A LEVEL MATHEMATICS B (MEI): Actual June 2025 PAST PAPER 2: Pure Mathematics and
Statistics. All Assessment Questions & Mark Scheme [OCR H640/02]
Exam Resource Summary
The A Level Mathematics B (MEI) June 2025 Pure Mathematics and Statistics Paper (OCR H640/02)
combines the full official examination paper with its detailed mark scheme to create a focused and
practical revision resource. This paper assesses students’ understanding of core topics in pure
mathematics, including algebra, calculus, and trigonometry, alongside statistical concepts such as
probability distributions, hypothesis testing, and data interpretation. By presenting each question
alongside its marking criteria, the resource provides clear guidance on examiner expectations, method
accuracy, and the structured reasoning required for high-band responses. This integrated format supports
targeted revision, strengthens problem-solving and analytical skills, and equips students for success in the
May/June 2026 OCR A Level Mathematics B (MEI) Pure Mathematics and Statistics examination.
OCR is an exempt Charity
Turn over
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
S = 1 n^a + lh = 1 n"2a +^n - 1hd,
n 2 2
Geometric series
a^1 - rnh
Sn = 1 - r
a
S = for r 1 1
3 1- r
Binomial series
^a + bhn = an + nC1JaNn-1b + nC2 a n-2b2 +f+ nCr a n-rbr +f+ bn ^n e Nh,
n n n!
where Cr = n Cr = K O =
L P r!^n - rh!
r
n^n - 1h 2 n^n - 1h f ^n - r + 1h r
^1 + xhn = 1 + nx + x +f+ x +f ^ x 1 1, n e Rh
2! r!
Differentiation
f^xh f l^xh
tan kx k sec2kx
sec x sec x tan x
cot x -cosec2x
cosec x -cosec x cot x
v du - u dv
u dy
Quotient Rule y = v , = dx 2 dx
dx v
Differentiation from first principles
f^x + hh - f^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
d dx = ln f^xh + c
e f ^ xh
n 1 n +1
; f l^xhaf^xhk dx =n + 1af^xhk + c
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i ≈ i , cos i ≈ 1 - 12i 2 , tan i ≈ i where i is measured in radians
© OCR 2025 H640/02 Jun25
, 3
Trigonometric identities
sin^A ! Bh = sin A cos B ! cos A sin B
cos^A ! Bh = cos A cos B " sin A sin B
tan A ! tan B a
tan^A ! Bh = A ! B ! ^k + 1hrk
1 " tan A tan B 2
Numerical methods
b-a
Trapezium rule: ; b y dx ≈ 1 h"^y + y h + 2^y + y +f+ y h,, where h =
a
2 0 n 1 2 n-1 n
f^xnh
The Newton-Raphson iteration for solving f^xh = 0: x n +1 = xn -
f l^xnh
Probability
P^A j Bh = P^Ah +P^Bh - P^A k Bh
P^A k Bh
P^A k Bh = P^AhP^B Ah = P^BhP^A Bh or P^A Bh =
P^Bh
Sample variance
2 1 2 ^/ xih2 2
s = n 1 Sxx where Sxx = /^xi - -xh = / x2i - n = / xi2 - n-x
-
Standard deviation, s = variance
The binomial distribution
If X + B^n, ph then P^X = rh = nCr p r q n-r where q = 1 - p
Mean of X is np
Hypothesis testing for the mean of a Normal distribution
J v2N X -n
2
If X + N^n, v h then X + NKn, O and ~ N^0, 1h
n v n
L P
Percentage points of the Normal distribution
p 10 5 2 1
1 p% 1 p%
z 1.645 1.960 2.326 2.576 2 2
z
Kinematics
Motion in a straight line Motion in two dimensions
v = u + at v = u + at
s = ut + 21 at2 s = ut + 12 at2
s = 12^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12 at2 s = vt - 12 at2
© OCR 2025 H640/02 Jun25 Turn over
, 4
Section A (23 marks)
1 The equation of a circle is (x - 4) 2 + ( y + 5) 2 - 64 = 0.
(a) State the coordinates of the centre of the circle. [1]
(b) State the radius of the circle. [1]
2 Determine the coefficient of x3 in the expansion of (1 + 2x) 12. [2]
3 The diagram shows triangle ABC.
C Not to scale
A B
AB = 4.7 cm, AC = 6.9 cm and BC = 11.2 cm.
