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2025 OCR A Level Further Mathematics B (MEI)
Y435/01 Extra Pure
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Friday 20 June 2025 – Afternoon A
Level Further Mathematics B (MEI) Y435/01
Extra Pure
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer Booklet. If
you need extra space use the lined pages at the end of the Printed Answer Booklet. The
question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be given
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [F/508/5599] OCR is an exempt Charity
DC (SL) 351624/2 Turn over
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2
1 A binary operation, ○, is defined on real numbers a and b by a ○ b = 3ab.
(a) Prove that ○ is associative over the real numbers R. [2]
(b) Determine the value of e, the identity element for ○. [2]
(c) Explain why R is closed under ○. [1]
(d) Explain why R under ○ does not form a group. Justify your answer. [2]
2 (a) Determine the general solution of the recurrence relation 25un +2 - 35un +1 + 12un = c, n H 0,
where c is a constant. [5]
The manager of a company is using the recurrence relation in part (a) to model the number of
containers held by the company using the sequence u0, u1, u2, … which satisfies this recurrence
relation.
The quantity u0 represents the number of containers held by the company at the start of
January 2000. The quantity un , for n H 1, represents the number of containers held by the
company at the end of month n, where the first month is January 2000. If necessary, the quantity
un is rounded to the nearest integer.
The company holds 20 containers at the start of January 2000 and 70 containers at the end of
January 2000.
(b) In an initial model the manager uses c = –10.
(i) Determine the particular solution for un when c = –10. [3]
(ii) The model predicts a single peak value for un .
Find, by direct calculation, the largest number of containers held by the company
according to the model. [2]
(iii) Show that the model in part (b)(i) breaks down in the 19th month. [3]
(c) In a second model the manager uses c = 10 instead of c = –10.
Show that the second model predicts that eventually the number of containers held by the
company will be a constant k, where k is to be determined. [2]
© OCR 2025 Y435/01 Jun25
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3
3 M is the group ({0, 1, 2, 3, 4, 5, 6, 7, 8}, + 9 ) where + 9 denotes the binary operation of addition
modulo 9.
(a) Write down the identity element of M. Justify your answer. [1]
(b) By considering the order of M, explain why there can be no subgroup of M of order 5. [2]
You are given that M is a cyclic group.
(c) Find all possible generators of M. [2]
You are given that there is only one proper, non-trivial subgroup of M, denoted by (H, + 9 ).
(d) (i) Write down H. [1]
(ii) Show that there can be no other proper, non-trivial subgroups of M. [2]
4 A surface, S, is defined in 3-D by z = f(x, y) where f(x, y) = x3 - 12xy2 + 96y2 + 30.
You are given that the point (0, 0, 30) on S is a stationary point.
(a) Determine the coordinates of any other stationary points on S. [6]
(b) By sketching the section z = f(x, 0) on the coordinate axes provided in the Printed Answer
Booklet, determine the nature of the stationary point (0, 0, 30) on S. [2]
The normal to S at the point where x = 3 and y = 2 passes through the point (a, 482, 295) where
a is a constant.
(c) Determine the value of a. [5]
The contour given by z = 542 comprises a straight line and a curve. The curved portion of the
contour is denoted by C. You are given that there are two points on C where the x coordinate is 8.
(d) Determine the values of the y coordinates at these two points on C. [3]
Turn over for question 5
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4
5 In this question you must show detailed reasoning.
K J a b 0N
The matrix A is given by
- b a 0OO where a and b are constants with 2a + b2 = 1 and a ! !1.
K
K O
L 0 0 1P
(a) (i) By finding the characteristic equation of A in terms of a, show that, for any value of a,
1 is always one of the eigenvalues of A. [3]
(ii) By showing that the other eigenvalues of A are a ! a2 - 1 , prove that the other
eigenvalues of A are not 1. [3]
(b) Hence show that either some of the eigenvalues of A must be non-real or some of the
elements of A must be non-real. [3]
You are now given the following information.
• a and b are both real.
• A represents a rotation by an acute angle in 3-D space.
• e is a real eigenvector of A associated with the eigenvalue 1.
(c) (i) Explain the significance of e, and the fact that its associated eigenvalue is 1, in relation
to the transformation that A represents. [2]
(ii) Explain geometrically why the other eigenvectors of A can not be real. [1]
(iii) Find a vector equation for the axis of rotation of the transformation that A represents.
