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Cambridge International Examinations

Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/

Paper 1 Pure Mathematics 1 May/June 2023

MARK SCHEME

Maximum Mark : 75

PRE-STANDARDISATION

Cambridge Assessment International Education – Generic Marking Principles

These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the

mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles.

GENERIC MARKING PRINCIPLE 1:

Marks must be awarded in line with:

 the specific content of the mark scheme or the generic level descriptors for the question

 the specific skills defined in the mark scheme or in the generic level descriptors for the question

 the standard of response required by a candidate as exemplified by the standardisation scripts.

GENERIC MARKING PRINCIPLE 2:

Marks awarded are always whole marks (not half marks, or other fractions).

GENERIC MARKING PRINCIPLE 3:

Marks must be awarded positively :

 marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of

the syllabus and mark scheme, referring to your Team Leader as appropriate

 marks are awarded when candidates clearly demonstrate what they know and can do

 marks are not deducted for errors

 marks are not deducted for omissions

 answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as

indicated by the mark scheme. The meaning, however, should be unambiguous.

GENERIC MARKING PRINCIPLE 4:

Rules must be applied consistently e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors.

GENERIC MARKING PRINCIPLE 5:

Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited

according to the quality of the candidate responses seen).

GENERIC MARKING PRINCIPLE 6:

Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors

in mind.

PRE-STANDARDISATION

Mathematics Specific Marking Principles

1. Unless a particular method has been specified in the question, full marks may be awarded for any correct method. However, if a calculation is required then

no marks will be awarded for a scale drawing.

2. Unless specified in the question, non-integer answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the

degree of accuracy is not affected.

3. Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points.
4. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw).
5. Where a candidate has misread a number or sign in the question and used that value consistently throughout, provided that number does not alter the

difficulty or the method required, award all marks earned and deduct just 1 A or B mark for the misread.

6. Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear.

PRE-STANDARDISATION

Mark Scheme Notes

The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific

reasons outside the scope of these notes.

Types of mark

M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors

in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the

formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct

application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a

correct answer.

A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the

associated method mark is earned (or implied).

B Mark for a correct result or statement independent of method marks.

DM or DB When a part of a question has two or more “method” steps, the M marks are generally independent unless the scheme specifically says

otherwise; and similarly, when there are several B marks allocated. The notation DM or DB is used to indicate that a particular M or B mark

is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier

marks are implied and full credit is given.

FT Implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks

are given for correct work only.

• A or B marks are given for correct work only (not for results obtained from incorrect working) unless follow through is allowed (see abbreviation FT

above).

• For a numerical answer, allow the A or B mark if the answer is correct to 3 significant figures or would be correct to 3 significant figures if rounded

(1 decimal place for angles in degrees).

• The total number of marks available for each question is shown at the bottom of the Marks column.
• Wrong or missing units in an answer should not result in loss of marks unless the guidance indicates otherwise.
• Square brackets [ ] around text or numbers show extra information not needed for the mark to be awarded.

PRE-STANDARDISATION

Abbreviations

AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent

AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid)

CAO Correct Answer Only (emphasising that no “follow through” from a previous error is allowed)

CWO Correct Working Only

ISW Ignore Subsequent Working

SOI Seen Or Implied

SC Special Case (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be

varied in the light of a particular circumstance)

WWW Without Wrong Working

AWRT Answer Which Rounds To

PRE-STANDARDISATION

Guidance For Examiners

Preparation For Marking

1. Please familiarise yourself with the RMA3 guide entitled Examiners’ Instructions on screen ( alternatively called Instructions for examiners

(marking on-screen)). This may be found among Assistant Examiner RM Assessor3 online marking resources through the training site:

https://training.cambridgeassessment.org.uk/login/index.php

2. If this is the first time you have marked using RMA3 please familiarise yourself with the RM Assessor3 How To Guides on the training site_._ If you

have marked using RMA3 before you may wish to refamiliarise yourself these guides.

3. Work through the question paper and provisional mark scheme. There will be further amendments and clarifications to this mark scheme at the

Standardisation Meeting.

4. Try to be clear in your own mind of what is required for each mark, and in particular for M (method) marks.
5. Browse scripts in RMA3 – see section C3 of the Instructions for examiners (marking on-screen). These should be available approximately 9 days

after the examination is taken, however it is worth checking after about 7 days. Please look at about 10 scripts and refer any queries/issues to

your Team Leader for consideration at the standardisation team meeting.

Standardisation

1. A FINAL mark scheme will be issued at/after the Standardisation meeting. You must use this version of the mark scheme for your marking.
2. Mark the 10 practice scripts and check them, ensuring the Comments Box is open – see Section C4 of the Instructions for examiners (marking

on-screen).

