OCR A Level Mathematics A H240/03 Pure Mathematics
and Mechanics Combined Question paper & Marking
Scheme A+ June 2025
Oxford Cambridge and RSA
Thursday 19 June 2025 – Afternoon
A Level Mathematics A
H240/03 Pure Mathematics and Mechanics
Time allowed: 2 hours
You must have:
• the Printed Answer Booklet
• 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 m s–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 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.
Turn over
, 2
Formulae
A Level Mathematics A (H240)
Arithmetic series
S = 1 n^a + lh = 1 n"2a +^n - 1hd,
n 2 2
Geometric series
a^1 - rnh
Sn =
1 -r
a for r 1 1
S3=
1 -r
Binomial series
^a + bhn = an + nC1 a n - 1b + nC2 a n - 2b2 + f + nCr a n - rbr + f + bn ^n e Nh
JnN= n!
n Cr = n Cr = Kr O
wher r!^n - rh!
e
LP
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
u dy v du - u dv
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
Ipnatretgsration by ;u dx = uv - ; v dx
dx dx
Small angle approximations
© OCR 2025 H240/03 Jun25
, 3
1
sin i ≈ i, cos i ≈ 1 - i , tan i ≈ i where i is measured in radians
2
2
© OCR 2025 H240/03 Jun25 Turn over
, 4
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 = 1 " tan A tan B aA ! B ! ^k + 1hrk
2
Numerical methods
b-a
Trapezium rule: y 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 U Bh = P^Ah +P^Bh - P^A + Bh
P^A + Bh
P^A + Bh = P^AhP^B Ah = P^BhP^A Bh or P^A Bh =
P^Bh
Standard deviation
2 2
xh f ^x xh
= n or =
n
The binomial distribution
JnN
x ^1 - phn-x , mean of X is np, variance of X is np^1 - ph
If X + B^n, ph then P^X = xh = K Op
x
LP
Hypothesis test 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
nL n
Pv
Percentage points of the normal distribution
If Z has a normal distribution with mean 0 and variance 1 then, for each value of p, the table gives the
value of z such that P^Z G zh = p.
p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995
z 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291
Kinematics
Motion in a straight line Motion in two dimensions
v = u + at s = vt - 1 at2
s = ut + 1 at2
2
s = 12^u + vht
v2 = u2 + 2as
© OCR 2025 2 H240/03 Jun25
and Mechanics Combined Question paper & Marking
Scheme A+ June 2025
Oxford Cambridge and RSA
Thursday 19 June 2025 – Afternoon
A Level Mathematics A
H240/03 Pure Mathematics and Mechanics
Time allowed: 2 hours
You must have:
• the Printed Answer Booklet
• 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 m s–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 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.
Turn over
, 2
Formulae
A Level Mathematics A (H240)
Arithmetic series
S = 1 n^a + lh = 1 n"2a +^n - 1hd,
n 2 2
Geometric series
a^1 - rnh
Sn =
1 -r
a for r 1 1
S3=
1 -r
Binomial series
^a + bhn = an + nC1 a n - 1b + nC2 a n - 2b2 + f + nCr a n - rbr + f + bn ^n e Nh
JnN= n!
n Cr = n Cr = Kr O
wher r!^n - rh!
e
LP
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
u dy v du - u dv
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
Ipnatretgsration by ;u dx = uv - ; v dx
dx dx
Small angle approximations
© OCR 2025 H240/03 Jun25
, 3
1
sin i ≈ i, cos i ≈ 1 - i , tan i ≈ i where i is measured in radians
2
2
© OCR 2025 H240/03 Jun25 Turn over
, 4
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 = 1 " tan A tan B aA ! B ! ^k + 1hrk
2
Numerical methods
b-a
Trapezium rule: y 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 U Bh = P^Ah +P^Bh - P^A + Bh
P^A + Bh
P^A + Bh = P^AhP^B Ah = P^BhP^A Bh or P^A Bh =
P^Bh
Standard deviation
2 2
xh f ^x xh
= n or =
n
The binomial distribution
JnN
x ^1 - phn-x , mean of X is np, variance of X is np^1 - ph
If X + B^n, ph then P^X = xh = K Op
x
LP
Hypothesis test 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
nL n
Pv
Percentage points of the normal distribution
If Z has a normal distribution with mean 0 and variance 1 then, for each value of p, the table gives the
value of z such that P^Z G zh = p.
p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995
z 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291
Kinematics
Motion in a straight line Motion in two dimensions
v = u + at s = vt - 1 at2
s = ut + 1 at2
2
s = 12^u + vht
v2 = u2 + 2as
© OCR 2025 2 H240/03 Jun25
