A2 June 2024 Paper 1 Q8

OCR MEICurrent spec10 marksInduction Matrices

(a) Specify fully the transformation T of the plane associated with the matrix \(\mathbf{M}\), where \(\mathbf{M} = \begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}\) and \(\lambda\) is a non-zero constant. [2]

(b)

(i) Find \(\det\mathbf{M}\). [1]

(ii) Deduce two properties of the transformation T from the value of \(\det\mathbf{M}\). [2]

(c) Prove that \(\mathbf{M}^n = \begin{pmatrix} 1 & n\lambda \\ 0 & 1 \end{pmatrix}\), where \(n\) is a positive integer. [4]

(d) Hence specify fully a single transformation which is equivalent to \(n\) applications of the transformation T. [1]

Mark scheme (a)

Scheme	Marks	AO
Shear	M1	1.2
with \(x\)-axis fixed, mapping (0, 1) to (\(\lambda\), 1)	A1	1.1
	[2]

Notes

M1: Do not accept “sheaf”

A1: Accept \(x\)-axis is a line of invariant points (not an invariant line only). “Shear parallel to \(x\)-axis” is insufficient.
Accept alternative mappings, e.g. \((1, 1)\) to \((1 + \lambda, 1)\).
Do not accept shear factor.

Mark scheme (b)

Scheme	Marks	AO
(i) 1	B1	1.1
	[1]
(ii) Preserves area	B1	1.2
Preserves orientation	B1	1.2
	[2]

Notes

(b)(ii)

B1: FT their determinant. Condone area scale factor = 1.

B1: FT their determinant. Do not accept “orientation not reversed”.

Mark scheme (c)

Scheme	Marks	AO
When \(n = 1\) \(\left[\mathbf{M}^1 =\right] \begin{pmatrix} 1 & 1 \times \lambda \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}\) so true when \(n = 1\)	B1	2.1
[Assume result holds for \(n = k\)] \(\left[\mathbf{M}^{k+1} =\right] \begin{pmatrix} 1 & k\lambda \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}\)	M1	2.1
\(\left[\mathbf{M}^{k+1} =\right] \begin{pmatrix} 1 & \lambda + k\lambda \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & \lambda(1 + k) \\ 0 & 1 \end{pmatrix}\)	A1*	2.2a
So true for \(n = 1\) and if true for \(n = k\) then true for \(n = k + 1\), so true for all \(n\)	A1dep	2.4
	[4]

Notes

B1: \(1 \times \lambda\) must be seen. “True when \(n = 1\)” could appear later.

M1: \(\mathbf{M}^{k+1} = \mathbf{M}^k\mathbf{M}\) or \(\mathbf{M}^{k+1} = \mathbf{M}\mathbf{M}^k\) could be used.

A1*: Required matrix with intermediate step seen

A1dep: \(n = 1\) must have been considered

Mark scheme (d)

Scheme	Marks	AO
A shear with \(x\)-axis fixed, mapping (0,1) to (\(n\lambda\), 1)	B1	1.1
	[1]

Notes

B1: Accept \(x\)-axis is a line of invariant points (not an invariant line only).
Accept alternative mappings, e.g. \((1, 1)\) to \((1 + n\lambda, 1)\).
Do not accept shear factor