Cambridge Assessment International Education

Cambridge International Advanced Level

MATHEMATICS 9709/73
Paper 7 May/June 2019
MARK SCHEME
Maximum Mark: 50

Published

This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the
examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the
details of the discussions that took place at an Examiners’ meeting before marking began, which would have
considered the acceptability of alternative answers.

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Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for
Teachers.

Cambridge International will not enter into discussions about these mark schemes.

Cambridge International is publishing the mark schemes for the May/June 2019 series for most
Cambridge IGCSE™, Cambridge International A and AS Level and Cambridge Pre-U components, and
some Cambridge O Level components.

This document consists of 12 printed pages.

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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.

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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.

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Mark Scheme Notes

Marks are of the following three types:

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.

• 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 (or dep*) 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.

• The symbol 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 and B marks are not given for fortuitously “correct” answers or results obtained from incorrect
working.

• Note: B2 or A2 means that the candidate can earn 2 or 0.

B2/1/0 means that the candidate can earn anything from 0 to 2.

The marks indicated in the scheme may not be subdivided. If there is genuine doubt whether a candidate has earned a mark, allow the
candidate the benefit of the doubt. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a
correct form of answer is ignored.

• Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise.

• For a numerical answer, allow the A or B mark if a value is obtained which is correct to 3 s.f., or which would be correct to 3 s.f. if rounded (1
d.p. in the case of an angle). As stated above, 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 10.

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The following abbreviations may be used in a mark scheme or used on the scripts:

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 – often written by a ‘fortuitous’ answer

ISW Ignore Subsequent Working

SOI Seen or implied

SR Special Ruling (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)

Penalties

MR –1 A penalty of MR –1 is deducted from A or B marks when the data of a question or part question are genuinely misread and the object
and difficulty of the question remain unaltered. In this case all A and B marks then become “follow through” marks. MR is not applied
when the candidate misreads his own figures – this is regarded as an error in accuracy. An MR –2 penalty may be applied in particular
cases if agreed at the coordination meeting.

PA –1 This is deducted from A or B marks in the case of premature approximation. The PA –1 penalty is usually discussed at the meeting.

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Question Answer Marks Guidance

1 ×0.6 M1 Recognisable value of z

0.6 ± z 0.4100

z = 2.326 B1 2.326 to 2.329

0.486 to 0.714 (3 sf) A1 Must be an interval

Question Answer Marks Guidance

2 50 ( 4361 − x 2 ) = 9.62 M1 or ( 4361 − ( Σx )2 ) = 9.62 BOD regarding symbols used

49 50 49 50×49

x 2 = 4361 49 = 77.7924
- 9.62× 50 A1 (Σx)2 = 4361 × 50 – 9.62 × 50 × 49 = 194481or Σx =441 ( Σx ) or ( x ) must be
50
correctly identified

x = 8.82 (3 sf) A1 SC use of ‘biased’ leading to 8.81 B1

Question Answer Marks Guidance

3(i) D more likely to be chosen B1 oe, e.g. P(D) > P(A) e.g. P(A)=P(B)=P(C)=1/6 P(D)=1/2 no contradictions

3(ii) Reject scores of 5 or 6 B1 or other correct: choose D when the score is 4

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Question Answer Marks Guidance

3(iii) AB AC AD BC BD CD B1

Allocate as follows: B1 or similar

1: AB; 2: AC; 3: AD; 4: BC; 5: BD 6: CD

Question Answer Marks Guidance

4 Total ~ N(1208, .....) B1

Var(total) (= 10×1.2 + 20×0.7 (+ 0)) = 26 B1 May be implied by next line

± 1200−"1208" (= –1.569) M1 FT their mean and var of total mass, e.g. allow 1200 and 11.24
"26" (from 10 × 1.22 + 20 × 0.72)

1 – Φ(“1.569”) M1 Correct area consistent with their working

= 0.0583 (3 sf) A1

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Question Answer Marks Guidance

