| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Easy -1.2 This is a standard textbook linear programming question requiring routine graphical methods: plotting linear inequalities, identifying the feasible region, and using the objective line method to find maxima at vertices. It involves no novel problem-solving or proof, just mechanical application of D1 techniques with straightforward constraints and objective functions. |
| Spec | 7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Lines \(y=5\), \(x=4\) drawn | B1 | |
| Line \(x+y=30\) drawn | B1 | |
| Line \(2x+y=40\) drawn | B1 | |
| Line \(y=\frac{1}{2}x\) drawn | B1 | |
| Feasible region correct | B1 | CAO |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Max at \((16,8)=56\) | M1 | Extreme point within \(\frac{1}{2}\) square of their region |
| A1 | ||
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Max at \((4,26)=82\) | M1 | Extreme point within \(\frac{1}{2}\) square of their region |
| A1 | ||
| Total: 2 |
## Question 2:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Lines $y=5$, $x=4$ drawn | B1 | |
| Line $x+y=30$ drawn | B1 | |
| Line $2x+y=40$ drawn | B1 | |
| Line $y=\frac{1}{2}x$ drawn | B1 | |
| Feasible region correct | B1 | CAO |
| **Total: 5** | | |
### Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Max at $(16,8)=56$ | M1 | Extreme point within $\frac{1}{2}$ square of their region |
| | A1 | |
| **Total: 2** | | |
### Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Max at $(4,26)=82$ | M1 | Extreme point within $\frac{1}{2}$ square of their region |
| | A1 | |
| **Total: 2** | | |
---2 [Figure 1, printed on the insert, is provided for use in this question.]\\
The feasible region of a linear programming problem is represented by
$$\begin{aligned}
x + y & \leqslant 30 \\
2 x + y & \leqslant 40 \\
y & \geqslant 5 \\
x & \geqslant 4 \\
y & \geqslant \frac { 1 } { 2 } x
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item On Figure 1, draw a suitable diagram to represent these inequalities and indicate the feasible region.
\item Use your diagram to find the maximum value of $F$, on the feasible region, in the case where:
\begin{enumerate}[label=(\roman*)]
\item $F = 3 x + y$;
\item $F = x + 3 y$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D1 2008 Q2 [9]}}