| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.8 This is a standard textbook linear programming problem with straightforward constraint formulation from a simple word problem. Part (a) is given, parts (b-d) follow routine D1 procedures (formulate, graph, find optimal vertex), and part (e) requires minimal additional work. The context is simple, all constraints are linear and obvious, and no novel problem-solving insight is required—just methodical application of the graphical LP algorithm taught in D1. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(4x + 2y \leq 5 \times 4 \times 60\) | B1 | Condone \(=\) — Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x \geq 40,\quad y \geq 40\) | B1 | Both |
| \(x + y \geq 120\) | ||
| \(x + y \leq 400\) | B1 | Both |
| \((P =)\ 3x + y\) | B1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Graph region | B1 | \(x \geq 40,\ y \geq 40\) |
| B1 | \(120 \leq x + y \leq 400\) | |
| B1 | \(2x + y \leq 600\) | |
| B1 | Correct FR | |
| B1 | Correct OL — Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Extreme points | M1 | |
| Max at \(x = 280,\ y = 40\) | A1 | |
| \(P = 840 + 40 = £880\) | B1 | SC: \((280, 20)\) scores \(\frac{1}{3}\) — Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Max at \((200, 200) \to (40, 360)\) | M1 | |
| Profit £800 | A1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| No. of combinations: \(200 - 40 = 160\) | B1 | |
| \(160 + 1 = 161\) | B1 | Total: 2 |
## Question 8(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $4x + 2y \leq 5 \times 4 \times 60$ | B1 | Condone $=$ — Total: 1 |
## Question 8(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x \geq 40,\quad y \geq 40$ | B1 | Both |
| $x + y \geq 120$ | | |
| $x + y \leq 400$ | B1 | Both |
| $(P =)\ 3x + y$ | B1 | Total: 3 |
## Question 8(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Graph region | B1 | $x \geq 40,\ y \geq 40$ |
| | B1 | $120 \leq x + y \leq 400$ |
| | B1 | $2x + y \leq 600$ |
| | B1 | Correct FR |
| | B1 | Correct OL — Total: 5 |
## Question 8(d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Extreme points | M1 | |
| Max at $x = 280,\ y = 40$ | A1 | |
| $P = 840 + 40 = £880$ | B1 | SC: $(280, 20)$ scores $\frac{1}{3}$ — Total: 3 |
## Question 8(e)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Max at $(200, 200) \to (40, 360)$ | M1 | |
| Profit £800 | A1 | Total: 2 |
## Question 8(e)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| No. of combinations: $200 - 40 = 160$ | B1 | |
| $160 + 1 = 161$ | B1 | Total: 2 |8 [Figure 2, printed on a separate sheet, is provided for use in this question.]\\
A bakery makes two types of pizza, large and medium.\\
Every day the bakery must make at least 40 of each type.\\
Every day the bakery must make at least 120 in total but not more than 400 pizzas in total.\\
Each large pizza takes 4 minutes to make, and each medium pizza takes 2 minutes to make. There are four workers available, each for five hours a day, to make the pizzas.
The bakery makes a profit of $\pounds 3$ on each large pizza sold and $\pounds 1$ on each medium pizza sold.\\
Each day, the bakery makes and sells $x$ large pizzas and $y$ medium pizzas.\\
The bakery wishes to maximise its profit, $\pounds P$.
\begin{enumerate}[label=(\alph*)]
\item Show that one of the constraints leads to the inequality
$$2 x + y \leqslant 600$$
\item Formulate this situation as a linear programming problem.
\item On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and an objective line.
\item Use your diagram to find the maximum daily profit.
\item The bakery introduces a new pricing structure in which the profit is $\pounds 2$ on each large pizza sold and $\pounds 2$ on each medium pizza sold.
\begin{enumerate}[label=(\roman*)]
\item Find the new maximum daily profit for the bakery.
\item Write down the number of different combinations that would give the new maximum daily profit.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D1 2005 Q8 [18]}}