UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the May/June 2010 question paper for the guidance of teachers 9709 MATHEMATICS 9709/11 Paper 11, maximum raw mark 75 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 must be read in conjunction with the question papers and the report on the examination. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.maxpapers.com
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/11 Paper 11, maximum raw mark 75
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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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Page 4 Mark Scheme: Teachers’ version Syllabus Paper
8 (i) y-step ÷ x-step = 2 → m = 1 (ii) Eqn of AC )3(22+ xy Eqn of BC )15(22 xy Sim eqns 42+ xy , 7+xy
→ C (–1, 6) (iii) M is (9, 10) Perp gradient is −½ → ,292 + xy 7+xy
Sim eqns → D (5, 12)
M1 A1
[2] M1 A1√
A1√
A1
[4] B1 M1 M1 A1
[4]
Gradient = y-step ÷ x step used co Correct form of one of lines. √ to his m
√ to his m
co co Use of m1m2 = −1 Solve sim eqns for their BC & perp. bis co
9 (i) 71222
+xx = 11)3(2 2x 3 × B1
[3] B1 for each value – accept if a, b, c not specifically quoted.
(ii) Range of f [ − 11 B1√ [1]
√ to his “c”. allow > or [.
(iii) 71222
+xx < 21
→ 141222
xx or
32)3(2 2<x
M1
3-term quadratic to 0 or 32)3(2 2
<x
→ end-values of 7 or −1 → −1 < x < 7
A1 A1
[3]
Correct end-values co
(iv) gf(x) = kxx ++ )7122(2 2 = 0 Use of b2 − 4ac → 242 − 16(14 + k) → k = 22
M1 A1
M1 A1
[4]
Puts f into g. co.
Used correctly with quadratic co.
10 OA i + 3j + 3k, OC = 3i − j + k.
(i) OB OA + OC = 4i + 2j + 4k B1 co
Unit vector = 6
1 (4i + 2j + 4k) M1 A1√ Divides by the modulus. √ on OB .
[3]
(ii) AC = OC − OA = 2i − 4j − 2k B1 co
AC .OB 8 − 8 − 8 = −8 M1 Use of x1 x2 + y1 y2 + z1 z2
OB = 6; 24AC M1 Correct method for a modulus.
−8 = 6 × 24 × cos θ M1 Connected correctly provided ,OB AC used
θ = 105.8° → 74.2° A1 co (accept acute or obtuse) [5]
(iii) OA = 19 or OC = 11 Perimeter = 2(√ + √) → 15.4
B1 M1 A1
[3]
Used as a length. co (accept 15.3)
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/12 Paper 12, maximum raw mark 75
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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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Page 4 Mark Scheme: Teachers’ version Syllabus Paper
→ A (−1, 9) and B (3, 1) Area under line = ½ × 4 × 10 or [ ]27 xx from −1 to 3.
M1 DM1A1 M1
y (or x) removed completely. Soln of quadratic. Both points correct. Uses any valid method – integration or area of trapezium etc.
Area under curve =
3
)2(3
x
M1
A1
Any attempt at integration.
Correct integration in either form.
or
+ xx
x
423
2
3
from −1 to 3
→ 10⅔.
[ok to use ( )∫ xyy d 21
– marks the same]
M1
A1 [8]
Correct use of limits in an integral.
co
10 ( ) xxy 4323
6
1
(i) x
y
d
d= 2
6
1 )32(3×× x × 2 − 4
B2,1
B1 [3]
Everything but the “×2”
For the “×2”, even if B0 given above.
(ii) x = 0, y = 6
27 ,
xy 56
27+ → xy 1092 +
B1
M1 A1 [3]
For correct y value
Must be using calculus for m. co. (ok unsimplified)
(iii) 4)32(2
x ( > 0)
→ x = 2½ or ½ → x > 2½, x < ½.
M1
DM1 A1
[3]
Links x
y
d
d with 0
Method for quadratic – lead to 2 answers Correct set of values.
