GCSE
Further Mathematics A
Y531/01: Pure Core
*1919766141*
AS 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 page 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 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [H/508/5496] DC (PQ) 357889/2
OCR is an exempt Charity
Turn over
1 (a) The complex number z is such that z = 7 and arg( )z = 2 2. radians.
Express z in cartesian form. [3
]
(b) Use an algebraic method to determine the exact square roots of 1+^4 3hi. [5
]
, 2
JK 2NO JK-4NO
2 Two vectors, a and b, are given by a = -
K O K 6 O
K 133 OP and b=L K p OP where p is a
constant. L
(a) Find expressions in terms of p for each of the following.
• a.b
• a#b
[3
]
(b) Hence or otherwise find the value of p in each of the following cases.
• a and b are perpendicular
• a and b are parallel
[2
]
3 The roots of the equation 2x2+ + =3x 5 0 are denoted by a and b.
(a) Write down the value of a b+ and the value of ab. [2
]
(b) Using the answers to part (a) determine the value of each of the following.
• a b2+ 2
• a1 +b1 [4
]
4 Two transformations, TA and TB, are represented by matrices A and B respectively.
The matrix A is given by A =JKKL01 10NOOP.
(a) (i) Describe the transformation TA. [1]
(ii) Explain geometrically why A-1 = A. [1]
L P
(b) Describe the transformation TB. [2]
The transformation TC is equivalent to TA followed by TB.
(c) Determine the single matrix which represents TC. [2]
© OCR 2025 Y531/01 Jun25
, 3
2
The matrix B is given by B=1 JKK13 -13NOO.
5 The locus L is defined by L ="z|z !C, z- +(20 15i)G 7,.
(a) On the Argand diagram in the Printed Answer Booklet, sketch and label L.
[2] (b) Determine the value of z ! L for which the value of z is smallest. Give your
answer in
cartesian form. [3]
(c) Determine the largest value of arg(z) for z ! L. [3]
JK 16NO JK 2NO JK 3NO
JK 1NO
m
6 The equations of two lines, l1 and l2, are l1|r = -LKK 13OOP+ - KKL-2719OPO and l2|
n
r =KKL-1010OPO+ KLK1010OPO. (a) Show that l1 and l2 intersect at a single
point, P, giving the coordinates of P. [5]
O is the origin of the coordinate system. The point Q lies on the line segment OP.
(b) Comment on the claim that the distance OQ is less than 100. [2]
Turn over
7 Prove by induction that n! 2 20 for all integers n H 52.
n
[5]
© OCR 2025 Y531/01 Jun25
Further Mathematics A
Y531/01: Pure Core
*1919766141*
AS 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 page 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 4 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [H/508/5496] DC (PQ) 357889/2
OCR is an exempt Charity
Turn over
1 (a) The complex number z is such that z = 7 and arg( )z = 2 2. radians.
Express z in cartesian form. [3
]
(b) Use an algebraic method to determine the exact square roots of 1+^4 3hi. [5
]
, 2
JK 2NO JK-4NO
2 Two vectors, a and b, are given by a = -
K O K 6 O
K 133 OP and b=L K p OP where p is a
constant. L
(a) Find expressions in terms of p for each of the following.
• a.b
• a#b
[3
]
(b) Hence or otherwise find the value of p in each of the following cases.
• a and b are perpendicular
• a and b are parallel
[2
]
3 The roots of the equation 2x2+ + =3x 5 0 are denoted by a and b.
(a) Write down the value of a b+ and the value of ab. [2
]
(b) Using the answers to part (a) determine the value of each of the following.
• a b2+ 2
• a1 +b1 [4
]
4 Two transformations, TA and TB, are represented by matrices A and B respectively.
The matrix A is given by A =JKKL01 10NOOP.
(a) (i) Describe the transformation TA. [1]
(ii) Explain geometrically why A-1 = A. [1]
L P
(b) Describe the transformation TB. [2]
The transformation TC is equivalent to TA followed by TB.
(c) Determine the single matrix which represents TC. [2]
© OCR 2025 Y531/01 Jun25
, 3
2
The matrix B is given by B=1 JKK13 -13NOO.
5 The locus L is defined by L ="z|z !C, z- +(20 15i)G 7,.
(a) On the Argand diagram in the Printed Answer Booklet, sketch and label L.
[2] (b) Determine the value of z ! L for which the value of z is smallest. Give your
answer in
cartesian form. [3]
(c) Determine the largest value of arg(z) for z ! L. [3]
JK 16NO JK 2NO JK 3NO
JK 1NO
m
6 The equations of two lines, l1 and l2, are l1|r = -LKK 13OOP+ - KKL-2719OPO and l2|
n
r =KKL-1010OPO+ KLK1010OPO. (a) Show that l1 and l2 intersect at a single
point, P, giving the coordinates of P. [5]
O is the origin of the coordinate system. The point Q lies on the line segment OP.
(b) Comment on the claim that the distance OQ is less than 100. [2]
Turn over
7 Prove by induction that n! 2 20 for all integers n H 52.
n
[5]
© OCR 2025 Y531/01 Jun25
