ComplexNumbers 47

COMPLEX NUMBERS

1. IfA(z1),B(z

2),C(z

3)aretheverticesofanequilateraltriangleABC,valueof

argz z z

z z2 3 1

3 2

2 + - -

FHG

IKJ isequalto:

(A) p/4 (B) p/2 (C) p/3 (D) p/6

2. IfA(z1),B(z

2),C(z

3)arethreepointsintheArgandPlanewhere|z

1+z

2|=||z

1||z

2||and

|(1i)z1+iz

3|=|z

1|+|z

3z

1|,then

(A)A,BandClieonafixedcirclewithcenter2zz 21 + (B)A,BandCarecollinearpoints

(C)ABCformanequilateraltriangle (D)ABCfromanobtuseangletriangle

3. Thepointinthecomplexplaneatwhichthecurvesarg(z3i)= 43p

andarg(2z+12i)= 4 p

intersect,is

(A)

49

,43

(B)

47

,43

(C)

47

,42

(D)

74

,76

4. If1

2

z11

z5ispurelyimaginary,then

21

21

z3z2z3z2

- +

is

(A) 3337 (B) 5

11 (C)1 (D)115

5. If ( ) nn 2i3 = + ,wherenisaninteger,then(A)nisamultipleof5 (B)nisamultipleof6(C)nisamultipleof10 (D)noneofthese

6. If|z|=2and2z

2z

zz

zz

31

21

+ -

= - -

thenz1,z

2,z

3willbetheverticesof

(A)equilateraltriangle (B)rightangledtriangle(C)acuteangledtriangle (D)noneofthese

7. If a isanonrealcomplexnumberand 0xx2 = a + a + hasarealroot g ,then

(A) a + a = g (B) ][2 a + a = g (C) 1 = g (D)noneofthese

8. Ifonetheverticesofthesquarecircumscribingthecircle 2|1z| = - is i32+ .Findtheotherverticesofthesquare.

9. Letz1andz

2berootsoftheequationz2+pz+q=0,wherethecoefficientspandqmaybe

complexnumbers.LetAandBrepresentz1andz

2inthecomplexplane.If = AOB a 0and

OA=OB,whereOistheorigin.Provethatp2=4 22q cos

aFHG IKJ.

10. UseDeMoiverstheoremtosolveequation2 2 3 1 3 14x i = - + +d i d i .

ComplexNumbers 48

ANSWERKEY

1. B 2. A 3. A 4. C

5. B 6. B

7. C

8. ( ) ( ) i13,3i,i31 - + - + - 10. cos sinr i r p p48 48

+ wherer=5,29,53and77