D1 June 2019 Q6

EdexcelOld spec10 marksLinear Programming

Figure 4: feasible region R, a quadrilateral with vertices A, B, C and D, on axes x from -3 to 6 and y from 0 to 8 — Figure 4

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 \(D(-2, 1)\).

(a) Determine the inequality that defines the boundary of \(R\) that passes through vertices \(A\) and \(C\), leaving your answer with integer coefficients only. (3)

The objective is to maximise \(P = 5x + y\)

(b) Find the coordinates of the optimal vertex and the corresponding value of \(P\). (3)

The objective is changed to maximise \(Q = kx + y\)

Mark scheme (a)

Scheme	Marks
\(A(4,7), B(5,3), C(-1,5), D(-2,1)\)
Equation through \(AC\) e.g. \(\dfrac{y-7}{5-7} = \dfrac{x-4}{-1-4}\) or \(y - 7 = \left(\dfrac{5-7}{-1-4}\right)(x-4)\) or \(y - 5 = \left(\dfrac{7-5}{4-(-1)}\right)(x+1)\)	M1
\(5y - 2x = 27\) (oe)	A1
\(5y - 2x \leqslant 27\)	A1
	(3)

Notes

a1M1: Correct method for finding the equation of the line through \(A\) and \(C\) – a correct equation can imply this mark – condone one sign error only

a1A1: Correct equation (any correct form (allow unsimplified or simplified incorrectly) – condone any inequality sign or equals)

a2A1: CAO (any equivalent form provided coefficients are integers)

Mark scheme (b)

Scheme	Marks
Point testing \(A\) and \(B\) or objective line (with gradient of \(-5\))	M1
At \(B(5, 3)\), \(P = 28\) at \(A(4, 7)\), \(P = 27\) so optimal point is \(B\) with \(P = 28\)	A1 A1
	(3)

Notes

b1M1: Correct objective line drawn (gradient of \(-5\) - acceptable minimum length is from \((0,1)\) to \((0.2,0)\)) or testing both \(A\) and \(B\) in the correct objective function

b1A1: CAO (\(B\) or (5, 3))

b2A1: \(P = 28\) (allow if seen in working for \(B\))

Mark scheme (c)

Scheme	Marks
\(A \gt B \Rightarrow 4k + 7 \gt 5k + 3\) or objective line argument	M1
\(k \lt 4\)	A1
\(A \gt C \Rightarrow 4k + 7 \gt -k + 5\) or objective line argument	M1
\(k \gt -\dfrac{2}{5}\)	A1
	(4)
	10 marks

Notes

c1M1: Put expression for \(A\) > expression for \(B\) (accept any inequality or equals) or considers gradient of line segment through \(A\) and \(B\) with \(-k\). Condone \(x\) for \(k\) for the M mark only

c1A1: \(k \lt 4\)

c2M1: Put expression for \(A\) > expression for \(C\) (accept any inequality or equals) or considers gradient of line segment through \(A\) and \(C\) with \(-k\). Condone \(x\) for \(k\) for the M mark only

c2A1: \(k \gt -\dfrac{2}{5}\)

Ignore any consideration of vertex \(D\)