OCR A Level Further Mathematics A
Y541/01 Combined Question paper &
Marking Scheme A+ June 2025
Oxford Cambridge and RSA
Tuesday 3 June 2025 – Afternoon
A Level Further Mathematics A
Y541/01 Pure Core 2
Time allowed: 1 hour 30 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
Mathematics A
• 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 non-exact numerical answers correct to 3 significant figures unless a different degree
of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is needed
use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
Turn over
, 32
41 In this question you must show detailed reasoning.
KJ 2ON KJ -1NO
Vectors a and b are given by a and b = .
= K -4O K 3O
3O K O
L P L-2P
Determine, in either order
• a.b
• a×b. [3]
2 You are given that - 7 - 5i is one root of the equation x3 + 10x2 + 18x - 296 = 0.
(a) Write down another complex root of the equation x3 + 10x2 + 18x - 296 = 0. [1]
(b) Using your answer to part (a), express x3 + 10x2 + 18x - 296 as a product of a real linear
factor and a real quadratic factor. [3]
JK1 - 3NO KJ- 3 - 3NO
3 Matrices A and B are given by A = K and B = K .
4L 8OP L1 2O
P
(a) Find the matrix AB. [1]
(b) Verify that det(AB) = det (A) # det(B). [2]
(c) Use matrices A and B to demonstrate that matrix multiplication is not commutative. [2]
The transformation represented by matrix A is denoted by T.
(d) Show that the point (2, -5) is not an invariant point under T. [2]
(e) Find the matrix which represents the inverse transformation of T. [2]
© OCR 2025 Y541/01 Jun25
, 32
41 In this question you must show detailed reasoning.
Determine the sum of all cube numbers from 216 to 512 000 inclusive. [4]
5 In this question you must show detailed reasoning.
(a) Use an algebraic method to determine the two square roots of - 3 + (4 7) i.
Give your answers in the form a + bi where a and b are exact. [5]
(b) State the relationship between the two arguments of the two square roots found in part (a).
[1]
6 One of the regions bounded by two polar curves, C1 and C2 , is used to model the face of a
flat earring.
The polar equations of C1 and C2 are
C1: r = 2i
C2: r = i2
where 0 G i G r.
The curves C1 and C2 are shown in the diagram below with the region used to model the earring
shaded.
C2
C1
O Initial line
You are given that C1 and C2 intersect at the pole O.
(a) Find the other point of intersection of C1 and C2 . Give your answer in polar coordinates. [2]
(b) In this question you must show detailed reasoning.
© OCR 2025 Y541/01 Jun25 Turn over
, 32
41 In this question you must show detailed reasoning.
Determine the area of the face of the earring. [4]
© OCR 2025 Y541/01 Jun25
Y541/01 Combined Question paper &
Marking Scheme A+ June 2025
Oxford Cambridge and RSA
Tuesday 3 June 2025 – Afternoon
A Level Further Mathematics A
Y541/01 Pure Core 2
Time allowed: 1 hour 30 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
Mathematics A
• 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 non-exact numerical answers correct to 3 significant figures unless a different degree
of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is needed
use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
Turn over
, 32
41 In this question you must show detailed reasoning.
KJ 2ON KJ -1NO
Vectors a and b are given by a and b = .
= K -4O K 3O
3O K O
L P L-2P
Determine, in either order
• a.b
• a×b. [3]
2 You are given that - 7 - 5i is one root of the equation x3 + 10x2 + 18x - 296 = 0.
(a) Write down another complex root of the equation x3 + 10x2 + 18x - 296 = 0. [1]
(b) Using your answer to part (a), express x3 + 10x2 + 18x - 296 as a product of a real linear
factor and a real quadratic factor. [3]
JK1 - 3NO KJ- 3 - 3NO
3 Matrices A and B are given by A = K and B = K .
4L 8OP L1 2O
P
(a) Find the matrix AB. [1]
(b) Verify that det(AB) = det (A) # det(B). [2]
(c) Use matrices A and B to demonstrate that matrix multiplication is not commutative. [2]
The transformation represented by matrix A is denoted by T.
(d) Show that the point (2, -5) is not an invariant point under T. [2]
(e) Find the matrix which represents the inverse transformation of T. [2]
© OCR 2025 Y541/01 Jun25
, 32
41 In this question you must show detailed reasoning.
Determine the sum of all cube numbers from 216 to 512 000 inclusive. [4]
5 In this question you must show detailed reasoning.
(a) Use an algebraic method to determine the two square roots of - 3 + (4 7) i.
Give your answers in the form a + bi where a and b are exact. [5]
(b) State the relationship between the two arguments of the two square roots found in part (a).
[1]
6 One of the regions bounded by two polar curves, C1 and C2 , is used to model the face of a
flat earring.
The polar equations of C1 and C2 are
C1: r = 2i
C2: r = i2
where 0 G i G r.
The curves C1 and C2 are shown in the diagram below with the region used to model the earring
shaded.
C2
C1
O Initial line
You are given that C1 and C2 intersect at the pole O.
(a) Find the other point of intersection of C1 and C2 . Give your answer in polar coordinates. [2]
(b) In this question you must show detailed reasoning.
© OCR 2025 Y541/01 Jun25 Turn over
, 32
41 In this question you must show detailed reasoning.
Determine the area of the face of the earring. [4]
© OCR 2025 Y541/01 Jun25