11 f : xx sin34a (i) 2sin34 x → sin x = ⅔ → x = 0.730 or 2.41 (ii)
(iii) k < 1, k > 7.
(iv) A = 2
3π.
(v) sin x = ⅓ – or using inverse g 1(3) = 2.80
M1 A1 A1√
[3] B1 B1
[2] B1 B1
[2] B1
[1] M1A1
[2]
Makes sin x the subject + solution. co. √ for π − first answer. Must be 1 complete oscillation. Shape and position correct, in 1st quadrant, curve not lines. B1 for k = 1, 7, B1 for answer Or B1 for k < 1, B1 for k > 7 co M1 for soln of 3 = 4 − 3sin x or inverse.
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/13 Paper 13, maximum raw mark 75
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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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Page 4 Mark Scheme: Teachers’ version Syllabus Paper
8 (i) Mid-point of AC = (2, 3) Gradient of AC = 1/3 Gradient of BD = –3 Equation y – 3 = –3(x – 2) (ii) If x = 0, y = 9, B (0, 9) Vector move D (4, –3)
(iii) AC = 40
BD = 160 Area = 40 (or by matrix method M2 A1)
B1 M1 A1
[3] B1√ M1 A1
[3]
M1 M1 A1
[3]
Co Use of mlm2 = –1 Co √ on his equation. Valid method. co.
Correct use on either AC or BD, Full and correct method. co
9 x
xy4
+
(i) 54
+
x
x → A (1, 5), B(4, 5)
2
41
d
d
xx
y
= 0 when x = 2, M (2, 4).
B1 B1
M1
DM1 A1 [5]
co. co.
Differentiates.
Setting to 0. co.
(ii) Vol of cylinder = π52.3
Vol under curve = xy d 2∫π
B1 M1
Any valid method. Attempt at integrating y2
Integral = x
x
x
816
3
3
+ A2, 1, 0 Allow if no π present.
Uses his limits “1 to 4” → πππ 185775
DM1 A1
[6]
Using his limits. co.
10 f : 14822
+xxx a (i) y + kx = 12, Sim Eqns. → 2x2 – 8x + kx + 2 = 0 Use of b2 – 4ac
→ (k – 8)2 =16 → k = 12 or 4. (ii) 2x2 – 8x + 14 = 2(x – 2)2 + 6 (iii) Range of f [ 6. (iv) Smallest A = 2 (v) Makes x the subject Order of operations correct.
M1 A1 M1 A1
[4] B1×3
[3] B1√
[1] B1√
[1] M1 M1
Complete elimination of y (or x) Uses b2 – 4ac on eqn = 0, no “x” in a, b, c. co.co √ for c or from calculus. √ to answer to (ii). Could interchange x, y first. Order must be correct.
( ) 22
6g
1+
x
x
A1
[3]
co
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/21 Paper 21, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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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Page 4 Mark Scheme: Teachers’ version Syllabus Paper
[The M1 is dependent on at least one of the B marks being obtained.]
