w
w
ap
eP
m
e
tr
.X
w
om
.c
s
er
Cambridge International Examinations
CambridgeInternationalGeneralCertificateofSecondaryEducation
* 9 9 3 7 8 4 8 4 1 4 *
0606/12
ADDITIONAL MATHEMATICS
Paper1
May/June 2015
2 hours
CandidatesanswerontheQuestionPaper.
AdditionalMaterials:
Electroniccalculator
READ THESE INSTRUCTIONS FIRST
WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin.
Writeindarkblueorblackpen.
YoumayuseanHBpencilforanydiagramsorgraphs.
Donotusestaples,paperclips,glueorcorrectionfluid.
DONOTWRITEINANYBARCODES.
Answerallthequestions.
Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalplaceinthecaseof
anglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion.
Theuseofanelectroniccalculatorisexpected,whereappropriate.
Youareremindedoftheneedforclearpresentationinyouranswers.
Attheendoftheexamination,fastenallyourworksecurelytogether.
Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestion.
Thetotalnumberofmarksforthispaperis80.
Thisdocumentconsistsof16printedpages.
DC(CW/JG)91463/2
©UCLES2015
[Turn over
2
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=
Binomial Theorem
()
()
b 2 − 4 ac
2a
−b
()
()
n
n
n
(a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,
n
n!
where n is a positive integer and r =
(n – r)!r!
2. TRIGONOMETRY
Identities
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A
Formulae for ∆ABC
a
b
c
sin A = sin B = sin C
a2 = b2 + c2 – 2bc cos A
∆=
© UCLES 2015
1
bc sin A
2
0606/12/M/J/15
3
1 Given that the graph of
of k.
2 Show that
© UCLES 2015
y = (2k + 5) x 2 + kx + 1
does not meet the x-axis, find the possible values
[4]
tan i + cot i
= sec i .
cosec i
[4]
0606/12/M/J/15
[Turn over
4
3 J4
Find the inverse of the matrix K
L5
© UCLES 2015
2N
O and hence solve the simultaneous equations
3P
4x + 2y - 8 = 0 ,
5x + 3y - 9 = 0 .
0606/12/M/J/15
[5]
5
4
B
A
1.7
rad O
12 cm
2.4
rad
C
The diagram shows a circle, centre O, radius 12 cm. The points A, B and C lie on the circumference of
this circle such that angle AOB is 1.7 radians and angle AOC is 2.4 radians.
(i) Find the area of the shaded region.
[4]
(ii) Find the perimeter of the shaded region.
[5]
© UCLES 2015
0606/12/M/J/15
[Turn over
6
5 (a) A security code is to be chosen using 6 of the following:
• the letters A, B and C
• the numbers 2, 3 and 5
• the symbols * and $.
None of the above may be used more than once. Find the number of different security codes that
may be chosen if
(i) there are no restrictions,
[1]
(ii) the security code starts with a letter and finishes with a symbol,
[2]
(iii) the two symbols are next to each other in the security code.
[3]
© UCLES 2015
0606/12/M/J/15
7
(b) Two teams, each of 4 students, are to be selected from a class of 8 boys and 6 girls. Find the
number of different ways the two teams may be selected if
(i) there are no restrictions,
[2]
(ii) one team is to contain boys only and the other team is to contain girls only.
[2]
© UCLES 2015
0606/12/M/J/15
[Turn over
8
6 A particle moves in a straight line such that its displacement, x m, from a fixed point O after t s, is
given by x = 10 ln ^t 2 + 4h - 4t .
(i) Find the initial displacement of the particle from O.
[1]
(ii) Find the values of t when the particle is instantaneously at rest.
[4]
© UCLES 2015
0606/12/M/J/15
9
(iii) Find the value of t when the acceleration of the particle is zero.
© UCLES 2015
0606/12/M/J/15
[5]
[Turn over
10
7
A
4a
B
X
D
b
7a
C
In the diagram AB = 4a, BC = b and DC = 7a. The lines AC and DB intersect at the point X. Find, in
terms of a and b,
(i) DA,
[1]
(ii) DB.
[1]
Given that AX = mAC , find, in terms of a, b and m,
(iii) AX ,
[1]
(iv) DX .
[2]
© UCLES 2015
0606/12/M/J/15
11
Given that DX = nDB ,
(v) find the value of m and of n .
© UCLES 2015
[4]
0606/12/M/J/15
[Turn over
12
8 (i) Find y ^10e 2x + e -2xh dx .
(ii) Hence find
y-k (10e2x + e-2x) dx in terms of the constant k.
(iii) Given that
y-k (10e2x + e-2x) dx =- 60 , show that 11e2k - 11e-2k + 120 = 0 .
© UCLES 2015
[2]
k
k
0606/12/M/J/15
[2]
[2]
13
(iv) Using a substitution of y = e 2k or otherwise, find the value of k in the form a ln b, where
a and b are constants.
© UCLES 2015
0606/12/M/J/15
[3]
[Turn over
14
9 r
A curve has equation y = 4x + 3 cos 2x . The normal to the curve at the point where x = meets
4
the x- and y-axes at the points A and B respectively. Find the exact area of the triangle AOB,
where O is the origin.
[8]
© UCLES 2015
0606/12/M/J/15
15
10 (a) Solve
2 cos 3x = sec 3x for 0° G x G 120° .
[3]
3 cosec 2 y + 5 cot y - 5 = 0 for 0° G y G 360° .
[5]
(b) Solve
Question 10(c) is printed on the next page.
© UCLES 2015
0606/12/M/J/15
[Turn over
16
(c) Solve
J
rN
2 sin Kz + O = 1 for 0 G z G 2r radians.
3P
L
[4]
Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonable
efforthasbeenmadebythepublisher(UCLES)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwill
bepleasedtomakeamendsattheearliestpossibleopportunity.
Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedonlineintheCambridgeInternational
ExaminationsCopyrightAcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadatwww.cie.org.ukafter
theliveexaminationseries.
CambridgeInternationalExaminationsispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocal
ExaminationsSyndicate(UCLES),whichisitselfadepartmentoftheUniversityofCambridge.
© UCLES 2015
0606/12/M/J/15