OCR A Level Mathematics A H240/01 Pure Mathematics Combined Question
Paper and Mark Scheme
Oxford Cambridge and RSA
A Level Mathematics A
H240/01 Pure Mathematics
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 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 8 pages.
ADVICE
• Read each question carefully before you start your answer.
, 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
S3 = for r 1 1
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
JN n!
n C = C = Kn O =
where r n r r r!^n - rh!
L P
n^n - 1h 2 n^n - 1hf^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 + 1af^xhk + c
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i ≈ i , cos i ≈ 1 - 12i2 , tan i ≈ i where i is measured in radians
© OCR 2025 H240/01 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 + 12hrk
1 " tan A tan B
Numerical methods
b b-a
Trapezium rule: y 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 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
= n or =
n
The binomial distribution
JnN
If X + B^n, ph then P^X = xh = K O p x ^1 - phn-x , mean of X is np, variance of X is np^1 - ph
x
L P
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
n v n
L P
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 v = u + at
s = ut + 21 at2 s = ut + 12 at2
s = 21^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12 at2 s = vt - 12 at2
© OCR 2025 H240/01 Jun25 Turn over
, 4
1 Express each of the following in the form pxq , where p and q are constants.
(a) 2
4 [1]
x
(b) ^5x xh3 [2]
(c) 2x3 # 8x5 [2]
(d) 13
[3]
x5^27x6h
2
y
O x
The diagram shows the curve y = ax (x - b), where a and b are constants and b 2 0.
(a) Given that the curve has a stationary point at x = 3, state the value of b. [1]
(b) Given also that the stationary point at x = 3 is a maximum, state what can be deduced about
the value of a. [1]
(c) Find the y-coordinate of the stationary point, giving your answer in terms of a. [1]
(d) State the range of values of x for which the curve is increasing. [1]
2
3 (a) A sequence has terms u1 , u2 , u3 , … defined by u1 = 3 and un +1 = u n - 5 for n H 1.
(i) Find the values of u2 , u3 and u4 . [2]
(ii) Describe the behaviour of the sequence. [1]
16
(b) The second, third and fourth terms of a geometric progression are 12, 8 and 3 .
Determine the sum to infinity of this geometric progression. [3]
© OCR 2025 H240/01 Jun25
Paper and Mark Scheme
Oxford Cambridge and RSA
A Level Mathematics A
H240/01 Pure Mathematics
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 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 8 pages.
ADVICE
• Read each question carefully before you start your answer.
, 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
S3 = for r 1 1
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
JN n!
n C = C = Kn O =
where r n r r r!^n - rh!
L P
n^n - 1h 2 n^n - 1hf^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 + 1af^xhk + c
dv du
Integration by parts ; u dx = uv - ; v dx
dx dx
Small angle approximations
sin i ≈ i , cos i ≈ 1 - 12i2 , tan i ≈ i where i is measured in radians
© OCR 2025 H240/01 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 + 12hrk
1 " tan A tan B
Numerical methods
b b-a
Trapezium rule: y 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 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
= n or =
n
The binomial distribution
JnN
If X + B^n, ph then P^X = xh = K O p x ^1 - phn-x , mean of X is np, variance of X is np^1 - ph
x
L P
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
n v n
L P
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 v = u + at
s = ut + 21 at2 s = ut + 12 at2
s = 21^u + vht s = 12^u + vht
v2 = u2 + 2as
s = vt - 12 at2 s = vt - 12 at2
© OCR 2025 H240/01 Jun25 Turn over
, 4
1 Express each of the following in the form pxq , where p and q are constants.
(a) 2
4 [1]
x
(b) ^5x xh3 [2]
(c) 2x3 # 8x5 [2]
(d) 13
[3]
x5^27x6h
2
y
O x
The diagram shows the curve y = ax (x - b), where a and b are constants and b 2 0.
(a) Given that the curve has a stationary point at x = 3, state the value of b. [1]
(b) Given also that the stationary point at x = 3 is a maximum, state what can be deduced about
the value of a. [1]
(c) Find the y-coordinate of the stationary point, giving your answer in terms of a. [1]
(d) State the range of values of x for which the curve is increasing. [1]
2
3 (a) A sequence has terms u1 , u2 , u3 , … defined by u1 = 3 and un +1 = u n - 5 for n H 1.
(i) Find the values of u2 , u3 and u4 . [2]
(ii) Describe the behaviour of the sequence. [1]
16
(b) The second, third and fourth terms of a geometric progression are 12, 8 and 3 .
Determine the sum to infinity of this geometric progression. [3]
© OCR 2025 H240/01 Jun25