7 (i) Make a recognisable sketch of a relevant graph, e.g. y = 2 – x B1
Sketch an appropriate second graph, e.g. y = e2x, and justify the given statement B1 [2]
(ii) Consider sign of e2x – (2 – x) at x = 0 and x = 0.5, or equivalent M1
Complete the argument correctly with correct calculations A1 [2]
(iii) Show that e2x = 2 – x is equivalent to x = 2
1ln(2 – x), or vice versa B1 [1]
(iv) Use the iterative formula correctly at least once M1
Obtain final answer 0.27 A1
Show sufficient iterations to justify its accuracy to 2 d.p., or show there is a sign change
in the interval (0.265, 0.275) A1 [3]
8 (i) Use quotient rule M1
Obtain correct derivative in any form A1
Obtain given result correctly A1 [3]
(ii) State cot2 x ≡ –1 + cos ec2x , or equivalent B1
Obtain integral –x – cotx (f.t. on signs in the identity) B1√
Substitute correct limits correctly M1
Obtain given answer A1 [4]
(iii) Use trig formulae to convert integrand to xk
2
sin
1 where k = ±2, or ±1 M1
Obtain given answer 2
1 cos ec2x correctly A1
Obtain answer –2
1cot x + c, or equivalent B1 [3]
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/22 Paper 22, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
(iii) Show that given equation is equivalent to )ln2( xx or vice versa B1 [1]
(iv) Use the iterative formula correctly at least once M1
Obtain final answer 1.31 A1
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change
in the interval (1.305, 1.315) B1 [3]
7 (i) Substitute x = 3 and equate to 30 M1
Substitute x = –1 and equate to 18 M1
Obtain a correct equation in any form A1
Solve a relevant pair of equations for a or for b M1
Obtain a = 1 and b = –13 A1 [5]
(ii) Either show that f(2) = 0 or divide by (x – 2), obtaining a remainder of zero B1
Obtain quadratic factor 2x2 + 5x – 3 B1
Obtain linear factor 2x – 1 B1
Obtain linear factor x + 3 B1
[Condone omission of repetition that x – 2 is a factor.]
[If linear factors 2x – 1, x + 3 obtained by remainder theorem or inspection, award B2 + B1.] [4]
8 (i) Use correct sin(A – B) and cos(A – B) formulae M1
Substitute exact values for sin 30° etc. M1
Obtain given answer correctly A1 [3]
(ii) State xx sec2
1sin3 B1
Rearrange to sin 2x = k, where k is a non-zero constant M1
Carry out evaluation of
3
1sin
2
1 1 M1
Obtain answer 17.6° A1
Carry out correct method for second answer M1
Obtain remaining 3 answers from 17.6°, 72.4°, 197.6°, 252.4° and no others in the
range A1 [6]
[Ignore answers outside the given range]
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/23 Paper 23, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
(iii) Show that given equation is equivalent to )ln2( xx or vice versa B1 [1]
(iv) Use the iterative formula correctly at least once M1
Obtain final answer 1.31 A1
Show sufficient iterations to justify its accuracy to 2 d.p. or show there is a sign change
in the interval (1.305, 1.315) B1 [3]
7 (i) Substitute x = 3 and equate to 30 M1
Substitute x = –1 and equate to 18 M1
Obtain a correct equation in any form A1
Solve a relevant pair of equations for a or for b M1
Obtain a = 1 and b = –13 A1 [5]
(ii) Either show that f(2) = 0 or divide by (x – 2), obtaining a remainder of zero B1
Obtain quadratic factor 2x2 + 5x – 3 B1
Obtain linear factor 2x – 1 B1
Obtain linear factor x + 3 B1
[Condone omission of repetition that x – 2 is a factor.]
[If linear factors 2x – 1, x + 3 obtained by remainder theorem or inspection, award B2 + B1.] [4]
8 (i) Use correct sin(A – B) and cos(A – B) formulae M1
Substitute exact values for sin 30° etc. M1
Obtain given answer correctly A1 [3]
(ii) State xx sec2
1sin3 B1
Rearrange to sin 2x = k, where k is a non-zero constant M1
Carry out evaluation of
3
1sin
2
1 1 M1
Obtain answer 17.6° A1
Carry out correct method for second answer M1
Obtain remaining 3 answers from 17.6°, 72.4°, 197.6°, 252.4° and no others in the
range A1 [6]
[Ignore answers outside the given range]
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/31 Paper 31, maximum raw mark 75
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/32 Paper 32, maximum raw mark 75
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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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Page 4 Mark Scheme: Teachers’ version Syllabus Paper
Use correct method for solving an equation of the form 2x = a, where a > 0 M1 Obtain answer x = 0.585 A1
OR: State an appropriate iterative formula, e.g. xn + 1 = ln((2 nx + 6) / 5) / ln 2 B1
Use the iterative formula correctly at least once M1 Obtain answer x = 0.585 A1 Show that the equation has no other root but 0.585 A1 [4] [For the solution 0.585 with no relevant working, award B1 and a further B1 if 0.585 is shown to
be the only root.]
