Question 8 - A-Level Maths

OCR MEI Further Pure Core AS 2022 June — Question 8 12 marks

Exam Board	OCR MEI
Module	Further Pure Core AS (Further Pure Core AS)
Year	2022
Session	June
Marks	12
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear transformations
Type	Decompose matrix into transformation sequence
Difficulty	Standard +0.8 This is a multi-part Further Maths question requiring determinant calculation, eigenvalue/invariant line analysis, and matrix decomposition into a sequence of transformations. While each individual part uses standard techniques (determinant for area, solving for invariant lines, matrix multiplication), the combination and the conceptual understanding needed to decompose M as MN^(-1) = (rotation)(shear)^(-1) requires solid matrix transformation knowledge beyond standard A-level.
Spec	4.03d Linear transformations 2D: reflection, rotation, enlargement, shear 4.03g Invariant points and lines 4.03h Determinant 2x2: calculation 4.03i Determinant: area scale factor and orientation 4.03o Inverse 3x3 matrix

8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta \\ \sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).

Show that T leaves areas unchanged for all values of \(\theta\).
Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T . The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
1. Find \(\mathbf { M N } ^ { - 1 }\).
2. Hence describe fully a sequence of two transformations of the plane that is equivalent to T . \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
  If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
  For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
  OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.

Show mark scheme Show mark scheme source

Question 8:

Answer	Marks	Guidance
8	(a)	det M = cos (2sin + cos) − sin (2cos −sin)
= cos2 + sin2 = 1 so T preserves area	M1

A1

Answer	Marks
[2]	2.1

2.2a

Answer	Marks	Guidance
8	(b)	c o s 2 c o s s in 0 0     −     

=

s in 2 s in c o s y y    + 

 (2cos  − sin  )y = 0 [for all y]

 2cos  = sin 

 tan  = 2

Answer	Marks
 = 1.11 rads	M1

A1

M1

A1

Answer	Marks
[4]	3.1a

1.1 3.1a

Answer	Marks
3.2a	c o s 2 c o s s in 0 0     −     

condone = for this M1…

s in 2 s in c o s y y    +

… but not this A1

solved using tan = sin/cos

cao 1.11 or better condone 63.4

Answer	Marks	Guidance
8	(c)	(i)

N−1 =

0 1

cos −sin

M N−1 =

 

Answer	Marks
sin cos	B1

B1

Answer	Marks
[2]	1.1

1.1

Answer	Marks	Guidance
8	(c)	(ii)

with x-axis fixed mapping (0,1) to (2,1)

followed by

a rotation

Answer	Marks
[about O] anticlockwise through 	M1

A1

M1

A1

Answer	Marks
[4]	2.2a

1.1 2.2a

Answer	Marks
1.1	not ‘scale factor 2’ or ‘shear factor 2’

Maximum 3 out of 4 if order is wrong

PMT

Need to get in touch?

If you ever have any questions about OCR qualifications or services (including administration, logistics and teaching) please feel free to get in

touch with our customer support centre.

Call us on

01223 553998

Alternatively, you can email us on

support@ocr.org.uk

For more information visit

ocr.org.uk/qualifications/resource-finder

ocr.org.uk

Twitter/ocrexams

/ocrexams

/company/ocr

/ocrexams

OCR is part of Cambridge University Press & Assessment, a department of the University of Cambridge.

2022 Oxford Cambridge and RSA Examinations is a Company Limited by Guarantee. Registered in England. Registered office

The Triangle Building, Shaftesbury Road, Cambridge, CB2 8EA.

Registered company number 3484466. OCR is an exempt charity.

OCR operates academic and vocational qualifications regulated by Ofqual, Qualifications Wales and CCEA as listed in their

qualifications registers including A Levels, GCSEs, Cambridge Technicals and Cambridge Nationals.

OCR provides resources to help you deliver our qualifications. These resources do not represent any particular teaching method

we expect you to use. We update our resources regularly and aim to make sure content is accurate but please check the OCR

website so that you have the most up-to-date version. OCR cannot be held responsible for any errors or omissions in these

resources.

