| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2007 |
| Session | January |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Easy -1.3 This is a standard D1 linear programming question requiring routine skills: interpreting inequality notation (strict vs non-strict), translating word problems into inequalities, graphing constraints, and identifying feasible region vertices. All techniques are textbook exercises with no problem-solving insight required, making it easier than average A-level maths. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables |
| Answer | Marks |
|---|---|
| To show a strict inequality | B1 (1) |
| Answer | Marks |
|---|---|
| There must be fewer than 10 children / There must be between 2 and 10 adults, inclusive | B1, B2, 1, 0 (3) |
| Answer | Marks |
|---|---|
| \(2x + 3y \geq 24\) / \(x \leq 2y\) | B1, B1 (2) |
| Answer | Marks |
|---|---|
| [Graph showing feasible region with constraints marked] | BW (3+4), 11₂ |
| B1W (x + 3y), B1W (x + 3y), B1 (filling) | |
| B1 (8) |
| Answer | Marks |
|---|---|
| minimum: 0 children, 8 adults, -8 passengers / maximum: 9 children, 10 adults, -19 passengers | M1, A1, B1, B1 (4), [16] |
## 7(a)
| To show a strict inequality | B1 (1) |
## 7(b)
| There must be fewer than 10 children / There must be between 2 and 10 adults, inclusive | B1, B2, 1, 0 (3) |
## 7(c)
| $2x + 3y \geq 24$ / $x \leq 2y$ | B1, B1 (2) |
## 7(d)
| [Graph showing feasible region with constraints marked] | BW (3+4), 11₂ |
| | B1W (x + 3y), B1W (x + 3y), B1 (filling) |
| | B1 (8) |
## 7(e)
| minimum: 0 children, 8 adults, -8 passengers / maximum: 9 children, 10 adults, -19 passengers | M1, A1, B1, B1 (4), [16] |\includegraphics{figure_6}
The captain of the \textit{Malde Mare} takes passengers on trips across the lake in her boat.
The number of children is represented by $x$ and the number of adults by $y$.
Two of the constraints limiting the number of people she can take on each trip are
$$x < 10$$
and
$$2 \leq y \leq 10$$
These are shown on the graph in Figure 6, where the rejected regions are shaded out.
\begin{enumerate}[label=(\alph*)]
\item Explain why the line $x = 10$ is shown as a dotted line. (1)
\item Use the constraints to write down statements that describe the number of children and the number of adults that can be taken on each trip. (3)
\end{enumerate}
For each trip she charges £2 per child and £3 per adult. She must take at least £24 per trip to cover costs.
The number of children must not exceed twice the number of adults.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use this information to write down two inequalities. (2)
\item Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R. (4)
\item Use your graph to determine how many children and adults would be on the trip if the captain takes:
\begin{enumerate}[label=(\roman*)]
\item the minimum number of passengers,
\item the maximum number of passengers.
\end{enumerate} (4)
\end{enumerate}
(Total 14 marks)
\hfill \mbox{\textit{Edexcel D1 2007 Q7}}