2 Integrate by parts and reach ∫±± xxxxx dcos2cos2
M1
Obtain ∫+ xxxxx dcos2cos2
, or equivalent A1
Complete the integration, obtaining xxxxx cos2sin2cos2
++ , or equivalent A1
Substitute limits correctly, having integrated twice M1 Obtain the given answer correctly A1 [5] 3 (i) State or imply sin a = 4/5 B1 Use sin(A – B) formula and substitute for cos a and sin a M1
Obtain answer )334(10
1, or exact eqivalent A1 [3]
(ii) Use tan 2A formula and substitute for tan a, or use sin 2A and cos 2A formulae, substitute sin a and cos a, and divide M1
Obtain 7
242tan a , or equivalent A1
Use tan(A + B) formula with A = 2a , B = a and substitute for tan 2a and tan a M1
Obtain 117
443tan a A1 [4]
4 (i) Use correct quotient or product rule M1 Obtain correct derivative in any form A1 Equate derivative to zero and solve for x M1
Obtain the given answer correctly A1 [4]
(ii) Use the iterative formula correctly at least once M1 Obtain final answer 4.49 A1
Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show that there is a sign change in the interval (4.485, 4.495) A1 [3] 5 (i) Substitute
2
1x , equate to zero and obtain a correct equation, e.g.
02
1
4
5
4
1++ ba B1
Substitute x = 2 and equate to 9 M1 Obtain a correct equation, e.g. 922016 ++ ba A1 Solve for a or for b M1 Obtain a = 4 and b = 3 A1 [5]
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Page 5 Mark Scheme: Teachers’ version Syllabus Paper
(ii) Attempt division by 2x + 1 reaching a partial quotient of kxx +2 M1
Obtain quadratic factor 322+ xx A1
Obtain factorisation )1)(3)(12( ++ xxx A1 [3] [The M1 is earned if inspection has an unknown factor of fexx ++
2 and an equation in e and/or
f, or if two coefficients with the correct moduli are stated without working.] [If linear factors are found by the factor theorem, give B1 + B1 for (x 1) and (x + 3), and then B1
for the complete factorisation.]
6 (i) EITHER: State or imply x
y
y d
d1 as derivative of ln y B1
State correct derivative of LHS, e.g. x
y
y
xy
d
dln + B1
Differentiate RHS and obtain an expression for x
y
d
d M1
Obtain given answer A1
OR 1: State x
xy
12ln
+, or equivalent, and differentiate both sides M1
State correct derivative of LHS, e.g. x
y
y d
d1 B1
State correct derivative of RHS, e.g. 2/1 x B1
Rearrange and obtain given answer A1 OR 2: State )/12exp( xy + , or equivalent, and attempt differentiation by chain rule M1
State correct derivative of RHS, e.g. 2/)/12exp( xx+ B1 + B1 Obtain given answer A1 [4] [The B marks are for the exponential term and its multiplier.] (ii) State or imply
2
1x when y = 1 B1
Substitute and obtain gradient of 4 B1√ Correctly form equation of tangent M1 Obtain final answer y + 4x + 1 = 0, or equivalent A1 [4] 7 (i) Separate variables correctly and attempt integration of both sides B1 Obtain term tan x B1 Obtain term t2
2
1e B1
Evaluate a constant or use limits x = 0, t = 0 in a solution containing terms a tan x and be 2t M1 Obtain correct solution in any form, e.g. t
x2
2
1
2
1etan A1
Rearrange as )e(tan 2
2
1
2
11 tx , or equivalent A1 [6]
(ii) State that x approaches )(tan
2
11 B1 [1]
(iii) State that t2
e1 increases and so does the inverse tangent, or state that xt 22cose is
positive B1 [1]
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Page 6 Mark Scheme: Teachers’ version Syllabus Paper
8 (i) EITHER: State a correct expression for zor 2
z , e.g. 22 )2(sin)2cos1( θθ ++ B1
