OCR A Level Further Mathematics A Pure Core
1 (Y540/01)
Oxford Cambridge and RSA
May 2026 – Afternoon
A Level Further Mathematics A
Y540/01 Pure Core 1
Time allowed: 1 hour 30 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
Mathematics A
• 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 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 ms–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 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
, 2
J5 0N
1 (a) A matrix M is given by M = K0 5O.
L P
Describe the transformation represented by M. [2]
(b) Write down the 2 # 2 matrix that represents a rotation of 90° anticlockwise about the origin.
[1]
(c) Write down the 3 # 3 matrix that represents a reflection in the x–z plane. [1]
dy
2 (a) Given that y = cosh-1b 1 xl, find in terms of x. [1]
3 dx
(b) Determine an equation of the normal to the curve y = cosh-13b1 xl at the point where x = 5.
Give your answer in the form ax + by = c +ln d , where a, b, c and d are integers. [4]
, 3
80
3 The region R1 is bounded by the curve y = , the line x = k and the coordinate axes, as
shown in Fig. 1.
Fig. 1
y
R1
O x
x=k
(a) In this question you must show detailed reasoning.
40
Given that the area of R1 is r, determine the value of k. [3]
3
80
The region R is bounded by the curve y = , the line y = 20 and the y-axis as shown in
2
Fig. 2.
Fig. 2
y
y = 20
R2
O x
(b) Find, in an exact form, the volume of the solid formed when R2 is rotated by 2r radians
about the y-axis. [3]
Turn over
, 4
4 The equation 5x3 - 4x2 + 10 = 0 has roots a, b and c.
(a) Write down the values of a + b + c, ab + bc + ca and abc. [2]
(b) By expanding (ab + bc + ca) 2, determine the value of a2 b2 + b2 c2 + c2 a2. [3]
(c) By expanding a suitable expression, find the value of a2 + b2 + c2. [2]
(d) Hence find a cubic equation with integer coefficients that has roots a2, b2, and c2. [2]
KJ O
N
5 A vector equation of the plane P is r = 1 + J K3 NO KJ 1ON
m +n .
1 K O K O K1O
4 O 1
3 -1 0
LP L P L P
(a) Verify that a cartesian equation of P1 is x - y + 2z = 3. [1]
For some real constant a, cartesian equations of planes P2 and P3 are
P2| x - 3z = 1
P3| ax - y - z = 4
(b) By considering a suitable matrix, show that P1 , P2 and P3 intersect at a single point for all
values of a except a = 2. [3]
(c) Use the matrix from part (b) to find the coordinates of the point of intersection of P1 , P2
and P3 in the case where a = 3. [2]
(d) In the case where a = 2, determine the geometrical arrangement of P1, P2 and P3. [2]
6 In this question you must show detailed reasoning.
J1 1 N +g + J 1 # 1 N for n e Z+ .
The series S is given by S = # 1 N J 1
n n K O+K # O K O
L 5 15P L 15 25P L10n - 5 10n + 5P
1
(a) Use the method of differences to show that, for all n e Z , Sn 1 50.
+
[5]
Let S3 = ln i"m3S n .
(b) Given that, for some value of k, S3 = 1 + S , find the value of k. [3]
2450 k
© OCR 2025 Y540/01 Jun25
1 (Y540/01)
Oxford Cambridge and RSA
May 2026 – Afternoon
A Level Further Mathematics A
Y540/01 Pure Core 1
Time allowed: 1 hour 30 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
Mathematics A
• 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 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 ms–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 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
, 2
J5 0N
1 (a) A matrix M is given by M = K0 5O.
L P
Describe the transformation represented by M. [2]
(b) Write down the 2 # 2 matrix that represents a rotation of 90° anticlockwise about the origin.
[1]
(c) Write down the 3 # 3 matrix that represents a reflection in the x–z plane. [1]
dy
2 (a) Given that y = cosh-1b 1 xl, find in terms of x. [1]
3 dx
(b) Determine an equation of the normal to the curve y = cosh-13b1 xl at the point where x = 5.
Give your answer in the form ax + by = c +ln d , where a, b, c and d are integers. [4]
, 3
80
3 The region R1 is bounded by the curve y = , the line x = k and the coordinate axes, as
shown in Fig. 1.
Fig. 1
y
R1
O x
x=k
(a) In this question you must show detailed reasoning.
40
Given that the area of R1 is r, determine the value of k. [3]
3
80
The region R is bounded by the curve y = , the line y = 20 and the y-axis as shown in
2
Fig. 2.
Fig. 2
y
y = 20
R2
O x
(b) Find, in an exact form, the volume of the solid formed when R2 is rotated by 2r radians
about the y-axis. [3]
Turn over
, 4
4 The equation 5x3 - 4x2 + 10 = 0 has roots a, b and c.
(a) Write down the values of a + b + c, ab + bc + ca and abc. [2]
(b) By expanding (ab + bc + ca) 2, determine the value of a2 b2 + b2 c2 + c2 a2. [3]
(c) By expanding a suitable expression, find the value of a2 + b2 + c2. [2]
(d) Hence find a cubic equation with integer coefficients that has roots a2, b2, and c2. [2]
KJ O
N
5 A vector equation of the plane P is r = 1 + J K3 NO KJ 1ON
m +n .
1 K O K O K1O
4 O 1
3 -1 0
LP L P L P
(a) Verify that a cartesian equation of P1 is x - y + 2z = 3. [1]
For some real constant a, cartesian equations of planes P2 and P3 are
P2| x - 3z = 1
P3| ax - y - z = 4
(b) By considering a suitable matrix, show that P1 , P2 and P3 intersect at a single point for all
values of a except a = 2. [3]
(c) Use the matrix from part (b) to find the coordinates of the point of intersection of P1 , P2
and P3 in the case where a = 3. [2]
(d) In the case where a = 2, determine the geometrical arrangement of P1, P2 and P3. [2]
6 In this question you must show detailed reasoning.
J1 1 N +g + J 1 # 1 N for n e Z+ .
The series S is given by S = # 1 N J 1
n n K O+K # O K O
L 5 15P L 15 25P L10n - 5 10n + 5P
1
(a) Use the method of differences to show that, for all n e Z , Sn 1 50.
+
[5]
Let S3 = ln i"m3S n .
(b) Given that, for some value of k, S3 = 1 + S , find the value of k. [3]
2450 k
© OCR 2025 Y540/01 Jun25
