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CAMBRIDGE INTERNATIONAL EXAMINATIONS
s
er
Cambridge Ordinary Level
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MARK SCHEME for the May/June 2015 series
4037 ADDITIONAL MATHEMATICS
4037/11
Paper 1, maximum raw mark 80
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 O Level – May/June 2015
Syllabus
4037
Paper
11
Abbreviations
awrt
cao
dep
FT
isw
oe
rot
SC
soi
www
1
answers which round to
correct answer only
dependent
follow through after error
ignore subsequent working
or equivalent
rounded or truncated
Special Case
seen or implied
without wrong working
(i)
180° or π radians or 3.14 radians ( or better)
B1
(ii)
2
B1
(iii) (a)
4
y
3
B1
y = sin 2 x all correct
B1
for either
↑↓↑starting at their highest value and
ending at their lowest value
Or
a curve with highest value at y = 3
and lowest value at y = −1
completely correct graph
2
(b)
1
x
45
90
135
180
−1
−2
−3
−4
B1
(iv)
2
(i)
3
B1
tan θ =
(8 + 5 2 )(4 − 3 2 )
(4 + 3 2 )(4 − 3 2 )
32 − 24 2 + 20 2 − 30
16 − 18
= 1 + 2 2 cao
M1
=
attempt to obtain tan θ and rationalise.
Must be convinced that no calculators
are being used
A1
© Cambridge International Examinations 2015
Page 3
(ii)
Mark Scheme
Cambridge O Level – May/June 2015
sec 2 θ = 1 + tan 2 θ
(
= 1 + −1 + 2 2
)
2
=1+1− 4 2 + 8
= 10 − 4 2
(
) (
2
AC 2 = 4 + 3 2 + 8 + 5 2
=
)
148 + 104 2
(4 + 3 2 )
2
attempt to simplify, must be convinced
no calculators are being used.
×
(
Need to expand − 1+ 2 2
terms
)
2
as 3
M1
34 − 24 2
34 − 24 2
DM1
A1
(i)
64 + 192 x 2 + 240 x 4 + 160 x 6
(ii)
(64 + 192 x
2
DM1
2
= 10 − 4 2
3
attempt to use sec 2 θ = 1 + tan 2 θ , with
their answer to (i)
)
= 148 + 104 2
148 + 104 2
sec 2 θ =
2
4+3 2
(
Paper
11
M1
A1
Alternative solution:
Syllabus
4037
)
6
9 

+ 240 x 4 1 − 2 + 4 
x 
 x
Terms needed 64 − (192 × 6) + (240 × 9)
= 1072
B3,2,1,0 –1 each error
B1
M1
3 

expansion of 1 − 2 
 x 
2
attempt to obtain 2 or 3 terms using
their (i)
A1
© Cambridge International Examinations 2015
Page 4
4
(a)
(b)
Mark Scheme
Cambridge O Level – May/June 2015
 4 − 4k
X2 = 
 2k
−8 

− 4k 
B2,1,0
Use of AA–1 = I
1
 5
 a 1  6 − 6   1 0 
 = 



 b 5  − 2 1   0 1 
 3 3 
Any 2 equations will give a = 2, b = 4
Alternative method 1:
1
 5
1  5 − 1  6 − 6 


=
5a − b  b a   − 2 1 


 3 3 
Compare any 2 terms to give a = 2, b = 4
M1
Syllabus
4037
Paper
11
–1 each incorrect element
use of AA-1 = I and an attempt to
obtain at least one equation.
A1,A1
M1
correct attempt to obtain A-1 and
comparison of at least one term.
A1,A1
Alternative method 2:
Inverse of
5
1  5 − 1  2 1 

=

6  − 4 2   4 5 
M1
A1,A1
reasoning and attempt at inverse
3x − 1 = x(3x − 1) + x 2 − 4 or
2
 y +1  y +1
y =
 −4
y + 
 3   3 
4 x 2 − 4 x − 3 = 0 or 4 y 2 − 4 y − 35 = 0
(2 x − 3)(2 x + 1) = 0 or (2 y − 7 )(2 y + 5) = 0
3
1
leading to x = , x = − and
2
2
7
5
y= ,y=−
2
2
1
1


