3D.5.1 Formulating linear programs Quiz: Pearson Edexcel Further Maths, Unit 39

The 20 questions

1. The objective function in a linear programme is

   • the list of constraints
   • the set of non-negativity conditions
   • the linear expression to be maximised or minimised
   • the graph of the feasible region

2. Constraints in a linear programme are written as

   • quadratic equations
   • linear inequalities or equations
   • logarithms
   • integrals

3. Non-negativity conditions state that the variables are

   • integers only
   • less than one
   • summing to one
   • greater than or equal to zero

4. A slack variable is introduced for

   • a less-than-or-equal-to constraint
   • a greater-than-or-equal-to constraint
   • the objective function
   • an equality constraint

5. A surplus variable is introduced for

   • an equality constraint
   • the objective function
   • a greater-than-or-equal-to constraint
   • a less-than-or-equal-to constraint

6. An artificial variable is needed for

   • the objective function
   • a non-negativity condition
   • a less-than-or-equal-to constraint
   • a greater-than-or-equal-to or an equality constraint

7. The feasible region of a linear programme is

   • the set of points that maximise the objective
   • the set of points satisfying all constraints and non-negativity conditions
   • the single line of the objective
   • the region outside all the constraints

8. A factory makes chairs (x) and tables (y), with profit 10x + 15y. Which is the objective?

   • Minimise P = 10x + 15y
   • Maximise 2x + 3y
   • Maximise x + y
   • Maximise P = 10x + 15y

9. Each chair takes 2 hours of labour, each table takes 3 hours, and 60 hours are available. Which constraint models this?

   • 2x - 3y <= 60
   • 2x + 3y <= 60
   • 2x + 3y >= 60
   • 2x + 3y = 60

10. At least twice as many Y as X is required. Which constraint is correct?

    • y >= 2x
    • x >= 2y
    • y <= 2x
    • 2y = x

11. Write 3x + 4y <= 24 in standard form using a slack variable s1.

    • 3x + 4y + s1 = 0
    • 3x + 4y + s1 = 24
    • 3x + 4y - s1 = 24
    • 3x + 4y = 24 + s1

12. Write x + y >= 5 in standard form using a surplus variable s.

    • x + y - s = 5
    • x + y + s = 5
    • x - y - s = 5
    • x + y - s = -5

13. In the tableau for a maximisation with constraints 4x + 5y + 3z <= 16 and 5x + 4y + 6z <= 24, how many slack variables appear?

    • 1
    • 0
    • 3
    • 2

14. Is the point (4, 4) feasible for 3x + 2y <= 20, 2x + 5y <= 35 and x + y <= 5?

    • No, because x and y are equal
    • No, it violates x + y <= 5
    • Yes, it lies on the boundary 3x + 2y = 20
    • No, it violates 2x + 5y <= 35

15. A bakery must use at most twice as many A as B. Which constraint models this?

    • x <= 2y
    • x >= 2y
    • x = 2y
    • 2x <= y

16. Maximise P = 5x + 4y with machine hours 2x + y <= 10 and labour x + 2y <= 8. What is P at (2, 3)?

    • 18
    • 23
    • 22
    • 26

17. Why are slack variables introduced when solving a linear programme?

    • to remove negative values
    • to count the constraints
    • to make the objective non-linear
    • to turn each inequality into an equation, showing the unused amount of a resource

18. A diet needs at least 40 units of a nutrient. Food X gives 4 units per kg and food Y gives 8 units per kg. Which constraint is correct?

    • 4x + 8y = 40
    • 4x + 8y >= 40
    • 4x + 8y <= 40
    • 8x + 4y >= 40

19. Why does a surplus variable enter the standard form with a negative sign?

    • it ensures the variables are integers
    • it is always equal to the slack variable
    • a >= constraint has its left side exceed the right, so subtracting the surplus turns it into an equation
    • it makes the objective smaller

20. Which extra variables are needed to write x + 2y >= 6 in standard form?

    • a slack variable only
    • two slack variables
    • no extra variable
    • a surplus variable (subtracted) and an artificial variable (added)

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