Find the size of angle CAB. Give your answer correct to 3 significant figures. [2]
4 (a) Use the factor theorem to show that (x - 3) is a factor of 4x3 - 8x2 - 11x - 3. [1]
(b) Hence show that 4x3 - 8x2 - 11x - 3 = (x - 3)(ax + b) 2 , where a and b are integers to be
determined. [2]
(c) Sketch the graph of y = 4x3 - 8x2 - 11x - 3 on the axes in the Printed Answer Booklet. [2]
5 Prove that the sum of the first n positive odd numbers is a square number. [3]
© OCR 2025 H640/02 Jun25
Statistics. All Assessment Questions & Mark Scheme [OCR H640/02]
Exam Resource Summary
The A Level Mathematics B (MEI) June 2025 Pure Mathematics and Statistics Paper (OCR H640/02)
combines the full official examination paper with its detailed mark scheme to create a focused and
practical revision resource. This paper assesses students’ understanding of core topics in pure
mathematics, including algebra, calculus, and trigonometry, alongside statistical concepts such as
probability distributions, hypothesis testing, and data interpretation. By presenting each question
alongside its marking criteria, the resource provides clear guidance on examiner expectations, method
accuracy, and the structured reasoning required for high-band responses. This integrated format supports
targeted revision, strengthens problem-solving and analytical skills, and equips students for success in the
May/June 2026 OCR A Level Mathematics B (MEI) Pure Mathematics and Statistics examination.
OCR is an exempt Charity
Turn over
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
S = 1 n^a + lh = 1 n"2a +^n - 1hd,
n 2 2
Geometric series
a^1 - rnh
Sn = 1 - r
a
S = for r 1 1
3 1- r
Binomial series
^a + bhn = an + nC1JaNn-1b + nC2 a n-2b2 +f+ nCr a n-rbr +f+ bn ^n e Nh,
n n n!
where Cr = n Cr = K O =
L P r!^n - rh!
r
n^n - 1h 2 n^n - 1h f ^n - r + 1h r
^1 + xhn = 1 + nx + x +f+ x +f ^ x 1 1, n e Rh
2! r!
Differentiation
f^xh f l^xh
tan kx k sec2kx
sec x sec x tan x
cot x -cosec2x
cosec x -cosec x cot x
v du - u dv
u dy
Quotient Rule y = v , = dx 2 dx
dx v
Differentiation from first principles
f^x + hh - f^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
d dx = ln f^xh + c
e f ^ xh
n 1 n +1
; f l^xhaf^xhk dx =n + 1af^xhk + c
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i ≈ i , cos i ≈ 1 - 12i 2 , tan i ≈ i where i is measured in radians
© OCR 2025 H640/02 Jun25
, 3
Trigonometric identities
sin^A ! Bh = sin A cos B ! cos A sin B
cos^A ! Bh = cos A cos B " sin A sin B
tan A ! tan B a
tan^A ! Bh = A ! B ! ^k + 1hrk
1 " tan A tan B 2
Numerical methods
b-a
Trapezium rule: ; b y dx ≈ 1 h"^y + y h + 2^y + y +f+ y h,, where h =
a
2 0 n 1 2 n-1 n
f^xnh
The Newton-Raphson iteration for solving f^xh = 0: x n +1 = xn -
f l^xnh
Probability
P^A j Bh = P^Ah +P^Bh - P^A k Bh
P^A k Bh
P^A k Bh = P^AhP^B Ah = P^BhP^A Bh or P^A Bh =
P^Bh
Sample variance
2 1 2 ^/ xih2 2
s = n 1 Sxx where Sxx = /^xi - -xh = / x2i - n = / xi2 - n-x
-
Standard deviation, s = variance
The binomial distribution
If X + B^n, ph then P^X = rh = nCr p r q n-r where q = 1 - p
Mean of X is np
Hypothesis testing for the mean of a Normal distribution
J v2N X -n
2
If X + N^n, v h then X + NKn, O and ~ N^0, 1h
n v n
L P
Percentage points of the Normal distribution
p 10 5 2 1
1 p% 1 p%
z 1.645 1.960 2.326 2.576 2 2
z
Kinematics
Motion in a straight line Motion in two dimensions
v = u + at v = u + at
s = ut + 21 at2 s = ut + 12 at2
s = 12^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12 at2 s = vt - 12 at2
© OCR 2025 H640/02 Jun25 Turn over
, 4
Section A (23 marks)
1 The equation of a circle is (x - 4) 2 + ( y + 5) 2 - 64 = 0.
(a) State the coordinates of the centre of the circle. [1]
(b) State the radius of the circle. [1]
2 Determine the coefficient of x3 in the expansion of (1 + 2x) 12. [2]
3 The diagram shows triangle ABC.
C Not to scale
A B
AB = 4.7 cm, AC = 6.9 cm and BC = 11.2 cm.
Find the size of angle CAB. Give your answer correct to 3 significant figures. [2]
4 (a) Use the factor theorem to show that (x - 3) is a factor of 4x3 - 8x2 - 11x - 3. [1]
(b) Hence show that 4x3 - 8x2 - 11x - 3 = (x - 3)(ax + b) 2 , where a and b are integers to be
determined. [2]
(c) Sketch the graph of y = 4x3 - 8x2 - 11x - 3 on the axes in the Printed Answer Booklet. [2]
5 Prove that the sum of the first n positive odd numbers is a square number. [3]
© OCR 2025 H640/02 Jun25
