A LEVEL FURTHER MATHEMATICS A: Actual June 2025 PAST PAPER 1: Pure Core 2. All
Assessment Questions & Mark Scheme [OCR Y541/01]
Exam Resource Summary
The A Level Further Mathematics A June 2025 Pure Core 2 Paper (OCR Y541/01) integrates the full
official examination paper with its detailed mark scheme to provide a focused and practical revision
resource. This paper assesses students’ understanding of advanced pure mathematics topics, including
complex numbers, matrices, vectors, calculus, and algebraic methods, 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 combined format supports targeted revision, strengthens analytical and
computational skills, and prepares students for success in the May/June 2026 OCR A Level Further
Mathematics A Pure Core 2 examination.
OCR is an exempt Charity
Turn over
, 2
1 In this question you must show detailed reasoning.
JK 2NO K N
J O
Vectors a and b are given by a and b = -1 .
= K-4O K 3O
K O K O
3
L P L-2P
Determine, in either order
• a.b
• a×b. [3]
2 You are given that -7 - 5i is one root of the equation x3 + 10x2 + 18x - 296 = 0.
(a) Write down another complex root of the equation x3 + 10x2 + 18x - 296 = 0. [1]
(b) Using your answer to part (a), express x3 + 10x2 + 18x - 296 as a product of a real linear
factor and a real quadratic factor. [3]
JK1 -3NO JK- 3 -3NO
3 Matrices A and B are given by A = K O and B = K 1 2O.
L 4 8P L P
(a) Find the matrix AB. [1]
(b) Verify that det(AB) = det (A) # det(B). [2]
(c) Use matrices A and B to demonstrate that matrix multiplication is not commutative. [2]
The transformation represented by matrix A is denoted by T.
(d) Show that the point (2, -5) is not an invariant point under T. [2]
(e) Find the matrix which represents the inverse transformation of T. [2]
© OCR 2025 Y541/01 Jun25
, 3
4 In this question you must show detailed reasoning.
Determine the sum of all cube numbers from 216 to 512 000 inclusive. [4]
5 In this question you must show detailed reasoning.
(a) Use an algebraic method to determine the two square roots of -3 + (4 7) i.
Give your answers in the form a + bi where a and b are exact. [5]
(b) State the relationship between the two arguments of the two square roots found in part (a). [1]
6 One of the regions bounded by two polar curves, C1 and C2 , is used to model the face of a flat
earring.
The polar equations of C1 and C2 are
C1: r = 2i
C2: r = i2
where 0 G i G r.
The curves C1 and C2 are shown in the diagram below with the region used to model the earring
shaded.
C2
C1
O Initial line
You are given that C1 and C2 intersect at the pole O.
(a) Find the other point of intersection of C1 and C2 . Give your answer in polar coordinates. [2]
(b) In this question you must show detailed reasoning.
Determine the area of the face of the earring. [4]
© OCR 2025 Y541/01 Jun25 Turn over
, 4
7 In this question you must show detailed reasoning.
-4x2 + 5x - 17
(a) Express in partial fractions. [6]
x3 - x2 + 3x - 3
y
3
-4x2 + 5x - 17 d x .
(b) Hence determine the exact value of [4]
3 x3 - x2 + 3x - 3
8 A function f (x) is defined by f (x) = xsinh 2x.
(a) d2n f d0 f
Prove by induction that n
= 4 (xsinh 2x + ncosh 2x) for n H 0 where is defined as
d x2n d x0
being equal to f (x). [6]
(b) Using the formula given in part (a), determine the exact value of the coefficient of x8 in the
Maclaurin series for xsinh 2x. [3]
(c) Use the Maclaurin series for ex to verify your answer to part (b). [3]
© OCR 2025 Y541/01 Jun25
Assessment Questions & Mark Scheme [OCR Y541/01]
Exam Resource Summary
The A Level Further Mathematics A June 2025 Pure Core 2 Paper (OCR Y541/01) integrates the full
official examination paper with its detailed mark scheme to provide a focused and practical revision
resource. This paper assesses students’ understanding of advanced pure mathematics topics, including
complex numbers, matrices, vectors, calculus, and algebraic methods, 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 combined format supports targeted revision, strengthens analytical and
computational skills, and prepares students for success in the May/June 2026 OCR A Level Further
Mathematics A Pure Core 2 examination.
OCR is an exempt Charity
Turn over
, 2
1 In this question you must show detailed reasoning.
JK 2NO K N
J O
Vectors a and b are given by a and b = -1 .
= K-4O K 3O
K O K O
3
L P L-2P
Determine, in either order
• a.b
• a×b. [3]
2 You are given that -7 - 5i is one root of the equation x3 + 10x2 + 18x - 296 = 0.
(a) Write down another complex root of the equation x3 + 10x2 + 18x - 296 = 0. [1]
(b) Using your answer to part (a), express x3 + 10x2 + 18x - 296 as a product of a real linear
factor and a real quadratic factor. [3]
JK1 -3NO JK- 3 -3NO
3 Matrices A and B are given by A = K O and B = K 1 2O.
L 4 8P L P
(a) Find the matrix AB. [1]
(b) Verify that det(AB) = det (A) # det(B). [2]
(c) Use matrices A and B to demonstrate that matrix multiplication is not commutative. [2]
The transformation represented by matrix A is denoted by T.
(d) Show that the point (2, -5) is not an invariant point under T. [2]
(e) Find the matrix which represents the inverse transformation of T. [2]
© OCR 2025 Y541/01 Jun25
, 3
4 In this question you must show detailed reasoning.
Determine the sum of all cube numbers from 216 to 512 000 inclusive. [4]
5 In this question you must show detailed reasoning.
(a) Use an algebraic method to determine the two square roots of -3 + (4 7) i.
Give your answers in the form a + bi where a and b are exact. [5]
(b) State the relationship between the two arguments of the two square roots found in part (a). [1]
6 One of the regions bounded by two polar curves, C1 and C2 , is used to model the face of a flat
earring.
The polar equations of C1 and C2 are
C1: r = 2i
C2: r = i2
where 0 G i G r.
The curves C1 and C2 are shown in the diagram below with the region used to model the earring
shaded.
C2
C1
O Initial line
You are given that C1 and C2 intersect at the pole O.
(a) Find the other point of intersection of C1 and C2 . Give your answer in polar coordinates. [2]
(b) In this question you must show detailed reasoning.
Determine the area of the face of the earring. [4]
© OCR 2025 Y541/01 Jun25 Turn over
, 4
7 In this question you must show detailed reasoning.
-4x2 + 5x - 17
(a) Express in partial fractions. [6]
x3 - x2 + 3x - 3
y
3
-4x2 + 5x - 17 d x .
(b) Hence determine the exact value of [4]
3 x3 - x2 + 3x - 3
8 A function f (x) is defined by f (x) = xsinh 2x.
(a) d2n f d0 f
Prove by induction that n
= 4 (xsinh 2x + ncosh 2x) for n H 0 where is defined as
d x2n d x0
being equal to f (x). [6]
(b) Using the formula given in part (a), determine the exact value of the coefficient of x8 in the
Maclaurin series for xsinh 2x. [3]
(c) Use the Maclaurin series for ex to verify your answer to part (b). [3]
© OCR 2025 Y541/01 Jun25