3. Mark the 10 standardisation scripts and submit them to your Team Leader- see Section C5 of the Instructions for examiners (marking on-

screen).

4. Your Team Leader will feed back to you and approve you to proceed with marking or ask you to mark a further 10 scripts.

PRE-STANDARDISATION

Marking Guidance

1. If you are in doubt about applying the mark scheme consult your Team Leader, preferably using the RMA3 messaging system.
2. There must be evidence that all of a candidate’s work has been marked. Any blank pages must be checked for responses by scrolling down and

then each annotated with a or. Similarly, if you receive a script with Additional Objects please annotate each page with

to show that you have checked these pages.

3. Responses that are not awarded either 0 or full marks should be appropriately annotated.
4. Although the scheme will allow for the majority of likely methods, full marks are to be given for a correct result from any correct method, with

equivalent sub-marks for equivalent stages. The abbreviation AEF in the mark scheme indicates that any equivalent form of the answer is

equally acceptable. Be alert for correct but unfamiliar or unexpected methods — often signalled by a correct result following an apparently

incorrect method. Such work must be carefully assessed. (This does not however apply if candidates have been directed to answer a question

by a particular method.) Extra attention to the working is also advisable when the final result is given in the question, indicated by the

abbreviation AG in the mark scheme.

5. Where a candidate has crossed out a complete part of a question, it should be marked provided that it has not been replaced. If two or more

methods are offered, mark the most complete method. Most complete means the method that has progressed the furthest and is nearest to

obtaining a final answer. If two or more complete methods are offered, mark the ‘final’ method. The final method is either towards the bottom of

the answer space or to the right-hand side of the answer space. If a candidate’s answer to an item is in two parts, it is important that it is marked

as a whole and appropriately annotated.

6. Award NR (No Response)

 if there is nothing written at all for a question (part), or

 if there is only a comment which does not in any way relate to the question being asked (e.g. ‘can’t do’, ‘don’t know’)

 if there is only a mark which is not an attempt at the question (e.g. a dash, a question mark).

Award 0 marks

 if there is an attempt that earns no credit, including copying out the question.

BP SEEN

SEEN

PRE-STANDARDISATION

7. Occasionally a candidate will genuinely misread a number in a question and use that value consistently throughout. Indicate such a misread by

the MR annotation. Provided that the object and demand of the question remains unaltered, M marks may still be awarded and where A and B

marks are available one A mark or one B mark is not awarded. This is only applied once in a whole question.

MR is not applied when the candidate misreads his or her own figures – this is regarded as an error in accuracy and marked accordingly.

Example scenario 1: If a candidate misreads a value in part (a) and then continues to use the same misread value in part (b) , then the misread

rule is only applied to part (a) and all marks in part (b) are available (followed through).

Example scenario 2: If a candidate misreads a value and uses it in only one part of a question e.g. in part (a) the value is misread but not in

part (b) or vice versa, then the misread rules are applied to part (a) and part (b) is marked as normal in line with the mark scheme.

8. When a part of a question has two or more “method” steps, the M marks are generally independent unless the scheme specifically says

otherwise; and similarly when there are several B marks allocated. When two or more steps are run together by the candidate, the earlier marks

are implied and full credit is given.

9. Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifies otherwise.
10. Full correct working is required in order to gain accuracy marks. Candidates are required to present the mathematical steps leading to a solution

using mathematical notation and not calculator commands. EXCEPTION: For SEAB Higher syllabuses, unsupported answers from a graphing

calculator are allowed unless a question specifically states otherwise.

11. Candidates are required to give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees,

unless a different level of accuracy is specified in the question. For international syllabuses, credit is withheld only once in a paper for answers

which are not stated to the minimum accuracy required (minimum accuracy being where an answer is given to 2 significant figures, rather than 3

significant figures).

12. For a non-exact numerical answer whose required accuracy is not specified in the mark scheme, allow the A or B mark if a value is obtained

which is correct to 3 significant figures or which would be correct to 3 significant figures if rounded (1 decimal place in the case of an angle). An

A or B mark is not given if a correct numerical answer arises fortuitously from incorrect working. For Mechanics questions, allow A or B marks for

correct answers which arise from taking g equal to 9.8 or 9.81 instead of 10ms

• 2

.

Report

Please send a brief report on the work of your candidates to the Principal Examiner by the date specified by the Principal Examiner, copied to

your Team Leader. Your report should cover candidates’ responses to each question (or the set of questions given to you by the Principal

Examiner) and any other matters that you wish to draw to the attention of Cambridge International or Centres. Consult previous years’ reports for the

type of comments that are appropriate. These are available on the Cambridge International website for international syllabuses.

PRE-STANDARDISATION

Annotations In RMA

Abbreviation Annotation in RMA3 Meaning

Correct work.