[2]
END OF QUESTION PAPER
Oxford Cambridge and RSA
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used
in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is
produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
© OCR 2025 Y435/01 Jun25
2025 OCR
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2025 OCR A Level Further Mathematics B (MEI)
Y435/01 Extra Pure
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Friday 20 June 2025 – Afternoon A
Level Further Mathematics B (MEI) Y435/01
Extra Pure
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a scientific or graphical calculator
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer Booklet. If
you need extra space use the lined pages at the end of the Printed Answer Booklet. The
question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be given
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• The marks for each question are shown in brackets [ ].
• This document has 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [F/508/5599] OCR is an exempt Charity
DC (SL) 351624/2 Turn over
2025 OCR
Download
A Levelthe
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2
1 A binary operation, ○, is defined on real numbers a and b by a ○ b = 3ab.
(a) Prove that ○ is associative over the real numbers R. [2]
(b) Determine the value of e, the identity element for ○. [2]
(c) Explain why R is closed under ○. [1]
(d) Explain why R under ○ does not form a group. Justify your answer. [2]
2 (a) Determine the general solution of the recurrence relation 25un +2 - 35un +1 + 12un = c, n H 0,
where c is a constant. [5]
The manager of a company is using the recurrence relation in part (a) to model the number of
containers held by the company using the sequence u0, u1, u2, … which satisfies this recurrence
relation.
The quantity u0 represents the number of containers held by the company at the start of
January 2000. The quantity un , for n H 1, represents the number of containers held by the
company at the end of month n, where the first month is January 2000. If necessary, the quantity
un is rounded to the nearest integer.
The company holds 20 containers at the start of January 2000 and 70 containers at the end of
January 2000.
(b) In an initial model the manager uses c = –10.
(i) Determine the particular solution for un when c = –10. [3]
(ii) The model predicts a single peak value for un .
Find, by direct calculation, the largest number of containers held by the company
according to the model. [2]
(iii) Show that the model in part (b)(i) breaks down in the 19th month. [3]
(c) In a second model the manager uses c = 10 instead of c = –10.
Show that the second model predicts that eventually the number of containers held by the
company will be a constant k, where k is to be determined. [2]
© OCR 2025 Y435/01 Jun25
2025 OCR
Download
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3
3 M is the group ({0, 1, 2, 3, 4, 5, 6, 7, 8}, + 9 ) where + 9 denotes the binary operation of addition
modulo 9.
(a) Write down the identity element of M. Justify your answer. [1]
(b) By considering the order of M, explain why there can be no subgroup of M of order 5. [2]
You are given that M is a cyclic group.
(c) Find all possible generators of M. [2]
You are given that there is only one proper, non-trivial subgroup of M, denoted by (H, + 9 ).
(d) (i) Write down H. [1]
(ii) Show that there can be no other proper, non-trivial subgroups of M. [2]
4 A surface, S, is defined in 3-D by z = f(x, y) where f(x, y) = x3 - 12xy2 + 96y2 + 30.
You are given that the point (0, 0, 30) on S is a stationary point.
(a) Determine the coordinates of any other stationary points on S. [6]
(b) By sketching the section z = f(x, 0) on the coordinate axes provided in the Printed Answer
Booklet, determine the nature of the stationary point (0, 0, 30) on S. [2]
The normal to S at the point where x = 3 and y = 2 passes through the point (a, 482, 295) where
a is a constant.
(c) Determine the value of a. [5]
The contour given by z = 542 comprises a straight line and a curve. The curved portion of the
contour is denoted by C. You are given that there are two points on C where the x coordinate is 8.
(d) Determine the values of the y coordinates at these two points on C. [3]
Turn over for question 5
© OCR 2025 Y435/01 Jun25 Turn over
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4
5 In this question you must show detailed reasoning.
K J a b 0N
The matrix A is given by
- b a 0OO where a and b are constants with 2a + b2 = 1 and a ! !1.
K
K O
L 0 0 1P
(a) (i) By finding the characteristic equation of A in terms of a, show that, for any value of a,
1 is always one of the eigenvalues of A. [3]
(ii) By showing that the other eigenvalues of A are a ! a2 - 1 , prove that the other
eigenvalues of A are not 1. [3]
(b) Hence show that either some of the eigenvalues of A must be non-real or some of the
elements of A must be non-real. [3]
You are now given the following information.
• a and b are both real.
• A represents a rotation by an acute angle in 3-D space.
• e is a real eigenvector of A associated with the eigenvalue 1.
(c) (i) Explain the significance of e, and the fact that its associated eigenvalue is 1, in relation
to the transformation that A represents. [2]
(ii) Explain geometrically why the other eigenvectors of A can not be real. [1]
(iii) Find a vector equation for the axis of rotation of the transformation that A represents.
[2]
END OF QUESTION PAPER
Oxford Cambridge and RSA
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used
in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is
produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
© OCR 2025 Y435/01 Jun25
2025 OCR
Download
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