (equivalently \(150x + 300y \leq 1800\)) | M1A1 | Correct terms; accept \(=\) here; accept swapped coefficients. cao does not need to be simplified. (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.9x + 1.2y \leq 9 \Rightarrow 3x + 4y \leq 30\) | M1 | Correct terms; must deal with cm/m correctly; accept \(=\) here |
| A1 cso | Answer given (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (You need to buy) at least 2 large cupboards | B1 | cao: 'at least' and '2' and 'large' (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Capacity \(C\) and \(140\%C\); total is \(Cx + \frac{140}{100}Cy\) | M1 | Must see engagement with 140% in some way |
| Simplifies to \(7y + 5x\) | A1 cso | Answer given (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correctly drawing \(y = 2\) | B1 | |
| Correctly drawing \(3x + 4y = 30\) | B1 | |
| Correctly drawing \(x + 2y = 12\) | B1 | ft only if swapped coefficients in (a): \((6,0)\) \((2,8)\) |
| Lines labelled and drawn with a ruler | B1 | At least 2 lines labelled including one diagonal line |
| Shading correct | B1 | |
| Region \(R\) identified correctly | B1 | (6 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Consider points and value of \(5x + 7y\), or draw a clear profit line | M1A1 | At least 2 points tested or objective line drawn with correct gradient |
| \((7,2) \to 49\); \((7\tfrac{1}{3}, 2) \to 50\tfrac{2}{3}\); \((6,3) \to 51\); \((0,6) \to 42\); \((0,2) \to 14\) | A1 | 3 points correctly tested or objective line correct and distinct/labelled |
| Best option: buy 6 standard cupboards and 3 large cupboards | A1 | Accept \((6,3)\) if very clearly selected (4 marks total) |
## Question 7:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $x + 2y \leq 12$ (equivalently $150x + 300y \leq 1800$) | M1A1 | Correct terms; accept $=$ here; accept swapped coefficients. cao does not need to be simplified. (2 marks) |
**Part (b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.9x + 1.2y \leq 9 \Rightarrow 3x + 4y \leq 30$ | M1 | Correct terms; must deal with cm/m correctly; accept $=$ here |
| | A1 cso | Answer given (2 marks) |
**Part (c):**
| Answer | Mark | Guidance |
|--------|------|----------|
| (You need to buy) at least 2 large cupboards | B1 | cao: 'at least' and '2' and 'large' (1 mark) |
**Part (d):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Capacity $C$ and $140\%C$; total is $Cx + \frac{140}{100}Cy$ | M1 | Must see engagement with 140% in some way |
| Simplifies to $7y + 5x$ | A1 cso | Answer given (2 marks) |
**Part (e):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Correctly drawing $y = 2$ | B1 | |
| Correctly drawing $3x + 4y = 30$ | B1 | |
| Correctly drawing $x + 2y = 12$ | B1 | ft only if swapped coefficients in (a): $(6,0)$ $(2,8)$ |
| Lines labelled and drawn with a ruler | B1 | At least 2 lines labelled including one diagonal line |
| Shading correct | B1 | |
| Region $R$ identified correctly | B1 | (6 marks total) |
**Part (f):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Consider points and value of $5x + 7y$, or draw a clear profit line | M1A1 | At least 2 points tested or objective line drawn with correct gradient |
| $(7,2) \to 49$; $(7\tfrac{1}{3}, 2) \to 50\tfrac{2}{3}$; $(6,3) \to 51$; $(0,6) \to 42$; $(0,2) \to 14$ | A1 | 3 points correctly tested or objective line correct and distinct/labelled |
| Best option: buy 6 standard cupboards and 3 large cupboards | A1 | Accept $(6,3)$ if very clearly selected (4 marks total) |
The images you've shared appear to be mostly blank/white pages with only publication/contact information visible on the second page. There is no mark scheme content visible in these images.
Could you please share the actual mark scheme pages? The pages containing the questions, answers, mark allocations, and guidance notes would be needed to extract the content you're looking for.7. You are in charge of buying new cupboards for a school laboratory.
The cupboards are available in two different sizes, standard and large.\\
The maximum budget available is $\pounds 1800$. Standard cupboards cost $\pounds 150$ and large cupboards cost $\pounds 300$.\\
Let $x$ be the number of standard cupboards and $y$ be the number of large cupboards.
\begin{enumerate}[label=(\alph*)]
\item Write down an inequality, in terms of $x$ and $y$, to model this constraint.\\
(2)
The cupboards will be fitted along a wall 9 m long. Standard cupboards are 90 cm long and large cupboards are 120 cm long.
\item Show that this constraint can be modelled by
$$3 x + 4 y \leqslant 30$$
You must make your reasoning clear.
Given also that $y \geqslant 2$,
\item explain what this constraint means in the context of the question.
The capacity of a large cupboard is $40 \%$ greater than the capacity of a standard cupboard. You wish to maximise the total capacity.
\item Show that your objective can be expressed as
$$\text { maximise } 5 x + 7 y$$
\item Represent your inequalities graphically, on the axes in your answer booklet, indicating clearly the feasible region, R.
\item Find the number of standard cupboards and large cupboards that need to be purchased. Make your method clear.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2010 Q7 [17]}}