Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 9 3 3 4 9 5 4 2 0 2 *
0606/22
ADDITIONAL MATHEMATICS
October/November 2015
Paper 2
2 hours
Candidates answer on the Question Paper.
Additional Materials:
Electronic calculator
READ THESE INSTRUCTIONS FIRST
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
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DO NOT WRITE IN ANY BARCODES.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of
angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 80.
This document consists of 16 printed pages.
DC (SLM) 115963
© UCLES 2015
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Mathematical Formulae
1. ALGEBRA
Quadratic Equation
Fortheequationax2+bx+c=0,
x=
Binomial Theorem
()
()
−b
b 2 − 4 ac
2a
()
()
n
n
n
(a+b)n=an+ 1 an–1b+ 2 an–2b2+…+ r an–rbr+…+bn,
n
wherenisapositiveintegerand r =
n!
(n–r)!r!
2. TRIGONOMETRY
Identities
sin2A+cos2A=1
sec2A=1+tan2A
cosec2A=1+cot2A
Formulae for ∆ABC
a
b
c
sinA = sinB = sinC
a2=b2+c2–2bccosA
1
2
∆= bcsinA
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1
Itisgiventhat f (x) = 4x 3 - 4x 2 - 15x + 18 .
(i) Showthat x + 2 isafactorof f (x) .
[1]
(ii) Hencefactorise f (x) completelyandsolvetheequation f (x) = 0 .
[4]
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2
(i) Find,inthesimplestform,thefirst3termsoftheexpansionof (2 - 3x) 6 ,inascendingpowers
ofx.
[3]
(ii) Findthecoefficientof x 2 intheexpansionof(1 + 2x) (2 - 3x) 6 .
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[2]
5
3
5 - 10
6
mandc
mrespectively.All
Relativetoanorigin O,pointsA, BandChavepositionvectorsc m,c
4
12
- 18
distancesaremeasuredinkilometres.Amandrivesataconstantspeeddirectlyfrom AtoB in20minutes.
(i) Calculatethespeedinkmh–1atwhichthemandrivesfromAtoB.
[3]
Henowdrivesdirectlyfrom BtoCatthesamespeed.
(ii) Findhowlongittakeshimtodrivefrom BtoC.
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4
2 -1
1 -2 4
m,calculate2BA.
5 p and B = c
(a) Giventhat A = f 3
-2
3 0
7
4
(b) ThematricesCandDaregivenby C = c
1
-1
2
3
m and D = c
6
1
[3]
-2
m.
4
(i) FindC–1.
[2]
(ii) Hencefindthematrix XsuchthatCX + D = I,whereIistheidentitymatrix.
[3]
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5
(a) Solvethefollowingequationstofindpandq.
8 q - 1 # 2 2p + 1 = 4 7
9 p - 4 # 3 q = 81
(b) Solvetheequation lg (3x - 2) + lg (x + 1) = 2 - lg 2 .
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[4]
[5]
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6
y
5
4
3
2
1
0
π
2
0
Thefigureshowspartofthegraphof
π
3π
2
2π
x
y = a + b sin cx .
(i) Findthevalueofeachoftheintegersa,bandc.
[3]
Usingyourvaluesofa,bandc find
(ii)
dy ,
dx
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r
(iii) theequationofthenormaltothecurveat ` , 3j .
2
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7
r cm
8 cm
h cm
6 cm
Acone,ofheight8cmandbaseradius6cm,isplacedoveracylinderofradiusrcmandheighthcm
andisincontactwiththecylinderalongthecylinder’supperrim.Thearrangementissymmetricaland
thediagramshowsaverticalcross-sectionthroughthevertexofthecone.
(i) Usesimilartrianglestoexpresshintermsof r.
[2]
(ii) Henceshowthatthevolume,Vcm3,ofthecylinderisgivenbyV = 8rr 2 - 43 rr 3 .
[1]
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(iii) Giventhatrcanvary,findthevalueofr whichgivesastationaryvalueofV.Findthisstationary
valueofVintermsofπanddetermineitsnature.
[6]
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8
Solutions to this question by accurate drawing will not be accepted.
TwopointsAandBhavecoordinates(–3,2)and(9,8)respectively.
(i) FindthecoordinatesofC, thepointwherethelineABcutsthey-axis.
[3]
(ii) FindthecoordinatesofD,themid-pointofAB.
[1]
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(iii) FindtheequationoftheperpendicularbisectorofAB.
[2]
TheperpendicularbisectorofABcutsthey-axisatthepointE.
(iv) FindthecoordinatesofE.
[1]
(v) ShowthattheareaoftriangleABEisfourtimestheareaoftriangle ECD.
[3]
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9 Solvethefollowingequations.
(i)
4 sin 2x + 5 cos 2x = 0 for 0c G x G 180c
[3]
(ii)
cot 2 y + 3 cosec y = 3 for 0c G y G 360c
[5]
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(iii)
1
r
cos ` z + j =- for 0 G z G 2r radians,givingeachanswerasamultipleofπ
4
2
[4]
Question 10 is printed on the next page.
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10 A particle is moving in a straight line such that its velocity, vms–1, t seconds after passing a
fixedpointOisv = e 2t - 6e -2t - 1.
(i) Findanexpressionforthedisplacement,sm,fromOoftheparticle aftert seconds.
[3]
(ii) Usingthesubstitutionu = e 2t , orotherwise,findthetimewhentheparticleisatrest.
[3]
(iii) Findtheaccelerationatthistime.
[2]
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