Cambridge International AS Level
MATHEMATICS
Paper 2 Pure Mathematics MARK SCHEME Maximum Mark: 50
9709/22 March 2020
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.
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 March 2020 series for most Cambridge
IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.
This document consists of 11 printed pages.
© UCLES 2020
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9709/22 Cambridge International AS Level – Mark Scheme March 2020
PUBLISHED
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.
© UCLES 2020
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9709/22 Cambridge International AS Level – Mark Scheme
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March 2020
Mathematics-Specific Marking Principles 1 2 3 4 5 6
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.
Unless specified in the question, answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the degree of accuracy is not affected.
Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points.
Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw). Where a candidate has misread a number 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 mark for the misread.
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.
© UCLES 2020 Page 3 of 11
9709/22 Cambridge International AS Level – Mark Scheme March 2020
PUBLISHED 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
A
B
DM or DB FT © UCLES 2020 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.
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).
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 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.
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.
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9709/22 Cambridge International AS Level – Mark Scheme March 2020
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Abbreviations
AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent
AG
CAO
CWO
ISW
SOI
SC
WWW
AWRT © UCLES 2020 Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) Correct Answer Only (emphasising that no “follow through” from a previous error is allowed) Correct Working Only Ignore Subsequent Working Seen Or Implied 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)
Without Wrong Working Answer Which Rounds To
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Guidance Question 1 Answer Express first term as 2sinθcos30+2cosθsin30 Divide by cosθ to produce linear equation in tanθ Obtain tanθ=Obtain 87.4 62−3Marks B1 M1 A1 or 22.39… A1 Or greater accuracy 87.44297… 4 Answer Marks M1 A1 A1 AG necessary detail needed 3 B1FT Following their quotient from part (a) M1 A1 3 Guidance Question 2(a) Carry out division as far as 4x+k Obtain quotient 4x−3 Confirm remainder is 18 2(b) State or imply equation is (4x−3)(x2+5x+6)=0 Attempt solution of cubic equation to find three real roots Obtain −3,−2, 34 © UCLES 2020 Page 6 of 11
9709/22 Cambridge International AS Level – Mark Scheme
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Guidance Question Answer Marks *M1 For non-zero constant k A1 *M1 OE A1 OE without logarithms 3 Integrate to obtain kln(2x−5) Apply limits to obtain ln(6a−5)−ln(2a−5)=ln7 2Apply subtraction law for logarithms Obtain equation 6a−57= 2a−52Solve equation for a Obtain a= DM1 A1 25 26 © UCLES 2020 Page 7 of 11
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Guidance Question Answer Marks B1 4 Differentiate −y2 to obtain −2ydy dx−8dy 2y+3dxDifferentiate 4ln(2−y+3) to obtain Attempt differentiation of all terms B1 M1 Dependent on appearance of at least one dy dxSubstitute 3,x=y=−1 to find numerical value of dy dxM1 Obtain dy=3 dxA1 Obtain equation y=3x−10 A1 OE 6 © UCLES 2020 Page 8 of 11
9709/22 Cambridge International AS Level – Mark Scheme
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Guidance Question Answer Marks M1 A1 5(a) Draw two V-shaped graphs with one vertex on negative x-axis and one vertex on positive x-axis Draw correct graphs related correctly to each other State correct coordinates −2k,2k,32k,3k A1 Either given on axes or stated separately 3 B1 5(b) State or imply non-modulus equation (x+2k)2=(2x−3k)2 or pair of linear equations Attempt solution of 3-term quadratic equation or pair of linear equations Obtain x=1k,3Obtain y=7k,3x=5k y=7k M1 A1 A1 If A0A0, award A1 for one pair of correct coordinates 4 M1 A1 5(c) Relate 2t to larger value of x from part (b) Apply logarithms to obtain t=ln(5k) ln2OE such as log10(5k) or log2(5k) log102 2 © UCLES 2020 Page 9 of 11
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Guidance Question Answer Marks *M1 A1 OE DM1 6(a) Differentiate using the product rule Obtain 3x2e0.2x+0.2x3e0.2x Equate first derivative to 15 and rearrange to x=... 75e−0.2xConfirm x= 15+xA1 AG – necessary detail needed 6(b) 75e−0.2x or equivalent for 1.7 and 1.8 Consider sign of x−15+x4 M1 Obtain −0.08... and 0.03... or equivalents and justify conclusion 6(c) Use iterative process correctly at least once Obtain final answer 1.771 Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [1.7705,1.7715] A1 2 M1 Answer required to exactly 4 sf A1 A1 3 © UCLES 2020 Page 10 of 11
9709/22 Cambridge International AS Level – Mark Scheme
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March 2020
Guidance Question Answer Marks M1 A1 7(a) Solve equation y=0 to find value of x 7Obtain π 6 7(b) Attempt first derivative using chain rule Obtain 2 M1 OE A1 OE dy=8sinxcosx+8cosx dxSubstitute value from part (a) to find gradient −23 7(c) Express integrand in the form k1+k2cos2x+k3sinx Obtain correct 5−2cos2x+8sinx Integrate to obtain 5x−sin2x−8cosx Apply limits 0 and their value from part (a) correctly Obtain A1 Or exact equivalent 3 *M1 A1 OE. Allow unsimplified A1 DM1 A1 357π+3+8 or exact equivalent 62 5 © UCLES 2020 Page 11 of 11
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