w
w
ap
eP
m
e
tr
.X
w
CAMBRIDGE INTERNATIONAL EXAMINATIONS
om
.c
s
er
Cambridge International Advanced Subsidiary and Advanced Level
MARK SCHEME for the May/June 2015 series
9709 MATHEMATICS
9709/11
Paper 1, 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 should be read in conjunction with the question paper and the Principal Examiner
Report for Teachers.
Cambridge will not enter into discussions about these mark schemes.
Cambridge is publishing the mark schemes for the May/June 2015 series for most
Cambridge IGCSE®, Cambridge International A and AS Level components and some
Cambridge O Level components.
® IGCSE is the registered trademark of Cambridge International Examinations.
Page 2
Mark Scheme
Cambridge International AS/A Level – May/June 2015
Syllabus
9709
Paper
11
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.
© Cambridge International Examinations 2015
Page 3
Mark Scheme
Cambridge International AS/A Level – May/June 2015
Syllabus
9709
Paper
11
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.
© Cambridge International Examinations 2015
Page 4
Mark Scheme
Cambridge International AS/A Level – May/June 2015
1
Syllabus
9709
Paper
11
θ is obtuse , sin θ = k
(i)
cos θ = −√(1 – k²)
B1
cao
[1]
sin θ
used
cos θ
k
→ tan θ = −
aef
(1 − k 2 )
(ii)
tan θ =
(iii)
sin (θ + π) = −k
M1
A1
Used, attempt at cosine seen in (i)
Ft for their cosine as a function of k
[2] only, from part (i)
B1
cao
[1]
2
y = 2x² , X(−2, 0) and P(p, 0)
(i)
A=
1
2
× (2 + p) × 2p² (= 2p² + p³)
2
dA
= 4p + 3p²
dp
dA d A dp
= 0.02 × 20 = 0.4
=
×
dt dp d t
dA
dp
dp
or
= 4p
+ 3 p2
dt
dt
dt
(ii)
M1 A1
Attempt at base and height in terms
[2]
bh
of p and use of
B1
cao
M1 A1
any correct method, cao
[3]
(1 − x) 2 (1 + 2 x) 6 .
3
(i) (a)
(1 − x) 6 = 1 – 6x + 15x²
B2,1
−1 each error
[2]
(b)
(ii)
(1 + 2 x) 6 = 1 + 12x + 60x²
B2,1
Product of (a) and (b) with >1 term
→ 60 – 72 + 15 = 3
Must be 2 or more products
M1
DM1A1 M1 exactly 3 products. cao,
[3] condone 3x2
−1 each error
[2] SC B1 only, in each part, for all 3
correct descending powers
SC only one penalty for omission
of the ‘1’ in each expansion
© Cambridge International Examinations 2015
Page 5
Syllabus
9709
Paper
11
 3
6
 k 
uuur   uuur  


OA =  0  , OB =  −3  , OC =  − 2k 
 2k − 3 
 −4 
2


 
 
4
(i)
(ii)
5
Mark Scheme
Cambridge International AS/A Level – May/June 2015
(i)
OA∙ OB = 18 – 8 = 10
Modulus of OA = 5, of OB = 7
 10 
Angle AOB = cos−1   aef
 35 
10
2
or
→
35
7
M1
Use of x1x2 + y1y2 + z1z2
M1
All linked with modulus
cao, (if angle given, no penalty),
correct angle implies correct cosine
 3 
 
