2025 OCR A Level Further Mathematics B (MEI)
Y420/01 Core Pure
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Thursday 22 May 2025 ʹ 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.
OCR is an exempt Charity
Turn over
QP
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Section A (33 marks)
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6
1 The complex number z satisfies the equation z + 2iz* +1 - 4i = 0 .
You are given that z = x +iy, 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 2x
-
y + 2z = 5 and x + 2y + z = 8.
[4]
3 Using standard summation formulae, show that, for integers n H
1,
1
#
3 + 2
#
4 + ... + n
#
(n + 2) = 1 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
JKa
1
N
O
KJ0
-
1NO
M = K0 1O and N = K1
O
and a and b are non-zero constants. [6]
L
P
L
bP
5 The cubic equation 2x3 - 3x + 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]
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a
dx
11 The lines l1 and l2 have equations
l : x
-
3 = y + 1 = z
-
2
l : r = 2i + j + 3k + m
(-
i + cj + 2k)
1 a b 1 2
where a, b and c are constants.
(a) In the case where c = 0 and l1 and l2 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 l1 and l2 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
-
z
3
+ cz
2
+ dz + 18 = 0 are
a
, 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 dy - 2y = 2 + x2.
(a) Show that the integrating factor for this differential equation is x-2. [3]
(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]
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© OCR 2025 Y420/01 Jun25
10
-
6 cos 4
14 (a) By using the definition of cosh x and sinh x in terms of ex and
e
-
x , show that
cosh
2
x + sinh
2
x
/
cosh 2x.
[2]
JKcosh
x sinh xNO
(b) The transformation T of the plane has associated matrix M, where M = Ksinh
x
cosh
O
and
x
2
0.
L
xP
Show that T transforms the unit square with coordinates (0, 0), (1, 0), (0, 1) and (1, 1) to a
rhombus of unit area. [6]
(c) You are given that the length of each side of the rhombus is 2 units.
Determine the exact value of x. Give your answer in logarithmic form. [2]
15 (a) Show that ^3 - e4iih^3 - e-4iih = a + b cos 4i, where a and b are integers to be determined. [2]
The infinite series C and S are defined as follows.
C
=
cos
i
+
1 cos
5
i
+
1 cos
9
i
+
1 cos
13
i
+
...
3 9 27
S
=
sin
i
+
1 sin
5
i
+
1 sin
9
i
+
1 sin
13
i
+
...
3 9 27
(b) Show that C iS 3eii . [4]
+
= 3
-
e
4ii
(c)
Hence show that C = 9 cos
i
- 3 cos 3
i
.
[3]
i
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© OCR 2025 Y420/01 Jun25
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Y420/01 Mark Scheme June 2025
4
Rubric Error Responses ± Optional Questions
Where candidates have a choice of question across a whole paper or a whole section and have provided more answers than requir ed, then
all responses are marked and the highest mark allowable within the rubric is given. Enter a mark for each question answered into RM
Assessor, which will select the highest mark from those awarded. (The underlying assumption is that the candidate has penalised
themselves by attempting more questions than necessary in the time allowed.)
Multiple-Choice Question Responses
When a multiple-choice question has only a single, correct response and a candidate provides two responses (even if one of these
responses is correct), then no mark should be awarded (as it is not possible to determine which was the first response select ed by the
candidate).
When a question requires candidates to select more than one option/multiple options, then local marking arrangements need to ensure
consistency of approach.
Contradictory Responses
When a candidate provides contradictory responses, then no mark should be awarded, even if one of the answers is correct.
Short Answer Questions (requiring only a list by way of a response, usually worth only one mark per response)
Where candidates are required to provide a set number of short answer responses then only the set number of responses should be
marked. The response space should be marked from left to right on each line and then line by line until the required number o f responses
have been considered. The remaining responses should not then be marked. Examiners will have to apply judgement as to whether a
µVHFRQG UHVSRQVH¶ on a line is a development of the µILUVW UHVSRQVH¶ rather than a separate, discrete response. (The underlying assumption
is that the candidate is attempting to hedge their bets and therefore getting undue benefit rather than engaging with the question and giving
the most relevant/correct responses.)
Short Answer Questions (requiring a more developed response, worth two or more marks)
If the candidates are required to provide a description of, say, three items or factors and four items or factors are provide d, then mark on a
similar basis ± that is downwards (as it is unlikely in this situation that a candidate will provide more than one response in each section of
the response space).
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Y420/01 Mark Scheme June 2025
5
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 D µQHZVWDUW¶ RUVLPSO\D SRRUO\H[SUHVVHG continuation of t he first
response.
6. 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 WKHDQQRWDWLRQµ6((1¶WRFRQILUPWKDWWKHZRUNKDVEHHQVHHQ and mark any
responses using the annotations in section 11.
7. 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., µFDQ¶W GR¶ µGRQ¶W NQRZ¶
- 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).
8. 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.
9. 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 .
10. For answers marked by levels of response: Not applicable in F501
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
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