Incorrect work.

M Method mark.

A

Accuracy mark, awarded for a correct final answer or intermediate stage. They depend on the

preceding M mark(s), hence M0 A1 is not possible.

B Mark awarded for a correct result or statement independent of method marks.

SC

Special case or specific wrong solution for which marks may be awarded, only when indicated in

mark scheme.

CWO or

NFWW

Correct Working Only or Not From Wrong Working, especially where ‘fortuitous’ answers are seen.

FT Work can be followed through after an error.

ISW Ignore subsequent working (after correct answer obtained).

MR Misread.

BOD Benefit of the doubt is given to the candidate.

Omission sign.

Pre Premature approximation – this applies to A or B marks only.

SF Error in number of significant figures.

Seen SEEN To show that blank pages have been checked for any candidate work.

BP BP

To show that blank pages have been checked for any candidate work.

O See other solution

PRE-STANDARDISATION

Question Answer Marks Guidance

𝑥

2

− 𝑘𝑥 + 2 = 3 𝑥 − 2 𝑘 → 𝑥

2

− 𝑥

(

𝑘 + 3

)

+

(

2 + 2 𝑘

)

[= 0 ]

M1 3 - term quadratic

𝑏

2

− 4 𝑎𝑐 = (𝑘 + 3 )

2

− 8 ( 1 + 𝑘) M

= (𝑘 − 1 )

2

A

≥ 0 Hence will meet for all values of k A

2 Stretch → ( 2 𝑥)

2

− 2 ( 2 𝑥) + 5 M

Reflection → (− 2 𝑥)

2

− 2 (− 2 𝑥) + 5 M1 FT on their stretch

Stretch → 3

{

(− 2 𝑥)

2

− 2

(

− 2 𝑥

)

}

M1 FT on their (stretch plus reflection)

12 𝑥

2

+ 12 𝑥 + 15 A

d𝑦

d𝑥

= {

1

60

( 3 𝑥 + 1 ) × 2 } × { 3 } B1 B

1

10

(

3 𝑥 + 1

)

= 1 M1 Equate their

d𝑦

d𝑥

to 1

𝑥 = 3 A

4(a) 5. 00 + 20 × 0. 02 or 5. 02 + 19 × 0. 02 M

5.40 A

PRE-STANDARDISATION

Question Answer Marks Guidance

4(b)

𝑟 =

= 1. 004

B

5. 00 × (𝑡ℎ𝑒𝑖𝑟 1. 004 )

20

or 5. 02 × (their1. 004 )

19

M

5.42 A

5 𝑟

2

= ( 7 + 2 )

2

+ ( 12 − 5 )

2

M1 Expect 130

Equation of circle is (𝑥 + 2 )

2

+ (𝑦 − 5 )

2

= 130 A1 FT FT their 130

(𝑥 + 2 )

2

+ (− 2 𝑥 + 21 )

2

= 130 M1 Substitute 𝑦 = − 2 𝑥 + 26

5 𝑥

2

− 80 𝑥 + 315

[

= 0

]

→ [ 5 ](𝑥 − 9 )(𝑥 − 7 ) M

Factorisation OE must be seen

𝑥 = 9 , 𝑦 = 8 OR (9, 8) A1 A

7C1(

𝑥

𝑎

)

6

(

𝑎

𝑥

2

) 7C2(

𝑥

𝑎

)

5

(

𝑎

𝑥

2

)

2

B1 B

Coefficients 𝑥

4

& 𝑥. Can be seen in an expansion

(

𝑎

5

)

(

𝑎

3

)

= 3 M1 A1 OE. Allow extraneous 𝑥

4

and x at this stage

𝑎

2

=

A1 SOI (implied by 𝑎 =

1

3

)

𝑎 = ±

A

PRE-STANDARDISATION

Question Answer Marks Guidance

7(a) tan 𝜃 sin 𝜃 = 1 → sin

2

𝜃 = cos 𝜃

M1 Multiply by cos 𝜃

1 − cos

2

𝜃 = cos 𝜃 or cos

2

𝜃 + cos 𝜃 − 1 [= 0 ] M

[cos 𝜃 =]

− 1 ±

√

M1 Must be seen. Expect 0.

