Lesson DD3-DD4

DD3-DD4 The Simplex algorithm Quiz: AQA Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DD3-DD4, The Simplex algorithm: 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 the Simplex method, what is a slack variable?

    • A variable that is always negative in any feasible solution
    • A variable that replaces a non-negativity constraint
    • A variable added to an inequality to turn it into an equation, representing the unused amount of a resource
    • A variable that measures the value of the objective function

  2. When maximising with the Simplex method, how is the entering variable chosen?

    • It is the variable with the largest value in the right-hand column
    • It is the variable with the largest ratio of its column entry to its right-hand side
    • It is the variable whose coefficient in the objective row is the most positive
    • It is the variable whose coefficient in the objective row is the most negative

  3. In the Simplex method, how is the leaving variable chosen?

    • It is the row giving the smallest non-negative ratio of right-hand side to a positive entry in the pivot column
    • It is the row in which the objective value is zero
    • It is the row with the most negative entry in the pivot column
    • It is the row with the largest right-hand side value

  4. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. In the initial tableau with slack variables s1 and s2, what are the values of s1 and s2 when x = y = 0?

    • s1 = 6 and s2 = 24, with P = 0
    • s1 = 4 and s2 = 6, with P = 5
    • s1 = 24 and s2 = 6, with P = 0
    • s1 = 0 and s2 = 0, with P = 24

  5. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. In the initial tableau, which variable should enter the basis first, and why?

    • x, since its coefficient -5 in the objective row is the most negative
    • y, since it has the smaller coefficient in the objective function
    • s1, since it is a slack variable
    • s2, since it is attached to the smaller constraint

  6. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. Using the ratio test on the first pivot, which row leaves the basis?

    • Both rows leave the basis together
    • The s1 row, since its ratio 24/6 = 4 is smaller than 6/1 = 6
    • No row leaves, since x has a negative coefficient in the objective row
    • The s2 row, since its ratio 6/1 = 6 is smaller than 24/6 = 4

  7. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. After the first pivot, x = 4 - (2/3)y - (1/6)s1. What is the value of x when y = 0 and s1 = 0?

    • 3
    • 24
    • 4
    • 6

  8. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. What is the optimal value of P?

    • 24
    • 21
    • 18
    • 20

  9. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. At which point (x, y) is the optimum reached?

    • (2.4, 1.8)
    • (3, 1.5)
    • (4, 0)
    • (0, 3)

  10. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. At the optimum, which constraints hold with equality?

    • Only x + 2y ≤ 6 holds with equality
    • Only 6x + 4y ≤ 24 holds with equality
    • Both 6x + 4y ≤ 24 and x + 2y ≤ 6 hold with equality
    • Neither constraint holds with equality at the optimum

  11. Maximise P = 5x + 4y subject to 6x + 4y ≤ 24 and x + 2y ≤ 6, with x ≥ 0 and y ≥ 0. At the optimum, what is the value of the slack variable s1, and what does it mean?

    • s1 = 3, so the first constraint is not binding
    • s1 = 24, so the first resource is not used at all
    • s1 = 0, so the first resource is fully used
    • s1 = 6, so six units of the first resource are unused

  12. In the Simplex method for maximisation, when is the optimal solution reached?

    • When every basic variable has the value zero
    • After exactly as many pivots as there are constraints
    • When no coefficient in the objective row is negative, so no entering variable can increase P
    • When the entering variable has no positive entry in its column

  13. A maximisation Simplex tableau has a pivot column with no positive entries. What does this indicate?

    • The problem has a unique optimum at the origin
    • The objective function is unbounded, so there is no finite maximum
    • The problem has no feasible solution
    • All slack variables must be zero at the optimum

  14. Why are slack variables introduced for ≤ constraints in the Simplex method?

    • To make the objective function linear
    • To reduce the number of constraints in the problem
    • To allow the decision variables to take negative values
    • To convert each inequality into an equation, so that an initial basic feasible solution with all decision variables zero is available

  15. What does it mean if a slack variable is zero at the optimum of a linear programme?

    • The objective value is zero at the optimum
    • The variable is absent from the optimal solution and can be ignored
    • The constraint it is attached to is binding, so its resource is fully used
    • The constraint has been removed from the problem entirely

  16. A maximisation problem has three ≤ constraints and two decision variables. How many slack variables are introduced?

    • 2
    • 5
    • 1
    • 3

  17. In a Simplex tableau, the pivot column has entries 2 and -1 in two constraint rows, with right-hand sides 8 and 4. Which row leaves the basis?

    • The second row, since its pivot entry -1 is negative
    • Neither row, because the pivot column contains a negative entry
    • The second row, since its right-hand side 4 is smaller
    • The first row, with pivot entry 2, since its ratio 8/2 = 4 is the only valid positive ratio

  18. In the initial tableau for maximising P = 3x + 2y, the objective row reads P - 3x - 2y = 0. What does the coefficient -3 for x mean?

    • Increasing x by 1 increases P by 3
    • Increasing x by 1 leaves P unchanged
    • Increasing x by 1 decreases P by 3
    • The variable x must be zero in every feasible solution

  19. Which statement about the Simplex method is correct?

    • Each pivot moves from one vertex of the feasible region to an adjacent vertex, and the method stops at the optimal vertex
    • The method can only be used when all constraints are equalities
    • Each pivot moves to a vertex that is further from the optimum
    • The method examines every point inside the feasible region before stopping

  20. At an optimal Simplex tableau for maximisation, a non-basic variable has coefficient 0 in the objective row. What does this indicate?

    • The problem is infeasible
    • The variable has reached its upper bound, so P is unbounded
    • The variable must be removed from the model
    • Increasing that variable does not change P, so there may be alternative optimal solutions

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