A-LevelFurther MathematicsMatrices9231/12 · Oct/Nov 2023 · Q38 Marks

The matrix M is given by M = beginpmatrix k & 0 1 & 1 endpmatrix, where k is a constant and k ≠ 0 and k ≠ 1.

(a)The matrix M represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.[2]
(b)The unit square in the x−y plane is transformed by M onto parallelogram OPQR. Find, in terms of k, the area of parallelogram OPQR and the matrix which transforms OPQR onto the unit square.[3]
(c)Show that the line through the origin with gradient 1/(k-1) is invariant under the transformation in the x-y plane represented by M.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[B2] Shear followed by a stretch. Award B1 if given in the wrong order.
[B1] |OPQR| = |det M| = |k|
[M1 A1] M⁻¹ = 1/k beginpmatrix 1 & 0 -1 & k endpmatrix
[B1] Sets y = 1/(k-1)x.
[M1 A1] Shows that Y = 1/(k-1)X. (Using X=kx, Y=x+y, and substituting y=1/(k-1)x gives Y = x+x/(k-1) = x(1+1/(k-1)) = x(k/(k-1)). Since x=X/k, Y = X/k(k/(k-1)) = X/(k-1). Thus, Y=1/(k-1)X. AG.) OR (Alternative method: [B1] Transforms (x/y) to (X/Y) = (kx/x+y). [M1 A1] Uses y=mx and Y=mX. mkx=x+mx implies m(k-1)x=x implies m=1/(k-1). AG.)

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.