AB = b – a =  − 3 
 6 
 
B1
allow for a – b
k² + 4k² + (2k – 3)² = 9 + 9 + 36
M1
→ 9k² −12k −45( = 0)
5
→ k= 3 or k = −
3
DM1
Correct use of moduli using their
AB
obtains 3 term quadratic.
cao
24 = r + r + rθ
24 − 2r
→ θ=
r
1
24r
− r 2 = 12r – r². aef, ag
A=
r²θ =
2
2
A1
[3]
A1
[4]
(May not use ߠ)
M1
M1A1
[3]
Attempt at s = rθ linked with 24
and r
Uses A formula with θ as f(r). cao
(ii)
( A =)36 − (r − 6) 2
B1 B1
cao
[2]
(iii)
Greatest value of A = 36
B1
Ft on (ii).
(r = 6) → θ = 2
B1
cao, may use calculus or the
discriminant on 12r − r 2
[2]
© Cambridge International Examinations 2015
Page 6
6
(i)
(ii)
Mark Scheme
Cambridge International AS/A Level – May/June 2015
y − 2t = −2( x − 3t )( y + 2 x = 8t )
Set x to 0 → B(0, 8t)
Set y to 0 → A(4t, 0)
→ Area = 16t²
m=
1
1
→ y − 2t = ( x − 3t )(2 y = x + t )
2
Set y to 0 → C (−t, 0)
Midpoint of CP is (t, t)
This lies on the line y = x.
Paper
11
M1
Unsimplified or equivalent forms
M1
A1
Attempt at both A and B, then using
cao
[3]
cao
Unsimplified or equivalent forms
co
B1
2
Syllabus
9709
M1
A1
correctly shown.
A1
[4]
7
(a)
1
2
, ar³ =
3
9
2
aef
→ r=
3
ar² =
Substituting → a =
→ S∞ =
(b)
3
4
1
3
M1
3
4
= 2 14 aef
4a = a + 4d → 3a = 4d
360 = S5 =
5
(2a + 4d ) or 12.5a
2
→ a = 28.8º aef
Largest = a + 4d or 4a = 115.2º aef
Any valid method, seen or implied.
Could be answers only.
Both a and r
A1
Correct formula with r < 1 , cao
M1 A1
[4]
B1
May be implied in
360 = 5 / 2(a + 4a)
M1
Correct Sn formula or sum of 5
terms
A1
B1
cao, may be implied
(may use degrees or radians)
[4]
© Cambridge International Examinations 2015
Page 7
Mark Scheme
Cambridge International AS/A Level – May/June 2015
Syllabus
9709
Paper
11
1 
2 
8
f : x⟼5 + 3cos  x  for 0 ø x ø 2π.
(i)
1 
2 
2
1 
cos  x  =
3
2 
5 + 3cos  x  = 7
1
2
x = 0.84
x = 1.68 only, aef
1  2
Makes cos  x  =
2  3
B1
M1A1
Looks up cos−1 first, then ×2
[3]
(in given range)
(ii)
B1
B1
y always +ve, m always –ve.
from (0, 8) to (2π, 2) (may be
[2] implied)
2
(iii)
No turning point on graph or 1:1
B1
cao, independent of graph in (ii)
[1]
(iv)
1 
2 
y = 5 + 3cos  x 
M1
Tries to make x subject.
Order; −5, ÷3, cos−1, ×2
 x −5
x = 2cos−1  3 
M1
Correct order of operations
A1
cao
[3]
© Cambridge International Examinations 2015
Page 8
Mark Scheme
Cambridge International AS/A Level – May/June 2015
Syllabus
9709
Paper
11
y = x 3 + px 2
9
(i)
dy
= 3x² + 2px
dx
Sets to 0 → x = 0 or −
2p
3
 2 p 4 p3 

→ (0, 0) or  −
,
 3 27 
(ii)
cao
M1
Sets differential to 0
A1 A1
cao cao, first A1 for any correct
[4] turning point or any correct pair of
x values. 2nd A1 for 2 complete
TPs
d2 y
= 6x + 2p
dx 2
M1
Other methods include; clear
demonstration of sign change of
gradient, clear reference to the
shape of the curve
At (0, 0) → 2p +ve Minimum
A1
www
 2p 4p
At  −
,
 3 27
A1
3
(iii)
B1

 → −2p –ve Maximum

y = x 3 + px 2 + px → 3x² + 2px + p (= 0)
2
Uses b − 4 ac
→ 4p² − 12p< 0
→ 0 <p< 3 aef
[3]
B1
M1
Any correct use of discriminant
cao (condone ø)
A1
[3]
© Cambridge International Examinations 2015
Page 9
Mark Scheme
Cambridge International AS/A Level – May/June 2015
y=
10
(i)
dy
−4
× 3 aef
=
3
dx (3 x + 4) 2
3
2
Perpendicular m(x=0) =
2
3
2
( x − 0)
3
 20 
Meets x = 4 at B  4, 
 3 
Eqn of normal y − 4 =
∫
Paper
11
8
3x + 4
→ m(x=0) = −
(ii)
Syllabus
9709
B1
B1
Without the “×3”
For “×3” even if 1st B mark lost.
M1
Use of m1m2 = −1 after attempting
dy
to find
(x=0)
dx
M1
Unsimplified line equation
A1
cao
[5]
8 (3 x + 4))
8
÷3
dx =
1
(3 x + 4)
B1 B1
Without “÷3”. For “÷3”
M1 A1
Correct use of correct limits. cao
M1
Correct method for area of
trapezium
2
Limits from 0 to 4 → Area P =
Area Q = Trapezium – P
Area of Trapezium =
1
20 
64
4 + × 4 =
2
3 
3
→ Areas of P and Q are both
32
3
32
3
A1
All correct.
[6]
© Cambridge International Examinations 2015