2025 OCR A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Thursday 12 June 2025 – Afternoon
A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Time allowed: 2 hours
You must have:
• the Printed Answer Booklet
• a scientific or graphical calculator
QP
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 your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 100.
• The marks for each question are shown in brackets [ ].
• This document has 16 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [603/1002/9] OCR is an exempt Charity
DC (DE/CT) 353315/4 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 + nC1 aJ n-N1b + nC2 a n-2b2 +f+ nCr a n- rbr +f+ bn ^n e Nh,
n n n!
where Cr = n Cr = K O =
r
L P r!^n - rh!
n ^n - 1 h 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 +a1f^xhk + c
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i ≈ i , cos i ≈ 1 - 1 i 2 ,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 = xn -
n +1 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 = / x2 i- = / x2i - n-x
- n
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 v ~ N^0, 1h
L n n
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 + 1
2 at2 s = ut + 12at2
s = 12^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12at2 s = vt - 12at2
© 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
H640/02 Pure Mathematics and Statistics
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Thursday 12 June 2025 – Afternoon
A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Time allowed: 2 hours
You must have:
• the Printed Answer Booklet
• a scientific or graphical calculator
QP
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 your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 100.
• The marks for each question are shown in brackets [ ].
• This document has 16 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [603/1002/9] OCR is an exempt Charity
DC (DE/CT) 353315/4 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 + nC1 aJ n-N1b + nC2 a n-2b2 +f+ nCr a n- rbr +f+ bn ^n e Nh,
n n n!
where Cr = n Cr = K O =
r
L P r!^n - rh!
n ^n - 1 h 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 +a1f^xhk + c
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i ≈ i , cos i ≈ 1 - 1 i 2 ,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 = xn -
n +1 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 = / x2 i- = / x2i - n-x
- n
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 v ~ N^0, 1h
L n n
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 + 1
2 at2 s = ut + 12at2
s = 12^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12at2 s = vt - 12at2
© 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
