| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2002 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.8 This is a standard textbook linear programming problem requiring routine formulation of inequalities from word problem constraints, graphical construction of feasible region, and finding optimal solution at vertices. All steps are algorithmic with no novel insight required, making it easier than average A-level material, though the multi-part structure and graphical work prevent it from being trivial. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables |
| Answer | Marks |
|---|---|
| Chemical A: \(5x + y \geq 10\) | B1 |
| Chemical B: \(2x + 2y \geq 12 \Rightarrow x + y \geq 6\) | B1 |
| Chemical C: \(\frac{1}{2}x + 2y > 6 \Rightarrow x + 4y \geq 12\) | B1 |
| \(x \geq 0\), \(y \geq 0\) - sum correct | B1, (4) |
| Answer | Marks |
|---|---|
| [Graph showing feasible region with shaded area] | B1, B1, B1, (3) |
| Answer | Marks |
|---|---|
| \(T = 2x + 3y\) | B1, (1) |
| Answer | Marks |
|---|---|
| Profit line or point testing (>3) | M1, A1 |
| \(x = 4\), \(y = 2\), \(T = 14\) | A1, A1, (4) |
| Answer | Marks |
|---|---|
| Three (or more) variables e.g. A blend of three fertilisers x, y and z | M1, A1, (2) |
## (a)
Chemical A: $5x + y \geq 10$ | B1 |
Chemical B: $2x + 2y \geq 12 \Rightarrow x + y \geq 6$ | B1 |
Chemical C: $\frac{1}{2}x + 2y > 6 \Rightarrow x + 4y \geq 12$ | B1 |
$x \geq 0$, $y \geq 0$ - sum correct | B1, (4) |
## (b)
[Graph showing feasible region with shaded area] | B1, B1, B1, (3) |
## (c)
$T = 2x + 3y$ | B1, (1) |
## (d)
Profit line or point testing (>3) | M1, A1 |
$x = 4$, $y = 2$, $T = 14$ | A1, A1, (4) |
## (e)
Three (or more) variables e.g. A blend of three fertilisers x, y and z | M1, A1, (2) |
---Two fertilizers are available, a liquid $X$ and a powder $Y$. A bottle of $X$ contains 5 units of chemical $A$, 2 units of chemical $B$ and $\frac{1}{2}$ unit of chemical $C$. A packet of $Y$ contains 1 unit of $A$, 2 units of $B$ and 2 units of $C$. A professional gardener makes her own fertilizer. She requires at least 10 units of $A$, at least 12 units of $B$ and at least 6 units of $C$.
She buys $x$ bottles of $X$ and $y$ packets of $Y$.
\begin{enumerate}[label=(\alph*)]
\item Write down the inequalities which model this situation. [4]
\item On the grid provided construct and label the feasible region. [3]
\end{enumerate}
A bottle of $X$ costs £2 and a packet of $Y$ costs £3.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{2}
\item Write down an expression, in terms of $x$ and $y$, for the total cost $£T$. [1]
\item Using your graph, obtain the values of $x$ and $y$ that give the minimum value of $T$. Make your method clear and calculate the minimum value of $T$. [4]
\item Suggest how the situation might be changed so that it could no longer be represented graphically. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2002 Q5 [14]}}