Use double angle formulae throughout or Pythagoras M1 Obtain given answer 2cos θ correctly A1 State a correct expression for tangent of argument, e.g. )2cos1/(2(sin θθ + B1
Use double angle formulae to express it in terms of cos θ and sin θ M1 Obtain tan θ and state that the argument is θ A1 OR: Use double angle formulae to express z in terms of cos θ and sin θ M1 Obtain a correct expression, e.g. θθθθ cossin2sincos1
22i++ A1
Convert the expression to polar form M1 Obtain )sin(coscos2 θθθ i+ A1
State that the modulus is 2 cosθ A1 State that the argument is θ A1 [6] (ii) Substitute for z and multiply numerator and denominator by the conjugate of z, or equivalent M1 Obtain correct real denominator in any form A1 Identify and obtain real part equal to
2
1 A1 [3]
9 (i) State or imply a correct normal vector to either plane, e.g. 3i + 2j + 4k or ai + j + k B1 Equate scalar product of normals to zero and obtain an equation in a, e.g. 3a + 2 + 4 = 0 M1 Obtain a = 2 A1 [3] (ii) Express general point of the line in component form, e.g. (λ , 1 + 2λ , 1 + 2λ) B1 Either substitute components in the equation of p and solve for λ , or substitute components and the value of a in the equation of q and solve for λ M1* Obtain λ = 1 for point A A1 Obtain λ = 2 for point B A1 Carry out correct process for finding the length of AB M1(dep*) Obtain answer AB = 3 A1 [6] [The second M mark is dependent on both values of λ being found by correct methods.]
10 (i) EITHER: Divide by denominator and obtain quadratic remainder M1 Obtain A = 1 A1 Use any relevant method to obtain B, C or D M1 Obtain one correct answer A1 Obtain B = 2, C = 1 and D = 3 A1 OR: Reduce RHS to a single fraction and equate numerators, or equivalent M1 Obtain A = 1 A1 Use any relevant method to obtain B, C or D M1
Obtain one correct answer A1 Obtain B = 2, C = 1 and D = 3 A1 [5] [SR: If A = 1 stated without working give B1.]
(ii) Integrate and obtain )12ln(2
31ln2+ x
x
xx , or equivalent B3√
(The f.t. is on A, B, C, D. Give B2√ if only one error in integration; B1√ if two.) Substitute limits correctly in the complete integral M1 Obtain given answer correctly following full and exact working A1 [5]
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/33 Paper 33, maximum raw mark 75
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/41 Paper 41, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
[4(15 – 0.05 × 152) = B/152] M1 For using v1(15) = v2(15)
B = 3375 A1
[5]
AG
(ii) s2(t) = Bt 1/(–1) (+ C) B1
[–3375/15 + C = 225] M1 For using s2(15) = 225 to find C
Distance travelled is [450 – 3375/t] m
(for t [ 15)
A1
[3]
(iii) [450 – 3375/t = 315] M1 For attempting to solve s2(t) = 315
[v = 3375/252] M1 For substituting into v = 3375/t2
Speed is 5.4 ms 1 A1
[3]
Alternative for 7(ii)
s = )
15
1
t
1(3375dt3375
15
2−−=∫
t
t
= 225 – 3375/t
B1
Distance travelled = 225 + (225 – 3375/t) M1
Distance travelled is [450 – 3375/t] m
(for t [ 15)
A1
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/42 Paper 42, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
[4(15 – 0.05 × 152) = B/152] M1 For using v1(15) = v2(15)
B = 3375 A1
[5]
AG
(ii) s2(t) = Bt 1/(–1) (+ C) B1
[–3375/15 + C = 225] M1 For using s2(15) = 225 to find C
Distance travelled is [450 – 3375/t] m
(for t [ 15)
A1
[3]
(iii) [450 – 3375/t = 315] M1 For attempting to solve s2(t) = 315
[v = 3375/252] M1 For substituting into v = 3375/t2
Speed is 5.4 ms 1 A1
[3]
Alternative for 7(ii)
s = )
15
1
t
1(3375dt3375
