AS LEVEL FURTHER MATHEMATICS A: Actual May 2025 PAST PAPER 1: Pure Core. All
Assessment Questions & Mark Scheme [OCR Y531/01]
Exam Resource Summary
The AS Level Further Mathematics A May 2025 Pure Core Paper (OCR Y531/01) combines the full
official examination paper with its detailed mark scheme to provide a focused and practical revision
resource. This paper assesses students’ understanding of core pure mathematics topics, including algebra,
functions, trigonometry, calculus, and coordinate geometry, emphasizing problem-solving and logical
reasoning. By presenting each question alongside its marking criteria, the resource offers clear guidance
on examiner expectations, method accuracy, and the structured working required to achieve high-band
marks. This integrated format supports targeted revision, strengthens analytical and computational skills,
and prepares students for success in the May/June 2026 OCR AS Level Further Mathematics A Pure
Core examination.
OCR is an exempt Charity
Turn over
, 2
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]
J 2N J-4N
K O K O
2 Two vectors, a and b, are given by a = KK-3OO and b = K 6O where p is a constant.
13 K pO
L P L P
(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.
• a2 + b2
1+1
• a b
[4]
© OCR 2025 Y531/01 Jun25
, 3
4 Two transformations, TA and TB, are represented by matrices A and B respectively.
J0 1N
The matrix A is given by A = K
0O
.
1
L P
(a) (i) Describe the transformation TA. [1]
(ii) Explain geometrically why A-1 = A. [1]
1 J 1 - 3N
The matrix B is given by B = K O.
2L 3 1 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]
5 The locus L is defined by L = "z | z e C, z - (20 + 15i) ≤ 7,.
(a) On the Argand diagram in the Printed Answer Booklet, sketch and label L. [2]
(b) Determine the value of z e 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 e L. [3]
J 16N J 2N J 3N J 1N
6 The equations of two lines, l and l , are l | r K O + mK O and l | r = K O + n K O.
1 2 1 = KK-1OO KK-27OO 2 K
K 10O
O K
K 10OO
3
L P L-19P L-10P L10P
(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]
© OCR 2025 Y531/01 Jun25 Turn over
Assessment Questions & Mark Scheme [OCR Y531/01]
Exam Resource Summary
The AS Level Further Mathematics A May 2025 Pure Core Paper (OCR Y531/01) combines the full
official examination paper with its detailed mark scheme to provide a focused and practical revision
resource. This paper assesses students’ understanding of core pure mathematics topics, including algebra,
functions, trigonometry, calculus, and coordinate geometry, emphasizing problem-solving and logical
reasoning. By presenting each question alongside its marking criteria, the resource offers clear guidance
on examiner expectations, method accuracy, and the structured working required to achieve high-band
marks. This integrated format supports targeted revision, strengthens analytical and computational skills,
and prepares students for success in the May/June 2026 OCR AS Level Further Mathematics A Pure
Core examination.
OCR is an exempt Charity
Turn over
, 2
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]
J 2N J-4N
K O K O
2 Two vectors, a and b, are given by a = KK-3OO and b = K 6O where p is a constant.
13 K pO
L P L P
(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.
• a2 + b2
1+1
• a b
[4]
© OCR 2025 Y531/01 Jun25
, 3
4 Two transformations, TA and TB, are represented by matrices A and B respectively.
J0 1N
The matrix A is given by A = K
0O
.
1
L P
(a) (i) Describe the transformation TA. [1]
(ii) Explain geometrically why A-1 = A. [1]
1 J 1 - 3N
The matrix B is given by B = K O.
2L 3 1 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]
5 The locus L is defined by L = "z | z e C, z - (20 + 15i) ≤ 7,.
(a) On the Argand diagram in the Printed Answer Booklet, sketch and label L. [2]
(b) Determine the value of z e 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 e L. [3]
J 16N J 2N J 3N J 1N
6 The equations of two lines, l and l , are l | r K O + mK O and l | r = K O + n K O.
1 2 1 = KK-1OO KK-27OO 2 K
K 10O
O K
K 10OO
3
L P L-19P L-10P L10P
(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]
© OCR 2025 Y531/01 Jun25 Turn over
