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) 355352/5
OCR is an exempt Charity
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*1869096913*
4
© OCR 2025
Y420/01 Jun25
2
+
Section B (111 marks)
7 In this question you must show detailed reasoning.
By first expressing 1 in partial fractions, show that
y
3
1
dx = 1 ln n , where m and n
x
2
-
4
3
x
2
-
4
m
are integers to be determined. [8]
8
The function f (x) is defined as f (x) = ln(1 + x), for x
2
-
1.
(a) Prove by mathematical induction that the nth derivative of f(x), f (n)(x), for all n H 1, is given
by f
(n)
(x)
=
(-
1)
n+1
(n
-
1)!
(1 + x)
n . [4]
(b) Hence prove that ln(1 + x) = x - x x3
2 3
-
... +
(-
1)
n+1
x
n
n
+ ... for
-
1
1
x
G
1.
[You are not required to show this series for ln(1 + x) converges for -1
1
x
G
1.]
[3]
5
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© OCR 2025
Y420/01 Jun25
5
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
= k
r
, 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
2 2
0
x
2
-
x + 1
dx. Give your answer in exact form. [4]
B
C
O
A
l
D
E
y
9
© OCR 2025
Y420/01 Jun25
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
-0.2t
).
[4]
(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.
d2h dh
dt
2
+ 0.3 dt + 0.02h = 0.4
(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
10
© OCR 2025
Y420/01 Jun25
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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.
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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
3
MARKING INSTRUCTIONS
PREPARATION FOR MARKING
RM ASSESSOR
1. 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.
2. Make sure that you have read and understood the mark scheme and the question paper for this unit. These are available in RM Assessor
3. Log-in to RM Assessor and mark the required number of practice responses (“scripts”) and the required number of standardisation
responses.
MARKING
1. Mark strictly to the mark scheme.
2. Marks awarded must relate directly to the marking criteria.
3. 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.
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messaging system.
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alternative response has been provided, examiners may give candidates the benefit of the doubt and mark the crossed-out response where
legible.