15
2−−=∫
t
t
= 225 – 3375/t
B1
Distance travelled = 225 + (225 – 3375/t) M1
Distance travelled is [450 – 3375/t] m
(for t [ 15)
A1
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/43 Paper 43, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/51 Paper 51, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
7 (i) [0.25v(dv/dx) = –(5 – x)] B1 For using Newton’s second law and
a = v(dv/dx)
]5)dx(x4vdv[∫ ∫
M1
For separating variables and attempting
to integrate
v2/2 = 4(x – 5)2/2 (+ A)
v2 = 4(x – 5) 2
Selects correct square root to obtain v = 10 – 2x
A1
M1
A1
A1
[6]
For using v(0) = 10
Any correct expression in x
AG
(ii) [ ∫x210
dx= ∫dt ] M1
For using v = dx/dt and separating
variables
–2
1 ln(10 – 2x) = t(–2
1 lnB) A1
B = 10 (or equivalent)
x = 5(1 – e 2t)
0 < e 2t < 1 for all t → x < 5 for all t
A1
B1ft
B1
[5]
ft x = (B/2)(1 – e 2t)
AG
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/52 Paper 52, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
7 (i) [0.25v(dv/dx) = –(5 – x)] B1 For using Newton’s second law and
a = v(dv/dx)
]5)dx(x4vdv[∫ ∫
M1
For separating variables and attempting
to integrate
v2/2 = 4(x – 5)2/2 (+ A)
v2 = 4(x – 5) 2
Selects correct square root to obtain v = 10 – 2x
A1
M1
A1
A1
[6]
For using v(0) = 10
Any correct expression in x
AG
(ii) [ ∫x210
dx= ∫dt ] M1
For using v = dx/dt and separating
variables
–2
1 ln(10 – 2x) = t(–2
1 lnB) A1
B = 10 (or equivalent)
x = 5(1 – e 2t)
0 < e 2t < 1 for all t → x < 5 for all t
A1
B1ft
B1
[5]
ft x = (B/2)(1 – e 2t)
AG
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/53 Paper 53, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
6 (i) 0.5vdv/dx = –3v1/2 M1 Newton’s 2nd Law with a = vdv/dx
∫2/1
v dv = – ∫6dx M1 Separates variables and integrates
v3/2/(3/2) = –6x (+ c)
x = 0, v = 9 hence c = 18
v3/2 = 3(18 – 6x)/2
v = (27 – 9x)2/3 AG
A1
M1
A1
[5]
Or uses limits
(ii) dx/dt = (27 – 9x)2/3 M1 0.5dv/dt = –3v1/2
∫ dx 9x)(27 32 / = ∫dt ∫2/1
v dv = – ∫ dt6
(27 – 9x)1/3/–3 = t (+ c)
t = 0, x = 0 hence c = –1
t = 0.5, x = 2.625
A1ft
M1
A1
[4]
v1/2 = –3t + c
t = 0, v = 9 hence c = 3 and t = 0.5,
giving v = 2.25
v = 2.25, x = 2.625
7 (i) 0.4v2/2 + 24x2/(2×3)
0.4g(3 + x) + 0.4×22/2
v2 = 64 + 20x – 20x2 AG
M1
A2
A1
[4]
PE, EE, KE terms
–1 each error to zero
(ii) 2vdv/dx = 20 – 40x = 0
x = 0.5
v = 8.31
M1
A1ft
A1
[3]
0.4g = 24x/3
(iii) 20x2 – 20x – 64 = 0
x = 2.357
T = 24×2.357/3
T = 18.9
M1
A1
M1
A1
[4]
And attempts to solve
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/61 Paper 61, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/62 Paper 62, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/63 Paper 63, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
Considering all three options Any one option correct Correct answer
OR Considering all three options
Correct fraction
Correct answer
(ii) colas together, no restrictions, 5!/2!2! = 30 ways colas together and green tea together, 4!/2! = 12 ways 30 – 12 = 18 ways. OR1 Attempt to list
OR2 182
343
×
×
M1 A1 M1 A1 A1 M1A1 M1A1 A1
M1
A1 M1 A1 A1
[5]
Considering all colas together, or 5! seen Correct answer Considering all colas tog and all green tea tog, or 4! seen Correct answer Correct final answer OR1 10 or more, 12 or more correct 14 or more, 16 or more correct 18 correct
OR2 Considering all colas together, or 3! seen
3 ways for colas and orange juice Considering green teas not together 4 × 3 or (4 × 3)/2 Correct final answer
