GCSE
Further Mathematics A
Y535/01: Additional pure mathematics
*1920006910*
AS Level
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 page 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 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.
© OCR 2025 [603/1329/8] DC (ST) 357834/4
OCR is an exempt Charity
Turn over
1 Use standard divisibility tests for small numbers to show that 2 573 208 is
divisible by 792. [4]
JK1NO JK 2NO JK 11NO
, 2
2 The vectors p, q and r are such that p =KKL37OOP, q =LKK-94OPO and r =KLK-
12OOP.
(a) (i) Determine the value of the integer k for which p#q = kr. [2]
##
(ii) Use the result of part (a)(i) to explain geometrically why (p q)
r = 0. [1] Relative to the origin O, the points P and Q have
position vectors p and q respectively.
(b) Show that the area of triangle OPQ can be written in the form n n-1 where
n is a positive integer to be determined. [3]
3 The surface S has equation z = 2x y2 -6y3+ +3x 4 for all real values of x and y.
(a) (i) State the equation of the section of S cut by the plane y = 1. [1]
(ii) Sketch the section of S cut by the plane y = 1. Give the coordinates
of the points of
intersection with the axes.
(b) Determine the coordinates of all stationary points of S. [6]
(c) The contour C of S is given by z = 37.
2
Find the coordinates of the unique point on C where 22yz2 +222y x2z = 0.
4 The binary operation * is defined on the set A = {1, 3, 5, 7, 9} by x * y = x + y + 3
(mod 10) for all x, y ! A.
(a) (i) Show that * is associative on the elements of A. [2]
(ii) Complete the Cayley table for (A, *) given in the Printed Answer
Booklet. [2]
(iii) Hence show that (A, *) forms a group, G. [4]
(b) (i) State the order of each non-identity element of G. [1]
(ii) List all subgroups of G, and explain why there are no others. [2]
© OCR 2025 Y535/01 Jun25
, 3
(iii) Explain whether G is cyclic. [1]
(iv) Explain whether G is abelian. [1] 5 Let f( )n = +3n 4n for
all positive integers n.
Prove by induction that f( )n is a multiple of 7 for all odd integers n H 1.
6 An investment company offers customers a three-year investment scheme.
The scheme is modelled by the recurrence system
I I I I
0 =a and n+1 = 0 000625. n
2
-0 625. n for nH 0.
In is the value in pounds of the scheme n years after its start, and £a is the integer
value of the initial sum invested.
(a) Determine the minimum initial sum invested that guarantees that the value
of the scheme
increases every year.
(b) The investment company decides to make the following change in the
scheme.
At the end of each year, the value of In is always rounded down to the
nearest pound. (i) Write down a modified recurrence formula that takes
this change into account. [1] (ii) A customer invests £2700 in the three-
year investment scheme.
Work out how much less the customer’s investment would be worth after
three years under the new scheme compared to the old scheme. [6]
7 Consider the two arithmetic sequences An = +{2n 7:n !N} and Bn = +{3n 1:n !N}.
Let h = hcf(A Bn, n) for each chosen value of n.
(a) (i) Find a value of n for which h = 1. [1]
(ii) Find a value of n for which h = 19.
(b) Determine all the values of n for which h = 19. Give your answer as an
expression in terms of an integer k. [2]
© OCR 2025 Y535/01 Jun25
Further Mathematics A
Y535/01: Additional pure mathematics
*1920006910*
AS Level
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 page 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 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.
© OCR 2025 [603/1329/8] DC (ST) 357834/4
OCR is an exempt Charity
Turn over
1 Use standard divisibility tests for small numbers to show that 2 573 208 is
divisible by 792. [4]
JK1NO JK 2NO JK 11NO
, 2
2 The vectors p, q and r are such that p =KKL37OOP, q =LKK-94OPO and r =KLK-
12OOP.
(a) (i) Determine the value of the integer k for which p#q = kr. [2]
##
(ii) Use the result of part (a)(i) to explain geometrically why (p q)
r = 0. [1] Relative to the origin O, the points P and Q have
position vectors p and q respectively.
(b) Show that the area of triangle OPQ can be written in the form n n-1 where
n is a positive integer to be determined. [3]
3 The surface S has equation z = 2x y2 -6y3+ +3x 4 for all real values of x and y.
(a) (i) State the equation of the section of S cut by the plane y = 1. [1]
(ii) Sketch the section of S cut by the plane y = 1. Give the coordinates
of the points of
intersection with the axes.
(b) Determine the coordinates of all stationary points of S. [6]
(c) The contour C of S is given by z = 37.
2
Find the coordinates of the unique point on C where 22yz2 +222y x2z = 0.
4 The binary operation * is defined on the set A = {1, 3, 5, 7, 9} by x * y = x + y + 3
(mod 10) for all x, y ! A.
(a) (i) Show that * is associative on the elements of A. [2]
(ii) Complete the Cayley table for (A, *) given in the Printed Answer
Booklet. [2]
(iii) Hence show that (A, *) forms a group, G. [4]
(b) (i) State the order of each non-identity element of G. [1]
(ii) List all subgroups of G, and explain why there are no others. [2]
© OCR 2025 Y535/01 Jun25
, 3
(iii) Explain whether G is cyclic. [1]
(iv) Explain whether G is abelian. [1] 5 Let f( )n = +3n 4n for
all positive integers n.
Prove by induction that f( )n is a multiple of 7 for all odd integers n H 1.
6 An investment company offers customers a three-year investment scheme.
The scheme is modelled by the recurrence system
I I I I
0 =a and n+1 = 0 000625. n
2
-0 625. n for nH 0.
In is the value in pounds of the scheme n years after its start, and £a is the integer
value of the initial sum invested.
(a) Determine the minimum initial sum invested that guarantees that the value
of the scheme
increases every year.
(b) The investment company decides to make the following change in the
scheme.
At the end of each year, the value of In is always rounded down to the
nearest pound. (i) Write down a modified recurrence formula that takes
this change into account. [1] (ii) A customer invests £2700 in the three-
year investment scheme.
Work out how much less the customer’s investment would be worth after
three years under the new scheme compared to the old scheme. [6]
7 Consider the two arithmetic sequences An = +{2n 7:n !N} and Bn = +{3n 1:n !N}.
Let h = hcf(A Bn, n) for each chosen value of n.
(a) (i) Find a value of n for which h = 1. [1]
(ii) Find a value of n for which h = 19.
(b) Determine all the values of n for which h = 19. Give your answer as an
expression in terms of an integer k. [2]
© OCR 2025 Y535/01 Jun25
