Thursday 18 June 2026 – Afternoon
A Level Mathematics B (MEI)
H640/03 Pure Mathematics and Comprehension
Time allowed: 2 hours
GCE
*
You must have:
• the Printed Answer Booklet
• the Insert
QP
• a scientific or graphical calculator Mathematics B MEI
H640/03: Pure Mathematics and Comprehension
*
A Level
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 Mark Scheme for June 2026
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.
• You can write on this Question Paper but it will not be sent for marking.
INFORMATION
• The total mark for this paper is 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
Cambridge 2026 [603/1002/9] Cambridge is an exempt Charity
DC (ST/TC) 366168/4 Turn over Cambridge
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
Cambridge is a leading UK awarding body, providing a wide range of qualifications to
S n = 12 n ^a + lh = 12 n "2a + ^n - 1h d , meet the needs of candidates of all ages and abilities. Cambridge qualifications include
AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills,
Geometric series Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT,
a ^1 - r nh business, languages, teaching/training, administration and secretarial skills.
Sn =
1-r
It is also responsible for developing new specifications to meet national requirements and the
a
S3 = for r 1 1 needs of students and teachers. Cambridge is a not-for-profit organisation; any surplus
1-r
made is invested back into the establishment to help towards the development of qualifications
Binomial series and support, which keep pace with the changing needs of today’s society.
^a + bhn = a n + n C1 a n - 1 b + n C2 a n - 2 b 2 + f + n Cr a n - r b r + f + b n ^n ! Nh, This mark scheme is published as an aid to teachers and students, to indicate the requirements
JnN of the examination. It shows the basis on which marks were awarded by examiners. It does not
n!
where C r = n C r = KK OO =
n
indicate the details of the discussions which took place at an examiners’ meeting before marking
L P r! ^n - rh !
r
commenced.
n ^n - 1h 2 n ^n - 1h f ^n - r + 1h r
^1 + xhn = 1 + nx + x +f+ x +f ^ x 1 1, n ! Rh All examiners are instructed that alternative correct answers and unexpected approaches in
2! r!
candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills
Differentiation demonstrated.
f ^xh f l^xh Mark schemes should be read in conjunction with the published question papers and the report
on the examination.
tan kx k sec 2 kx
sec x sec x tan x Cambridge 2026
cot x - cosec 2 x
cosec x - cosec x cot x
du dv
v -u
u dy dx dx
Quotient Rule y = , =
v dx v 2
Differentiation from first principles
f ^x + hh - f ^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
dd dx = ln f ^xh + c
e f ^xh
; f l^xhaf ^xhk dx = n + 1 af ^xhk + c
n 1 n+1
Integration by parts ; u dx = uv - ; v dx
dv du
dx dx
Small angle approximations
sin i . i , cos i . 1 - 12 i 2 , tan i . i where i is measured in radians
Cambridge 2026 H640/03 Jun26
Cambridge
, 3 H640/03 Mark Scheme June 2026
Trigonometric identities
sin ^A ! Bh = sin A cos B ! cos A sin B Marking Instructions
cos ^A ! Bh = cos A cos B " sin A sin B
tan ^A ! Bh = aA ! B ! ^k + 12h rk
tan A ! tan B Preparation For Marking
1 " tan A tan B 1. RM Assessor
Numerical methods • Access and complete the on-screen marking training packages: Examiner Training (RMA3).
Trapezium rule: ; y dx . 12 h "^y 0 + ynh + 2 ^y 1 + y2 + f + yn - 1h, , where h =
b
b-a
n • Read the mark scheme and question paper for this component or unit.
f ^x nh
a
The Newton‑Raphson iteration for solving f ^xh = 0: xn + 1 = x n -
f l^x nh • The mark scheme and question paper are available in RM Assessor or on your Component Page if you use the Training Platform for standardisation.
Probability • Log in to RM Assessor and mark the required number of practice scripts and the required number of standardisation scripts.
