Oxford Cambridge and
RSA Examinations GCE
Further Mathematics
AY535/01: Additional
pure mathematics AS
Level question paper
and marking scheme
(merged)
, Oxford Cambridge and RSA
Friday 16 June 2023 – Afternoon
AS Level Further Mathematics A
Y535/01 Additional Pure Mathematics
Time allowed: 1 hour 15 minutes
* 9 9 7 5 5 4 4 6 3 4 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for AS 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 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 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [H/508/5501] OCR is an exempt Charity
DC (ST) 328404/3 Turn over
, 2
1 (a) Express 205 in the form 7q + r for positive integers q and r, with 0 G r 1 7 . [1]
(b) Given that 7 | ^205 # 8666h, use the result of part (a) to justify that 7 | 8666. [2]
2 For all positive integers n, the terms of the sequence "u n, are given by the formula
u n = 3n 2 + 3n + 7 (mod 10) .
(a) Show that u n + 5 = u n for all positive integers n. [2]
(b) Hence describe the behaviour of the sequence, justifying your answer. [2]
3 A surface has equation z = x 2 y 2 - 3xy + 2x + y for all real values of x and y.
Determine the coordinates of all stationary points of this surface. [6]
4 The equation of line l can be written in either of the following vector forms.
• r = a + mb , where m ! R
• (r - c) # d = 0
(a) Write down two equations involving the vectors a, b, c and d, giving reasons for your
answers. [4]
(b) Determine the value of a . (c # d) . [3]
5 (a) Express as a decimal (base-10) number the base-23 number 7119 23 . [2]
(b) Solve the linear congruence 7n + (mod 23) . [3]
(c) Let N = 10a + b and M = a + 7b , where a and b are integers and 0 G b G 9 .
(i) By considering 3N - 7M, prove that 23 | N if and only if 23 | M. [4]
(ii) Use a procedure based on this result to show that N = 711 965 is a multiple of 23. [2]
© OCR 2023 Y535/01 Jun23
, 3
6 When 10 6 of a certain type of bacteria are detected in a blood sample of an infected animal,
a course of treatment is started. The long-term aim of the treatment is to reduce the number of
bacteria in such a sample to under 10 000. At this level the animal’s immune system can fight the
infection for itself. Once treatment has started, if the number of bacteria in a sample is 10 000 or
more, then treatment either continues or restarts.
The model suggested to predict the progress of the course of treatment is based on the recurrence
2Pn n
system Pn + 1 = + for n H 0 , with P0 = 1000 , where Pn denotes the number of bacteria
n + 1 Pn
(in thousands) present in the animal’s body n days after the treatment was started.
The table below shows the values of Pn , for certain chosen values of n. Each value has been given
correct to 2 decimal places (where appropriate).
n 0 1 2 3 4 5 6 7 8 9
Pn 1000 2000 2000 1333.33 666.67 266.67 25.47 6.64 2.68
n 10 20 40 60 80 100 200 300 400
Pn 3.89 4.67 6.45 7.84 9.03 10.08 14.20 17.36 20.04
(a) Find the value of P6 correct to 2 decimal places. [2]
(b) Using the given values for P0 to P9 , and assuming that the model is valid,
(i) describe the effects of this course of treatment during the first 9 days, [1]
(ii) state the number of days after treatment is started when the animal’s own immune
system is expected to be able to fight the infection for itself. [1]
(c) (i) Using information from the above table, suggest a function f such that, for n 2 10 , f(n)
is a suitable approximation for Pn . [1]
(ii) Use your suggested function to estimate the number of days after treatment is started
when the animal may once again require medical intervention in order to help fight off
this bacterial infection. [1]
(iii) Using information from the above table and the recurrence relation, verify or correct the
estimate which you found in part (c)(ii). [2]
2Pn n
(d) One criticism of the system Pn + 1 = + , with P0 = 1000 , is that it gives non-integer
n + 1 Pn
values of Pn .
