A2 October 2021 Paper 1 Q9

OCR MEICurrent spec11 marksMatrices

9 The transformation T of the plane has associated matrix \(\mathbf{M}\), where \(\mathbf{M} = \begin{pmatrix} -1 & 0 \\ -2 & 1 \end{pmatrix}\).

(a) On the grid below, plot the image OA′B′C′ of the unit square OABC under the transformation T.

Grid from the Printed Answer Booklet: x and y from -4 to 4, showing the unit square OABC with O at the origin, A(1, 0), B(1, 1) and C(0, 1)

[2]

(b)

(i) Calculate the value of \(\det\mathbf{M}\). [1]

(ii) Explain the significance of the value of \(\det\mathbf{M}\) in relation to the image OA′B′C′. [2]

(i) Specify fully two transformations equivalent to T. [3]

(ii) Use matrices to verify your answer. [3]

Grid showing the unit square OABC dashed and its image: a parallelogram with vertices O, A'(-1, -2), B'(-1, -1) and C'(0, 1), where C' coincides with C — Mark scheme (a)

Scheme	Marks	AO
	M1 A1	1.1 1.1
	[2]

Scheme	Marks	AO
(i) \(\det\mathbf{M} = -1 \times 1 - 0 \times (-2) = -1\)	B1	1.1
	[1]
(ii) area is preserved orientation is reversed	B1 B1	1.1 1.1
	[2]

Mark scheme (c)

Scheme	Marks	AO
(i) Reflection in \(y\) axis then shear invariant line \(y\)-axis, mapping \((-1, 0)\) to \((-1, -2)\)	B1 M1 A1	3.1a 3.1a 1.1
	[3]
(ii) Reflection: \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)	B1	1.1
Shear: \(\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}\)	B1	1.1
\(\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ -2 & 1 \end{pmatrix}\)	B1	2.2a
	[3]

Notes

(c)(i)

A1: oe, e.g. mapping \((1, 0)\) to \((1, 2)\)

Alternatively

Scheme	Marks
Shear invariant line \(y\)-axis, mapping \((1, 0)\) to \((1, -2)\) then reflection in \(y\)-axis	M1 A1 B1
	[3]

(c)(ii)

Or shear \(\begin{pmatrix} 1 & 0 \\ -2 & 1 \end{pmatrix}\) first
then reflection \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
\(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ -2 & 1 \end{pmatrix}\)

B1: Must match the order described in ci for final mark