P ^A j Bh = P ^Ah + P ^Bh - P ^A k Bh
P ^A k Bh
P ^A k Bh = P ^Ah P ^B Ah = P ^Bh P ^A Bh or P ^A Bh =
P ^Bh
Marking
2. General Guidance
Sample variance
^/ xih2 • Mark strictly to the mark scheme.
S where S xx = /^xi - xh = / x i -
1 2
2
s = - 2
= / x 2i - nx- 2
n - 1 xx n
• Marks awarded must relate directly to the marking criteria.
Standard deviation, s = variance
• If you are in any doubt about applying the mark scheme, consult your Team Leader by phone, email or via the RM Assessor messaging system.
The binomial distribution
If X + B ^n, ph then P ^X = rh = n C r p r q n - r where q = 1 - p • It is essential that you meet the RM Assessor 50% and 100% batch deadlines. For traditional marking this will be 40% and 100%. If you experience
problems, contact your Team Leader without delay.
Mean of X is np
• Always check the pages (and additional objects if present) at the end of the response in case any answers have been continued there. If the candidate
Hypothesis testing for the mean of a Normal distribution has continued an answer there, then add the annotation ‘SEEN’ to confirm that the work has been seen and mark any responses using the annotations in
J N Section 11.
If X + N ^n, v 2h then X + N KKn, OO and
X -n
+ N ^0, 1h
v2
L n P v n
Percentage points of the Normal distribution 2
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
1 2
s = ut + 2 at s = ut + 12 at 2
s = 12 ^u + vh t s = 12 ^u + vh t
v 2 = u 2 + 2as
s = vt - 12 at 2 s = vt - 12 at 2
Cambridge 2026 H640/03 Jun26 Turn over
, 4 H640/03 Mark Scheme June 2026
Section A (60 marks) • The RM Assessor comments box is used by your Team Leader to explain the marking of the practice responses. Use these comments when checking
your practice responses. Do not use the comments box for any other reason.
1 (a) Express 3x 2 + 6x - 2 in the form a (x + p) 2 + q , where a, p and q are constants to be found. [3]
• Before the end of the marking period send a brief report on the performance of candidates to your Team Leader via email. The report should contain
(b) Explain why the minimum value of 3x 2 + 6x - 2 is q. [1] notes on strengths displayed as well as common errors or weaknesses. Constructive criticism of the question paper/mark scheme is also appreciated.
3. No Response and Crossed-out Answers
2 Two vectors, p and s, are given by
JK- 1NO JK2NO Using the No Response (NR) option. Only mark as NR if:
p = KK OO and s = KK OO, where y is a real variable.
L 3P L yP - the answer space is blank
- there is only a comment not related to the question (e.g., ‘can’t do’, ‘don’t know’)
(a) Find the angle that p makes with the positive x‑axis. [2]
- there is only a mark (e.g., a dash, a question mark) which is not an attempt at the question.
(b) Determine the value of y for which the magnitude of 3p + s is a minimum. [3]
Note: Enter 0 marks for an attempt that earns no credit (including copying out the question). Do not use NR.
Crossed-out answers
3 You are given that If a candidate has crossed out an answer and written a clear alternative, do not mark the crossed-out answer.
f (x) = 4 + x 2 on the domain "x: x ! R, x H 0, .
If a candidate has crossed out an answer and not written a clear alternative, mark the crossed-out answer if it is readable.
(a) Find the range of f. [1]
(b) Explain why the function g (x) = x 2 + 4, x ! R does not have an inverse. [1]
Find an expression for f - 1 (x) . [2]
(d) (i) Show that the quadratic equation x 2 + 4 = x has no real roots. [1]
(ii) Hence explain how you can be sure that the equation f (x) = f - 1 (x) has no real roots. [1]
(e) Sketch the graphs of y = f (x) and y = f - 1 (x) on the diagram in the Printed Answer Booklet.
[2]
3
4 This question is about the curve y = x 3 - 4x 2 - 3x + 2 .
(a) In this question you must show detailed reasoning.
