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2015 HIGHER SCHOOL CERTIFICATE EXAMINATION
Mathematics
General Instructions
•Readingtime–5minutes
•Workingtime–3hours
•Writeusingblackpen
•Board-approvedcalculatorsmaybeused
•Atableofstandardintegralsisprovidedatthebackofthispaper
•InQuestions11–16,showrelevantmathematicalreasoningand/orcalculations
2590
Total marks – 100
Section I Pages 2–6
10 marks
•AttemptQuestions1–10
•Allowabout15minutesforthissection
Section II Pages7–17
90 marks
•AttemptQuestions11–16
•Allowabout2hoursand45minutesforthissection
– 2 –
Section I
10 marksAttempt Questions 1–10Allow about 15 minutes for this section
Usethemultiple-choiceanswersheetforQuestions1–10.
1 Whatis0.005 233 59writteninscientificnotation,correctto4significantfigures?
(A) 5.2336×10–2
(B) 5.234×10–2
(C) 5.2336×10–3
(D) 5.234×10–3
2 Whatistheslopeofthelinewithequation 2x −4y +3=0?
(A) − 2
1(B) −
2
1(C)
2
(D) 2
3 Thefirstthreetermsofanarithmeticseriesare3,7and11.
Whatisthe15thtermofthisseries?
(A) 59
(B) 63
(C) 465
(D) 495
–3–
54 TheprobabilitythatMel’ssoccerteamwinsthisweekendis .
72
TheprobabilitythatMel’srugbyleagueteamwinsthisweekendis .3
Whatistheprobabilitythatneitherteamwinsthisweekend?
2(A)
21
10(B)
21
13(C)
21
19(D)
21
5 Usingthetrapezoidalrulewith4subintervals,whichexpressiongivestheapproximateareaunderthecurve y = xe x between x = 1 and x = 3?
1(A) (e1 + 6e1.5 + 4e2 +10e2.5 + 3e3
4)
1(B) (e1 + 3e1.5 + 4e2 + 5e2.5 + 3e3
4)
1(C) (e1 + 6e1.5 + 4e2 +10e2.5 + 3e3
2)
1(D) (e1 + 3e1.5 + 4e2 + 5e2.5 + 3e3
2)
–4–
6 Whatisthevalueofthederivativeof y =2sin3x–3tanx at x =0?
(A) –1
(B) 0
(C) 3
(D) –9
7 The diagram shows the parabola y =4x − x2 meeting the line y = 2x at (0,0) and(2,4).
(4, 0)
(2, 4)
y = 2x
y = 4x – x2
(0, 0)
y
x
Whichexpressiongivestheareaoftheshadedregionboundedbytheparabolaandtheline?
⌠ 2
(A) x2⎮ − 2x dx⌡0
⌠ 2
(B) 2x − x2⎮ dx⌡0
⌠ 4
(C) x2⎮ − 2x dx⌡0
⌠ 4
(D) 2x − x2⎮ dx⌡0
– 5 –
8 Thediagramshowsthegraphof y = e x (1+ x).
−1 O x
1
y
−2
Howmanysolutionsaretheretotheequation e x (1+ x) =1− x 2?
(A) 0
(B) 1
(C) 2
(D) 3
9 Aparticleismovingalongthex-axis.Thegraphshowsitsvelocityvmetrespersecondattimetseconds.
O
8
4 t
v
When t =0 thedisplacementxisequalto2metres.
Whatisthemaximumvalueofthedisplacementx ?
(A) 8m
(B) 14m
(C) 16m
(D) 18m
– 6 –
210 Thediagramshowstheareaunderthecurve y = from x =1 to x = d.x
1 d
y
xO
Whatvalueofdmakestheshadedareaequalto2?
(A) e
(B) e +1
(C) 2e
(D) e2
– 7 –
Section II
90 marksAttempt Questions 11–16Allow about 2 hours and 45 minutes for this section
Answereachquestionintheappropriatewritingbooklet.Extrawritingbookletsareavailable.
In Questions 11–16, your responses should include relevant mathematical reasoning and/orcalculations.
Question 11 (15marks)UsetheQuestion11WritingBooklet.
(a) Simplify 4x − (8− 6x). 1
(b) Factorisefully 3x2–27. 2
8(c) Express witharationaldenominator. 2
2 + 7
1 1 1(d) Findthelimitingsumofthegeometricseries1− + − +! . 2
4 16 64
5(e) Differentiate (ex + x) . 2
(f) Differentiate y = (x + 4)lnx . 2
π⌠ 4
(g) Evaluate⎮ cos2x dx. 2⌡0
⌠ x(h) Find⎮ dx. 2
⌡ x2 − 3
–8–
Question 12 (15marks)UsetheQuestion12WritingBooklet.
