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A-LevelFurther MathematicsMatrices9231/12 · May/June 2025 · Q49 Marks

The matrix M is given by M = ( (1, 2), (0, 1) ) ( (cosθ, -sinθ), (sinθ, cosθ) ) where 0 < θ < 2π.

  1. (a)The matrix M represents a sequence of two geometrical transformations in the x-y plane. State the type of each transformation, and make clear the order in which they are applied.[2]
  2. (b)Find the value of θ for which the transformation represented by M has a line of invariant points.[7]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

  1. [B2] Award B1 if given in the wrong order. [anticlockwise, centred at the origin, through an angle θ)], [x-axis fixed, with (0,1) mapped to (2,1).]
  2. [B1] M = ( (cosθ+2sinθ, -sinθ+2cosθ), (sinθ, cosθ) )
  3. [B1] Transforms (x,y) to (X,Y) = (x,y)
  4. [M1*] Uses X=x and Y=y.
  5. [A1] Combines equations correctly to eliminate y or x.
  6. [DM1 A1] Simplifies.
  7. [A1] θ = π/2

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About this question

Topic
Matrices, A-Level Further Mathematics (syllabus 9231). Structured question, 9 marks.
Source
A-Level Further Mathematics, May/June 2025 Paper 1 Variant 2, question 4.

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