Find the equation of the tangent to the curve at the point (-1, 0). [4]
(b) Determine the exact coordinates of the other point on the curve where the tangent is parallel
to the tangent found in part (a). [3]
Cambridge 2026 H640/03 Jun26
A Level Mathematics B (MEI)
H640/03 Pure Mathematics and Comprehension
Time allowed: 2 hours
GCE
*
You must have:
• the Printed Answer Booklet
• the Insert
QP
• a scientific or graphical calculator Mathematics B MEI
H640/03: Pure Mathematics and Comprehension
*
A Level
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 Mark Scheme for June 2026
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.
• You can write on this Question Paper but it will not be sent for marking.
INFORMATION
• The total mark for this paper is 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
Cambridge 2026 [603/1002/9] Cambridge is an exempt Charity
DC (ST/TC) 366168/4 Turn over Cambridge
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
Cambridge is a leading UK awarding body, providing a wide range of qualifications to
S n = 12 n ^a + lh = 12 n "2a + ^n - 1h d , meet the needs of candidates of all ages and abilities. Cambridge qualifications include
AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills,
Geometric series Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT,
a ^1 - r nh business, languages, teaching/training, administration and secretarial skills.
Sn =
1-r
It is also responsible for developing new specifications to meet national requirements and the
a
S3 = for r 1 1 needs of students and teachers. Cambridge is a not-for-profit organisation; any surplus
1-r
made is invested back into the establishment to help towards the development of qualifications
Binomial series and support, which keep pace with the changing needs of today’s society.
^a + bhn = a n + n C1 a n - 1 b + n C2 a n - 2 b 2 + f + n Cr a n - r b r + f + b n ^n ! Nh, This mark scheme is published as an aid to teachers and students, to indicate the requirements
JnN of the examination. It shows the basis on which marks were awarded by examiners. It does not
n!
where C r = n C r = KK OO =
n
indicate the details of the discussions which took place at an examiners’ meeting before marking
L P r! ^n - rh !
r
commenced.
n ^n - 1h 2 n ^n - 1h f ^n - r + 1h r
^1 + xhn = 1 + nx + x +f+ x +f ^ x 1 1, n ! Rh All examiners are instructed that alternative correct answers and unexpected approaches in
2! r!
candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills
Differentiation demonstrated.
f ^xh f l^xh Mark schemes should be read in conjunction with the published question papers and the report
on the examination.
tan kx k sec 2 kx
sec x sec x tan x Cambridge 2026
cot x - cosec 2 x
cosec x - cosec x cot x
du dv
v -u
u dy dx dx
Quotient Rule y = , =
v dx v 2
Differentiation from first principles
f ^x + hh - f ^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
dd dx = ln f ^xh + c
e f ^xh
; f l^xhaf ^xhk dx = n + 1 af ^xhk + c
n 1 n+1
Integration by parts ; u dx = uv - ; v dx
dv du
dx dx
Small angle approximations
sin i . i , cos i . 1 - 12 i 2 , tan i . i where i is measured in radians
Cambridge 2026 H640/03 Jun26
Cambridge
, 3 H640/03 Mark Scheme June 2026
Trigonometric identities
sin ^A ! Bh = sin A cos B ! cos A sin B Marking Instructions
cos ^A ! Bh = cos A cos B " sin A sin B
tan ^A ! Bh = aA ! B ! ^k + 12h rk
tan A ! tan B Preparation For Marking
1 " tan A tan B 1. RM Assessor
Numerical methods • Access and complete the on-screen marking training packages: Examiner Training (RMA3).
Trapezium rule: ; y dx . 12 h "^y 0 + ynh + 2 ^y 1 + y2 + f + yn - 1h, , where h =
b
b-a
n • Read the mark scheme and question paper for this component or unit.
f ^x nh
a
The Newton‑Raphson iteration for solving f ^xh = 0: xn + 1 = x n -
f l^x nh • The mark scheme and question paper are available in RM Assessor or on your Component Page if you use the Training Platform for standardisation.
Probability • Log in to RM Assessor and mark the required number of practice scripts and the required number of standardisation scripts.
P ^A j Bh = P ^Ah + P ^Bh - P ^A k Bh
P ^A k Bh
P ^A k Bh = P ^Ah P ^B Ah = P ^Bh P ^A Bh or P ^A Bh =
P ^Bh
Marking
2. General Guidance
Sample variance
^/ xih2 • Mark strictly to the mark scheme.
