| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Parametric objective analysis |
| Difficulty | Standard +0.8 This question requires finding a boundary inequality from two points (routine), then optimizing a linear objective (standard), but part (c) requires understanding how the gradient of the objective function relates to the gradients of the feasible region edges to determine when vertex A is uniquely optimal—this parametric analysis requires deeper conceptual understanding of linear programming beyond mechanical application. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06c Working with constraints: algebra and ad hoc methods7.06e Sensitivity analysis: effect of changing coefficients |
6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{87f0e571-e708-4ca9-adc7-4ed18e144d32-07_1502_1659_230_210}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows the constraints of a linear programming problem in $x$ and $y$, where $R$ is the feasible region.
The vertices of the feasible region are $A ( 4,7 ) , B ( 5,3 ) , C ( - 1,5 )$ and $D ( - 2,1 )$.
\begin{enumerate}[label=(\alph*)]
\item Determine the inequality that defines the boundary of $R$ that passes through vertices $A$ and $C$, leaving your answer with integer coefficients only.
The objective is to maximise $P = 5 x + y$
\item Find the coordinates of the optimal vertex and the corresponding value of $P$.
The objective is changed to maximise $Q = k x + y$
\item If $k$ can take any value, find the range of values of $k$ for which $A$ is the only optimal vertex.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2019 Q6 [10]}}