OCR
GCSE
Further Mathematics A
Y545/01: Additional Pure Mathematics
*1920573537*
A 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 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 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 2025 [D/508/5514] DC (DE) 359664/3
, OCR is an exempt Charity
Turn over
1 A student claims that the order of 5 modulo 37 is 5.
Without calculation, explain why this student’s claim must be incorrect.
(b) Show that the order of 5 modulo 37 is 36.
2 The points P, Q, R and S have coordinates (1, 2, 4), (3, 1, 6), (4, 8, -2) and (5, 5, 5)
respectively.
(a) Find, in exact form, the area of triangle PQR. [3]
(b) Find, in exact form, the volume of tetrahedron PQRS. [3]
(c) Use your answers to part (a) and part (b) to deduce the shortest distance from S to
the plane containing P, Q and R. Give your answer in exact form.
[You may use the formula V = Ah, where V is the volume of a pyramid, A is the area of
one
of its bases and h is the height relative to the base.]
3 Let N = +10a b and M = -a 3b , where a and b are positive integers and 0 G Gb
9.
(a) By considering 3N + M , prove that 31 | N if and only if 31 | M. [4]
(b) Use a procedure based on the result of part (a) to show that 31 | 7 167 758. [2]
4 In this question you must show detailed reasoning.
The surface S is given by the equation z = 4x2y -3xy2-5x for all real values of x and y.
(a) The point P(1, 1, -4) lies on S.
Find the equation of the tangent plane to S at P.
(b) The point Q is a stationary point of S with positive x- and y-coordinates.
(i) Find, in exact form, the coordinates of Q. [5]
(ii) Find the value of the determinant of H, the Hessian matrix of S, at Q.
[4]
, (iii) Deduce the nature of the stationary point Q. [1] 5 Determine the
/
solution of the linear congruence 3x 7 (mod 13). Give your answer in the
form x / a (mod 13), where 0 G a 1 13.
(b) Hence or otherwise, determine the solution of the simultaneous linear congruences
3x / 7 (mod 13), 7x / 3 (mod 31).
6 In this question, the set S is {2, 4, 6, 8}, and #10 is the operation of multiplication
modulo 10.
(a) The group G consists of the set S, under the operation #10.
(i) Complete the Cayley table for G given in the Printed Answer Booklet.
[1]
(ii) State the identity element of G. [1]
(iii) State the order of each non-identity element of G. [2]
0
(b) The group H consists of all 2#2 matrices of the form Ja NOO, where a, b ! S , under the
KKL0 bP
operation of matrix multiplication (using #10 to multiply elements of S ).
(i) State, with justification, the order of H. [2]
(ii) State also the identity element of H. [1]
(iii) State all the elements of H of order 2. [2]
(iv) Using the result of part (a)(iii), or otherwise, explain why all the (non-
identity) elements which do not have order 2 must have order 4. [3]
(v) State whether or not H is cyclic. You must give a reason for your answer.
[1]
7 In this question you must show detailed reasoning.
A curve is defined parametrically by x = cos3t , y = sin3t , for 0 G Gt r. When the curve is
rotated through 2r about the x-axis, a surface of revolution is formed with surface area A.
Determine the exact value of A.
GCSE
Further Mathematics A
Y545/01: Additional Pure Mathematics
*1920573537*
A 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 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 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 2025 [D/508/5514] DC (DE) 359664/3
, OCR is an exempt Charity
Turn over
1 A student claims that the order of 5 modulo 37 is 5.
Without calculation, explain why this student’s claim must be incorrect.
(b) Show that the order of 5 modulo 37 is 36.
2 The points P, Q, R and S have coordinates (1, 2, 4), (3, 1, 6), (4, 8, -2) and (5, 5, 5)
respectively.
(a) Find, in exact form, the area of triangle PQR. [3]
(b) Find, in exact form, the volume of tetrahedron PQRS. [3]
(c) Use your answers to part (a) and part (b) to deduce the shortest distance from S to
the plane containing P, Q and R. Give your answer in exact form.
[You may use the formula V = Ah, where V is the volume of a pyramid, A is the area of
one
of its bases and h is the height relative to the base.]
3 Let N = +10a b and M = -a 3b , where a and b are positive integers and 0 G Gb
9.
(a) By considering 3N + M , prove that 31 | N if and only if 31 | M. [4]
(b) Use a procedure based on the result of part (a) to show that 31 | 7 167 758. [2]
4 In this question you must show detailed reasoning.
The surface S is given by the equation z = 4x2y -3xy2-5x for all real values of x and y.
(a) The point P(1, 1, -4) lies on S.
Find the equation of the tangent plane to S at P.
(b) The point Q is a stationary point of S with positive x- and y-coordinates.
(i) Find, in exact form, the coordinates of Q. [5]
(ii) Find the value of the determinant of H, the Hessian matrix of S, at Q.
[4]
, (iii) Deduce the nature of the stationary point Q. [1] 5 Determine the
/
solution of the linear congruence 3x 7 (mod 13). Give your answer in the
form x / a (mod 13), where 0 G a 1 13.
(b) Hence or otherwise, determine the solution of the simultaneous linear congruences
3x / 7 (mod 13), 7x / 3 (mod 31).
6 In this question, the set S is {2, 4, 6, 8}, and #10 is the operation of multiplication
modulo 10.
(a) The group G consists of the set S, under the operation #10.
(i) Complete the Cayley table for G given in the Printed Answer Booklet.
[1]
(ii) State the identity element of G. [1]
(iii) State the order of each non-identity element of G. [2]
0
(b) The group H consists of all 2#2 matrices of the form Ja NOO, where a, b ! S , under the
KKL0 bP
operation of matrix multiplication (using #10 to multiply elements of S ).
(i) State, with justification, the order of H. [2]
(ii) State also the identity element of H. [1]
(iii) State all the elements of H of order 2. [2]
(iv) Using the result of part (a)(iii), or otherwise, explain why all the (non-
identity) elements which do not have order 2 must have order 4. [3]
(v) State whether or not H is cyclic. You must give a reason for your answer.
[1]
7 In this question you must show detailed reasoning.
A curve is defined parametrically by x = cos3t , y = sin3t , for 0 G Gt r. When the curve is
rotated through 2r about the x-axis, a surface of revolution is formed with surface area A.
Determine the exact value of A.
