Challenging +1.2 Part (a) is a standard graphical linear programming problem requiring plotting constraints and finding the optimal vertex. Part (b) introduces a parametric constraint y ≥ kx and asks when the optimal solution changes, requiring analysis of how this line intersects the feasible region and affects the objective function—this adds conceptual depth beyond routine exercises but remains within expected Further Maths problem-solving without requiring exceptional insight.
7 A linear programming problem is
Maximise \(P = 4 x + y\)
subject to
$$\begin{aligned}
3 x - y & \leqslant 30 \\
x + y & \leqslant 15 \\
x - 3 y & \leqslant 6
\end{aligned}$$
and \(x \geqslant 0 , y \geqslant 0\)
Use a graphical method to find the optimal value of \(P\), and the corresponding values of \(x\) and \(y\).
An additional constraint is introduced.
This constraint means that the value of \(y\) must be at least \(k\) times the value of \(x\), where \(k\) is a positive constant.
Determine the set of values of \(k\) for which the optimal value of \(P\) found in part (a) is unchanged.
Determine, in terms of \(k\), the values of \(x , y\) and \(P\) in the cases when the optimal solution is different from that found in part (a).
7 A linear programming problem is\\
Maximise $P = 4 x + y$\\
subject to
$$\begin{aligned}
3 x - y & \leqslant 30 \\
x + y & \leqslant 15 \\
x - 3 y & \leqslant 6
\end{aligned}$$
and $x \geqslant 0 , y \geqslant 0$
\begin{enumerate}[label=(\alph*)]
\item Use a graphical method to find the optimal value of $P$, and the corresponding values of $x$ and $y$.
An additional constraint is introduced.\\
This constraint means that the value of $y$ must be at least $k$ times the value of $x$, where $k$ is a positive constant.
\item \begin{enumerate}[label=(\roman*)]
\item Determine the set of values of $k$ for which the optimal value of $P$ found in part (a) is unchanged.
\item Determine, in terms of $k$, the values of $x , y$ and $P$ in the cases when the optimal solution is different from that found in part (a).
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS 2023 Q7 [12]}}