A2 October 2020 Q4

EdexcelCurrent spec10 marksLinear Programming

Figure 3: graph for 0 ≤ x ≤ 5.5 and 0 ≤ y ≤ 7 showing the lines 2y = 5x, y = x + 1 and 6x + 5y = 30, with the feasible region R the triangle between them — Figure 3

Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region.

(a) Write down the inequalities that define \(R\). (2)

The objective is to maximise \(P\), where \(P = 3x + y\)

(b) Obtain the exact value of \(P\) at each of the three vertices of \(R\) and hence find the optimal vertex, \(V\). (4)

The objective is changed to maximise \(Q\), where \(Q = 3x + ay\). Given that \(a\) is a constant and the optimal vertex is still \(V\),

Mark scheme (a)

Scheme	Marks	AO
\(2y \leqslant 5x,\ y \geqslant x + 1,\ 6x + 5y \leqslant 30\)	B2,1,0	1.1b 2.5
	(2)

Notes

(a) B1: Any two correct (accept strict inequalities) – accept equivalent inequalities

B1: CAO (accept equivalent inequalities but inequalities must not be strict)

Mark scheme (b)

Scheme	Marks	AO
\(\left(\frac{2}{3}, \frac{5}{3}\right), \left(\frac{60}{37}, \frac{150}{37}\right), \left(\frac{25}{11}, \frac{36}{11}\right)\)	B1 B1	1.1b 1.1b
\(\left(\frac{2}{3}, \frac{5}{3}\right) \to P = \frac{11}{3}\) \(\left(\frac{60}{37}, \frac{150}{37}\right) \to P = \frac{330}{37}\) \(\left(\frac{25}{11}, \frac{36}{11}\right) \to P = \frac{111}{11}\) so optimal vertex is \(\left(\frac{25}{11}, \frac{36}{11}\right)\)	M1 A1	2.1 2.2a
	(4)

Notes

(b) B1: One correct vertex (must be exact)

B1: All three correct vertices (must be exact)

M1: Testing all three of their vertices in the correct objective function

A1: Correct three values of \(P\) and correct optimal vertex either stated or clearly indicated on the graph

Mark scheme (c)

Scheme	Marks	AO
\(Q = 3x + ay\)
\(3\left(\frac{25}{11}\right) + \frac{36a}{11} \gt 3\left(\frac{60}{37}\right) + \frac{150a}{37}\)	M1	3.1a
\(\Rightarrow a \lt \frac{5}{2}\)	A1	2.2a
\(3\left(\frac{25}{11}\right) + \frac{36a}{11} \gt 3\left(\frac{2}{3}\right) + \frac{5a}{3}\)	M1	1.1b
\(\Rightarrow a \gt -3\)	A1	2.2a
	(4)
	(10 marks)

Notes

(c) M1: Their optimal point from (b) evaluated in \(Q\) compared to their \(\left(\frac{60}{37}, \frac{150}{37}\right)\) evaluated in \(Q\) (with correct inequality)

A1: \(a \lt \frac{5}{2}\)

M1: Their optimal point from (b) evaluated in \(Q\) compared to their \(\left(\frac{2}{3}, \frac{5}{3}\right)\) evaluated in \(Q\) (with correct inequality)

A1: \(a \gt -3\)