Lesson DD1-DD2

DD1-DD2 Formulating and solving linear programmes graphically Quiz: AQA Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DD1-DD2, Formulating and solving linear programmes graphically: 20 multiple choice questions for the AQA Further Maths (7367), Unit 5: Optional application 3: discrete mathematics, written with Revision Ninja.

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The 20 questions

  1. In a linear programme, what is the feasible region?

    • The set of points lying on the boundary of the objective function
    • The straight line along which the objective function takes a constant value
    • The set of points that give the largest value of the objective function
    • The set of points that satisfy all the constraints, including the non-negativity constraints

  2. What is the objective function in a linear programme?

    • The linear expression that is to be maximised or minimised
    • The region in which all the variables are non-negative
    • The point at which two constraint lines intersect
    • The inequalities that limit the values of the variables

  3. In the graphical method for a linear programme with two variables, where is the optimal solution found?

    • At the point of the feasible region furthest from the origin
    • At a vertex of the feasible region
    • At any point on the objective line
    • At the centre of the feasible region

  4. Maximise P = 3x + 4y subject to x + 2y ≤ 8, 3x + y ≤ 9, x ≥ 0 and y ≥ 0. What is the maximum value of P?

    • 20
    • 16
    • 9
    • 18

  5. Maximise P = 3x + 4y subject to x + 2y ≤ 8, 3x + y ≤ 9, x ≥ 0 and y ≥ 0. At which point is the maximum achieved?

    • (0, 4)
    • (2, 3)
    • (3, 0)
    • (2.5, 2)

  6. Minimise C = 2x + 5y subject to x + y ≥ 6, x + 3y ≥ 9, x ≥ 0 and y ≥ 0. What is the minimum value of C?

    • 16.5
    • 30
    • 18
    • 15

  7. Minimise C = 2x + 5y subject to x + y ≥ 6, x + 3y ≥ 9, x ≥ 0 and y ≥ 0. At which point is the minimum achieved?

    • (4.5, 1.5)
    • (0, 6)
    • (6, 0)
    • (9, 0)

  8. For the maximum problem Maximise P = 3x + 4y subject to x + 2y ≤ 8 and 3x + y ≤ 9, which constraints hold with equality at the optimum (2, 3)?

    • Only x + 2y ≤ 8 holds with equality
    • Both x + 2y ≤ 8 and 3x + y ≤ 9 hold with equality
    • Only 3x + y ≤ 9 holds with equality
    • Neither constraint holds with equality at the optimum

  9. A firm makes desks and chairs, earning £40 per desk and £25 per chair. Let x be the number of desks and y the number of chairs. Which expression is the objective function to be maximised?

    • P = 2x + y
    • P = (40/25)x + y
    • P = 40x + 25y
    • P = 40x + 25y + 60

  10. In the desk and chair problem, each desk needs 2 hours of machining and each chair needs 1 hour, with 60 hours available. Which constraint represents the machining time?

    • x + 2y ≤ 60
    • 2x + y = 60 exactly
    • 2x + y ≥ 60
    • 2x + y ≤ 60

  11. The desk and chair problem is: maximise P = 40x + 25y subject to 2x + y ≤ 60, y ≤ 25, x ≥ 0 and y ≥ 0. What is the maximum profit?

    • £1200
    • £1325
    • £1450
    • £1000

  12. In the desk and chair problem with objective P = 40x + 25y subject to 2x + y ≤ 60 and y ≤ 25, at which point is the maximum profit obtained?

    • x = 25, y = 10
    • x = 30, y = 0
    • x = 0, y = 25
    • x = 17.5, y = 25

  13. Which property of a feasible region guarantees that a linear programme has both a maximum and a minimum value?

    • The objective line is parallel to one of the constraint lines
    • The feasible region is bounded and non-empty
    • The objective function has only positive coefficients
    • The feasible region has no vertices

  14. In the graphical method, what does an objective line represent?

    • The boundary of the non-negativity constraints
    • A line of constant objective value, which is moved parallel until it leaves the feasible region
    • A line joining two vertices of the feasible region that have equal objective values
    • The line through the origin with gradient 1

  15. A constraint x + y ≥ 6 is part of a linear programme. Which side of the line x + y = 6 does the feasible region lie on?

    • The side with x + y less than 6
    • The line itself only, with no area
    • The side containing the origin
    • The side containing points with x + y greater than or equal to 6, away from the origin

  16. Maximise P = 2x + 3y subject to x + y ≤ 4, x + 3y ≤ 6, x ≥ 0 and y ≥ 0. What is the maximum value of P?

    • 9
    • 12
    • 8
    • 6

  17. Maximise P = 2x + 3y subject to x + y ≤ 4, x + 3y ≤ 6, x ≥ 0 and y ≥ 0. At which point is the maximum reached?

    • (3, 1)
    • (0, 2)
    • (2, 2)
    • (4, 0)

  18. Which statement about the feasible region of a linear programme and its optimum is correct?

    • An unbounded feasible region always has a maximum at the origin
    • The graphical method cannot be applied when the feasible region is unbounded
    • If the feasible region is unbounded, the objective may have no finite maximum, depending on the direction of the objective
    • An unbounded feasible region means the problem has no feasible solution

  19. Can a linear programme have infinitely many optimal solutions?

    • No, a linear programme always has a unique optimal vertex
    • Yes, but only when the feasible region is empty
    • Yes, if the objective line is parallel to an edge of the feasible region, so every point on that edge is optimal
    • No, because the objective line can never be parallel to a boundary line

  20. A linear programme has an empty feasible region. What is the outcome?

    • The optimum lies at infinity
    • The problem has a unique solution at the origin
    • The problem has no feasible solution, so there is no optimal value
    • The objective function takes the value zero at every point

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