Standard +0.8 This is a Further Maths linear programming question requiring understanding that the maximum occurs at a vertex when the objective function's gradient lies between the gradients of the two edges meeting at that vertex. Students must set up and solve inequalities comparing -a with the slopes of edges from (5,11) to the other vertices, then find the range of P values. This requires conceptual understanding beyond routine vertex testing, making it moderately challenging but still a standard Further Maths technique.
A linear programming problem is set up to maximise \(P = ax + y\) where \(a\) is a constant.
\(P\) is maximised subject to three linear constraints which form the triangular feasible region shown in the diagram below.
\includegraphics{figure_8}
The vertices of the triangle are \((1, 6)\), \((5, 11)\) and \((13, 9)\)
\(P\) is maximised at \((5, 11)\)
Find the range of possible values for \(P\)
[5 marks]
Question 8:
8 | Finds gradient of one constraint
line
or
Finds expressions for P at two
points of intersection | 3.1a | M1 | 11 6 5
gradient= =
5 1 4
−
11− 9 1
gradient= =
5 13 4
−
−
−
g𝑃𝑃ra=di𝑎𝑎e𝑎𝑎nt+ =𝑦𝑦 ⇒ 𝑦𝑦 = −𝑎𝑎𝑎𝑎+𝑃𝑃
−𝑎𝑎
5 1
4 4
− ≤ 𝑎𝑎 ≤
25 5
11 P 11
4 4
− + ≤ ≤ +
4.75 P 12.25
≤ ≤
Finds gradient of objective line
or
Finds correct expressions for P
at all points of intersection | 1.1b | A1
Finds the correct absolute value
of at least one critical value for a
or for P | 3.1a | M1
Finds both correct critical values
for a or for P | 1.1b | A1
Obtains correct range for P
Condone strict inequalities | 1.1b | A1
Total | 5
Paper total | 40
A linear programming problem is set up to maximise $P = ax + y$ where $a$ is a constant.
$P$ is maximised subject to three linear constraints which form the triangular feasible region shown in the diagram below.
\includegraphics{figure_8}
The vertices of the triangle are $(1, 6)$, $(5, 11)$ and $(13, 9)$
$P$ is maximised at $(5, 11)$
Find the range of possible values for $P$
[5 marks]
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2021 Q8 [5]}}