Oxford Cambridge and RSA
Examinations GCE Further
Mathematics B MEIY436/01: Further
pure with technology
A Level
Question paper and marking scheme
(merged)
, Oxford Cambridge and RSA
Monday 26 June 2023 – Afternoon
A Level Further Mathematics B (MEI)
Y436/01 Further Pure with Technology
Time allowed: 1 hour 45 minutes
* 9 0 5 2 2 5 2 8 5 7 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a computer with appropriate software
• 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 your final answers to a degree of accuracy that is appropriate to the context.
• 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 [K/508/5600] OCR is an exempt Charity
DC (ST/FC) 333410/5 R Turn over
, 2
1 A family of functions is defined as
x2
f (x) = ax + , x ! -1
1+x
where the parameter a is a real number. You may find it helpful to use a slider (for a) to
investigate the family of curves y = f (x) .
(a) (i) On the axes in the Printed Answer Booklet, sketch the curve y = f (x) in each of the
following cases.
• a = -2
• a = -1
• a=0 [3]
(ii) State a feature which is common to the curve in all three cases, a = -2, a = -1 and
a = 0. [1]
(iii) State a feature of the curve for the cases a = -2, a = -1 that is not a feature of the curve
in the case a = 0. [1]
(b) (i) Determine the equation of the oblique asymptote to the curve y = f (x) in terms of a. [3]
(ii) For b ! -1, 0, 1 let A be the point with coordinates (-b, f(-b)) and let B be the point
with coordinates (b, f(b)).
Show that the y-coordinate of the point at which the chord to the curve y = f (x) between
A and B meets the y-axis is independent of a. [3]
(iii) With y = f (x) , determine the range of values of a for which
• y H 0 for all x H 0
• y G 0 for all x H 0 [5]
(c) In the case of a = 0, the curve y = 4 f (x) has a cusp.
Find its coordinates and fully justify that it is a cusp. [5]
© OCR 2023 Y436/01 Jun23
, 3
2 Throughout this question (a, b, c) is a Pythagorean triple with the positive integers a, b, c ordered
such that a G b G c.
(a) Show that a 2 = b + c if and only if c = b + 1. [4]
(b) Create a program to find all the Pythagorean triples (a, b, c) such that a 2 = b + c and
c G 1000. Write out your program in full in the Printed Answer Booklet. [3]
(c) Write down the number of Pythagorean triples found by your program in (b). [1]
(d) Prove that there is no Pythagorean triple, (a, b, c), in which b 2 = a + c . [3]
3 Wilson’s theorem states that an integer p 2 1 is prime if and only if (p - 1) ! / - 1 (mod p) .
(a) Use Wilson’s theorem to show that 17! / 1 (mod 19) . [2]
(b) A prime number p is called a Wilson prime if (p - 1) ! / - 1 (mod p 2) .
For example, 5 is a Wilson prime because (5 - 1) ! / 24 / - 1 (mod 25) .
At the time of writing all known Wilson primes are less than 1000.
(i) Create a program to find all the known Wilson primes. Write out your program in full in
the Printed Answer Booklet. [4]
(ii) Use your program to find and write down all the known Wilson primes. [1]
(iii) Prove that if there is an integer solution m to the equation (p - 1) ! + 1 = m 2 where p is
prime, then p is a Wilson prime. [3]
© OCR 2023 Y436/01 Jun23 Turn over
Examinations GCE Further
Mathematics B MEIY436/01: Further
pure with technology
A Level
Question paper and marking scheme
(merged)
, Oxford Cambridge and RSA
Monday 26 June 2023 – Afternoon
A Level Further Mathematics B (MEI)
Y436/01 Further Pure with Technology
Time allowed: 1 hour 45 minutes
* 9 0 5 2 2 5 2 8 5 7 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• a computer with appropriate software
• 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 your final answers to a degree of accuracy that is appropriate to the context.
• 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 [K/508/5600] OCR is an exempt Charity
DC (ST/FC) 333410/5 R Turn over
, 2
1 A family of functions is defined as
x2
f (x) = ax + , x ! -1
1+x
where the parameter a is a real number. You may find it helpful to use a slider (for a) to
investigate the family of curves y = f (x) .
(a) (i) On the axes in the Printed Answer Booklet, sketch the curve y = f (x) in each of the
following cases.
• a = -2
• a = -1
• a=0 [3]
(ii) State a feature which is common to the curve in all three cases, a = -2, a = -1 and
a = 0. [1]
(iii) State a feature of the curve for the cases a = -2, a = -1 that is not a feature of the curve
in the case a = 0. [1]
(b) (i) Determine the equation of the oblique asymptote to the curve y = f (x) in terms of a. [3]
(ii) For b ! -1, 0, 1 let A be the point with coordinates (-b, f(-b)) and let B be the point
with coordinates (b, f(b)).
Show that the y-coordinate of the point at which the chord to the curve y = f (x) between
A and B meets the y-axis is independent of a. [3]
(iii) With y = f (x) , determine the range of values of a for which
• y H 0 for all x H 0
• y G 0 for all x H 0 [5]
(c) In the case of a = 0, the curve y = 4 f (x) has a cusp.
Find its coordinates and fully justify that it is a cusp. [5]
© OCR 2023 Y436/01 Jun23
, 3
2 Throughout this question (a, b, c) is a Pythagorean triple with the positive integers a, b, c ordered
such that a G b G c.
(a) Show that a 2 = b + c if and only if c = b + 1. [4]
(b) Create a program to find all the Pythagorean triples (a, b, c) such that a 2 = b + c and
c G 1000. Write out your program in full in the Printed Answer Booklet. [3]
(c) Write down the number of Pythagorean triples found by your program in (b). [1]
(d) Prove that there is no Pythagorean triple, (a, b, c), in which b 2 = a + c . [3]
3 Wilson’s theorem states that an integer p 2 1 is prime if and only if (p - 1) ! / - 1 (mod p) .
(a) Use Wilson’s theorem to show that 17! / 1 (mod 19) . [2]
(b) A prime number p is called a Wilson prime if (p - 1) ! / - 1 (mod p 2) .
For example, 5 is a Wilson prime because (5 - 1) ! / 24 / - 1 (mod 25) .
At the time of writing all known Wilson primes are less than 1000.
(i) Create a program to find all the known Wilson primes. Write out your program in full in
the Printed Answer Booklet. [4]
(ii) Use your program to find and write down all the known Wilson primes. [1]
(iii) Prove that if there is an integer solution m to the equation (p - 1) ! + 1 = m 2 where p is
prime, then p is a Wilson prime. [3]
© OCR 2023 Y436/01 Jun23 Turn over