51.8º, 308.2º

A

A1 FT

FT for 360º ‒ 1st solution

7(b)

tan 𝜃

sin 𝜃

−

sin 𝜃

tan 𝜃

=

sin 𝜃

sin 𝜃 cos 𝜃

−

sin 𝜃 cos 𝜃

sin 𝜃

=

cos 𝜃

− cos 𝜃

M1 Use tan 𝜃 =

sin 𝜃

cos 𝜃

=

1 − cos

2

𝜃

cos 𝜃

=

sin

2

𝜃

cos 𝜃

M1 Use 1 − cos

2

𝜃 = sin

2

𝜃

= tan 𝜃 sin 𝜃 A1 WWW

8(a)

tan 𝐵𝐷𝐶 =

𝑜𝑟 sin 𝐵𝐷𝐶 =

𝑜𝑟 cos 𝐵𝐷𝐶 =

M

𝐵𝐷𝐶 = 0. 927

[

]

→ 𝐴𝐷𝐶 = π − 0. 927

[

]

[= 2. 214 ] A1 Allow degrees

𝐴𝑟𝑐 𝐴𝐶 = 5 × 𝑡ℎ𝑒𝑖𝑟 2. 214 M

Expect 11.

𝐴𝐶 =

√

2

2

𝑜𝑟 2 × 5 × sin 1. 107

M1 Expect 8.

[Perimeter = 11. 07 + 8. 94 =] 20. 0 A

PRE-STANDARDISATION

Question Answer Marks Guidance

8(b) Sector 𝐴𝐶𝐷 = ½ × 5

2

× 2. 214

M1 Expect 27.

∆𝐴𝐷𝐶 = ½ × 5 × 4 M1 Expect 10

Shaded area = 27. 7 ‒ 10 = 17. 7 A

9(a) [𝑦] ≤ − 1 B1 Accept f(𝑥) ≤ − 1

9(b)

𝑦 = − 3 𝑥

2

+ 2 → 3 𝑥

2

= 2 − 𝑦 → 𝑥

2

=

2 −𝑦

3

M

𝑥 =

[

±

]

√

2 − 𝑦

→ f

− 1

(

𝑥

)

= {−}

{

√

2 − 𝑥

} A1 A

A1 for minus, A1 for √

2 −𝑥

3

9(c) fg

(

𝑥

)

= − 3 (−𝑥

2

− 1 )

2

+ 2 M

SOI

gf(𝑥) = −(− 3 𝑥

2

+ 2 )

2

− 1 M1 SOI

fg(𝑥)‒ gf(𝑥) + 8 = 0 → 6 𝑥

4

− 18 𝑥

2

+ 12 [= 0 ] A1 OE

[ 6 ](𝑥

2

− 1 )(𝑥

2

− 2 ) [= 0 ] M1 Must be seen

𝑥 = − 1 , −

√

2 only

A

10(a)

−

=

+ 𝑘 → 𝑘 = − 2

B1 AG

PRE-STANDARDISATION

Question Answer Marks Guidance

10(b)

𝑦 = 2 𝑥

1

2 − 2 𝑥 [+𝑐]

M1 A

‒ 1 = 4 − 8 + 𝑐 M1 Subsitute x = 4, y = − 1 ( c present) Expect c = 3

𝑦 = 2 𝑥

1

2 − 2 𝑥 + 3

A

10(c)

𝑥

− 1 / 2

− 2 = 0

M

Set 𝑡ℎ𝑒𝑖𝑟

d𝑦

d𝑥

to zero

𝑥 = 1 4

⁄

A

(¼, 3½)

A

10(d)

d

2

𝑦

d𝑥

2

= −

1

2

𝑥

−

3

2

B

< 0 (or ‒4) → hence Maximum B

11(a)

Gradient of AB =

2 −(− 1 )

5 − 2

M1 Expect 1

Equation of AB is 𝑦 − 2 = 1 (𝑥 − 5 ) or 𝑦 + 1 = 1 (𝑥 − 2 ) A1 Expect 𝑦 = 𝑥 − 3

PRE-STANDARDISATION

Question Answer Marks Guidance

11(b)

[

π

]

∫ 𝑥

2

d𝑦 =

[

π

]

∫

(

𝑦

2

)

2

d𝑦 = [π] ∫

(

𝑦

4

+ 2 𝑦

2

)

d𝑦 M

For curve: Attempt to square 𝑦

2

• 1 and attempt

integration

[π] (

𝑦

5

+

2 𝑦

3

+ 𝑦)

A2, 1, 0

[π] {

+

+ 2 − (−

−

− 1 )} M

Apply limits − 1 → 2. Expect 15

3

5

[π]

[π] ∫(𝑦 + 3 )

2

d𝑦 = [π] ∫(𝑦

2

• 6 𝑦 + 9 ) d𝑦 M

For line: Attempt to square 𝑦 + 3 and attempt

integration

[π] (

𝑦

3

+ 3 𝑦

2

+ 9 𝑦) A2, 1, 0

[π] {

+ 12 + 18 − (−

+ 3 − 9 )}

M1 Apply limits − 1 → 2. Expect 39[π]

Volume = π(39 ‒ 15

3

5

) = 23

2

5

π A