2025 OCR A Level Mathematics B (MEI)
H640/03 Pure Mathematics and Comprehension
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Thursday 19 June 2025 – Afternoon
A Level Mathematics B (MEI)
H640/03 Pure Mathematics and Comprehension
Time allowed: 2 hours
You must have:
• the Printed Answer Booklet
• the Insert
QP
• 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 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 75.
• The marks for each question are shown in brackets [ ].
• This document has 12 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [603/1002/9] OCR is an exempt Charity
DC (ST/TC) 356484/5 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
S3= for r 1 1
1-r
Binomial series
^a + bhn = an + nC1 a n-1 n n-2 2 n
JnN b + C2 a b +f+ Cr a b +f+ b
n-r r n ^n e Nh,
n n!
where Cr = n Cr = K O =
r
L P r!^n - rh!
n^n - 1h 2 n^n - 1h f ^n - r + 1h r ^ x 1 1, n e Rh
^1 + xhn = 1 + nx + x +f+ x +f
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 H640/03 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
tan^A ! Bh = aA ! B ! ^k + 1h2rk
1 " tan A tan B
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 at2 s = ut + 12at2
2
s = 12^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12at2 s = vt - 12at2
© OCR 2025 H640/03 Jun25 Turn over
, 4
Section A (60 marks)
1
1 (a) Find the first three terms in the binomial expansion of (1 + 3x) 2 . [3]
(b) State the range of values of x for which this expansion is valid. [1]
2 You are given that
3
f (x) = x on the domain {x : x d R, x C 0}
g(x) = x - 1 on the domain {x : x d R}.
Find gf(x). You must state the domain. [2]
3 The straight line with equation 2x + y = 6 crosses the x‑axis at A and the y‑axis at B.
(a) Draw the line with equation 2x + y = 6 on the grid in the Printed Answer Booklet. [1]
(b) Determine the equation of the perpendicular bisector of AB. [6]
(c) Points A and B are opposite vertices of a square of side a.
Determine the exact value of a. [3]
4 The first term of a geometric sequence is
6. The fourth term 9is 2 .
The sequence is infinite.
Find the sum of the associated series. [3]
© OCR 2025 H640/03 Jun25
H640/03 Pure Mathematics and Comprehension
Verified Question paper with Marking Scheme Attached
Oxford Cambridge and RSA
Thursday 19 June 2025 – Afternoon
A Level Mathematics B (MEI)
H640/03 Pure Mathematics and Comprehension
Time allowed: 2 hours
You must have:
• the Printed Answer Booklet
• the Insert
QP
• 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 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 75.
• The marks for each question are shown in brackets [ ].
• This document has 12 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [603/1002/9] OCR is an exempt Charity
DC (ST/TC) 356484/5 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
S3= for r 1 1
1-r
Binomial series
^a + bhn = an + nC1 a n-1 n n-2 2 n
JnN b + C2 a b +f+ Cr a b +f+ b
n-r r n ^n e Nh,
n n!
where Cr = n Cr = K O =
r
L P r!^n - rh!
n^n - 1h 2 n^n - 1h f ^n - r + 1h r ^ x 1 1, n e Rh
^1 + xhn = 1 + nx + x +f+ x +f
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 H640/03 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
tan^A ! Bh = aA ! B ! ^k + 1h2rk
1 " tan A tan B
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 at2 s = ut + 12at2
2
s = 12^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12at2 s = vt - 12at2
© OCR 2025 H640/03 Jun25 Turn over
, 4
Section A (60 marks)
1
1 (a) Find the first three terms in the binomial expansion of (1 + 3x) 2 . [3]
(b) State the range of values of x for which this expansion is valid. [1]
2 You are given that
3
f (x) = x on the domain {x : x d R, x C 0}
g(x) = x - 1 on the domain {x : x d R}.
Find gf(x). You must state the domain. [2]
3 The straight line with equation 2x + y = 6 crosses the x‑axis at A and the y‑axis at B.
(a) Draw the line with equation 2x + y = 6 on the grid in the Printed Answer Booklet. [1]
(b) Determine the equation of the perpendicular bisector of AB. [6]
(c) Points A and B are opposite vertices of a square of side a.
Determine the exact value of a. [3]
4 The first term of a geometric sequence is
6. The fourth term 9is 2 .
The sequence is infinite.
Find the sum of the associated series. [3]
© OCR 2025 H640/03 Jun25