Midpoint  , 
2
2


1
Perpendicular gradient = −
3
1
1
1
Perp bisector: y − = −  x − 
2
3
2
(3 y + x − 2 = 0)
M1
DM1
equate and attempt to obtain an
equation in 1 variable
forming a 3 term quadratic equation
and attempt to solve
A1
x values
A1
y values
B1
for midpoint, allow anywhere
M1
correct attempt to obtain the gradient
of the perpendicular, using AB
M1
straight line equation through the
midpoint; must be convinced it is a
perpendicular gradient.
allow unsimplified
A1
© Cambridge International Examinations 2015
Page 5
6
(i)
(ii)
(iii)
7
(i)
Mark Scheme
Cambridge O Level – May/June 2015
 1  a 15 b
f  = − + − 2 = 0
2 8 4 2
leading to a + 4b = 46
f (1) = a − 15 + b − 2 = 5
leading to a + b = 22
M1
A1
Paper
11
1
correct use of either f   or f(1)
2
paired correctly
both equations correct (allow
unsimplified)
giving b = 8 (AG), a = 14
M1,A1
M1 for solution of equations
A1 for both a and b. AG for b.
(2 x − 1)(7 x 2 − 4 x + 2)
M1,A1
M1 for valid attempt to obtain g(x), by
either observation or by algebraic long
division.
7 x 2 − 4 x + 2 = 0 has no real solutions as
b 2 < 4 ac
16 < 56
dy
=
dx
( x − 1)
) (
)
8x
− ln 4 x 2 + 3
4x2 + 2
( x − 1) 2
(
When x = 0, y = − ln 3 oe
dy
1
= − ln 3 so gradient of normal is
dx
ln 3
(allow numerical equivalent)
1
x
ln 3
10
11
or y = 0.910 x − 1.10 , or y = x −
11
10
(Allow y = 0.91x − 1.1 )
normal equation y + ln 3 =
(ii)
Syllabus
4037
when x = 0, y = − ln 3
cao
M1
use of b 2 − 4 ac
A1
correct conclusion; must be from a
correct g(x) or 2g(x) www
M1
B1
A1
differentiation of a quotient (or
product)
correct differentiation of ln 4 x 2 + 3
all else correct
B1
for y value
M1
valid attempt to obtain gradient of the
normal
M1
attempt at normal equation must be
using a perpendicular
A1
M1
when y = 0, x = (ln 3)
Area = ±0.66 or ±0.67 or awrt these
1
3
or (ln 3)
2
(
valid attempt at area
2
A1
© Cambridge International Examinations 2015
)
Page 6
8
Mark Scheme
Cambridge O Level – May/June 2015
Syllabus
4037
Paper
11
(i)
Range for f: y ≥ 3
Range for g: y ≥ 9
B1
B1
(ii)
x = −2 + y − 5
M1
attempt to obtain the inverse function
A1
B1
Must be correct form
for domain
M1
A1
attempt to use quadratic formula and
find inverse
must have + not ±
M1
correct order
g −1 (x ) = −2 + x − 5
Domain of g–1: x ≥ 9
Alternative method:
y2 + 4 y + 9 − x = 0
y=
(iii)
−4 + 16 − 4(9 − x )
2
( )
+ 2) + 5 = 41
Need g 3e 2 x
(3e
2x
2
4x
DM1
correct attempt to solve the equation
M1
dealing with the exponential correctly
in order to reach a solution for x
A1
Allow equivalent logarithmic forms
M1
correct use of g −1
2x
or 9e + 12e − 32 = 0
3e 2 x − 4 3e 2 x + 8 = 0
(
)(
)
leading to 3e 2 x + 2 = ±6 so x =
or e 2 x =
1 4
ln
2 3
4
1 4
so x = ln
3
2 3
Alternative method:
Using f ( x) = g −1 (41) , g −1 (41) = 4
1 4
leading to 3e 2 x = 4 , so x = ln
2 3
DM1
M1
A1
(iv)
g′( x) = 6e 2 x
g′(ln 4) = 96
B1
B1
dealing with g −1 (41) to obtain an
equation in terms of e 2 x
dealing with the exponential correctly
in order to reach a solution for x
Allow equivalent logarithmic forms
B1 for each
© Cambridge International Examinations 2015
Page 7
9
(i)
Mark Scheme
Cambridge O Level – May/June 2015
dy
= 3 x 2 − 10 x + 3
dx
Syllabus
4037
Paper
11
M1
for differentiation
dy
= 3,
dx
gradient of line also 3 so line is a tangent.
A1
comparing both gradients
Alternate method:
3 x + 10 = x 3 − 5 x 2 + 3 x + 10
M1
leading to x 2 = 0 , so tangent at x = 0
A1
attempt to deal with simultaneous
equations
obtaining x = 0
When x = 0, for curve
(ii)
(iii)
dy
= 0 , (3 x − 1)( x − 3) = 0
dx
1
x= , x=3
3
When
Area =
M1
A1,A1
equating gradient to zero and valid
attempt to solve
A1 for each
3
1
(10 + 19)3 − ∫ x3 − 5 x 2 + 3 x + 10dx
0
2
B1
area of the trapezium
=

