3.7-3.8 Solving three linear equations Quiz: Pearson Edexcel Further Maths, Unit 3

The 20 questions

1. Three planes in 3-D give a system of three linear equations. Which arrangement gives exactly one solution?

   • Two of the planes coincide
   • The three planes meet at a single point
   • The three planes are parallel and distinct
   • The three planes all contain a common line

2. What is a sheaf of planes in the context of three simultaneous linear equations?

   • Three planes that meet at exactly one point
   • Three planes forming a closed triangular prism
   • Three planes that are mutually parallel
   • Three planes that all contain one common line

3. Three distinct parallel planes form which geometrical configuration, and what does it mean for the solutions?

   • A sheaf, with infinitely many solutions
   • A prism, with no common point, so the system is inconsistent
   • A single point, so the system has a unique solution
   • Coincident planes, so the system has infinitely many solutions

4. In matrix form, a 3 x 3 system of equations is written as Ax = b. Which expression gives the solution when A is invertible?

   • x = A^T b
   • x = A b
   • x = b A^(-1)
   • x = A^(-1) b

5. If the coefficient matrix of a 3 x 3 system is singular, which statement is correct?

   • The system always has no solution at all
   • The system always has the solution x = 0
   • The system does not have a unique solution, so the inverse method cannot be used
   • The system always has exactly one solution found by the inverse matrix

6. Solve the system x + y + z = 6, x - y + z = 2, 2x + y - z = 1.

   • (1, 3, 2)
   • (3, 2, 1)
   • (2, 1, 3)
   • (1, 2, 3)

7. Solve the system with A = diag(1, 2, 4) and b = (3, 4, 8).

   • (1, 2, 2)
   • (3, 2, 8)
   • (3, 4, 8)
   • (3, 2, 2)

8. Solve the system x + 2y = 5, y + z = 3, x + z = 4.

   • (7/3, 5/3, 4/3)
   • (2, 1, 2)
   • (1, 2, 1)
   • (7/3, 4/3, 5/3)

9. Which system of three planes has infinitely many solutions?

   • x + y + z = 1, x - y = 0, 2x + 2y + 2z = 2
   • x + y + z = 1, x - y = 0, x + z = 5
   • x + y + z = 1, x - y = 0, x + y + 2z = 0
   • x + y + z = 1, x + y + z = 2, x - y = 0

10. The planes x + y + z = 1, x + y + z = 2 and x + y + z = 5 are considered together. What are the solutions?

    • No solution, because the planes are parallel and distinct
    • A unique solution (1, 2, 5)
    • Exactly the solution x = 1, y = 2, z = 2
    • Infinitely many solutions on a line

11. For which value of k is the coefficient matrix of x + y + kz = 1, x + ky + z = 1, kx + y + z = 1 singular? Choose the complete set of values.

    • k = 1 and k = -2
    • k = 1 only
    • k = -1 and k = 2
    • k = 0 and k = 2

12. A consistent 3 x 3 system has a singular coefficient matrix. Which description of its solution set is correct?

    • A line or a plane, so infinitely many solutions
    • Exactly one point
    • Exactly two points
    • The empty set

13. Solve the system with A = diag(2, 1, 1), b = (4, 2, 6).

    • (2, 2, 6)
    • (2, 2, 3)
    • (4, 2, 6)
    • (4, 1, 6)

14. The inverse of the diagonal matrix diag(1, 2, 4) is which matrix?

    • [[1, 0, 0], [0, 2, 0], [0, 0, 4]]
    • [[1, 0, 0], [0, 1/4, 0], [0, 0, 1/2]]
    • [[1, 0, 0], [0, 1/2, 0], [0, 0, 1/4]]
    • [[-1, 0, 0], [0, -2, 0], [0, 0, -4]]

15. Three planes are x = 1, y = 2 and x + y + z = 10. What is the value of z?

    • 13
    • 7
    • 10
    • 3

16. For which value of k is the coefficient matrix [[1, 1, 1], [1, 2, 3], [1, 3, k]] singular?

    • k = 3
    • k = -5
    • k = 5
    • k = 9

17. Three planes have normal vectors whose determinant is zero, and their equations are consistent. Which statement is correct?

    • The normal vectors are mutually orthogonal, and the planes meet at one point
    • The normal vectors are linearly dependent, and the planes share a common line
    • The normal vectors are linearly independent, and the planes form a prism
    • The constants on the right-hand side must all be zero for consistency

18. If A^(-1) = [[2, 1], [1, 1]] and Ax = (3, 2), what is x?

    • (8, 5)
    • (3, 2)
    • (1, -1)
    • (5, 8)

19. How many solutions does the system x + y + z = 3, x - y + 2z = 4, 2x + 3z = 7 have?

    • Infinitely many, lying on a line
    • Exactly two
    • Exactly one
    • None

20. Solve the system x + y = 3, y + z = 5, x + z = 4.

    • (1, 2, 3)
    • (1, 3, 2)
    • (2, 1, 3)
    • (3, 2, 1)

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