A-LevelFurther MathematicsMatrices9231/12 · Oct/Nov 2020 · Q413 Marks

The matrices A and B are given by A = (0 1; 1 0) and B = (1/2 -√3/2; √3/2 1/2)

(a)Give full details of the geometrical transformation in the x-y plane represented by A.[1]
(b)Give full details of the geometrical transformation in the x-y plane represented by B.[2]
(c)The triangle DEF in the x-y plane is transformed by AB onto triangle PQR. Show that the triangles DEF and PQR have the same area.[3]
(d)Find the matrix which transforms triangle PQR onto triangle DEF.[2]
(e)Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by AB.[5]

Mark scheme

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

[B1] Only mention reflection (and no other transformation). Reflection in the line y = x.
[B1] Writes 'rotation' or 'rotate' (and no other transformation) Rotation
[B1] Or 60° (π/3 anticlockwise about the origin).
[M1] Finds AB or uses product of determinants. Full marks here may be obtained by arguing that both reflection and rotation preserve [absolute] value of area and so also does their combination.
[B1] det AB = -1
[A1] Area of PQR = |-1| Area of DEF
[M1 A1] or using (AB)-1 = B-1A-1 (AB)^(-1) = (√3/2 1/2; 1/2 -√3/2)
[B1] Transforms (x;y) to (x';y'). Accept t (x;y) instead of (x;y)
[M1A1] Uses Y = mX . OE. Allow working with y = mx +c as long as c = 0 is stated explicitly. ½-½m√3=½m√3+½m²
[A1] 1-m√3=m√3+ m² ⇒ m² + 2m√3−1=0 ⇒ m = ±2-√3
[A1] OF y = (2-√3)x and y + (2+√3)x = 0

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