| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Optimal vertex with additional constraint |
| Difficulty | Standard +0.3 This is a standard D1 linear programming question requiring graphical methods (plotting constraints, identifying feasible region, using objective line method) and basic optimization. Part (f) adds mild challenge by requiring understanding of how a new constraint relates to existing vertices, but all techniques are routine for this specification. Slightly easier than average A-level due to being Decision Maths rather than pure, with straightforward algebraic manipulation. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
6. A linear programming problem in $x$ and $y$ is described as follows.
Minimise $C = 2 x + 3 y$\\
subject to
$$\begin{aligned}
x + y & \geqslant 8 \\
x & < 8 \\
4 y & \geqslant x \\
3 y & \leqslant 9 + 2 x
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Add lines and shading to Diagram 1 in the answer book to represent these constraints.
\item Hence determine the feasible region and label it R .
\item Use the objective line (ruler) method to find the exact coordinates of the optimal vertex, V, of the feasible region. You must draw and label your objective line clearly.
\item Calculate the corresponding value of $C$ at V .
The objective is now to maximise $2 x + 3 y$, where $x$ and $y$ are integers.
\item Write down the optimal values of $x$ and $y$ and the corresponding maximum value of $2 x + 3 y$.
A further constraint, $y \leqslant k x$, where $k$ is a positive constant, is added to the linear programming problem.
\item Determine the least value of $k$ for which this additional constraint does not affect the feasible region.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2015 Q6 [13]}}