Download 2025 OCR A Level Further Mathematics B (MEI) Y420/01 Core Pure May QUESTION PAPER and MARK and more Exams Mathematics in PDF only on Docsity!
2025 OCR A Level Further Mathematics B (MEI) Y420/01 Core Pure May
QUESTION PAPER and MARK SCHEME MERGED
Oxford Cambridge and RSA
Thursday 22 May 2025 – Afternoon
A Level Further Mathematics B (MEI)
Y420/01 Core Pure
Time allowed: 2 hours 40 minutes
You must have:
- the Printed Answer Booklet
- the Formulae Booklet for Further Mathematics B
(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 144.
- The marks for each question are shown in brackets [ ].
- This document has 12 pages.
ADVICE
- Read each question carefully before you start your answer.
QP
© OCR 2025 [Y/508/5592]
DC (DE/SW) 355352/
OCR is an exempt Charity
Turn over
Section A (33 marks)
© OCR 2025 Y420/01 Jun
1 The complex number z satisfies the equation z + 2i z * +1 - 4i = 0.
You are given that z = x +i y , where x and y are real numbers.
Determine the values of x and y. [4]
2 In this question you must show detailed reasoning.
Find the acute angle between the planes 2 x - y + 2 z = 5 and x + 2 y + z = 8. [4]
3 Using standard summation formulae, show that, for integers n H 1,
1 # 3 + 2 # 4 + ... + n # ( n + 2) =
n ( n + 1)( an + b ),
where a and b are integers to be determined. [5]
4 (a) You are given that M and N are non-singular 2 # 2 matrices.
Write down the product rule for the inverse matrices of M , N and MN. [1]
(b) Verify this rule for the matrices M and N , where
J
K
a 1 N O K
J 0 - 1
N
O
M = K
O and^ N^ =^ K
1
O and^ a^ and^ b^ are^ non-zero^ constants.^ [6]
L P L
b
P
5 The cubic equation 2 x
3
- 3 x + 4 = 0 has roots a , b and c.
Determine a cubic equation with integer coefficients whose roots are
1 ( a + 1),
1 ( b + 1) and
1 ( c + 1).
2 2 2
[4]
© OCR 2025 Y420/01 Jun25 Turn over
6 The figure below shows the curve with cartesian equation ( x
2
2
)
2
= xy.
y
x
(a) Show that the polar equation of the curve is r
2
= a sin bi , where a and b are positive
constants to be determined. [3]
(b) Determine the exact maximum value of r. [2]
(c) Determine the area enclosed by one of the loops. [4]
O
© OCR 2025 Y420/01 Jun25 Turn over
9 The figure below shows an Argand diagram with a regular pentagon ABCDE. The point A
represents the real number 1. The point B represents the complex number w.
Im
Re
(a) (i) Write down, in terms of w , the complex numbers represented by the points C, D and E.
[1]
(ii) Write down an equation whose roots are the complex numbers represented by the points
A, B, C, D and E. [1]
(iii) Show that the sum of these roots is zero. [2]
(b) (i) Find w. Give your answer in the form r (cos i +i sin i ), where r 2 0 and i = kr , where
k is a positive constant to be found. [1]
(ii) By considering the line segment AB, show that the length of each side of the pentagon is
2 sin
r
. [5]
10 In this question you must show detailed reasoning.
Evaluate
1
0 x
2
d x. Give your answer in exact form. [4]
B
C
O
A
l
D
E
y
© OCR 2025 Y420/01 Jun
a
d x
11 The lines l 1
and l 2
have equations
l :
x - 3
y + 1
z - 2 l : r = 2 i + j + 3 k + m (- i + c j + 2 k )
(^1) a b 1
2
where a , b and c are constants.
(a) In the case where c = 0 and l 1
and l 2
intersect at right angles, determine the coordinates of
the point of intersection of the two lines. [6]
(b) Now consider the case where c = 4 and lines l 1
and l 2
are parallel.
(i) Find the values of a and b. [3]
(ii) Determine the distance between the two lines. [5]
12 In this question you must show detailed reasoning.
The roots of the equation z
4
3
2
, b and - b.
Determine, in any order, the exact values of the following.
- The four roots of the equation
- The value of c
- The value of d [8]
13 The gradient of a curve y = f ( x ) satisfies the differential equation x
d y
2
.
(a) Show that the integrating factor for this differential equation is x
(b) You are given that the curve passes through the point (1, 0).
By solving this differential equation, determine the exact x - coordinate of the stationary point
on the curve y = f ( x ). [8]
© OCR 2025 Y420/01 Jun
16 In this question you must show detailed reasoning.
The diagram shows the curve with equation y =
x + 3
y
x
The region R, shown shaded in the diagram, is bounded by the curve, the x - axis, the y - axis, and
the line x = 4.
(a) Determine the area of R. Give your answer in the form p +ln q where p and q are integers to
be determined. [6]
The region R is rotated through 2 r radians about the x - axis.
(b) Determine the volume of the solid of revolution formed. Give your answer in the form
J K
J
K
c N O
N
O
r K a + b lnK
d
OO where a , b , c and d are integers to be determined. [6]
L L PP
R
O
x = 4
© OCR 2025 Y420/01 Jun
17 A researcher is modelling the height of a particular type of tree over its lifetime.
Data suggests that the maximum possible height of this type of tree over its lifetime is double the
height of the tree 5 years after planting.
