| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| 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 constraints, identifying the feasible region, and using the objective line method to find optimal vertices. While multi-part, it involves only mechanical application of well-practiced D1 techniques with no problem-solving insight required. |
| Spec | 7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Line \(y = mx\) through origin correct to 1 square | M1 | Must be correct to 1 square horizontally or vertically at origin |
| Line through \((0,0)\) and \((4,8)\) | A1 | |
| Line through \((0,0)\) and \((16,4)\) | A1 | |
| Line through \((15,8)\) and \((17,0)\) | B1 | |
| Line through \((4,8)\) and \((12,6)\) | B1 | |
| Feasible Region (FR) labelled | B1 | FR must have scored previous 5 marks and labelled region (condone no shading). Total: 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Max at \((4,8)\) | B1 | Coordinates must be stated explicitly |
| \(= 44\) | B1 | Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Max at \((16,4)\) | B1 | Coordinates must be stated explicitly |
| \(= 84\) | B1 | Total: 2 |
# Question 3:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Line $y = mx$ through origin correct to 1 square | M1 | Must be correct to 1 square horizontally or vertically at origin |
| Line through $(0,0)$ and $(4,8)$ | A1 | |
| Line through $(0,0)$ and $(16,4)$ | A1 | |
| Line through $(15,8)$ and $(17,0)$ | B1 | |
| Line through $(4,8)$ and $(12,6)$ | B1 | |
| Feasible Region (FR) labelled | B1 | FR must have scored previous 5 marks and labelled region (condone no shading). Total: 6 |
## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Max at $(4,8)$ | B1 | Coordinates must be stated explicitly |
| $= 44$ | B1 | Total: 2 |
## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Max at $(16,4)$ | B1 | Coordinates must be stated explicitly |
| $= 84$ | B1 | Total: 2 |
---3 [Figure 1, printed on the insert, is provided for use in this question.]\\
The feasible region of a linear programming problem is represented by the following:
$$\begin{aligned}
x \geqslant 0 , y & \geqslant 0 \\
x + 4 y & \leqslant 36 \\
4 x + y & \leqslant 68 \\
y & \leqslant 2 x \\
y & \geqslant \frac { 1 } { 4 } x
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item On Figure 1, draw a suitable diagram to represent the inequalities and indicate the feasible region.
\item Use your diagram to find the maximum value of $P$, stating the corresponding coordinates, on the feasible region, in the case where:
\begin{enumerate}[label=(\roman*)]
\item $P = x + 5 y$;
\item $\quad P = 5 x + y$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D1 2010 Q3 [10]}}