2025 OCR A Level Further Mathematics B (MEI)
Y436/01 Further Pure with Technology
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Monday 23 June 2025 – Afternoon A
Level Further Mathematics B (MEI) Y436/01
Further Pure with Technology
Time allowed: 1 hour 45 minutes
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 2025 [K/508/5600] OCR is an exempt Charity
DC (PQ/FC) 352593/3 Turn over
, 2
1 A family of curves is given by the cartesian equation
x2 y2
+ mxy + =1
a2 b2
where a, b and m are real numbers with a and b non-zero.
(a) In this part of the question a = 2 and b = 1.
(i) On the axes in the Printed Answer Booklet, sketch the curve in each of these
cases.
• m=0
• m=1 [3]
• m=2
(ii) State one feature of the curve for the case m = 0 that is not a feature of the curve in
the cases m = 1 and m = 2. [1]
For the remainder of this question m = 0.
(b) Verify that the parametric equations of the curve are
x (t) = a cos (t), y (t) = b sin (t),
where 0 G t 1 2r is a parameter. [1]
dy b
(c) Show that =- cot (t). [2]
dx a
(d) Show that the equation of the normal to the curve at the point with parameter t is
a a2 - b2m
y = tan(t) x - c sin(t). [5]
b b
(e) Show that the parametric equations of the envelope of the normal to the curve are
a2 - b2 b2 - a2m 3
x (t) = c a mcos3 (t), y (t) = c sin (t),
b
where 0 G t 1 2r is a parameter. [6]
(f) In this part of the question a = 2 and b = 1.
On the axes in the Printed Answer Booklet, sketch the envelope of the normal to the curve. [1]
2 2
(g) By considering the expression (ax (t)) + (by (t)) or otherwise, determine a cartesian
3 3
equation of the envelope of the normal to the curve. [2]
© OCR 2025 Y436/01 Jun25
, 3
2 (a) (i) Write down 710 (mod 1000). [1]
Fermat’s little theorem states that if p is a prime and x is an integer which is co-prime to p,
then x p- (mod p).
(ii) Explain why Fermat’s little theorem implies (mod 13). [1]
(iii) Determine 2(12q+1) (mod 13), where q is a positive integer. [2]
(b) In the rest of this question the highest common factor of positive integers m and n is
denoted by (m, n).
(i) Write down (354, 27). [1]
Euler’s totient function φ(n), where n is a positive integer, is defined to be the number of
integers m with 1 G m G n such that (m, n) = 1.
For example, φ (12) = 4 since 1, 5, 7 and 11 are all co-prime with 12, but 2, 3, 4, 6, 8, 9, 10,
and 12 all share a common factor greater than 1 with 12.
(ii) Create a program which returns the value of φ(n) for a given positive integer n.
Write out your program in full in the Printed Answer Booklet. [4]
(iii) Use your program to find φ(1000). [1]
Euler’s theorem states that if a and n are co-prime positive integers, then aφ(n) / 1 (mod n).
(iv) Determine 7(400r+10) (mod 1000), where r is a positive integer. [4]
(v) Using part (b)(iv), determine the tens digit of 72010. [2]
(c) Suppose that p H 3 is a prime number and that x and y are positive integers.
By considering the equation p2x + 1 = 22y modulo 4, or otherwise, prove there are no integer
solutions to the equation p2x + 1 = 22y . [5]
© OCR 2025 Y436/01 Jun25 Turn over
, 4
3 This question concerns the family of differential equations
dy
= x - y + aex sin (y) (**)
dx
where a is a constant.
(a) In this part of the question a = 0.
(i) Verify that y = 2e-x + x - 1 is the particular solution of (**) that satisfies y = 1 when x = 0.
[3]
The solution y = 2e-x + x - 1 has a minimum value at the point (m, n) where 0 1 m 1 1.
(ii) Find the exact value of m. [2]
(iii) Sketch the particular solution of (**) given in part (a)(i) for 0 G x G 3 on the axes in the
Printed Answer Booklet. [2]
(b) The figure below shows the tangent field for an unspecified value of a. A sketch of the
1
solution curve y = g(x) which passes through the point (0, 1) is shown for 0 G x G .
2
y
4
3
2
1
0 x
0 1 2 3
1
Continue the sketch of the solution curve for G x G 3 on the axes in the Printed Answer
2 [2]
Booklet.
