A-LevelFurther MathematicsMatrices9231/12 · May/June 2023 · Q412 Marks
The matrix M is given by M = ((a, b^2), (c^2, a)), where a, b, c are real constants and b ≠ 0.
- (a)Show that M does not represent a rotation about the origin.[2]
- (b)Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by M.[5]
- (c)It is given that M represents the sequence of two transformations in the x-y plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, x-axis fixed, with (0,1) mapped to (5, 1). Find M.[3]
- (d)The triangle DEF in the x−y plane is transformed by M onto triangle PQR. Given that the area of triangle DEF is 12 cm², find the area of triangle PQR.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.
- M1 Uses correct matrix for rotation.
- A1 AG
- B1 Transforms (x, y) to (X, Y).
- M1 A1 Uses y = mx and Y = mX .
- A1 c^2 + am = ma + b^2 m^2 ⇒ c^2 = b^2 m^2
- A1 y = (c/b)x and y = -(c/b)x
- M1 A1* Award M1 if matrices correct but order is wrong.
- DA1 Dep: previous A1
- M1 Using their M.
- A1FT
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