2025 OCR A Level Further Mathematics A
Y545/01 Additional Pure Mathematics Combined Question Paper & Final Marking Scheme
Oxford Cambridge and RSA
Friday 20 June 2025 – Afternoon
A Level Further Mathematics A
Y545/01 Additional Pure Mathematics
Time allowed: 1 hour 30 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
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 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 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [D/508/5514] OCR is an exempt Charity
DC (DE) 359664/3 Turn over
,*1920573537*
, 2
1 (a) A student claims that the order of 5 modulo 37 is 5.
Without calculation, explain why this student’s claim must be incorrect. [1]
(b) Show that the order of 5 modulo 37 is 36. [3]
2 The points P, Q, R and S have coordinates (1, 2, 4), (3, 1, 6), (4, 8, -2) and (5, 5, 5) respectively.
(a) Find, in exact form, the area of triangle PQR. [3]
(b) Find, in exact form, the volume of tetrahedron PQRS. [3]
(c) Use your answers to part (a) and part (b) to deduce the shortest distance from S to the plane
containing P, Q and R. Give your answer in exact form.
[You may use the formula V = 13Ah, where V is the volume of a pyramid, A is the area of one
of its bases and h is the height relative to the base.] [2]
3 Let N = 10a + b and M = a - 3b, where a and b are positive integers and 0 ≤ b ≤ 9.
(a) By considering 3N + M , prove that 31 | N if and only if 31 | M. [4]
(b) Use a procedure based on the result of part (a) to show that 31 | 7 167 758. [2]
4 In this question you must show detailed reasoning.
The surface S is given by the equation z = 4x2y - 3xy2 - 5x for all real values of x and y.
(a) The point P(1, 1, -4) lies on S.
Find the equation of the tangent plane to S at P. [4]
(b) The point Q is a stationary point of S with positive x- and y-coordinates.
(i) Find, in exact form, the coordinates of Q. [5]
(ii) Find the value of the determinant of H, the Hessian matrix of S, at Q. [4]
(iii) Deduce the nature of the stationary point Q. [1]
© OCR 2025 Y545/01 Jun25
Y545/01 Additional Pure Mathematics Combined Question Paper & Final Marking Scheme
Oxford Cambridge and RSA
Friday 20 June 2025 – Afternoon
A Level Further Mathematics A
Y545/01 Additional Pure Mathematics
Time allowed: 1 hour 30 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
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 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 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [D/508/5514] OCR is an exempt Charity
DC (DE) 359664/3 Turn over
,*1920573537*
, 2
1 (a) A student claims that the order of 5 modulo 37 is 5.
Without calculation, explain why this student’s claim must be incorrect. [1]
(b) Show that the order of 5 modulo 37 is 36. [3]
2 The points P, Q, R and S have coordinates (1, 2, 4), (3, 1, 6), (4, 8, -2) and (5, 5, 5) respectively.
(a) Find, in exact form, the area of triangle PQR. [3]
(b) Find, in exact form, the volume of tetrahedron PQRS. [3]
(c) Use your answers to part (a) and part (b) to deduce the shortest distance from S to the plane
containing P, Q and R. Give your answer in exact form.
[You may use the formula V = 13Ah, where V is the volume of a pyramid, A is the area of one
of its bases and h is the height relative to the base.] [2]
3 Let N = 10a + b and M = a - 3b, where a and b are positive integers and 0 ≤ b ≤ 9.
(a) By considering 3N + M , prove that 31 | N if and only if 31 | M. [4]
(b) Use a procedure based on the result of part (a) to show that 31 | 7 167 758. [2]
4 In this question you must show detailed reasoning.
The surface S is given by the equation z = 4x2y - 3xy2 - 5x for all real values of x and y.
(a) The point P(1, 1, -4) lies on S.
Find the equation of the tangent plane to S at P. [4]
(b) The point Q is a stationary point of S with positive x- and y-coordinates.
(i) Find, in exact form, the coordinates of Q. [5]
(ii) Find the value of the determinant of H, the Hessian matrix of S, at Q. [4]
(iii) Deduce the nature of the stationary point Q. [1]
© OCR 2025 Y545/01 Jun25
