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MA6102 Complex NumbersTest 1

Name: ________________ Class: ___M10_______ Duration: 50 minutes

Signature of Parent/Guardian: ________________________

INSTRUCTIONS TO STUDENTS:

Answer ALL questions in the space provided.

The use of calculators is NOT allowed.

The omission of essential working will result in the loss of marks.

The intended marks for each question is shown in brackets [ ].

This question paper consists of 6 printed pages including the cover page.

DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO

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1. (a) Simplify1 i

1 i

+

. Hence, find the smallest positive integer nsuch that

1 i1

1 i

n +

=

. [3]

(b) Find all the complex numbers z, in the form x+ yi, such that2

3i 1 3i.z z+ = + [4]

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2. Given that the equation

3 2 4 0z az bz + + =

where ,a b , has a root 1 i, find all the other roots of the equation. [4]

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3. The complex number zis such that 2z= and2

arg( )3

z

= . Find the modulus and principal

argument of( )

3*

8 cos isin4 4

z

+

. [5]

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4. The roots of the equation 2 2 4 0z z + = are denoted by 1zand 2z , where arg 1z> arg 2z .

Find1z and 2z in the exponential form

ire , giving the exact values of rand .

Find the value of*

1 1

*

2 2

.z z

z z [5]

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5. In an Argand diagram, the points A, Band Crepresent the complex numbers 2 + i, 1 + 3i and

p+ qirespectively.

(i) Mark the two points A and Bin an Argand diagram. [2]

(ii) If OABCis a parallelogram, find pand q. [2]

(iii) Show that, with the values obtained in (ii), OABCis a square. [2]

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Qn

1(a)

1 i 1 i

1 i 1 i

+ +

+Multiplying by the conjugate

i=

i 1 4n

n= =

(b)Substituting z= x+ yi into the equation and simplifying

( )2 2 3 3 i 1 3ix y y x + + = + Equating parts: 2 2 3 1,3 3x y y x + = =

Attempt to solve both equations involving elimination of unknown

1 or 1+3iz=

2 Method INotice 1 i+ is also a root.

Obtain a quadratic factor ( ) ( )1 i 1 iz z +

= 2 2 2z z + Attempt to find the 3rd root

Write ( ) ( )2 3 22 2 4z z z m z az bz + + = + + and compare constant termObtain 2m= and the 3rd root is 2Method IINotice 1 i+ is also a root.

Obtain a quadratic factor ( ) ( )1 i 1 iz z +

= 2 2 2z z + Attempt to find the 3rd root using long divisionObtain and the 3rd root is 2

3 Method I

Able to get

*

z from zUses the formula 11

2 2

zz

z z= correctly

Obtain modulus is 1

Uses the formula 12

1 2arg( ) arg( ) arg( ) 2z

zz z k= + correctly

Obtain the argument is4

Method II

Able to get *z from z

Attempt to use the exponential/polar form to evaluate( )

3*

8 cos isin4 4

z

+

Obtain9

i i4 4z e e

= =

Obtain modulus is 1

Obtain arg z=4

4 Applying quadratic formula to solve the equation

i3

1 2 ,z e

=

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i3

2 2 ,z e

=

i i 2 2* 3 3 i i1 1 3 3

*i i

2 2 3 3

2 2 2 2 2 2cos +isin cos +isin

3 3 3 32 2

z z e ee e

z ze e

= = =

3i=

5(i) Two points drawn

(ii) OC

// AB

AB

: ( )1 3i 2 i 1 2i+ + = +

1, 2p q= =

(iii) Show iOC OA=

which means that ,2

OA OC AOC

= =

using part (ii), it is a square.