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k Latt\ D"A,J%/P %il+*c lve/v -1" 1AndersonJunior CollegeIf2 Mathematics QCzl2Xll)ClassAssessment ComplexNumbers

l(a) Given thatz=*l+2i is a root of the equation 3 -22 z-a=O, find thevalueoftheconstant. tllHence,actorise (z) = zt * z' - z -a completelynto linear actors. t2l

(b) The omplexumberzssuchhat1

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4) PJC(2006) (modified)(a) Write down the five roots of the equation w5 =1. _

Henceshowthatthe roots of the ' ( 2+ z\)equationl*r) =l are/t \

zi tnl$ l ,wne"e= 0,+7,+.\ s '(b) The points Z and W n anArgand diagram represent the variable complex numbersand w respectively.On a singleArgand diagram, sketch he following loci.

(r) ar1(b6)=+ ,(ir) lw+zil=tV:,1

- Hence, ind the eastvalueaf lz -vl.

0

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,'TioL' '(r il^^/z- - 1r g. '

- l t ) . )= c-( e?. ' )3- a *( t2. ' \ t- C-.t+ 2.)-Cr-=J7;,."

)2 s-; -z-c\ - (z\.( Z - '- ( +2, z- C-r -2. ' ) 2 *1,

- ( t2, -( -2. ) C S) = rS56' ((z\ = = _ 6--, +2i) z-c-c*Z.))cz_3)

o=

I

7-aLL.l

-stL4-T9""2ao4Q,. >-o

w

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( S. ,2) -- -

- l = h'*z i+3 = 2(1 +- 2 ,-t2-: .+-+

2

( r , - t )

j+z+ ir ( t*"

w

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rz@a. +q' \ "( --=- T/ ( . (5 .' ,27 '

9-Cc)

k. () , (- --e-4;,e.4i e-qt

z ,'(, 'L

sK4-

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C*6to)--.-/a \

zq?

1o+2;{=r* G;u t2; l =

v:-l:=

Li (z-c-e l \7 = {,

\./

t_'J t

? Q-=o",71i-eJ--$:_Z- i- r =r*l =n14z- = , lG: tTi f ",p = t14 v

f , . r lez-L4.(16L e .( -e) r - to.

w

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.!b22

q a( 'a^-F4 e-Y:i

Z* (- l+t l a {z- c s+i)