OCR AS Level Further Mathematics B Y410/01
Combined Question paper & Marking Scheme A+
May 2025
Oxford Cambridge and RSA
Monday 12 May 2025 – Afternoon AS
Level Further Mathematics B (MEI) Y410/01 Core
Pure
Time allowed: 1 hour 15 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 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.
Turn over
, 2
1 The matrices A and B are given by
J1 3N J2 1N
A =K O and B = K .
L0 1P L0 1O
P
(a) Use A and B to show that matrix multiplication is not, in general, commutative. [2]
(b) Verify that A and B satisfy (AB) –1 = B–1A–1. [3]
2 In this question you must show detailed reasoning.
Find the acute angle between the vector 3i + 2j - k and the normal vector to the plane
2x + 3y + z = 6. [4]
3 The matrices M and N are given by
Ja - bN Jb - aN
M =K O and N = K O where a and b are positive constants.
Lb aP La bP
(a) Given that M2 = N, determine the exact values of a and b. [4]
(b) Hence state the transformations of the plane associated with matrices M and N. [3]
© OCR 2025 Y410/01 Jun25
, 3
J1 - 2 2N
K O
4 (a) The transformation T is represented by the matrix M = 2 1 0 .
K O
L1 2 - 1P
A shape S1 is mapped to a shape S2 by the transformation T.
Show that volume of S1 is the same as the volume of S2. [2]
(b) Three planes have equations
x - 2y + 2z = m,
2x + y = 2,
x + 2y - z = 0,
where m is a constant.
(i) Explain why the three planes intersect at a point for any value of m. [2]
(ii) Use a matrix method to determine, in terms of m, the coordinates of this point. [4]
5 In this question you must show detailed reasoning.
The complex number w is given by w = – 4 2 +^4 2 h i.
(a) (i) Find w . [2]
(ii) Find arg (w). [2]
The complex numbers z1 and z2 are given by z1 = a + i and z2 = 4 (cos i +i sin i), where a is a
positive real constant and - r 1 i G r.
(b) You are given that z1 z2 = w .
(i) Find the exact value of a. [3]
(ii) Find the value of i. Give your answer as an exact multiple of r. [3]
© OCR 2025 Y410/01 Jun25 Turn over
Combined Question paper & Marking Scheme A+
May 2025
Oxford Cambridge and RSA
Monday 12 May 2025 – Afternoon AS
Level Further Mathematics B (MEI) Y410/01 Core
Pure
Time allowed: 1 hour 15 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 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.
Turn over
, 2
1 The matrices A and B are given by
J1 3N J2 1N
A =K O and B = K .
L0 1P L0 1O
P
(a) Use A and B to show that matrix multiplication is not, in general, commutative. [2]
(b) Verify that A and B satisfy (AB) –1 = B–1A–1. [3]
2 In this question you must show detailed reasoning.
Find the acute angle between the vector 3i + 2j - k and the normal vector to the plane
2x + 3y + z = 6. [4]
3 The matrices M and N are given by
Ja - bN Jb - aN
M =K O and N = K O where a and b are positive constants.
Lb aP La bP
(a) Given that M2 = N, determine the exact values of a and b. [4]
(b) Hence state the transformations of the plane associated with matrices M and N. [3]
© OCR 2025 Y410/01 Jun25
, 3
J1 - 2 2N
K O
4 (a) The transformation T is represented by the matrix M = 2 1 0 .
K O
L1 2 - 1P
A shape S1 is mapped to a shape S2 by the transformation T.
Show that volume of S1 is the same as the volume of S2. [2]
(b) Three planes have equations
x - 2y + 2z = m,
2x + y = 2,
x + 2y - z = 0,
where m is a constant.
(i) Explain why the three planes intersect at a point for any value of m. [2]
(ii) Use a matrix method to determine, in terms of m, the coordinates of this point. [4]
5 In this question you must show detailed reasoning.
The complex number w is given by w = – 4 2 +^4 2 h i.
(a) (i) Find w . [2]
(ii) Find arg (w). [2]
The complex numbers z1 and z2 are given by z1 = a + i and z2 = 4 (cos i +i sin i), where a is a
positive real constant and - r 1 i G r.
(b) You are given that z1 z2 = w .
(i) Find the exact value of a. [3]
(ii) Find the value of i. Give your answer as an exact multiple of r. [3]
© OCR 2025 Y410/01 Jun25 Turn over
