| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Three-variable constraint reduction |
| Difficulty | Standard +0.8 This D1 question requires systematic elimination of a variable from three constraints, then solving a reduced 2D LP problem. While the algebraic manipulation is straightforward, students must recognize the strategy of using the equality to eliminate z, correctly handle the resulting 2D feasible region, and work backwards to find the complete solution. This is more conceptually demanding than typical D1 simplex or graphical LP questions. |
| Spec | 7.06f Integer programming: branch-and-bound method |
4. A linear programming problem in $x , y$ and $z$ is described as follows.
Maximise $\quad P = - x + y$\\
subject to
$$\begin{gathered}
x + 2 y + z \leqslant 15 \\
3 x - 4 y + 2 z \geqslant 1 \\
2 x + y + z = 14 \\
x \geqslant 0 , y \geqslant 0 , z \geqslant 0
\end{gathered}$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Eliminate $z$ from the first two inequality constraints, simplifying your answers.
\item Hence state the maximum possible value of $P$
Given that $P$ takes the maximum possible value found in (a)(ii),
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item determine the maximum possible value of $x$
\item Hence find a solution to the linear programming problem.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2022 Q4 [7]}}