| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Dual objective optimization |
| Difficulty | Moderate -0.8 This is a straightforward linear programming question requiring standard techniques: evaluating objective functions at vertices of a given feasible region and writing inequalities from a diagram. All procedures are routine for D1 with no problem-solving insight needed, making it easier than average but not trivial since it requires careful coordinate reading and multiple calculations. |
| Spec | 7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Max \(2x+3y\) at \((30, 70)\) | M1 | Extreme point |
| \(= 270\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Max \(3x+2y\) at \((60, 40)\) | M1 | Extreme point |
| \(= 260\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Min \(-2x+y\) at \((75, 10)\) | M1 | \(x=75\) |
| \(= -140\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x \geq 20,\quad y \geq 10\) | B1 B1 | OE |
| \(x + y \leq 100\) | OE | |
| \(2x + y \leq 160\) OE | M1 A1 | for gradient of \(-2\) |
| \(y \leq x + 40\) OE | M1 A1 | for positive gradient |
## Question 4:
### Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Max $2x+3y$ at $(30, 70)$ | M1 | Extreme point |
| $= 270$ | A1 | |
**Total: 2 marks**
### Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Max $3x+2y$ at $(60, 40)$ | M1 | Extreme point |
| $= 260$ | A1 | |
**Total: 2 marks**
### Part (a)(iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Min $-2x+y$ at $(75, 10)$ | M1 | $x=75$ |
| $= -140$ | A1 | |
**Total: 2 marks**
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x \geq 20,\quad y \geq 10$ | B1 B1 | OE |
| $x + y \leq 100$ | | OE |
| $2x + y \leq 160$ OE | M1 A1 | for gradient of $-2$ |
| $y \leq x + 40$ OE | M1 A1 | for positive gradient |
**Total: 6 marks**
---4 The diagram shows the feasible region of a linear programming problem.\\
\includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-04_1349_1395_408_294}
\begin{enumerate}[label=(\alph*)]
\item On the feasible region, find:
\begin{enumerate}[label=(\roman*)]
\item the maximum value of $2 x + 3 y$;
\item the maximum value of $3 x + 2 y$;
\item the minimum value of $- 2 x + y$.
\end{enumerate}\item Find the 5 inequalities that define the feasible region.
\end{enumerate}
\hfill \mbox{\textit{AQA D1 2006 Q4 [8]}}