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 – 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
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 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 2025 [Y/508/5592] OCR is an exempt Charity
DC (DE/SW) 355352/5 Turn over
, 2
Section A (33 marks)
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, 1 # 3 + 2 # 4 + ... + n # (n + 2) = 6 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 a 1NO J0 -1NO
M = KK O and N = KK and a and b are non-zero constants. [6]
0 1 1 bO
L P L P
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]
© OCR 2025 Y420/01 Jun25
, 3
6 The figure below shows the curve with cartesian equation (x2 + y2) 2 = xy.
y
O x
(a) Show that the polar equation of the curve is r2 = 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]
© OCR 2025 Y420/01 Jun25 Turn over
, 4
Section B (111 marks)
7 In this question you must show detailed reasoning.
3
By first expressing
that
1
in partial fractions, show y 1
dx =
1
ln n , where m and n
x2 - 4 3 x2 - 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
(-1) n+1 (n - 1)!
by f (n)(x) = . [4]
(1 + x) n
x 2 x3 (-1) n+1xn
(b) Hence prove that ln(1 + x) = x - 2 + 3 - ... + + ... for -1 1 x G 1.
n
[You are not required to show this series for ln(1 + x) converges for -1 1 x G 1.] [3]
© OCR 2025 Y420/01 Jun25
Y420/01 Core Pure
Verified Question paper with Marking Scheme Attached
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
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 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 2025 [Y/508/5592] OCR is an exempt Charity
DC (DE/SW) 355352/5 Turn over
, 2
Section A (33 marks)
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, 1 # 3 + 2 # 4 + ... + n # (n + 2) = 6 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 a 1NO J0 -1NO
M = KK O and N = KK and a and b are non-zero constants. [6]
0 1 1 bO
L P L P
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]
© OCR 2025 Y420/01 Jun25
, 3
6 The figure below shows the curve with cartesian equation (x2 + y2) 2 = xy.
y
O x
(a) Show that the polar equation of the curve is r2 = 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]
© OCR 2025 Y420/01 Jun25 Turn over
, 4
Section B (111 marks)
7 In this question you must show detailed reasoning.
3
By first expressing
that
1
in partial fractions, show y 1
dx =
1
ln n , where m and n
x2 - 4 3 x2 - 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
(-1) n+1 (n - 1)!
by f (n)(x) = . [4]
(1 + x) n
x 2 x3 (-1) n+1xn
(b) Hence prove that ln(1 + x) = x - 2 + 3 - ... + + ... for -1 1 x G 1.
n
[You are not required to show this series for ln(1 + x) converges for -1 1 x G 1.] [3]
© OCR 2025 Y420/01 Jun25