(a) Findthesolutionsof 2sinq =1 for 0≤ q ≤ 2p. 2
(b) ThediagramshowstherhombusOABC.
ThediagonalfromthepointA(7,11)tothepointCliesonthelinel1.
Theotherdiagonal,fromtheoriginOtothepointB,liesonthelinel2which
hasequation y = − x
3.
O
C
BD
A(7,11)
x
y1
2
NOT TO SCALEy = −
x
3
(i) Showthattheequationofthelinel1is y =3x –10. 2
(ii) Thelinesl1andl2intersectatthepointD.
FindthecoordinatesofD.
2
Question 12 continues on page 9
– 9 –
Question12(continued)
x2 + 3(c) Find ƒ ′(x) ,where ƒ (x) = .
x −12
(d) For what values of k does the quadratic equation x2–8x + k=0 have realroots?
2
x2 ⎛ 1⎞(e) Thediagramshowstheparabola y = withfocus S 0, .Atangenttothe2 ⎝ 2⎠
parabolaisdrawnat P 1,12
⎛ ⎞⎝ ⎠ .
O
y
x
y =x2
2
P 1,12
S 0,12
NOT TOSCALE
(i) FindtheequationofthetangentatthepointP. 2
(ii) Whatistheequationofthedirectrixoftheparabola? 1
(iii) ThetangentanddirectrixintersectatQ.
ShowthatQliesonthey-axis. 1
(iv) ShowthatrPQSisisosceles. 1
End of Question 12
–10–
Question 13 (15marks)UsetheQuestion13WritingBooklet.
(a) ThediagramshowsrABCwithsides AB =6cm, BC =4cmand AC =8cm.
6 cm
4 cm8 cm
C
BA
NOT TO SCALE
7 (i) Showthat cosA = . 1
8
(ii) ByfindingtheexactvalueofsinA,determinetheexactvalueofthearea 2ofrABC.
(b) (i) Findthedomainandrangeforthefunction ƒ (x) = 9 − x2 . 2
(ii) Onanumberplane,shadetheregionwherethepoints(x,y)satisfyboth 2
oftheinequalities y ≤ 9 − x2 and y ≥ x.
(c) Considerthecurve y = x3 − x2 − x + 3.
(i) Findthestationarypointsanddeterminetheirnature. 4
⎛ 1 70 ⎞ (ii) Giventhatthepoint P , liesonthecurve,provethatthereisa⎝ 3 27⎠pointofinflexionatP.
2
(iii) Sketch the curve, labelling the stationary points, point of inflexionand y-intercept.
2
–11–
Question 14 (15marks)UsetheQuestion14WritingBooklet.
(a) In a theme park ride, a chair is released from a height of 110metres andfalls vertically. Magnetic brakes are applied when the velocity of the chairreaches−37 metrespersecond.
110 m
Magnetic brakes NOT TOSCALE
Release point
Theheightof thechairat time t seconds isxmetres.Theaccelerationof thechairisgivenby x!! = −10 .Atthereleasepoint,t = 0,x =110and x! = 0 .
(i) Usingcalculus,showthat x = −5t2 +110. 2
(ii) Howfarhasthechairfallenwhenthemagneticbrakesareapplied? 2
Question 14 continues on page 12
–12–
Question14(continued)
(b) Weatherrecordsforatownsuggestthat:
5• ifaparticulardayiswet(W),theprobabilityofthenextdaybeingdryis
61
• ifaparticulardayisdry(D),theprobabilityofthenextdaybeingdryis .2
In a specific week Thursday is dry. The tree diagram shows the possibleoutcomesforthenextthreedays:Friday,SaturdayandSunday.
W
Friday Saturday Sunday
D
W
D
W
D
W
D
W
D
W
D
W
D
12
12
12
16
56
16
56
12
12
16
56
12
12
12
2 (i) ShowthattheprobabilityofSaturdaybeingdryis . 1
3
(ii) WhatistheprobabilityofbothSaturdayandSundaybeingwet? 2
(iii) WhatistheprobabilityofatleastoneofSaturdayandSundaybeingdry? 1
Question 14 continues on page 13
–13–
Question14(continued)
(c) Sam borrows $100 000 to be repaid at a reducible interest rate of 0.6% permonth.Let $An be the amount owing at the endof nmonths and$M be themonthlyrepayment.
(i) Showthat A2 = 100 000(1.006)2 − M (1+1.006) . 1
⎛ (1.006)n −1⎞ (ii) Showthat A = 100 000(1.006)n − Mn ⎜⎝ 0.006 ⎟ . 2
⎠
(iii) Sammakesmonthlyrepaymentsof$780.
Showthataftermaking120monthlyrepayments theamountowing is$68 500tothenearest$100.
