| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Dual objective optimization |
| Difficulty | Moderate -0.3 This is a standard D1 linear programming question requiring graphical representation of constraints, identification of feasible region, and optimization of an objective function at vertices. While it involves dual optimization (min and max), the techniques are routine textbook methods with no novel problem-solving required. The constraint system is straightforward with 4 inequalities, making it slightly easier than average A-level questions. |
| Spec | 7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Line \(8x + 3y = 480\) drawn correctly | B1 | |
| Line \(y = 4x\) drawn correctly | B1 | |
| Line \(8x + 7y = 560\) drawn correctly | B1 | Lines |
| Correct shading/feasible region identified | B1 | Shading |
| Region R found correctly | B1 | R found |
| Labels on graph correct | B1 | Labels |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Point testing or profit line method shown | M1 A1 | |
| Minimum point \((0, 80)\); Value of 80 | B1 A1 | |
| Maximum point \((24, 96)\); Value of 168 | B1 A1 |
# Question 7:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Line $8x + 3y = 480$ drawn correctly | B1 | |
| Line $y = 4x$ drawn correctly | B1 | |
| Line $8x + 7y = 560$ drawn correctly | B1 | Lines |
| Correct shading/feasible region identified | B1 | Shading |
| Region R found correctly | B1 | R found |
| Labels on graph correct | B1 | Labels |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Point testing or profit line method shown | M1 A1 | |
| Minimum point $(0, 80)$; Value of 80 | B1 A1 | |
| Maximum point $(24, 96)$; Value of 168 | B1 A1 | |
---7. A linear programming problem is modelled by the following constraints
$$\begin{aligned}
8 x + 3 y & \leqslant 480 \\
8 x + 7 y & \geqslant 560 \\
y & \geqslant 4 x \\
x , y & \geqslant 0
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Use the grid provided in your answer book to represent these inequalities graphically. Hence determine the feasible region and label it R .
The objective function, $F$, is given by
$$F = 3 x + y$$
\item Making your method clear, determine
\begin{enumerate}[label=(\roman*)]
\item the minimum value of the function $F$ and the coordinates of the optimal point,
\item the maximum value of the function $F$ and the coordinates of the optimal point.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2009 Q7 [12]}}