Suggest a modification that would correct this issue. [1]
© OCR 2023 Y535/01 Jun23 Turn over
RSA Examinations GCE
Further Mathematics
AY535/01: Additional
pure mathematics AS
Level question paper
and marking scheme
(merged)
, Oxford Cambridge and RSA
Friday 16 June 2023 – Afternoon
AS Level Further Mathematics A
Y535/01 Additional Pure Mathematics
Time allowed: 1 hour 15 minutes
* 9 9 7 5 5 4 4 6 3 4 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for AS 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 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 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2023 [H/508/5501] OCR is an exempt Charity
DC (ST) 328404/3 Turn over
, 2
1 (a) Express 205 in the form 7q + r for positive integers q and r, with 0 G r 1 7 . [1]
(b) Given that 7 | ^205 # 8666h, use the result of part (a) to justify that 7 | 8666. [2]
2 For all positive integers n, the terms of the sequence "u n, are given by the formula
u n = 3n 2 + 3n + 7 (mod 10) .
(a) Show that u n + 5 = u n for all positive integers n. [2]
(b) Hence describe the behaviour of the sequence, justifying your answer. [2]
3 A surface has equation z = x 2 y 2 - 3xy + 2x + y for all real values of x and y.
Determine the coordinates of all stationary points of this surface. [6]
4 The equation of line l can be written in either of the following vector forms.
• r = a + mb , where m ! R
• (r - c) # d = 0
(a) Write down two equations involving the vectors a, b, c and d, giving reasons for your
answers. [4]
(b) Determine the value of a . (c # d) . [3]
5 (a) Express as a decimal (base-10) number the base-23 number 7119 23 . [2]
(b) Solve the linear congruence 7n + (mod 23) . [3]
(c) Let N = 10a + b and M = a + 7b , where a and b are integers and 0 G b G 9 .
(i) By considering 3N - 7M, prove that 23 | N if and only if 23 | M. [4]
(ii) Use a procedure based on this result to show that N = 711 965 is a multiple of 23. [2]
© OCR 2023 Y535/01 Jun23
, 3
6 When 10 6 of a certain type of bacteria are detected in a blood sample of an infected animal,
a course of treatment is started. The long-term aim of the treatment is to reduce the number of
bacteria in such a sample to under 10 000. At this level the animal’s immune system can fight the
infection for itself. Once treatment has started, if the number of bacteria in a sample is 10 000 or
more, then treatment either continues or restarts.
The model suggested to predict the progress of the course of treatment is based on the recurrence
2Pn n
system Pn + 1 = + for n H 0 , with P0 = 1000 , where Pn denotes the number of bacteria
n + 1 Pn
(in thousands) present in the animal’s body n days after the treatment was started.
The table below shows the values of Pn , for certain chosen values of n. Each value has been given
correct to 2 decimal places (where appropriate).
n 0 1 2 3 4 5 6 7 8 9
Pn 1000 2000 2000 1333.33 666.67 266.67 25.47 6.64 2.68
n 10 20 40 60 80 100 200 300 400
Pn 3.89 4.67 6.45 7.84 9.03 10.08 14.20 17.36 20.04
(a) Find the value of P6 correct to 2 decimal places. [2]
(b) Using the given values for P0 to P9 , and assuming that the model is valid,
(i) describe the effects of this course of treatment during the first 9 days, [1]
(ii) state the number of days after treatment is started when the animal’s own immune
system is expected to be able to fight the infection for itself. [1]
(c) (i) Using information from the above table, suggest a function f such that, for n 2 10 , f(n)
is a suitable approximation for Pn . [1]
(ii) Use your suggested function to estimate the number of days after treatment is started
when the animal may once again require medical intervention in order to help fight off
this bacterial infection. [1]
(iii) Using information from the above table and the recurrence relation, verify or correct the
estimate which you found in part (c)(ii). [2]
2Pn n
(d) One criticism of the system Pn + 1 = + , with P0 = 1000 , is that it gives non-integer
n + 1 Pn
values of Pn .
Suggest a modification that would correct this issue. [1]
© OCR 2023 Y535/01 Jun23 Turn over