1
(iv) Immediately aftermaking the120th repayment,Sammakes aone-offpayment,reducingtheamountowingto$48 500.Theinterestrateandmonthlyrepaymentremainunchanged.
After how many more months will the amount owing be completelyrepaid?
3
End of Question 14
–14–
Question 15 (15marks)UsetheQuestion15WritingBooklet.
(a) The amount of caffeine, C, in the human body decreases according to theequation
dC =−0.14C,
dt
whereCismeasuredinmgandtisthetimeinhours.
− dC (i) Show that C = Ae 0.14t is a solution to =−0.14C, where A is
dtaconstant.1
When t =0, thereare130mgofcaffeineinLee’sbody.
(ii) FindthevalueofA. 1
(iii) WhatistheamountofcaffeineinLee’sbodyafter7hours? 1
(iv) WhatisthetimetakenfortheamountofcaffeineinLee’sbodytohalve? 2
Question 15 continues on page 15
–15–
Question15(continued)
(b) ThediagramshowsrABCwhichhas a right angle atC.ThepointD is themidpointofthesideAC.ThepointEischosenonABsuchthatAE = ED.ThelinesegmentEDisproducedtomeetthelineBCatF.
NOT TOSCALE
C
A
D
E
BF
Copyortracethediagramintoyourwritingbooklet.
(i) ProvethatrACB issimilartorDCF. 2
(ii) ExplainwhyrEFBisisosceles. 1
(iii) ShowthatEB = 3AE. 2
(c) Waterisflowinginandoutofarockpool.Thevolumeofwaterinthepoolattimet hoursisV litres.Therateofchangeofthevolumeisgivenby
dV = 80sin(0.5t) .dt
Attime t = 0,thevolumeofwaterinthepoolis1200 litresandisincreasing.
(i) Afterwhattimedoesthevolumeofwaterfirststarttodecrease? 2
(ii) Findthevolumeofwaterinthepoolwhen t = 3. 2
(iii) Whatisthegreatestvolumeofwaterinthepool? 1
End of Question 15
–16–
Question 16 (15marks)UsetheQuestion16WritingBooklet.
(a) The diagram shows the curve with equation y = x2 – 7x + 10. The curveintersectsthex-axisatpointsAandB.ThepointConthecurvehasthesamey-coordinateasthey-interceptofthecurve.
O A B
C
y
x
(i) Findthex-coordinatesofpointsAandB. 1
(ii) WritedownthecoordinatesofC. 1
⌠ 2
(iii) Evaluate x2⎮ ( − 7x +10)dx . 1⌡0
(iv) Hence,orotherwise,findtheareaoftheshadedregion. 2
Question 16 continues on page 17
–17–
Question16(continued)
(b) A bowl is formed by rotating the curve y =8loge (x –1) about the y-axisfor 0≤ y ≤6.
3
2
NOT TOSCALE
O
6
y
x
Findthevolumeofthebowl.Giveyouranswercorrectto1decimalplace.
(c) Thediagramshowsacylinderofradiusxandheighty inscribedinaconeofradiusRandheightH,whereRandHareconstants.
y
R
Hx
1 Thevolumeofaconeofradiusrandheighthis π r2h .
3
Thevolumeofacylinderofradiusrandheighthispr2h.
(i) Showthatthevolume,V,ofthecylindercanbewrittenasH
V = π x2 (R − x) .R
3
(ii) Byconsidering the inscribedcylinderofmaximumvolume,showthat4
thevolumeofanyinscribedcylinderdoesnotexceed ofthevolume9
ofthecone.
4
End of paper
– 20 –
STANDARD INTEGRALS
©2015BoardofStudies,TeachingandEducationalStandardsNSW
⌠ 1xn dx = xn+1
⎮ , n ≠ −1; x ≠ 0, if n < 0⌡ n +1
⌠ 1⎮ dx = ln x , x > 0⌡ x
⌠eax 1
⎮ dx = eax , a ≠ 0⌡ a
⌠ 1⎮ cosax dx = sinax , a ≠ 0⌡ a
⌠ 1⎮ sinax dx = − cosax , a ≠ 0⌡ a
⌠sec2 1
⎮ ax dx = tanax , a ≠ 0⌡ a
⌠ 1⎮ secax tanax dx = secax , a ≠ 0⌡ a
⌠ 1 1= tan−⎮ dx 1 x
, a ≠ 0⌡ 2 + a aa x2
⌠ 1 x⎮ dx = sin−1 , a > 0, − a < x < a⌡ a2 a− x2
⌠ 1⎮ dx = ln x + x2 − a2 , x > a > 0⌡ x2 − a2
( )
⌠ 1⎮ dx = ln(x + x2 + a2
⌡ x2 + a2)
NOTE : ln x = log xe , x > 0