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

The 20 questions

1. What is the region containing all points satisfying a linear programme's constraints called?

   • Constraint plane
   • Objective space
   • Optimal region
   • Feasible region

2. What is the linear expression to be maximised or minimised in a linear programme called?

   • Objective function
   • Constraint equation
   • Boundary line
   • Feasible function

3. Where does the optimal solution of a bounded linear programme always lie?

   • At the centre
   • At a vertex
   • Outside the boundary
   • At the origin

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. What term describes a constraint that holds as an exact equality at the optimal solution?

   • Non-binding constraint
   • Binding constraint
   • Redundant constraint
   • Objective line

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 non-empty feasible region ensures a linear programme has a maximum value?

    • Convexity
    • Boundedness
    • Unboundedness
    • Symmetry

14. In graphical linear programming, what is a line representing a constant objective value called?

    • Constraint line
    • Objective line
    • Optimal ray
    • Feasible boundary

15. Which side of the line x + y = 6 contains the feasible region for x + y ≥ 6?

    • Left of y-axis
    • Below the line
    • Containing the origin
    • Away from 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. What type of feasible region can result in an optimal objective value being infinitely large?

    • Bounded
    • Degenerate
    • Unbounded
    • Empty

19. When does a linear programme have infinitely many optimal solutions?

    • Objective parallel to axis
    • Unbounded feasible region
    • Objective parallel to edge
    • Multiple objective functions

20. What is the outcome of a linear programme with an empty feasible region?

    • Unbounded maximum
    • Infinitely many solutions
    • No feasible solution
    • Unique optimal solution

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