6 (i) for X: Median = 0.825 cm IQ range = 0.019 cm (0.833 – 0.814)
B1 B1
[2]
Correct median Correct IQ range
(ii) q = 4 r = 2 SR q = 0.824 and r = 0.852
B1 B1
[2] B1
Must be 4 and 2 not 3 and 1
(iii)
Y
X
0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 length in cm
B1 B1ft B1ft B1
[4]
Labels X, Y and length/cm, linear scale from 0.80 to 0.87 and both on one diagram Correct median and quartiles for X ft theirs must be a box Correct median and quartiles for Y ft theirs must be a box Whiskers correct no line through middle
(iv) Y has longer insects on average Y has larger range
B1 B1
[2]
Correct statement about lengths Correct statement about spreads
7 (i) 0.431 = σ
µ135
–0.842 = σ
µ127
σ = 6.29 µ = 132
B1
B1
M1
A1 A1
[5]
One ±z-value correct, accept 0.430
A second ±z-value correct
Solving two equations relating µ, σ, 135,
127 and their z-values (must be z-values) Correct answer accept 6.28 Correct answer
(ii) P(X < 145) =
<
284.6
3.132145P z
=P(z < 2.023) = 0.978
M1
M1 A1
[3]
Standardising no sq rt no cc
Correct use of normal tables Answer rounding to 0.978 or 0.979
(iii) p = 1/3 P(at least 2) = 1 – P(0, 1) = 1 – [ 71
1
88 )3/2()3/1(C)3/2( ×+ ] = 0.805
M1 A1 A1
[3]
Binomial expression with powers summing to 8 and 8Csomething. (any p) Correct unsimplified expression Answer rounding to 0.805
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/71 Paper 71, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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.
www.maxpapers.com
Page 4 Mark Scheme: Teachers’ version Syllabus Paper
Equation of correct form relating their λ and 0.99
Valid attempt to solve equation of correct form
by logs or trial and error
Correct answer
(SR Accept use of Binomial leading to n = 57)
7 (i) Type I error is made when we say the
number of white blood cells has decreased
when it hasn’t.
P(0) = e 5.2 = 0.005516
P(1) = e 5.2(5.2) = 0.02868 Σ < 0.10
P(2) = e 5.2(5.22/2) = 0.07458 Σ > 0.10
P(Type I error) = 0.0342
B1
M1
M1*
A1dep
[4]
Correct and relating to question
Evaluating at least 2 of P(X = 0, 1, 2)
Comparing their Σ 3 probs with 10% (must be Σ
probs)
Correct answer, dep on previous M
(ii) H0: λ = 5.2
H1: λ < 5.2
P(0+1+2) = 0.1087 > 10%
2 not in C Region.
Accept H0. Not enough evidence to say the
number of blood cells has decreased.
B1
M1
A1
[3]
Both hypotheses correct
Stating 2 is not in the critical region from above,
or evaluating P(0, 1, 2) and comparing with 10%
again
Correct conclusion no contradictions
(iii) P(Type II error) = 1 – P(0, 1)
= 1 – e 4.1(1 + 4.1)
= 0.915
B1
M1
A1
[3]
Identifying correct area
(indep) Some form of (Poisson) expression with
mean 4.1
Correct answer
www.maxpapers.com
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/72 Paper 72, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
www.maxpapers.com
Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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.
www.maxpapers.com
Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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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Page 4 Mark Scheme: Teachers’ version Syllabus Paper
Equation of correct form relating their λ and 0.99
Valid attempt to solve equation of correct form
by logs or trial and error
Correct answer
(SR Accept use of Binomial leading to n = 57)
7 (i) Type I error is made when we say the
number of white blood cells has decreased
when it hasn’t.
