◈ Oracle PrepSign Up Free
A-LevelFurther MathematicsMatrices9231/12 · Oct/Nov 2019 · Q1012 Marks

The matrix A is defined by A = [[1,5,1],[1,-2,-2],[2,3,θ]]

  1. (i)(a)Find the rank of A when θ ≠ −1.[3]
  2. (i)(b)Find the rank of A when θ = −1.[1]
  3. (ii)Solve the system of equations when θ ≠ −1: x + 5y + z = −1, x - 2y - 2z = 0, 2x + 3y + θz = 0.[3]
  4. (iii)Find the general solution when θ = −1.[3]
  5. (iv)Show that if θ = −1 and φ ≠ −1 then Ax = [[-1],[0],[φ]] has no solution.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

  1. [M1 A1] Reduces to echelon form. At least one row operation for M1.
  2. [A1]
  3. [B1]
  4. [M1] Uses reduced form of augmented matrix or eliminates variables from scratch.
  5. [A1] One correct.
  6. [A1] All three correct.
  7. [M1] Uses parameter.
  8. [A1 A1]
  9. [M1] Uses reduced form of augmented matrix or eliminates variables from scratch.
  10. [A1]

Write your answer and get it marked

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.

Sign up free →

About this question

Topic
Matrices, A-Level Further Mathematics (syllabus 9231). Structured question, 12 marks.
Source
A-Level Further Mathematics, Oct/Nov 2019 Paper 1 Variant 2, question 10.

Questions and mark schemes are from past examination papers and remain the copyright of the exam board. How Oracle Prep builds its content

Related Further Mathematics questions

MatricesMay/June 20176m

The matrix A, given by A = ( 1 2 -2 ) ( 6 4 -6 ) ( 6 5 -7 ) has eigenvalues 1, -1 and -2.

MatricesMay/June 201810m

The linear transformation T : R⁴ → R³ is represented by the matrix M, where M = ⎛ 1 2 α -1 ⎞ ⎜ 2 6 -3 -3 ⎟ ⎝…

MatricesMay/June 201812m

The line l₁ is parallel to the vector ai – j + k, where a is a constant, and passes through the point whose position …

IntegrationMay/June 20107m

At any point (x, y) on the curve C, dx/dt = t(t²+4) and dy/dt = -t√(4-t²), where the parameter t is such that 0 ≤ t ≤ 2.

Summation of seriesMay/June 20107m

The sum S_N is defined by S_N = ∑_{n=1}^N n³. Using the identity (n+1)⁶ – (n-1)⁶ = 6n⁵ + 5n³ + 3n, find S_N in terms …

IntegrationMay/June 20109m

Let I_n = ∫_1^e x(lnx)ⁿ dx, where n ≥ 1.

← Browse all past-paper questions

© 2026 Oracle Prep