© OCR 2025 Y436/01 Jun25
Y436/01 Further Pure with Technology
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Monday 23 June 2025 – Afternoon A
Level Further Mathematics B (MEI) Y436/01
Further Pure with Technology
Time allowed: 1 hour 45 minutes
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 2025 [K/508/5600] OCR is an exempt Charity
DC (PQ/FC) 352593/3 Turn over
, 2
1 A family of curves is given by the cartesian equation
x2 y2
+ mxy + =1
a2 b2
where a, b and m are real numbers with a and b non-zero.
(a) In this part of the question a = 2 and b = 1.
(i) On the axes in the Printed Answer Booklet, sketch the curve in each of these
cases.
• m=0
• m=1 [3]
• m=2
(ii) State one feature of the curve for the case m = 0 that is not a feature of the curve in
the cases m = 1 and m = 2. [1]
For the remainder of this question m = 0.
(b) Verify that the parametric equations of the curve are
x (t) = a cos (t), y (t) = b sin (t),
where 0 G t 1 2r is a parameter. [1]
dy b
(c) Show that =- cot (t). [2]
dx a
(d) Show that the equation of the normal to the curve at the point with parameter t is
a a2 - b2m
y = tan(t) x - c sin(t). [5]
b b
(e) Show that the parametric equations of the envelope of the normal to the curve are
a2 - b2 b2 - a2m 3
x (t) = c a mcos3 (t), y (t) = c sin (t),
b
where 0 G t 1 2r is a parameter. [6]
(f) In this part of the question a = 2 and b = 1.
On the axes in the Printed Answer Booklet, sketch the envelope of the normal to the curve. [1]
2 2
(g) By considering the expression (ax (t)) + (by (t)) or otherwise, determine a cartesian
3 3
equation of the envelope of the normal to the curve. [2]
© OCR 2025 Y436/01 Jun25
, 3
2 (a) (i) Write down 710 (mod 1000). [1]
Fermat’s little theorem states that if p is a prime and x is an integer which is co-prime to p,
then x p- (mod p).
(ii) Explain why Fermat’s little theorem implies (mod 13). [1]
(iii) Determine 2(12q+1) (mod 13), where q is a positive integer. [2]
(b) In the rest of this question the highest common factor of positive integers m and n is
denoted by (m, n).
(i) Write down (354, 27). [1]
Euler’s totient function φ(n), where n is a positive integer, is defined to be the number of
integers m with 1 G m G n such that (m, n) = 1.
For example, φ (12) = 4 since 1, 5, 7 and 11 are all co-prime with 12, but 2, 3, 4, 6, 8, 9, 10,
and 12 all share a common factor greater than 1 with 12.
(ii) Create a program which returns the value of φ(n) for a given positive integer n.
Write out your program in full in the Printed Answer Booklet. [4]
(iii) Use your program to find φ(1000). [1]
Euler’s theorem states that if a and n are co-prime positive integers, then aφ(n) / 1 (mod n).
(iv) Determine 7(400r+10) (mod 1000), where r is a positive integer. [4]
(v) Using part (b)(iv), determine the tens digit of 72010. [2]
(c) Suppose that p H 3 is a prime number and that x and y are positive integers.
By considering the equation p2x + 1 = 22y modulo 4, or otherwise, prove there are no integer
solutions to the equation p2x + 1 = 22y . [5]
© OCR 2025 Y436/01 Jun25 Turn over
, 4
3 This question concerns the family of differential equations
dy
= x - y + aex sin (y) (**)
dx
where a is a constant.
(a) In this part of the question a = 0.
(i) Verify that y = 2e-x + x - 1 is the particular solution of (**) that satisfies y = 1 when x = 0.
[3]
The solution y = 2e-x + x - 1 has a minimum value at the point (m, n) where 0 1 m 1 1.
(ii) Find the exact value of m. [2]
(iii) Sketch the particular solution of (**) given in part (a)(i) for 0 G x G 3 on the axes in the
Printed Answer Booklet. [2]
(b) The figure below shows the tangent field for an unspecified value of a. A sketch of the
1
solution curve y = g(x) which passes through the point (0, 1) is shown for 0 G x G .
2
y
4
3
2
1
0 x
0 1 2 3
1
Continue the sketch of the solution curve for G x G 3 on the axes in the Printed Answer
2 [2]
Booklet.
© OCR 2025 Y436/01 Jun25
