| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Discrete (Further Paper 3 Discrete) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Three-variable constraint reduction |
| Difficulty | Challenging +1.2 This is a standard three-variable linear programming problem requiring constraint formulation and simplex method application. While it involves more variables than typical A-level questions and requires understanding of slack variables and optimality conditions (Further Maths content), the setup is straightforward with clearly stated constraints and the solution method is algorithmic rather than requiring novel insight. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(3x + 2y + z \leq 360\) | B1 | First non-trivial inequality correct |
| \(40x + 20y + 5z \leq 2500\) | B1 | Second non-trivial inequality correct |
| \(x \geq 0,\ y \geq 0,\ z \geq 0\) | B1 | All three trivial inequalities stated |
| Answer | Marks | Guidance |
|---|---|---|
| Translate inequalities into simplex tableau with two slack variables | M1 (AO3.1a) | |
| Identify correct pivot from tableau | A1 (AO1.1b) | Underlined pivot is \(x\)-column, value 40 |
| Apply simplex algorithm correctly to modify two non-pivot rows | M1 (AO1.1a) | |
| Identify correct pivot in modified tableau | A1F (AO1.1b) | |
| Apply simplex algorithm correctly to modify two non-pivot rows again | M1 (AO1.1a) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\) | \(x\) | \(y\) |
| 1 | \(-80\) | \(-35\) |
| 0 | 3 | 2 |
| 0 | \(\underline{40}\) | 20 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\) | \(x\) | \(y\) |
| 1 | 0 | 5 |
| 0 | 0 | 0.5 |
| 0 | 1 | 0.5 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P\) | \(x\) | \(y\) |
| 1 | 0 | 9 |
| 0 | 0 | 0.8 |
| 0 | 1 | 0.4 |
| To maximise profit, the company should repair and sell 28 monitors, 0 hard drives and 276 keyboards each month | E1 (AO3.2a) | Correct interpretation required; objective row must be non-negative |
| Answer | Marks |
|---|---|
| The objective row of the final tableau being non-negative shows no further use of the simplex algorithm is required | R1 (AO2.4) |
| Answer | Marks |
|---|---|
| As \(y = 0\), enforce some hard drives to be repaired by introducing a new inequality, for instance \(y \geq 10\) | E1 (AO3.4) |
## Question 7(a):
$3x + 2y + z \leq 360$ | B1 | First non-trivial inequality correct
$40x + 20y + 5z \leq 2500$ | B1 | Second non-trivial inequality correct
$x \geq 0,\ y \geq 0,\ z \geq 0$ | B1 | All three trivial inequalities stated
---
## Question 7(b)(i):
Translate inequalities into simplex tableau with two slack variables | M1 (AO3.1a) |
Identify correct pivot from tableau | A1 (AO1.1b) | Underlined pivot is $x$-column, value 40
Apply simplex algorithm correctly to modify two non-pivot rows | M1 (AO1.1a) |
Identify correct pivot in modified tableau | A1F (AO1.1b) |
Apply simplex algorithm correctly to modify two non-pivot rows again | M1 (AO1.1a) |
**Initial tableau:**
| $P$ | $x$ | $y$ | $z$ | $s$ | $t$ | value |
|---|---|---|---|---|---|---|
| 1 | $-80$ | $-35$ | $-15$ | 0 | 0 | 0 |
| 0 | 3 | 2 | 1 | 1 | 0 | 360 |
| 0 | $\underline{40}$ | 20 | 5 | 0 | 1 | 2500 |
**After first pivot:**
| $P$ | $x$ | $y$ | $z$ | $s$ | $t$ | value |
|---|---|---|---|---|---|---|
| 1 | 0 | 5 | $-5$ | 0 | 2 | 5000 |
| 0 | 0 | 0.5 | $\underline{0.625}$ | 1 | $-0.075$ | 172.5 |
| 0 | 1 | 0.5 | 0.125 | 0 | 0.025 | 62.5 |
**Final tableau:**
| $P$ | $x$ | $y$ | $z$ | $s$ | $t$ | value |
|---|---|---|---|---|---|---|
| 1 | 0 | 9 | 0 | 8 | 1.4 | 6380 |
| 0 | 0 | 0.8 | 1 | 1.6 | $-0.12$ | 276 |
| 0 | 1 | 0.4 | 0 | $-0.2$ | 0.04 | 28 |
To maximise profit, the company should repair and sell 28 monitors, 0 hard drives and 276 keyboards each month | E1 (AO3.2a) | Correct interpretation required; objective row must be non-negative
---
## Question 7(b)(ii):
The objective row of the final tableau being non-negative shows no further use of the simplex algorithm is required | R1 (AO2.4) |
---
## Question 7(b)(iii):
As $y = 0$, enforce some hard drives to be repaired by introducing a new inequality, for instance $y \geq 10$ | E1 (AO3.4) |
---7 A company repairs and sells computer hardware, including monitors, hard drives and keyboards.
Each monitor takes 3 hours to repair and the cost of components is $\pounds 40$.
Each hard drive takes 2 hours to repair and the cost of components is $\pounds 20$.
Each keyboard takes 1 hour to repair and the cost of components is $\pounds 5$.
Each month, the business has 360 hours available for repairs and $\pounds 2500$ available to buy components.
Each month, the company sells all of its repaired hardware to a local computer shop.
Each monitor, hard drive and keyboard sold gives the company a profit of $\pounds 80 , \pounds 35$ and $\pounds 15$ respectively.
The company repairs and sells $x$ monitors, $y$ hard drives and $z$ keyboards each month.
The company wishes to maximise its total profit.
7
\begin{enumerate}[label=(\alph*)]
\item Find five inequalities involving $x , y$ and $z$ for the company's problem.\\[0pt]
[3 marks]\\
7
\item (i) Find how many of each type of computer hardware the company should repair and sell each month.\\
7 (b) (ii) Explain how you know that you had reached the optimal solution in part (b) (i).\\
7 (b) (iii) The local computer shop complains that they are not receiving one of the types of computer hardware that the company repairs and sells.
Using your answer to part (b) (i), suggest a way in which the company's problem can be modified to address the complaint.\\[0pt]
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Discrete Q7 [11]}}