Moderate -0.8 This is a standard textbook linear programming question from D1, requiring routine graphical methods (plotting constraints, shading feasible region, using objective line method) and solving simultaneous equations. While multi-part with 11 marks total, each step follows a well-practiced algorithm with no novel problem-solving or insight required—significantly easier than average A-level maths questions.
A linear programming problem in \(x\) and \(y\) is described as follows.
Maximise P = \(5x + 3y\)
subject to: \(x \geqslant 3\)
$$x + y \leqslant 9$$
$$15x + 22y \leqslant 165$$
$$26x - 50y \leqslant 325$$
Add lines and shading to Diagram 2 in the answer book to represent these constraints. Hence determine the feasible region and label it R. \hfill [4]
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. \hfill [2]
Calculate the exact coordinates of vertex V and hence calculate the corresponding value of P at V. \hfill [3]
The objective is now to minimise \(5x + 3y\), where \(x\) and \(y\) are integers.
Write down the minimum value of \(5x + 3y\) and the corresponding value of \(x\) and corresponding value of \(y\). \hfill [2]
[Graph showing feasible region with constraints and objective line]
B1, B1, B1, B1(R), B1, B1
(4), (2)
Lines must be long enough to define the correct feasible region and pass through one small square of the points stated:
- \(x + y = 9\) passes through (5, 4) and (9, 0) but in most cases check (0, 9) and (9, 0)
- \(26x - 50y = 325\) passes through (5, -3.9) and (10, -1.3) but in most cases check (0, -6.5) and (12.5, 0)
- \(15x + 22y = 165\) passes through \(\left(3, \frac{60}{11}\right)\) and \(\left(4, \frac{105}{22}\right)\) but in most cases check (0, 7.5) and (11, 0)
Any two lines correctly drawn. Any three lines correctly drawn. All four lines correctly drawn. Region, R, correctly labelled – not just implied by shading – dependent on scoring the first three marks in (a). Drawing the correct objective line on the graph, use line drawing tool to check if necessary. Line must not pass outside of a small square if extended from axis to axis. V labelled clearly on their graph. This mark is dependent on both the correct feasible region (but maybe not labelled) and the correct objective line.
5(c)
Answer
Marks
\(V\left(\frac{775}{76}, -\frac{91}{76}\right)\)
M1 A1
\(P = \frac{1801}{38}\)
A1
(3)
Candidates must have drawn either the correct objective line or its reciprocal. If they have drawn the correct objective line they must be solving \(x + y = 9\) and \(26x - 50y = 325\). If they have drawn the reciprocal objective line they must be solving \(x = 3\) and \(15x + 22y = 165\). Must get to either \(x = \ldots\) or \(y = \ldots\) (condone one error in the solving of the simultaneous equations).
The correct exact answer \(\left(\frac{775}{76}, -\frac{91}{76}\right)\), or for the reciprocal \(\left(3, \frac{60}{11}\right)\), can imply this mark.
cao \(\left(\frac{775}{76}, -\frac{91}{76}\right)\) or \(\left(10\frac{15}{76}, -1\frac{15}{76}\right)\) (coordinates must be exact) – if correct answer stated with no working seen then award M1A0 only (however, they can still earn the next A mark for the corresponding value of P at V). This mark is dependent on the correct feasible region (but maybe not labelled).
cao \(\frac{1801}{38}\) or \(47\frac{15}{38}\) (must be exact). This mark is dependent on the correct feasible region (but maybe not labelled).
5(d)
Answer
Marks
\(x = 3, y = -4\) minimum value is 3
B1 B1
(2)
cao \(x = 3, y = -4\) or \((3, -4)\); cao of 3
## 5(a)(b)
[Graph showing feasible region with constraints and objective line] | B1, B1, B1, B1(R), B1, B1 |
| | (4), (2) |
Lines must be long enough to define the correct feasible region and pass through one small square of the points stated:
- $x + y = 9$ passes through (5, 4) and (9, 0) but in most cases check (0, 9) and (9, 0)
- $26x - 50y = 325$ passes through (5, -3.9) and (10, -1.3) but in most cases check (0, -6.5) and (12.5, 0)
- $15x + 22y = 165$ passes through $\left(3, \frac{60}{11}\right)$ and $\left(4, \frac{105}{22}\right)$ but in most cases check (0, 7.5) and (11, 0)
Any two lines correctly drawn. Any three lines correctly drawn. All four lines correctly drawn. Region, R, correctly labelled – not just implied by shading – dependent on scoring the first three marks in (a). Drawing the correct objective line on the graph, use line drawing tool to check if necessary. Line must not pass outside of a small square if extended from axis to axis. V labelled clearly on their graph. This mark is dependent on both the correct feasible region (but maybe not labelled) and the correct objective line.
## 5(c)
$V\left(\frac{775}{76}, -\frac{91}{76}\right)$ | M1 A1 |
$P = \frac{1801}{38}$ | A1 |
| | (3) |
Candidates must have drawn either the correct objective line or its reciprocal. If they have drawn the correct objective line they must be solving $x + y = 9$ and $26x - 50y = 325$. If they have drawn the reciprocal objective line they must be solving $x = 3$ and $15x + 22y = 165$. Must get to either $x = \ldots$ or $y = \ldots$ (condone one error in the solving of the simultaneous equations).
The correct exact answer $\left(\frac{775}{76}, -\frac{91}{76}\right)$, or for the reciprocal $\left(3, \frac{60}{11}\right)$, can imply this mark.
cao $\left(\frac{775}{76}, -\frac{91}{76}\right)$ or $\left(10\frac{15}{76}, -1\frac{15}{76}\right)$ (coordinates must be exact) – if correct answer stated with no working seen then award M1A0 only (however, they can still earn the next A mark for the corresponding value of P at V). This mark is dependent on the correct feasible region (but maybe not labelled).
cao $\frac{1801}{38}$ or $47\frac{15}{38}$ (must be exact). This mark is dependent on the correct feasible region (but maybe not labelled).
## 5(d)
$x = 3, y = -4$ minimum value is 3 | B1 B1 |
| | (2) |
cao $x = 3, y = -4$ or $(3, -4)$; cao of 3
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A linear programming problem in $x$ and $y$ is described as follows.
Maximise P = $5x + 3y$
subject to: $x \geqslant 3$
$$x + y \leqslant 9$$
$$15x + 22y \leqslant 165$$
$$26x - 50y \leqslant 325$$
\begin{enumerate}[label=(\alph*)]
\item Add lines and shading to Diagram 2 in the answer book to represent these constraints. Hence determine the feasible region and label it R. \hfill [4]
\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. \hfill [2]
\item Calculate the exact coordinates of vertex V and hence calculate the corresponding value of P at V. \hfill [3]
\end{enumerate}
The objective is now to minimise $5x + 3y$, where $x$ and $y$ are integers.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Write down the minimum value of $5x + 3y$ and the corresponding value of $x$ and corresponding value of $y$. \hfill [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2018 Q5 [11]}}