P(0) = e 5.2 = 0.005516
P(1) = e 5.2(5.2) = 0.02868 Σ < 0.10
P(2) = e 5.2(5.22/2) = 0.07458 Σ > 0.10
P(Type I error) = 0.0342
B1
M1
M1*
A1dep
[4]
Correct and relating to question
Evaluating at least 2 of P(X = 0, 1, 2)
Comparing their Σ 3 probs with 10% (must be Σ
probs)
Correct answer, dep on previous M
(ii) H0: λ = 5.2
H1: λ < 5.2
P(0+1+2) = 0.1087 > 10%
2 not in C Region.
Accept H0. Not enough evidence to say the
number of blood cells has decreased.
B1
M1
A1
[3]
Both hypotheses correct
Stating 2 is not in the critical region from above,
or evaluating P(0, 1, 2) and comparing with 10%
again
Correct conclusion no contradictions
(iii) P(Type II error) = 1 – P(0, 1)
= 1 – e 4.1(1 + 4.1)
= 0.915
B1
M1
A1
[3]
Identifying correct area
(indep) Some form of (Poisson) expression with
mean 4.1
Correct answer
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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
GCE Advanced Subsidiary Level and GCE Advanced Level
MARK SCHEME for the May/June 2010 question paper
for the guidance of teachers
9709 MATHEMATICS
9709/73 Paper 73, maximum raw mark 50
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 must be read in conjunction with the question papers and the report on the examination.
• CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
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Page 2 Mark Scheme: Teachers’ version Syllabus Paper
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 √ 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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Page 3 Mark Scheme: Teachers’ version Syllabus Paper
The following abbreviations may be used in a mark scheme or used on the scripts:
AEF Any Equivalent Form (of answer is equally acceptable) AG Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear) 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 MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently
accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) 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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Page 4 Mark Scheme: Teachers’ version Syllabus Paper
1 H0: Pop prop = 1/3 (or unchanged) H1: Pop prop < 1/3 (or decreased)
(3
2 )20 + 30(3
2 )19(3
1 ) + 20C2(3
2 )18(3
1 )2
+ 20C3(3
2 )17(3
1 )3
= 0.0604/0.0605 comp “0.0604” with 0.025 No evidence that support decreased or support probably not decreased SC Use Of Normal Standardising with or without cc Obtains z = –1.502 Valid Comparison with z = –1.96 Correct conclusion
B1
M1
A1 M1 A1ft
[5] M1 A1 M1 A1ft
Accept p
Attempt Bin(20, ⅓) P(Y 3)
Allow one term omitted
For comparison of their 0.0604 Correct conclusion no contradictions
2 (i) 2-tail; H1: µ ¸ 35 B1 [1]
(ii) comp –1.75 with –1.645 (or 1.75 with 1.645) Evidence that µ is not 35 or reject µ = 35
M1 A1
[2]
Allow “Accept µ ¸ 35”. No contradictions
(iii) 8 B2 [2]
SR B1 for 4, 8.02, or 92%
3 (i) (Approx) normal mean 62
sd = 50
2.8 = 1.16 (3 sfs)
B1 B1
B1
[3]
or var = 50
2.82
= 1.34 (3 sfs)
(ii) "16.1"
6264 (= 1.725 or 1.724) M1 For standardising ÷ 50 essential (no CC)
1 – Φ(“1.725”) = (1 – 0.9577) = 0.0423 (3 sfs)
M1 A1
[3]
For correct area consistent with their mean
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Page 5 Mark Scheme: Teachers’ version Syllabus Paper