A-LevelFurther MathematicsMatrices9231/12 · May/June 2022 · Q717 Marks

The matrix A is given by A = beginpmatrix 1 & 2 & 3 4 & k & 6 7 & 8 & 9 endpmatrix.

(a)Find the set of values of k for which A is non-singular.[3]
(b)Given that A is non-singular, find, in terms of k, the entries in the top row of A⁻¹.[4]
(c)Given that B = beginpmatrix 1 & 0 & 0 0 & 1 & 0 endpmatrix, give an example of a matrix C such that BAC = beginpmatrix 2 & 1 k & 4 endpmatrix.[4]
(d)Find the set of values of k for which the transformation in the x-y plane represented by beginpmatrix 2 & 1 k & 4 endpmatrix has two distinct invariant lines through the origin.[6]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[M1 A1] Sets det A = 0. SOI
[A1]
[M1 A1] Finds one correct entry.
[A1]
[A1]
[M1 A1] Finds BA.
[M1] Sets C as 3x2 matrix.
[A1] Accept any a, b, c, d, e and f such that beginpmatrix a+2b+3c & d+2e+3f 4a+kb+6c & 4d+ke+6f endpmatrix = beginpmatrix 2 & 1 k & 4 endpmatrix
[B1] Transforms beginpmatrix x y endpmatrix to beginpmatrix X Y endpmatrix
[M1 A1] Uses y = mx and Y = mX.
[A1]
[M1] Sets discriminant positive.
[A1] k>-1 k≠8

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Topic: Matrices, A-Level Further Mathematics (syllabus 9231). Structured question, 17 marks.
Source: A-Level Further Mathematics, May/June 2022 Paper 1 Variant 2, question 7.

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