| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2016 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.8 This is a standard D1 linear programming question requiring routine graphical methods: plotting constraints, identifying the feasible region, using the objective line method, and finding vertex coordinates. While multi-part with several constraints, it follows a completely standard template with no novel problem-solving required—students drill this exact procedure repeatedly in D1. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| (a), (b) | Four graphs shown: (1) \(x = 3\) as vertical line, (2) Region R, (3) \(2cx - 50y = 325\), (4) \(15x + 2y = 165\) | B1 B1 B1 B1 (R) B1 B1 |
| (c) | \(V\left(\frac{775}{76}, -\frac{91}{76}\right)\) | M1 A1 |
| (d) | \(x = 3, y = -4\) minimum value is 3 | B1 B1 |
| Answer | Marks |
|---|---|
| a1B1 | Any two lines correctly drawn |
| a2B1 | Any three lines correctly drawn |
| a3B1 | All four lines correctly drawn |
| a4B1 | Region, R, correctly labelled – not just implied by shading – dependent on scoring the first three marks in this part. |
| b1B1 | Cao – for x and y |
| b2B1 | Cao (value of \(5x + 3y\)) |
| c1M1 | Simultaneous equations being used to find their V. Must get to \(x = \ldots\) and \(y = \ldots\) |
| c1A1 | Correct coordinates of V stated exactly |
| c2A1 | Correct value for P |
| d1B1 | Cao – for x and y |
| d2B1 | Cao (value of \(5x + 3y\)) |
(a), (b) | Four graphs shown: (1) $x = 3$ as vertical line, (2) Region R, (3) $2cx - 50y = 325$, (4) $15x + 2y = 165$ | B1 B1 B1 B1 (R) B1 B1 | (4) + (2) |
(c) | $V\left(\frac{775}{76}, -\frac{91}{76}\right)$ | M1 A1 | $P = \frac{1801}{38}$ | A1 | (3) |
(d) | $x = 3, y = -4$ minimum value is 3 | B1 B1 | (2) |
**Notes:**
| a1B1 | Any two lines correctly drawn |
| a2B1 | Any three lines correctly drawn |
| a3B1 | All four lines correctly drawn |
| a4B1 | Region, R, correctly labelled – not just implied by shading – dependent on scoring the first three marks in this part. |
| b1B1 | Cao – for x and y |
| b2B1 | Cao (value of $5x + 3y$) |
| c1M1 | Simultaneous equations being used to find their V. Must get to $x = \ldots$ and $y = \ldots$ |
| c1A1 | Correct coordinates of V stated exactly |
| c2A1 | Correct value for P |
| d1B1 | Cao – for x and y |
| d2B1 | Cao (value of $5x + 3y$) |
**Total: 11 marks**
---5. A linear programming problem in $x$ and $y$ is described as follows.
$$\begin{array} { l l }
\text { Maximise } & \mathrm { P } = 5 x + 3 y \\
\text { subject to: } & x \geqslant 3 \\
& x + y \leqslant 9 \\
& 15 x + 22 y \leqslant 165 \\
& 26 x - 50 y \leqslant 325
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R .
\item Use the objective line method to find the optimal vertex, V, of the feasible region. You must draw and label your objective line and label vertex V clearly.
\item Calculate the exact coordinates of vertex V and hence calculate the corresponding value of P at V .
The objective is now to minimise $5 x + 3 y$, where $x$ and $y$ are integers.
\item Write down the minimum value of $5 x + 3 y$ and the corresponding value of $x$ and corresponding value of $y$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2016 Q5 [11]}}