5 H0: Pop mean = 20 B1

H1: Pop mean ≠ 20 Accept µ

Σx (= 126.9 = 21.15) M1 Attempted

6 6 or 126.9 and 11.64 attempted

'21.15'−20 M1 Must have 6

1.94
6 120 − 126.9
or no mixed method
11.64

= 2.022 A1

2(1 - ɸ(‘2.022’) ) 2 (1 – '0.9784)' = 0.0432) (

M1 FT 2 × 1 −′ .9784' )
α = 4.32 (3 sf) A1 FT Allow 4.3 or 4, if correct working seen, or clearly implied, as far as 0.0216
FT their z, no error seen
One-tail test scores maximum 3/6

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Question Answer Marks Guidance

6(i) a M1
∫ x dx
3 2
3
a3 Attempt to integrate f(x) with limits 0 and a (condone missing )
0 a3
3 a
(= 33  x3  )
a   0

3 A1 3a3
= 3a3 − 0 or better seen
3a 3a3

= 1 Hence f is pdf for all a A1 Answer = 1 and comment

6(ii) 2 M1
∫ x dx
3 2 = 0.5 3
a3 Attempt to integrate f(x)=0.5, limits 0 and 2 oe, condone missing
0 a3
3  x3  2 = 0.5
a3  3  0

3 × 8 = 0.5 oe A1 23 3
a3 3 − 0 or better, condone missing 3
3 a

a3 = 16 or a = 3 16 A1 Convincingly obtained
Note: Attempt to verify 2.52, M1 as stated except not equated to 0.5.A1 as
( = 2.52 AG)
stated, A1 for evaluation to 0.499..apprx 0.5

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Question Answer Marks Guidance

6(iii) 2.52 a M1
3
16 ∫0 x3dx 3
or 16
∫0 x3dx Attempt integ xf(x), correct limits, condone missing
3
a3
3  x4  2.52
= 16 3  x4  a
or 16
 4  0  4  0

3 × 40.317
= 16 A1 2.524 3
4 − 0 or better, condone missing 3
4 a

= 1.89 (3 sf) A1

Question Answer Marks Guidance

7(i) Use of Po(2.8) M1 May be implied

1 – e-2.8(1 + 2.8 + 2.82

) ) M1 Any λ allowing one end error
2

= 0.531or 0.53(0) (3 sf) A1 SC Binomial 0.534 B1

7(ii) Use of Po(5.8) M1 May be implied

e-5.8 × 5.86 M1 Any λ

6!

= 0.16(0) (3 sf) A1

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Question Answer Marks Guidance

7(iii) Use of N(58, 58) M1 May be implied

or N(58, 55.38)

50.5 − '58' M1 Standardised with their values, allow wrong or incorrect cc

(= -0.985)
'58'

Φ('0.985') M1 Correct area consistent with their working

or Φ ("1.008 )

= 0.838 (3 sf) A1 or 0.843

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Question Answer Marks Guidance

8(i) H0: p = 1 B1
4
1
H1: p > 4

10
C6 ( 14 )6 ( 43 )4 + 10C7 ( 14 )7 ( 34 )3 + 10C8 ( 14 )8 ( 43 ) 2 + M1 Correct terms, allow one term incorrect or omitted or extra
10 ( 14 )9 ( 43 ) + ( 14 )10 or summing all correct terms from 0 to 5 allow one term incorrect or omitted or
extra

= 0.0197 A1 or 0.9803

comp '0.0197' with 0.01 M1 Valid comparison with 0.01

or valid comparison with 0.99

No evidence to conclude p > 1 A1 FT No contradictions

4
Use of two-tail test can score BOM1A1M1(comparison with 0.005) A0

8(ii) 10
C7 ( 14 )7 ( 34 )3 + 10C8 ( 14 )8 ( 43 ) 2 +10 ( 14 )9 ( 43 ) + ( 14 )10 M1 Their P(X . 6) − 10C ( 0.25)6 ( 0.75)4
6

P(Type I) = 0.00351 (3 sf) A1 Accept 0.00348 to 0.00351

8(iii) C.R is X ⩾ 7 M1 May be implied

P(Type II) = 1 – P(X ⩾ 7 | p = 3
5
)=

1– (10C7 ( 53 )7 ( 52 )3 + 10C8 ( 53 )8 ( 52 ) 2 +10 ( 53 )9 ( 52 ) + ( 53 )10 ) M1 Accept 1 – P(X ⩾ 8 | p = 3

) or 1 – P(X ⩾ 6 | p = 3
)
5 5

= 0.618 A1

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