Though we make every effort to check our resources, there may be contradictions between published support and the

specification, so it is important that you always use information in the latest specification. We indicate any specification changes

within the document itself, change the version number and provide a summary of the changes. If you do notice a discrepancy

between the specification and a resource, please contact us.

Whether you already offer OCR qualifications, are new to OCR or are thinking about switching, you can request more

information using our Expression of Interest form.

Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.

Question 8:
8 | (a) | det M = cos (2sin + cos) − sin (2cos −sin)
= cos2 + sin2 = 1 so T preserves area | M1
A1
[2] | 2.1
2.2a
8 | (b) | c o s 2 c o s s in 0 0     −     
=
s in 2 s in c o s y y    + 
 (2cos  − sin  )y = 0 [for all y]
 2cos  = sin 
 tan  = 2
 = 1.11 rads | M1
A1
M1
A1
[4] | 3.1a
1.1
3.1a
3.2a | c o s 2 c o s s in 0 0     −     
condone = for this M1…
s in 2 s in c o s y y    +
… but not this A1
solved using tan = sin/cos
cao 1.11 or better condone 63.4
8 | (c) | (i) |  1 − 2 
N−1 =
0 1
cos −sin
M N−1 =
 
sin cos | B1
B1
[2] | 1.1
1.1
8 | (c) | (ii) | T is a shear
with x-axis fixed mapping (0,1) to (2,1)
followed by
a rotation
[about O] anticlockwise through  | M1
A1
M1
A1
[4] | 2.2a
1.1
2.2a
1.1 | not ‘scale factor 2’ or ‘shear factor 2’
Maximum 3 out of 4 if order is wrong
PMT
Need to get in touch?
If you ever have any questions about OCR qualifications or services (including administration, logistics and teaching) please feel free to get in
touch with our customer support centre.
Call us on
01223 553998
Alternatively, you can email us on
support@ocr.org.uk
For more information visit
ocr.org.uk/qualifications/resource-finder
ocr.org.uk
Twitter/ocrexams
/ocrexams
/company/ocr
/ocrexams
OCR is part of Cambridge University Press & Assessment, a department of the University of Cambridge.
For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored. © OCR
2022 Oxford Cambridge and RSA Examinations is a Company Limited by Guarantee. Registered in England. Registered office
The Triangle Building, Shaftesbury Road, Cambridge, CB2 8EA.
Registered company number 3484466. OCR is an exempt charity.
OCR operates academic and vocational qualifications regulated by Ofqual, Qualifications Wales and CCEA as listed in their
qualifications registers including A Levels, GCSEs, Cambridge Technicals and Cambridge Nationals.
OCR provides resources to help you deliver our qualifications. These resources do not represent any particular teaching method
we expect you to use. We update our resources regularly and aim to make sure content is accurate but please check the OCR
website so that you have the most up-to-date version. OCR cannot be held responsible for any errors or omissions in these
resources.
Though we make every effort to check our resources, there may be contradictions between published support and the
specification, so it is important that you always use information in the latest specification. We indicate any specification changes
within the document itself, change the version number and provide a summary of the changes. If you do notice a discrepancy
between the specification and a resource, please contact us.
Whether you already offer OCR qualifications, are new to OCR or are thinking about switching, you can request more
information using our Expression of Interest form.
Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.

Show LaTeX source

8 A transformation T of the plane has matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta \\ \sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Show that T leaves areas unchanged for all values of $\theta$.
\item Find the value of $\theta$, where $0 < \theta < \frac { 1 } { 2 } \pi$, for which the $y$-axis is an invariant line of T .

The matrix $\mathbf { N }$ is $\left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)$.
\item \begin{enumerate}[label=(\roman*)]
\item Find $\mathbf { M N } ^ { - 1 }$.
\item Hence describe fully a sequence of two transformations of the plane that is equivalent to T .

\section*{END OF QUESTION PAPER}
}{www.ocr.org.uk}) after the live examination series.\\
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.\\
For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.\\
OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2022 Q8 [12]}}

This paper (8 questions)

View full paper

Q1 6 Q2 7 Q3 5 Q4 6 Q5 5 Q6 10 Q7 9 Q8 12