87  x 4 5 x 3 3 x 2
− −
+
+ 10 x 
2 4
3
2
0
M1
=
87  81
27

−  − 45 +
+ 30 
2  4
2

attempt to obtain the area enclosed by
the curve and the coordinate axes, by
integration
integration all correct
correct application of limits
(must be using their 3 from (ii) and 0)
3
A1
DM1
= 24.7 or 24.8
Alternative method:
(
3
A1
)
Area = ∫ (3x + 10) − x3 − 5 x 2 + 3x + 10 dx
0
3
= ∫ − x3 + 5 x 2dx
0
3
 x 4 5x3 
99
= −
+
 =
3 0 4
 4
10 (a)
sin 2 x =
1
4
sin x = (±)
1
2
B1
M1
A1
DM1
A1
M1
x = 30°, 150°, 210°, 330°
A1,A1
correct use of ‘Y–y’
attempt to integrate
integration all correct
correct application of limits
using cosecx =
1
and obtaining
sin x
sin x = …
A1 for one correct pair, A1 for another
correct pair with no extra solutions
© Cambridge International Examinations 2015
Page 8
(b)
Mark Scheme
Cambridge O Level – May/June 2015
(sec
2
)
3 y − 1 − 2 sec 3 y − 2 = 0
2
sec 3 y − 2 sec 3 y − 3 = 0
(sec 3 y + 1)(sec 3 y − 3) = 0
leading to cos 3 y = −1 , cos 3 y =
1
3
3 y = 180°, 540° 3 y = 70.5°, 289.5°, 430.5°
y = 60°, 180°, 23.5°, 96.5°, 143.5°
Alternative 1:
sec2 3 y − 2 sec 3 y − 3 = 0
2
leading to 3 cos 3 y + 2 cos 3 y − 1
(3 cos y − 1)(cos y + 1) = 0
Alternative 2:
sin 2 y
2
−
−2=0
2
cos y cos y
(1 − cos x)− 2 cos x − 2 cos
2
(c)
z−
2
Paper
11
M1
use of the correct identity
M1
attempt to obtain a 3 term quadratic
equation in sec 3y and attempt to solve
dealing with sec and 3y correctly
M1
A1,A1
A1
A1 for a correct pair, A1 for a second
correct pair, A1 for correct 5th solution
and no other within the range
M1
use of the correct identity
M1
attempt to obtain a quadratic equation
in cos 3y and attempt to solve
dealing with 3y correctly
A marks as above
M1
M1
use of the correct identity,
sin y
1
and sec y =
, then
tan y =
cos y
cos y
as before
M1
correct order of operations
x=0
π π 4π
= ,
3 3 3
2 π 5π
z=
,
or 2.09 or 2.1, 5.24
3 3
Syllabus
4037
A1,A1
A1 for a correct solution
A1 for a second correct solution and
no other within the range
© Cambridge International Examinations 2015