S where S xx = /^xi - xh = / x i -
1 2
2
s = - 2
= / x 2i - nx- 2
n - 1 xx n
• Marks awarded must relate directly to the marking criteria.
Standard deviation, s = variance
• If you are in any doubt about applying the mark scheme, consult your Team Leader by phone, email or via the RM Assessor messaging system.
The binomial distribution
If X + B ^n, ph then P ^X = rh = n C r p r q n - r where q = 1 - p • It is essential that you meet the RM Assessor 50% and 100% batch deadlines. For traditional marking this will be 40% and 100%. If you experience
problems, contact your Team Leader without delay.
Mean of X is np
• Always check the pages (and additional objects if present) at the end of the response in case any answers have been continued there. If the candidate
Hypothesis testing for the mean of a Normal distribution has continued an answer there, then add the annotation ‘SEEN’ to confirm that the work has been seen and mark any responses using the annotations in
J N Section 11.
If X + N ^n, v 2h then X + N KKn, OO and
X -n
+ N ^0, 1h
v2
L n P v n
Percentage points of the Normal distribution 2
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
1 2
s = ut + 2 at s = ut + 12 at 2
s = 12 ^u + vh t s = 12 ^u + vh t
v 2 = u 2 + 2as
s = vt - 12 at 2 s = vt - 12 at 2
Cambridge 2026 H640/03 Jun26 Turn over
, 4 H640/03 Mark Scheme June 2026
Section A (60 marks) • The RM Assessor comments box is used by your Team Leader to explain the marking of the practice responses. Use these comments when checking
your practice responses. Do not use the comments box for any other reason.
1 (a) Express 3x 2 + 6x - 2 in the form a (x + p) 2 + q , where a, p and q are constants to be found. [3]
• Before the end of the marking period send a brief report on the performance of candidates to your Team Leader via email. The report should contain
(b) Explain why the minimum value of 3x 2 + 6x - 2 is q. [1] notes on strengths displayed as well as common errors or weaknesses. Constructive criticism of the question paper/mark scheme is also appreciated.
3. No Response and Crossed-out Answers
2 Two vectors, p and s, are given by
JK- 1NO JK2NO Using the No Response (NR) option. Only mark as NR if:
p = KK OO and s = KK OO, where y is a real variable.
L 3P L yP - the answer space is blank
- there is only a comment not related to the question (e.g., ‘can’t do’, ‘don’t know’)
(a) Find the angle that p makes with the positive x‑axis. [2]
- there is only a mark (e.g., a dash, a question mark) which is not an attempt at the question.
(b) Determine the value of y for which the magnitude of 3p + s is a minimum. [3]
Note: Enter 0 marks for an attempt that earns no credit (including copying out the question). Do not use NR.
Crossed-out answers
3 You are given that If a candidate has crossed out an answer and written a clear alternative, do not mark the crossed-out answer.
f (x) = 4 + x 2 on the domain "x: x ! R, x H 0, .
If a candidate has crossed out an answer and not written a clear alternative, mark the crossed-out answer if it is readable.
(a) Find the range of f. [1]
(b) Explain why the function g (x) = x 2 + 4, x ! R does not have an inverse. [1]
Find an expression for f - 1 (x) . [2]
(d) (i) Show that the quadratic equation x 2 + 4 = x has no real roots. [1]
(ii) Hence explain how you can be sure that the equation f (x) = f - 1 (x) has no real roots. [1]
(e) Sketch the graphs of y = f (x) and y = f - 1 (x) on the diagram in the Printed Answer Booklet.
[2]
3
4 This question is about the curve y = x 3 - 4x 2 - 3x + 2 .
(a) In this question you must show detailed reasoning.
Find the equation of the tangent to the curve at the point (-1, 0). [4]
(b) Determine the exact coordinates of the other point on the curve where the tangent is parallel
to the tangent found in part (a). [3]
Cambridge 2026 H640/03 Jun26