It is given that, t years after planting a seed for this type of tree, the corresponding height of the
tree is h m, and that h = 0 when t = 0.
(a) The researcher first models the height of the tree by assuming that the rate of increase of h is
proportional to (20 - h ), with constant of proportionality 0.2.
(i) Write down the first order differential equation for this model. [1]
(ii) Show that this model predicts that the maximum possible height of the tree is 20 m. [1]
(iii) Show by integration that h = 20 (1 - e
(iv) Determine whether this model’s prediction for the height of the tree 5 years after
planting is consistent with the maximum possible height of the tree being 20 m. [2]
(b) The researcher refines the model for the height of the tree using the following second order
differential equation.
d
2
h d h
d t
2
d t
(i) Determine the general solution of this second order differential equation. [4]
(ii) Show that the refined model also predicts that the maximum possible height of the tree
is 20 m. [1]
Further research determines that the initial rate of growth of this type of tree is 2.9 metres per
year.
(iii) By applying the initial conditions to find the particular solution of this differential
equation, determine whether the refined model’s prediction for the height of the tree
5 years after planting is consistent with the maximum possible height of the tree
being 20 m. [5]
END OF QUESTION PAPER
© OCR 2025 Y420/01 Jun
BLANK PAGE
© OCR 2025 Y420/01 Jun
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.
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.
OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of
qualifications to meet the needs of candidates of all ages and abilities. OCR qualifications
include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals,
Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in
areas such as IT, business, languages, teaching/training, administration and secretarial skills.
It is also responsible for developing new specifications to meet national requirements and the
needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is
invested back into the establishment to help towards the development of qualifications and
support, which keep pace with the changing needs of today’s society.
This mark scheme is published as an aid to teachers and students, to indicate the requirements
of the examination. It shows the basis on which marks were awarded by examiners. It does not
indicate the details of the discussions which took place at an examiners’ meeting before marking
commenced.
All examiners are instructed that alternative correct answers and unexpected approaches in
candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills
demonstrated.
Mark schemes should be read in conjunction with the published question papers and the report
on the examination.
© OCR 2025
Oxford Cambridge and RSA Examinations
Y420/01 Mark Scheme June 2025
MARKING INSTRUCTIONS
PREPARATION FOR MARKING
RM ASSESSOR
- Make sure that you have accessed and completed the relevant training packages for on-screen marking: RM Assessor Online Training:
OCR Essential Guide to Marking.
- Make sure that you have read and understood the mark scheme and the question paper for this unit. These are available in RM Assessor
- Log-in to RM Assessor and mark the required number of practice responses (“scripts”) and the required number of standardisation
responses.
MARKING
- Mark strictly to the mark scheme.
- Marks awarded must relate directly to the marking criteria.
- The schedule of dates is very important. It is essential that you meet the RM Assessor 50% and 100% (traditional 40% Batch 1 and 100%
Batch 2) deadlines. If you experience problems, you must contact your Team Leader (Supervisor) without delay.
- If you are in any doubt about applying the mark scheme, consult your Team Leader by telephone , email or via the RM Assessor
messaging system.
- Crossed-Out Responses
Where a candidate has crossed out a response and provided a clear alternative then the crossed-out response is not marked. Where no
alternative response has been provided, examiners may give candidates the benefit of the doubt and mark the crossed-out response where
legible.
Y420/01 Mark Scheme June 2025
Longer Answer Questions (requiring a developed response)
Where candidates have provided two (or more) responses to a medium or high tariff question which only required a single (deve loped)
response and not crossed out the first response, then only the first response should be marked. Examiners will need to apply professional
judgement as to whether the second (or a subsequent) response is a ‘new start’ or simply a poorly expressed continuation of t he first
response.
- Always check the pages (and additional objects if present) at the end of the response in case any answers have been continued there. If
the candidate has continued an answer there, then add the annotation ‘SEEN’ to confirm that the work has been seen and mark any
responses using the annotations in section 11.
- There is a NR ( No Response ) option. Award NR (No Response):
- if there is nothing written at all in the answer space
- OR if there is a comment which does not in any way relate to the question ( e.g., ‘can’t do’, ‘don’t know’)
- OR if there is a mark (e.g., a dash, a question mark) which is not an attempt at the question.
Note: Award 0 marks – for an attempt that earns no credit (including copying out the question).
- The RM Assessor comments box is used by your Team Leader to explain the marking of the practice responses. Please refer to these
comments when checking your practice responses. Do not use the comments box for any other reason.
- Assistant Examiners will send a brief report on the performance of candidates to their Team Leader (Supervisor) via email by the end of
the marking period. The report should contain notes on particular strengths displayed as well as common errors or weaknes ses.
Constructive criticism of the question paper/mark scheme is also appreciated.
- For answers marked by levels of response: Not applicable in F
To determine the level – start at the highest level and work down until you reach the level that matches the answer
To determine the mark within the level , consider the following
Y420/01 Mark Scheme June 2025
Descriptor Award mark
On the borderline of this level and the one
below
At bottom of level
Just enough achievement on balance for this
level
Above bottom and either below middle or at middle of level (depending on number of marks
available)
Meets the criteria but with some slight
inconsistency
Above middle and either below top of level or at middle of level (depending on number of
marks available)
Consistently meets the criteria for this level At top of level
