A-LevelFurther MathematicsMatrices9231/12 · May/June 2020 · Q613 Marks

Let A = 2 0 1 1 .

(a)The transformation in the x-y plane represented by A⁻¹ transforms a triangle of area 30 cm² into a triangle of area d cm². Find the value of d.[3]
(b)Prove by mathematical induction that, for all positive integers n, An = 2n 0 2n-1 1 .[5]
(c)The line y = 2x is invariant under the transformation in the x-y plane represented by Aⁿ B, where B= 1 0 33 0 . Find the value of n.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[M1 A1] det A⁻¹ = (det A)⁻¹ = 1/2
[A1] d = 15
[B1] A = (2¹ 0; 2¹-1 1) so true when n = 1.
[B1] Assume true for n=k, so Aᵏ = (2ᵏ 0; 2ᵏ-1 1).
[M1 A1] Aᵏ⁺¹ = Aᵏ A = (2ᵏ 0; 2ᵏ-1 1) (2 0; 1 1) = (2ᵏ⁺¹ 0; 2ᵏ⁺¹-2+1 1) = (2ᵏ⁺¹ 0; 2ᵏ⁺¹-1 1)
[A1] So, it is also true for n = k+1. Hence, by induction, true for all positive integers.
[M1 A1] AⁿB = (2ⁿ 0; 2ⁿ-1 1) (1 0; 33 0) = (2ⁿ 0; 2ⁿ-1+33 0)
[B1] (2ⁿ 0; 2ⁿ+32 0) (x; 2x) = (2ⁿx; (2ⁿ+32)x)
[M1 A1] (2ⁿ+32)x = 2(2ⁿx) ⇒ 2ⁿ+32 = 2*2ⁿ (assuming x≠0)

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Topic: Matrices, A-Level Further Mathematics (syllabus 9231). Structured question, 13 marks.
Source: A-Level Further Mathematics, May/June 2020 Paper 1 Variant 2, question 6.

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