GCE
Further Mathematics A
Y545/01: Additional Pure Mathematics
A Level
OXFORD
CAMBRIDGE AND
RSA 2024
, Oxford Cambridge and RSA
Friday 21 June 2024 – 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 AnswerBooklet. 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 begiven
for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a differentdegree
of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value isneeded
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 is an exempt Charity
Turn over
, 2
1 (a) The number N has the base -10 form N = abba abba … abba, consisting of blocks of four
digits, as shown, where a and b are integers such that 1 G a 1 10 and 0 G b 1 10.
Use a standard divisibility test to show that N is always divisible by 11. [3]
(b) The number M has the base-n form M = cddc cddc … cddc, where n 2 11 and c and d are
integers such that 1 G c 1 n and 0 G d 1 n.
Show that M is always divisible by a number of the form k1 n + k2 , where k1 and k2 are
integers to be determined. [3]
2 A surface S has equation z = 4x y - y x + y2 for x, y H 0.
Determine the equation of the tangent plane to S at the point (1, 4, 20). Give your answer in the
form ax + by + cz = d where a, b, c and d are integers. [5]
3 Determine all integers x for which x / 1 (mod 7) and x / 22 (mod 37) and x / 7 (mod 67).
Give your answer in the form x = qn + r for integers n, q, r with q 2 0 and 0 G r 1 q. [6]
Jp - 1 N J2p + 4N
K O K O
4 The vectors a and b are given by a = Kq + 2 Oand b = 2q
K - 5 , where p, q and r are real numbers.
O
2r - 3 r+3
L P L P
(a) Given that b is not a multiple of a and that a # b = 0, determine all possible sets of values of
p, q and r. [3]
(b) You are given instead that b = ma , where m is an integer with m 2 1.
By writing each of p, q and r in terms of m, show that there is a unique value of m for which
p, q and r are all integers, stating this set of values of p, q and r. [7]
© OCR 2024 Y545/01 Jun24
Further Mathematics A
Y545/01: Additional Pure Mathematics
A Level
OXFORD
CAMBRIDGE AND
RSA 2024
, Oxford Cambridge and RSA
Friday 21 June 2024 – 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 AnswerBooklet. 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 begiven
for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a differentdegree
of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value isneeded
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 is an exempt Charity
Turn over
, 2
1 (a) The number N has the base -10 form N = abba abba … abba, consisting of blocks of four
digits, as shown, where a and b are integers such that 1 G a 1 10 and 0 G b 1 10.
Use a standard divisibility test to show that N is always divisible by 11. [3]
(b) The number M has the base-n form M = cddc cddc … cddc, where n 2 11 and c and d are
integers such that 1 G c 1 n and 0 G d 1 n.
Show that M is always divisible by a number of the form k1 n + k2 , where k1 and k2 are
integers to be determined. [3]
2 A surface S has equation z = 4x y - y x + y2 for x, y H 0.
Determine the equation of the tangent plane to S at the point (1, 4, 20). Give your answer in the
form ax + by + cz = d where a, b, c and d are integers. [5]
3 Determine all integers x for which x / 1 (mod 7) and x / 22 (mod 37) and x / 7 (mod 67).
Give your answer in the form x = qn + r for integers n, q, r with q 2 0 and 0 G r 1 q. [6]
Jp - 1 N J2p + 4N
K O K O
4 The vectors a and b are given by a = Kq + 2 Oand b = 2q
K - 5 , where p, q and r are real numbers.
O
2r - 3 r+3
L P L P
(a) Given that b is not a multiple of a and that a # b = 0, determine all possible sets of values of
p, q and r. [3]
(b) You are given instead that b = ma , where m is an integer with m 2 1.
By writing each of p, q and r in terms of m, show that there is a unique value of m for which
p, q and r are all integers, stating this set of values of p, q and r. [7]
© OCR 2024 Y545/01 Jun24
