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Tuanaki, Kaupae 3, 202091577M Te whakahāngai i te taurangi o ngā tau matatini

hei whakaoti rapanga

9.30 i te ata Rāhina 23 Whiringa-ā-rangi 2020 Whiwhinga: Rima

Paetae Kaiaka KairangiTe whakahāngai i te taurangi o ngā tau matatini hei whakaoti rapanga.

Te whakahāngai i te taurangi o ngā tau matatini mā te whakaaro whaipānga hei whakaoti rapanga.

Te whakahāngai i te taurangi o ngā tau matatini mā te whakaaro waitara hōhonu hei whakaoti rapanga.

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Me whakamātau koe i ngā tūmahi KATOA kei roto i tēnei pukapuka.

Tuhia ō mahinga KATOA.

Tirohia mēnā kei a koe te pukapuka Tikanga Tātai me ngā Tūtohi L3–CALCMF.

Mēnā ka hiahia whārangi atu anō koe mō ō tuhinga, whakamahia te (ngā) whārangi wātea kei muri o tēnei pukapuka, ka āta tohu ai i te tau tūmahi.

Tirohia mēnā e tika ana te raupapatanga o ngā whārangi 2 – 15 kei roto i tēnei pukapuka, ka mutu, kāore tētahi o aua whārangi i te takoto kau.

ME HOATU RAWA KOE I TĒNEI PUKAPUKA KI TE KAIWHAKAHAERE Ā TE MUTUNGA O TE WHAKAMĀTAUTAU.

Tohua tēnei pouaka mēnā kāore he tuhituhi i

roto i tēnei pukapuka

Tuanaki 91577M, 2020

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ANAKE

TŪMAHI TUATAHI

(a) Mēnākos=2+3i,ā,kot = 3 + k i,whiriwhiriateuaraokinast = 21 – i.

(b) Whiriwhiriate(ngā)uaraorkiakotahianaketeotingaotewhāritex2 + 4rx + r = 0.

(c) Whakaotiatewhāriteewhaiakemōxepāanakig.

2 x −5= 4x − g

2

QUESTION ONE

(a) If s = 2 + 3i andt = 3 + k i,findthevalueofk if st = 21 – i.

(b) Findthevalue(s)ofrsuchthattheequationx2 + 4rx + r=0hasonlyonesolution.

(c) Solvethefollowingequationforxintermsofg.

2 x −5= 4x − g

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Calculus 91577, 2020

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Tuanaki 91577M, 2020

(d) Tuhia k + ki1− i

+ 2k1+ i

kitanaāhuatinomāmā.

(e) MēnākoT = a − bia + bi

,ā,hetaupūmautūturuameb,hāponotiako1+T2

2T= a

2 − b2

a2 + b2.

4

(d) Write k + ki1− i

+ 2k1+ i

initssimplestpossibleform.

(e) GiventhatT = a − bia + bi

, where aandbarerealconstants,provethat1+T2

2T= a

2 − b2

a2 + b2.

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Calculus 91577, 2020

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Tuanaki 91577M, 2020

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TŪMAHI TUARUA

(a) Mēnākox–2hepāngao2x3 + qx2 – 17x–10,whiriwhiriateuaraoq.

(b) Whiriwhiriangāuarakatoaokkataeainako|5 + 3k i| = 13.

(c) Kotētahiongāotingao2z3 – 15z2 + bz–30=0koz=3+i(kobhetautūturu).

Whiriwhiriangāotingakē,meteuaraob.

6

QUESTION TWO

(a) Giventhatx–2isafactorof2x3 + qx2 – 17x–10,findthevalueofq.

(b) Findallpossiblevaluesofkgiventhat|5 + 3k i| = 13.

(c) Oneofthesolutionsof2z3 – 15z2 + bz–30=0isz = 3 + i (bisarealnumber).

Findtheothersolutions,andthevalueofb.

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Tuanaki 91577M, 2020

(d) Mēnākou = p + p i,ā,kov = –q + q i,inapmeqhetaupūmautūturutōrunga,

whiriwhiria a arg u

v⎛⎝⎜

⎞⎠⎟

.

(e) Whiriwhiriatewhāriteā-taungatukutuku(Cartesianequation)otehuanuiewhakaahuahiaanae|z + i|2 + |z – i|2 = 10.

Tuhiatōotingakiteāhuax2 + y2 = k.

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(d) Giventhatu = p + p iandv = –q + q i, where pandqarebothpositiverealconstants,

findarg u

v⎛⎝⎜

⎞⎠⎟

.

(e) FindtheCartesianequationofthelocusdescribedby|z + i|2 + |z – i|2 = 10.

Writeyoursolutionintheformx2 + y2 = k.

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Calculus 91577, 2020

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Tuanaki 91577M, 2020

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TŪMAHI TUATORU

(a) Mēnākou = 12k 3cis(π ) and v = 2kcis π3

⎛⎝⎜

⎞⎠⎟

meu = 12k 3cis(π ) and v = 2kcis π3

⎛⎝⎜

⎞⎠⎟,tuhiatetinouarao u

vkiteāhuaahuroa.

(b) Inakoz=5–iā,w=–2+3i,whakaaturiako|z|2 = 2|w|2.

(c) Mēnākoz = a + b i,inaamebhetautūturueharaitekore,whakaaturiako zzz + z

he tau tūturu.

10

QUESTION THREE

(a) If u = 12k 3cis(π ) and v = 2kcis π3

⎛⎝⎜

⎞⎠⎟,writetheexactvalueof u

vinpolarform.

(b) If z=5–iandw=–2+3i,showthat|z|2 = 2|w|2.

(c) Giventhatz = a + b i, where aandbarenon-zerorealnumbers,

showthat zzz + z

isarealnumber.

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Tuanaki 91577M, 2020

(d) Whakaotiatewhāritez4 = –16k8,inakokhetaupūmautūturu.

Tuhiaōotingakiteāhuaahuroaepāanakik.

(e) Mōngātaumatatiniumev,hāponotiamēnāko|u + v| = |u – v|,kātihepohewanoaihote uv .

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(d) Solvetheequationz4 = –16k8, where kisarealconstant.

Giveyoursolutionsinpolarformintermsofk.

(e) Forcomplexnumbersuandv,provethatif|u + v| = |u – v|,then uv ispurelyimaginary.

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Calculus 91577, 2020

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ANAKETAU TŪMAHI

He whārangi anō ki te hiahiatia.Tuhia te (ngā) tau tūmahi mēnā e tika ana.

Tuanaki 91577M, 2020

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15

Calculus 91577, 2020

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QUESTION NUMBER

Extra paper if required.Write the question number(s) if applicable.

Level 3 Calculus 202091577 Apply the algebra of complex numbers

in solving problems

9.30 a.m. Monday 23 November 2020 Credits: Five

Achievement Achievement with Merit Achievement with ExcellenceApply the algebra of complex numbers in solving problems.

Apply the algebra of complex numbers, using relational thinking, in solving problems.

Apply the algebra of complex numbers, using extended abstract thinking, in solving problems.

Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.

You should attempt ALL the questions in this booklet.

Show ALL working.

Make sure that you have the Formulae and Tables Booklet L3–CALCMF.

If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.

Check that this booklet has pages 2 – 15 in the correct order and that none of these pages is blank.

YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

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57

7M

English translation of the wording on the front cover